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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf,len=693
+page_content='THERMODYNAMICAL MODELING OF MULTIPHASE FLOW  SYSTEM WITH SURFACE TENSION AND FLOW  HAJIME KOBA  Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content=' We consider the governing equations for the motion of the viscous  fluids in two moving domains and an evolving surface from both energetic  and thermodynamic points of view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content=' We make mathematical models for multi-  phase flow with surface flow by our energetic variational and thermodynamic  approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='  More precisely, we apply our energy densities, the first law of  thermodynamics, and the law of conservation of total energy to derive our  multiphase flow system with surface tension and flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content=' We study the conserva-  tive forms and conservation laws of our system by using the surface transport  theorem and integration by parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content=' Moreover, we investigate the enthalpy, the  entropy, the Helmholtz free energy, and the Gibbs free energy of our model by  applying the thermodynamic identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content=' The key idea of deriving surface tension  and viscosities is to make use of both the first law of thermodynamics and our  energy densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content=' Introduction  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content=' Moving Domains, Surfaces and Notations  We are interested in a mathematical modeling of a soap bubble floating in the  air.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content=' When we focus on a soap bubble, we can see the fluid flow in the bubble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content=' We  2020 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content=' 80M30, 35Q79, 76-10, 80-10, 35A15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='  Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content=' Multiphase flow, Surface tension, Surface flow, Mathematical model-  ing, First law of thermodynamics, Energetic variational approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='  This work was partly supported by the Japan Society for the Promotion of Science (JSPS)  KAKENHI Grant Number JP21K03326.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='  1  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='02860v1  [math-ph]  7 Jan 2023  PA,PB, Ps: Density  n2  UA, UB, Us: Velocity  μA, μB, μs, 入A,入B, 入s: Viscosity  nr  TA,TB, s: Pressure  I(t)  A,OB,Os: Temperature  A(t)  B(t)  eA, eB, es: Internal energy  KA, KB, Ks: Thermal conductivity  hA,hB,hs: Enthalpy  SA, SB, Ss: Entropy  2  AB  = A(t) UI(t) U 2B(t2  HAJIME KOBA  call the fluid flow in the bubble a surface flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content=' We can consider a surface flow  as a fluid-flow on an evolving surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content=' To make a mathematical model for a soap  bubble floating in the air, we have to study the dependencies among fluid-flows in  two moving domains and surface flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content=' We consider the governing equations for  the motion of the viscous fluids in the two moving domains and surface from both  energetic and thermodynamic points of view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content=' More precisely, we apply the first law  of thermodynamics and our energy densities to derive our multiphase flow system  with surface tension and flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='  Let us first introduce fundamental notations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content=' Let t ≥ 0 be the time variable,  and x(= t(x1, x2, x3)) ∈ R3 the spatial variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='  Fix T > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='  Let Ω ⊂ R3 be  a bounded domain with a smooth boundary ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='  The symbol nΩ = nΩ(x) =  t(nΩ  1 , nΩ  2 , nΩ  3 ) denotes the unit outer normal vector at x ∈ ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='  Let ΩA(t)(=  {ΩA(t)}0≤t<T ) be a bounded domain in R3 with a moving boundary Γ(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content=' Assume  that Γ(t)(= {Γ(t)}0≤t<T ) is a smoothly evolving surface and is a closed Riemannian  2-dimensional manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content=' The symbol nΓ = nΓ(x, t) = t(nΓ  1, nΓ  2, nΓ  3) denotes the unit  outer normal vector at x ∈ Γ(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content=' For each t ∈ [0, T), assume that ΩA(t) ⋐ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content=' Set  ΩB(t) = Ω \\ ΩA(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content=' It is clear that Ω = ΩA(t) ∪ Γ(t) ∪ ΩB(t) (see Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content=' Set  ΩA,T =  �  0<t<T  {ΩA(t) × {t}}, ΩB,T =  �  0<t<T  {ΩB(t) × {t}},  ΓT =  �  0<t<T  {Γ(t) × {t}}, ΩT = Ω × (0, T), ∂ΩT = ∂Ω × (0, T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='  In this paper we assume that the fluids in ΩA,T , ΩB,T , and ΓT are compressible  ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='  Let us state physical notations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='  For ♯ = A, B, S, let ρ♯ = ρ♯(x, t), v♯ =  v♯(x, t) = t(v♯  1, v♯  2, v♯  3), π♯ = π♯(x, t), θ♯ = θ♯(x, t), e♯ = e♯(x, t), κ♯ = κ♯(x, t)  and µ♯ = µ♯(x, t), λ♯ = λ♯(x, t) be the density, the velocity, the pressure, the  temperature, the internal energy, the thermal conductivity, and two viscosities of  the fluid in Ω♯(t), where ΩS(t) := Γ(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content=' The symbols h♯ = h♯(x, t), ς♯ = ς♯(x, t),  F H  ♯  = F H  ♯ (x, t), and F G  ♯ = F G  ♯ (x, t) denote the enthalpy, the entropy, the Helmholtz  free energy, and the Gibbs free energy of the fluid in Ω♯(t), respectively (see Figure  1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content=' We call µ♯ the share viscosity and µ♯+λ♯ the dilatational viscosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content=' In particular,  we often call µS the surface share viscosity, µS+λS the surface dilatational viscosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='  We assume that ρ♯, v♯, π♯, θ♯, e♯, κ♯, µ♯, λ♯, h♯, ς♯, F H  ♯ , and F G  ♯ are smooth functions  in R4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='  Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content=' We call vS a total velocity, and πS a total pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content=' Total velocity  means that vS can be divided into surface velocity uS and motion velocity wS, that  is, vS = uS + wS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content=' Total pressure means one that includes surface pressure and  tension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content=' In this paper, we focus on the total velocity and the total pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='  Let us introduce several operators and notations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content=' For each f = f(x, t) ∈ C1(R4)  and V = V (x, t) = t(V1, V2, V3) ∈ [C1(R4)]3, DA  t f := ∂tf + (vA · ∇)f, DB  t f :=  ∂tf+(vB·∇)f, DS  t f := ∂tf+(vS·∇)f, gradf := ∇f, divV := ∇·V , gradΓf := ∇Γf,  divΓV := ∇Γ·V , (V ·∇)f := V1∂1f +V2∂2f +V3∂3f, (V ·∇Γ)f := V1∂Γ  1 f +V2∂Γ  2 f +  V3∂Γ  3 f, where ∇ := t(∂1, ∂2, ∂3), ∂i := ∂/∂xi, ∂t := ∂/∂t, ∇Γ := t(∂Γ  1 , ∂Γ  2 , ∂Γ  3 ), and  ∂Γ  i f := �3  j=1(δij −nΓ  i nΓ  j )∂jf = ∂if −nΓ  i (nΓ ·∇)f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content=' Define the orthogonal projection  MULTIPHASE FLOW SYSTEM WITH SURFACE TENSION AND FLOW  3  PΓ to a tangent space by  PΓ = PΓ(x, t) = I3×3 − nΓ ⊗ nΓ =  �  �  1 − nΓ  1nΓ  1  −nΓ  1nΓ  2  −nΓ  1nΓ  3  −nΓ  2nΓ  1  1 − nΓ  2nΓ  2  −nΓ  2nΓ  3  −nΓ  3nΓ  1  −nΓ  3nΓ  2  1 − nΓ  3nΓ  3  �  � ,  and the mean curvature HΓ in the direction nΓ by HΓ = HΓ(x, t) = −divΓnΓ,  where I3×3 is the 3×3 identity matrix, and ⊗ denotes the tensor product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content=' It is easy  to check that PΓnΓ = t(0, 0, 0) and PΓ∇f = ∇Γf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='  Let us explain the key restrictions on the boundaries ∂ΩT and ΓT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content=' We assume  that  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='1)  �  �  �  �  �  vB = t(0, 0, 0)  on ∂ΩT ,  vA · nΓ = vB · nΓ = vS · nΓ  on ΓT ,  PΓvA = PΓvB = rPΓvS  on ΓT ,  �  (nΩ · ∇)θB = 0  on ∂ΩT ,  θA = θB = θS  on ΓT ,  where r ∈ {0, 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content=' We call PΓvA = PΓvB = rPΓvS a slip boundary condition if r = 1  and a no-slip boundary condition if r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content=' Note that we do not consider phase  transition in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='  This paper has three purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='  The first purpose is to derive the following  multiphase flow system with surface tension and flow:  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='2)  �  �  �  �  �  DA  t ρA + (divvA)ρA = 0  in ΩA,T ,  DB  t ρB + (divvB)ρB = 0  in ΩB,T ,  DS  t ρS + (divΓvS)ρS = 0  on ΓT ,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='3)  �  �  �  �  �  ρADA  t eA + (divvA)πA = divqA + eDA  in ΩA,T ,  ρBDB  t eB + (divvB)πB = divqB + eDB  in ΩB,T ,  ρSDS  t eS + (divΓvS)πS = divΓqS + eDS + qB · nΓ − qA · nΓ  on ΓT ,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='4)  �  �  �  �  �  ρADA  t vA = divTA  in ΩA,T ,  ρBDB  t vB = divTB  in ΩB,T ,  ρSDS  t vS = divΓTS + �TBnΓ − �TAnΓ  on ΓT ,  where  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='5)  �  �  �  �  �  qA = qA(θA) := κAgradθA,  qB = qB(θB) := κBgradθB,  qS = qS(θS) := κSgradΓθS,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='6)  �  �  �  �  �  eDA = eDA(vA) := µA|D(vA)|2 + λA|divvA|2,  eDB = eDB(vB) := µB|D(vB)|2 + λB|divvB|2,  eDS = eDS(vS) := µS|DΓ(vS)|2 + λS|divΓvS|2,  �  �  �  �  �  D(vA) := {(∇vA) + t(∇vA)}/2,  D(vB) := {(∇vB) + t(∇vB)}/2,  DΓ(vS) := {(PΓ∇ΓvS) + t(PΓ∇ΓvS)}/2,  4  HAJIME KOBA  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='7)  �  �  �  �  �  TA = TA(vA, πA) := µAD(vA) + λA(divvA)I3×3 − πAI3×3,  TB = TB(vB, πB) := µBD(vB) + λB(divvB)I3×3 − πBI3×3,  TS = TS(vS, πS) := µSDΓ(vS) + λS(divΓvS)PΓ − πSPΓ,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='8)  �  �  �  �  �  �  �  �  �  �  �  �TA = �TA(vA, πA) =  �  µAnΓ · (nΓ · ∇)vA + λA(divvA) − πA if r = 0,  TA(vA, πA) if r = 1,  �TB = �TB(vB, πB) =  �  µBnΓ · (nΓ · ∇)vB + λB(divvB) − πB if r = 0,  TB(vB, πB) if r = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='  Here |D(vA)|2 = D(vA) : D(vA), |D(vB)|2 = D(vB) : D(vB), and |DΓ(vS)|2 =  DΓ(vS) : DΓ(vS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='  The symbol : denotes the Frobenius inner product, that is,  M : N = �3  i,j=1[M]ij[N]ij, where M, N are two 3×3 matrices, and [M]ij denotes  the (i, j)-component of the matrix M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content=' We call qA, qB, qS the heat fluxes, eDA, eDB,  eDS the energy densities for the energy dissipation due to the viscosities, D(vA),  D(vB) strain rate tensors, DΓ(vS) a surface strain tensor, TA, TB stress tensors, and  TS a surface stress tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content=' We often call TS the surface stress tensor determined by  the Boussinesq-Scriven law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content=' More precisely, under the restrictions ( 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='1) we apply  our energy densities and thermodynamic approaches to derive ( 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='2)-( 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content=' See  Section 4 for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='  Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content=' (i) Using nΓ · nΓ = 1, PΓ∇f = ∇Γf, (nΓ · ∇Γ)f = 0, and HΓ =  −divΓnΓ, we easily check that  divΓ(πSPΓ) = gradΓπS + πSHΓnΓ,  2DΓ(vS) = PΓ{(∇vS) + t(∇vS)}PΓ = PΓ{(∇ΓvS) + t(∇ΓvS)}PΓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='  Note that 2[DΓ(vS)]ij = ∂Γ  i vS  j + ∂Γ  j vS  i − nΓ  i (nΓ · ∂Γ  j vS) − nΓ  j (nΓ · ∂Γ  i vS),  ∇vS =  �  �  ∂1vS  1  ∂2vS  1  ∂3vS  1  ∂1vS  2  ∂2vS  2  ∂3vS  2  ∂1vS  3  ∂2vS  3  ∂3vS  3  �  � , ∇ΓvS =  �  �  ∂Γ  1 vS  1  ∂Γ  2 vS  1  ∂Γ  3 vS  1  ∂Γ  1 vS  2  ∂Γ  2 vS  2  ∂Γ  3 vS  2  ∂Γ  1 vS  3  ∂Γ  2 vS  3  ∂Γ  3 vS  3  �  � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='  We often call πSHΓnΓ surface tension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='  (ii) If the fluids in ΩA,T , ΩB,T , ΓT are barotropic fluids, then we can write  �  �  �  �  �  πA = πA(ρA) = ρAp′  A(ρA) − pA(ρA),  πB = πB(ρB) = ρBp′  B(ρB) − pB(ρB),  πS = πS(ρS) = ρSp′  S(ρS) − pS(ρS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='  Here pA, pB, pS are three C1-functions, p′ = p′(r) = dp/dr(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content=' See Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='4,  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='5, and Section 4 for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='  The second purpose is to study the conservative forms and conservation laws of  system ( 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='2)-( 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content=' In fact, if we set DN  t f = ∂tf +(vS ·nΓ)(nΓ ·∇)f, and the total  energy E♯ = E♯(x, t) by E♯ = ρ♯|v♯|2/2 + ρ♯e♯, then we can write our system as the  conservative form:  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='9)  �  �  �  �  �  ∂tρA + div(ρAvA) = 0  in ΩA,T ,  ∂tρB + div(ρBvB) = 0  in ΩB,T ,  DN  t ρS + divΓ(ρSvS) = 0  on ΓT ,  MULTIPHASE FLOW SYSTEM WITH SURFACE TENSION AND FLOW  5  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='10)  �  �  �  �  �  ∂tEA + div(EAvA − qA − TAvA) = 0  in ΩA,T ,  ∂tEB + div(EBvB − qB − TBvB) = 0  in ΩB,T ,  DN  t ES + divΓ(ESvS − qS − TSvS) = ES  on ΓT ,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='11)  �  �  �  �  �  ∂t(ρAvA) + div(ρAvA ⊗ vA − TA) = t(0, 0, 0)  in ΩA,T ,  ∂t(ρBvB) + div(ρBvB ⊗ vB − TB) = t(0, 0, 0)  in ΩB,T ,  DN  t (ρSvS) + divΓ(ρSvS ⊗ vS − TS) = �TBnΓ − �TAnΓ  on ΓT ,  where  ES := �TBnΓ · vS − �TAnΓ · vS + qB · nΓ − qA · nΓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='  Moreover, any solution to system ( 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='1)-( 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='4) satisfies that for t1 < t2,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='12)  �  ΩA(t2)  ρA(x, t2) dx +  �  ΩB(t2)  ρB(x, t2) dx +  �  Γ(t2)  ρS(x, t2) dH2  x  =  �  ΩA(t1)  ρA(x, t1) dx +  �  ΩB(t1)  ρB(x, t1) dx +  �  Γ(t1)  ρS(x, t1) dH2  x,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='13)  �  ΩA(t2)  EA dx +  �  ΩB(t2)  EB dx +  �  Γ(t2)  ES dH2  x  =  �  ΩA(t1)  EA dx +  �  ΩB(t1)  EB dx +  �  Γ(t1)  ES dH2  x,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='14)  �  ΩA(t2)  1  2ρA|vA|2 dx +  �  ΩB(t2)  1  2ρB|vB|2 dx +  �  Γ(t2)  1  2ρS|vS|2 dH2  x  +  � t2  t1  �  ΩA(t)  eDA dxdt +  � t2  t1  �  ΩB(t)  eDB dxdt +  � t2  t1  �  Γ(t)  eDS dH2  xdt  =  �  ΩA(t1)  1  2ρA|vA|2 dx +  �  ΩB(t1)  1  2ρB|vB|2 dx +  �  Γ(t1)  1  2ρS|vS|2 dH2  x  +  � t2  t1  �  ΩA(t)  (divvA)πA dxdt +  � t2  t1  �  ΩB(t)  (divvB)πB dxdt  +  � t2  t1  �  Γ(t)  (divΓvS)πS dH2  xdt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='  Here dH2  x denotes the 2-dimensional Hausdorff measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content=' Under some assumptions  (see Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='8), any solution to system ( 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='1)-( 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='3) satisfies that for t1 < t2  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='15)  �  ΩA(t2)  ρAvA dx +  �  ΩB(t2)  ρBvB dx +  �  Γ(t2)  ρSvS dH2  x  =  �  ΩA(t1)  ρAvA dx +  �  ΩB(t1)  ρBvB dx +  �  Γ(t1)  ρSvS dH2  x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='  We often call ( 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='12), ( 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='13), ( 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='14), and ( 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='15), the law of conservation of mass,  the law of conservation of total energy, the energy law of our system, and the law  of conservation of momentum, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content=' See Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='8 and Section 5 for  details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='  6  HAJIME KOBA  Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content=' From DS  t f = DN  t f + (vS · ∇Γ)f for f ∈ C1(R4), we see that  DN  t (ρSf) + divΓ(ρSfvS) = ρSDS  t f,  DN  t (ρSvS) + divΓ(ρSvS ⊗ vS) = ρSDS  t vS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='  Since ρS, vS ∈ C1(R4) in this paper, we can define DN  t  for ρS, ρSvS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='  The third purpose is to investigate the thermodynamic potential such as the  enthalpy h♯, the entropy ς♯, the Helmholtz free energy F H  ♯ , and the Gibbs free  energy F G  ♯ of the fluid in Ω♯(t), where ♯ = A, B, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content=' Assume that (ρ♯, θ♯) are positive  functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content=' Set the enthalpy h♯ by h♯ = e♯ + π♯/ρ♯.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content=' Then  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='16)  �  �  �  �  �  ∂t(ρAhA) + div(ρAhAvA − qA) = eDA + DA  t πA,  ∂t(ρBhB) + div(ρBhBvB − qB) = eDB + DB  t πB,  DN  t (ρShS) + divΓ(ρShSvS − qS) = eDS + DS  t πS + qB · nΓ − qA · nΓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='  Suppose that the thermodynamic identity (Gibbs [9]): D♯  te♯ = θ♯D♯  tς♯ −π♯D♯  t(1/ρ♯)  holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content=' Then  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='17)  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  ∂t(ρAςA) + div  �  ρAςAvA − qA  θA  �  =  eDA  θA + qA·gradθA  θ2  A  ,  ∂t(ρBςB) + div  �  ρBςBvB − qB  θB  �  =  eDB  θB + qB·gradθB  θ2  B  ,  DN  t (ρSςS) + divΓ  �  ρSςSvS − qS  θS  �  =  eDS  θS + qS·gradΓθS  θ2  S  + qB·nΓ−qA·nΓ  θS  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='  Set the Helmholtz free energy F H  ♯  by F H  ♯  = e♯ − θ♯ς♯.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content=' Then  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='18)  �  �  �  �  �  ρADA  t F H  A + ρAςADA  t θA = −(divvA)πA,  ρBDB  t F H  B + ρBςBDB  t θB = −(divvB)πB,  ρSDS  t F H  S + ρSςSDS  t θS = −(divΓvS)πS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='  Set the Gibbs free energy F G  ♯  by F G  ♯ = h♯ − θ♯ς♯.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content=' Then  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='19)  �  �  �  �  �  ρADA  t F G  A + ρAςADA  t θA = DA  t πA,  ρBDB  t F G  B + ρBςBDB  t θB = DB  t πB,  ρSDS  t F G  S + ρSςSDS  t θS = DS  t πS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='  See Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='9 and Section 6 for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='  Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content=' (i) Since  TA(vA, πA) : D(vA) = eDA − (divvA)πA,  TB(vB, πB) : D(vB) = eDB − (divvB)πB,  TS(vS, πS) : DΓ(vS) = eDS − (divΓvS)πS,  it follows from ( 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='18) to see that  �  �  �  �  �  ρADA  t F H  A + ρAςADA  t θA − TA(vA, πA) : D(vA) = −eDA,  ρBDB  t F H  B + ρBςBDB  t θB − TB(vB, πB) : D(vB) = −eDB,  ρSDS  t F H  S + ρSςSDS  t θS − TS(vS, πS) : DΓ(vS) = −eDS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='  MULTIPHASE FLOW SYSTEM WITH SURFACE TENSION AND FLOW  7  Note that PΓ : DΓ(vS) = divΓvS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='  (ii) Set DAf := ρADA  t f, DBf := ρBDB  t f, DSf := ρSDS  t f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content=' Then  �  �  �  �  �  DAeA = θADAςA − πADA(1/ρA),  DBeB = θBDBςB − πBDB(1/ρB),  DSeS = θSDSςS − πSDS(1/ρS),  �  �  �  �  �  DAhA = θADAςA + (1/ρA)DAπA,  DBhB = θBDBςB + (1/ρB)DBπB,  DShS = θSDSςS + (1/ρS)DSπS,  �  �  �  �  �  DAF H  A = −ςADAθA − πADA(1/ρA),  DBF H  B = −ςBDBθB − πBDB(1/ρB),  DSF H  S = −ςSDSθS − πSDS(1/ρS),  �  �  �  �  �  DAF G  A = −ςADAθA + (1/ρA)DAπA,  DBF G  B = −ςBDBθB + (1/ρB)DBπB,  DSF G  S = −ςSDSθS + (1/ρS)DSπS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='  We often call 1/ρ♯ a specific volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='  Let us explain the main difficulties in the derivation of our multiphase flow system  with surface tension and flow, and the key ideas to overcome these difficulties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content=' The  main difficulties are to derive the viscous terms of the system, to derive the surface  tension from a theoretical point of view, and to derive the dependencies among  fluid-flows in two moving domains and surface flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content=' To overcome these difficulties,  we apply the first law of thermodynamics (Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='4), our energy densities  (Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='2), and the conservation law of total energy to derive equations ( 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='3)  and ( 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content=' See Section 4 for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='  Let us mention the study of surface flow (interfacial flow).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content=' Boussinesq [5] first  discovered the existence of surface flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content=' Scriven [22] considered their surface stress  tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content=' Slattery [23] investigated some properties of the surface stress tensor de-  termined by the Boussinesq-Scriven law (see TS in ( 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='7)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content=' Then many researchers  have studied surface flow (see Slattery-Sagis-Oh [24] and Gatignol-Prud’homme [8]  for the study of interfacial phenomena).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='  Let us state derivations of the governing equations for the motion of the viscous  fluid on manifolds and surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content=' Taylor [26] introduced their surface stress tensor  to make their incompressible viscous fluid system on a manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content=' Mitsumatsu-Yano  [20] applied their energetic variational approach to derive their incompressible vis-  cous fluid system on a manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content=' Arnaudon-Cruzeiro [2] made use of their stochastic  variational approach to derive their incompressible viscous fluid system on a man-  ifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content=' Koba-Liu-Giga [18] employed their energetic variational approach and the  generalized Helmholtz-Weyl decomposition on a closed surface to derive their in-  compressible fluid systems on an evolving closed surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content=' Koba [14, 15] applied their  energetic variational approaches and the first law of thermodynamics to derive their  compressible fluid flow systems on an evolving closed surface and an evolving sur-  face with a boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content=' This paper modifies and improves the methods in [14, 15] to  derive our multiphase flow system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='  Now we mention results for modeling of multiphase flow system with surface  flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='  Bothe-Pr¨uss [4] made their multiphase flow system with surface flow by  using the surface stress tensor determined by the Boussinesq-Scriven law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content=' Koba  [17] derived the inviscid multiphase flow system with surface flow by applying a  geometric variational approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='  This paper derives our multiphase flow system  from a thermodynamic point of view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content=' Therefore, our modeling methods are different  from ones in [4] and [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='  Finally, we introduce some results and textbooks related to this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content=' Hyon-  Kwak-Liu [13] and Koba-Sato [19] applied their energetic variational approaches  to derive and study their complex and non-Newtonian fluid systems in domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='  8  HAJIME KOBA  Feireisl [7] studied the motion of the viscous fluid in a domain from a thermody-  namic point of view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content=' We refer the readers to Gyarmati [12] and Gurtin-Fried-Anand  [11] for the theory of thermodynamics, Chapter XIII in Angel [1] for thermody-  namical potential such as internal energy, enthalpy, entropy, and free energies, and  Pr¨uss-Simonett [21] for several elliptic and parabolic equations on hypersurfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='  The outline of this paper is as follows: In Section 2, we first introduce the  transport theorems and the energy densities for our model, and then we state the  main results of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content=' In Section 3, we make use of the transport theorems to  derive the first law of thermodynamics, and apply integration by parts to calculate  variations of our dissipation energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content=' In Section 4, we apply our thermodynamic  approaches to make mathematical models for multiphase flow with surface tension  and flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content=' In Section 5, we study the conservation and energy laws of our system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='  In Section 6, we investigate the thermodynamic potential for our system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content=' Main Results  We first introduce the transport theorems and the energy densities for our mul-  tiphase flow system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content=' Then we state the main results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='1 (ΩT is flowed by the velocity fields (vA, vB, vS)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content=' We say that ΩT  is flowed by the velocity fields (vA, vB, vS) if for each 0 < t < T, f ∈ C1(R4), and  Λ ⊂ Ω,  d  dt  �  ΩA(t)∩Λ  f(x, t) dx =  �  ΩA(t)∩Λ  {DA  t f + (divvA)f} dx,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='1)  d  dt  �  ΩB(t)∩Λ  f(x, t) dx =  �  ΩB(t)∩Λ  {DB  t f + (divvB)f} dx,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='2)  d  dt  �  Γ(t)∩Λ  f(x, t) dH2  x =  �  Γ(t)∩Λ  {DS  t f + (divΓvS)f} dH2  x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='3)  Here D♯  tf = ∂tf +(v♯·∇)f, divΓvS = ∂Γ  1 vS  1 +∂Γ  2 vS  2 +∂Γ  3 vS  3 , ∂Γ  j f = ∂jf −nΓ  j (nΓ·∇)f,  where ♯ = A, B, S, and j = 1, 2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='  We often call ( 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='1), ( 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='2) the transport theorems, and ( 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='2) the surface transport  theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content=' The derivation of the surface transport theorem can be founded in [3, 10,  6, 18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content=' Throughout this paper we assume that ΩT is flowed by the velocity fields  (vA, vB, vS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='2 (Energy densities).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content=' Set  �  �  �  �  �  eKA = ρA|vA|2/2,  eKB = ρB|vB|2/2,  eKS = ρS|vS|2/2,  �  �  �  �  �  eDA = µA|D(vA)|2 + λA|divvA|2,  eDB = µB|D(vB)|2 + λB|divvB|2,  eDS = µS|DΓ(vS)|2 + λS|divΓvS|2,  �  �  �  �  �  eWA = (divvA)πA,  eWB = (divvB)πB,  eWS = (divΓvS)πS,  �  �  �  �  �  eQA = κA|gradθA|2,  eQB = κB|gradθB|2,  eQS = κS|gradΓθS|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='  We call eK♯ the kinetic energy, eD♯ the energy density for the energy dissipation due  to the viscosities (µ♯, λ♯), eW♯ the power density for the work done by the pressure  π♯, and eQ♯ the energy density for the energy dissipation due to thermal diffusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='  MULTIPHASE FLOW SYSTEM WITH SURFACE TENSION AND FLOW  9  See [19], [14], and Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='5 in [15] for mathematical validity of the energy densi-  ties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content=' Applying the energy densities, the restrictions ( 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='1), and our thermodynamic  approaches, we derive system ( 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='2)-( 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='4) in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='  We now state the main results of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content=' From Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='1, we have  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='3 (Continuity equations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content=' Assume that for each 0 < t < T and  Λ ⊂ Ω,  d  dt  � �  ΩA(t)∩Λ  ρA(x, t) dx +  �  ΩB(t)∩Λ  ρB(x, t) dx +  �  Γ(t)∩Λ  ρS(x, t) dH2  x  �  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='  Then (ρA, ρB, ρS) satisfies ( 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='  The proof of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='3 is left for the readers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='4 (First law of thermodynamics).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content=' Let �QA, �QB, �QS ∈ C(R4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content=' Assume  that (ρA, ρB, ρS) satisfies ( 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content=' Then  (i) Suppose that for every 0 < t < T and Λ ⊂ Ω,  d  dt  �  ΩA(t)∩Λ  ρAeA dx =  �  ΩA(t)∩��  { �QA − (divvA)πA} dx,  d  dt  �  ΩB(t)∩Λ  ρBeB dx =  �  ΩB(t)∩Λ  { �QB − (divvB)πB} dx,  d  dt  �  Γ(t)∩Λ  ρSeS dH2  x =  �  Γ(t)∩Λ  { �QS − (divΓvS)πS} dH2  x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='  Then  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='4)  �  �  �  �  �  ρADA  t eA + (divvA)πA = �QA in ΩA,T ,  ρBDB  t eB + (divvB)πB = �QB in ΩB,T ,  ρSDS  t eS + (divΓvS)πS = �QS on ΓT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='  (ii) Let pA, pB, pS ∈ C1(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content=' Suppose that for every 0 < t < T and Λ ⊂ Ω,  d  dt  �  ΩA(t)∩Λ  {ρAeA − pA(ρA)} dx =  �  ΩA(t)∩Λ  �QA dx,  d  dt  �  ΩB(t)∩Λ  {ρBeB − pB(ρB)} dx =  �  ΩB(t)∩Λ  �QB dx,  d  dt  �  Γ(t)∩Λ  {ρSeS − pS(ρS)} dH2  x =  �  Γ(t)∩Λ  �QS dH2  x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='  Then  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='5)  �  �  �  �  �  ρADA  t eA + (divvA)ΠA = �QA in ΩA,T ,  ρBDB  t eB + (divvB)ΠB = �QB in ΩB,T ,  ρSDS  t eS + (divΓvS)ΠS = �QS on ΓT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='  Here  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='6)  �  �  �  �  �  ΠA = ΠA(ρA) = ρAp′  A(ρA) − pA(ρA),  ΠB = ΠB(ρB) = ρBp′  B(ρB) − pB(ρB),  ΠS = ΠS(ρS) = ρSp′  S(ρS) − pS(ρS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='  10  HAJIME KOBA  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content=' (i) We can write system ( 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='4) as follows:  �  �  �  �  �  DAeA = �QA − πADA(1/ρA),  DBeB = �QB − πBDB(1/ρB),  DSeS = �QS − πSDS(1/ρS),  where D♯f = ρ♯D♯  tf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content=' Therefore, we call Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='4 the first law of thermody-  namics in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content=' See also (ii) in Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='  (ii) The pressures (ΠA, ΠB, ΠS) derived from the assertion (ii) of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='4 cor-  respond to the pressures derived from an energetic variational approach (see [17]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='  Next we consider the variation of our dissipation energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content=' Let r ∈ {0, 1} and 0 <  t < T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content=' Let ϕA, ϕB, ϕS ∈ [C∞(R3)]3 and ψA, ψB, ψS ∈ C∞(R3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content=' For −1 < ε < 1,  vε  A := vA +εϕA, vε  B := vB +εϕB, vε  S := vS +εϕS, θε  A := θA +εψA, θε  B := θB +εψB,  and θε  S := θS + εψS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content=' We call (vε  A, vε  B, vε  S, θε  A, θε  B, θε  S) variations of (vA, vB, vS, θA,  θB, θS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content=' From ( 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='1), for −1 < ε < 1, we assume that  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='7)  �  �  �  �  �  vε  B = t(0, 0, 0)  on ∂Ω,  vε  A · nΓ = vε  B · nΓ = vε  S · nΓ  on Γ(t),  PΓvε  A = PΓvε  B = rPΓvε  S  on Γ(t),  �  (nΩ · ∇)θε  B = 0  on ∂Ω,  θε  A = θε  B = θε  S  on Γ(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='  Then we have  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='8)  �  �  �  �  �  ϕB = t(0, 0, 0)  on ∂Ω,  ϕA · nΓ = ϕB · nΓ = ϕS · nΓ  on Γ(t),  PΓϕA = PΓϕB = rPΓϕS  on Γ(t),  and  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='9)  �  (nΩ · ∇)ψB = 0  on ∂Ω,  ψA = ψB = ψS  on Γ(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='  For each variation (vε  A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content=' vε  B,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content=' vε  S,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content=' θε  A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content=' θε  B,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content=' θε  S),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='  ED[vε  A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content=' vε  B,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content=' vε  S] :=  �  ΩA(t)  �  − µA  2 |D(vε  A)|2 − λA  2 |divvε  A|2  �  dx  +  �  ΩB(t)  �  − µB  2 |D(vε  B)|2 − λB  2 |divvε  B|2  �  dx  +  �  Γ(t)  �  − µS  2 |DΓ(vε  S)|2 − λS  2 |divΓvε  S|2  �  dH2  x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='  EW [vε  A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content=' vε  B,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content=' vε  S] :=  �  ΩA(t)  (divvε  A)πA dx  +  �  ΩB(t)  (divvε  B)πB dx +  �  Γ(t)  (divΓvε  S)πS dH2  x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='  and  ET D[θε  A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content=' θε  B,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content=' θε  S] :=  �  ΩA(t)  �  − κA  2 |gradθε  A|2  �  dx  +  �  ΩB(t)  �  − κB  2 |gradθε  B|2  �  dx +  �  Γ(t)  �  − κS  2 |gradΓθε  S|2  �  dH2  x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='  MULTIPHASE FLOW SYSTEM WITH SURFACE TENSION AND FLOW  11  Set ED+W [·] = ED[·] + EW [·].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content=' We call ED the energy dissipation due to viscosities,  EW the work done by pressures, and ET D the energy dissipation due to thermal  diffusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='6 (Forces derived from variation of dissipation energies).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content=' Let r ∈  {0, 1}, 0 < t < T, and FA, FB, FS ∈ [C(R3)]3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content=' Assume that for every ϕA, ϕB, ϕS ∈  [C∞(R3)]3 satisfying ( 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='8),  d  dε  ����  ε=0  ED+W [vε  A, vε  B, vε  S] =  �  ΩA(t)  FA·ϕA dx+  �  ΩB(t)  FB·ϕB dx+  �  Γ(t)  FS·ϕS dH2  x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='  Then  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='10)  �  �  �  �  �  FA = divTA(vA, πA)  in ΩA(t),  FB = divTB(vB, πB)  in ΩB(t),  FS = divΓTS(vS, πS) + �TB(vB, πB)nΓ − �TA(vA, πA)nΓ  on Γ(t),  where (TA, TB, TS) and ( �TA, �TB) are defined by ( 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='7) and ( 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='8), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='7 (Endothermic energies derived from thermal diffusion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content=' Let 0 <  t < T, and QA, QB, QS ∈ C(R3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content=' Assume that for every ψA, ψB, ψS ∈ C∞(R3)  satisfying ( 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='9),  d  dε  ����  ε=0  ET D[θε  A, θε  B, θε  S] =  �  ΩA(t)  QAψA dx +  �  ΩB(t)  QBψB dx +  �  Γ(t)  QSψS dH2  x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='  Then  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='11)  �  �  �  �  �  QA = div(κA∇θA)  in ΩA(t),  QB = div(κB∇θB)  in ΩB(t),  QS = divΓ(κS∇ΓθS) + κB(nΓ · ∇)θB − κA(nΓ · ∇)θA  on Γ(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='  Finally, we state the conservation laws and thermodynamic potential for our  multiphase flow system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='8 (Conservative forms and conservation laws).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content=' Let r ∈ {0, 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content=' Then  (i) Any solution to system ( 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='2)-( 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='4) satisfies ( 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='9)-( 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='  (ii) Any solution to system ( 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='1)-( 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='4) satisfies ( 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='12)-( 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='  (iii) Assume that r = 1, and that for 0 < t < T  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='12)  �  ∂Ω  {µBD(vB)nΩ + λB(divvB)nΩ − πBnΩ} dH2  x = t(0, 0, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='  Then any solution to ( 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='1)-( 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='4) satisfies ( 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='9 (Thermodynamic potential).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content=' Let ♯ = A, B, S, and r ∈ {0, 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content=' Sup-  pose that ρ♯ and θ♯ are positive functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content=' Set h♯ = e♯ + π♯/ρ♯, F H  ♯  = e♯ − θ♯ς♯,  F G  ♯  = h♯ − θ♯ς♯.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='  Assume that e♯ satisfies D♯  te♯ = θ♯D♯  tς♯ − π♯D♯  t(1/ρ♯).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='  Then  ( 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='16)-( 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='19) hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='  We prove Theorems 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='4, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='6, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='7 in Section 3, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='8 in Section 5, and  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='9 in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content=' In Section 4 we derive system ( 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='2)-( 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='4) by our ther-  modynamic approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='  12  HAJIME KOBA  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content=' Application of Transport Theorems and Integration by Parts  We apply the transport theorems (Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='1) and several formulas for inte-  gration by parts (Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='1) to prove Theorems 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='4, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='6, and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='1 (Formulas for integration by parts).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='  Fix 0 ≤ t < T and j = 1, 2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='  Then for every f, g ∈ C1(R3),  �  ΩA(t)  (∂jf)g dx = −  �  ΩA(t)  f(∂jg) dx +  �  Γ(t)  fgnΓ  j dH2  x,  �  ΩB(t)  (∂jf)g dx = −  �  ΩB(t)  f(∂jg) dx +  �  ∂Ω  fgnΩ  j dH2  x −  �  Γ(t)  fgnΓ  j dH2  x,  �  Γ(t)  (∂Γ  j f)g dH2  x = −  �  Γ(t)  f(∂Γ  j g) dH2  x −  �  Γ(t)  HΓfgnΓ  j dH2  x,  where HΓ = −divΓnΓ and ∂Γ  j f = ∂jf − nΓ  j (nΓ · ∇)f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='  Here nΓ = nΓ(x, t) =  t(nΓ  1, nΓ  2, nΓ  3) denotes the unit outer normal vector at x ∈ Γ(t) and nΩ = nΩ(x) =  t(nΩ  1 , nΩ  2 , nΩ  3 ) the unit outer normal vector at x ∈ ∂Ω (see Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='  Applying the Gauss divergence theorem and the surface divergence theorem (Sec-  tion 9 in [25], Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='3 in [16]), we can prove Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='  We now make use of the transport theorems to prove Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='  Proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content=' We only prove (ii) since the proof of (i) is similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content=' We as-  sume that (ρA, ρB, ρS) satisfies ( 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content=' Let �QA, �QB, �QS ∈ C(R4) and pA, pB, pS ∈  C1(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content=' We first show that for every 0 < t < T and Λ ⊂ Ω,  d  dt  �  ΩA(t)∩Λ  {ρAeA − pA(ρA)} dx =  �  ΩA(t)∩Λ  {ρADA  t eA + (divvA)ΠA} dx,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='1)  d  dt  �  ΩB(t)∩Λ  {ρBeB − pB(ρB)} dx =  �  ΩB(t)∩Λ  {ρBDB  t eB + (divvB)ΠB} dx,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='2)  d  dt  �  Γ(t)∩Λ  {ρSeS − pS(ρS)} dH2  x =  �  Γ(t)∩Λ  {ρSDS  t eS + (divΓvS)ΠS} dH2  x,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='3)  where (ΠA, ΠB, ΠS) is defined by ( 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content=' We only derive ( 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content=' Using the surface  transport theorem ( 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='3) and ( 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='2), we check that for 0 < t < T and Λ ⊂ Ω  d  dt  �  Γ(t)∩Λ  {ρSeS−pS(ρS)} dH2  x =  �  Γ(t)∩Λ  {(DS  t ρS)eS+ρSDteS+ρSeS(divΓvS)} dH2  x  +  �  Γ(t)∩Λ  {−DS  t ρSp′  S(ρS) − pS(ρS)(divΓvS)} dH2  x  =  �  Γ(t)∩Λ  {ρSDS  t eS + (ρSp′  S(ρS) − pS(ρS))(divΓvS)} dH2  x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='  MULTIPHASE FLOW SYSTEM WITH SURFACE TENSION AND FLOW  13  Thus, we see ( 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content=' We assume that for every 0 < t < T and Λ ⊂ Ω,  d  dt  �  ΩA(t)∩Λ  {ρAeA − pA(ρA)} dx =  �  ΩA(t)∩Λ  �QA dx,  d  dt  �  ΩB(t)∩Λ  {ρBeB − pB(ρB)} dx =  �  ΩB(t)∩Λ  �QB dx,  d  dt  �  Γ(t)∩Λ  {ρSeS − pS(ρS)} dH2  x =  �  Γ(t)∩Λ  �QS dH2  x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='  Applying ( 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='1)-( 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='3), we have ( 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content=' Therefore, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='4 is proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='  □  Let us apply integration by parts to prove Theorems 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='6 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='  Proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content=' Let r ∈ {0, 1} and 0 < t < T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content=' Let ϕA, ϕB, ϕS ∈ [C∞(R3)]3  satisfying ( 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content=' A direct calculation gives  d  dε  ����  ε=0  ED[vε  A, vε  B, vε  S] = −  �  ΩA(t)  {µAD(vA) : D(ϕA) + λA(divvA)(divϕA)} dx  −  �  ΩB(t)  {µBD(vB) : D(ϕB) + λB(divvB)(divϕB)} dx  −  �  Γ(t)  {µSDΓ(vS) : DΓ(ϕS) + λS(divΓvS)(divΓϕS)} dH2  x,  and  d  dε  ����  ε=0  EW [vε  A, vε  B, vε  S]  =  �  ΩA(t)  (divϕA)πA dx +  �  ΩB(t)  (divϕB)πB dx +  �  Γ(t)  (divΓϕS)πS dH2  x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='  Using integration by parts (Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='1), we find that  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='4)  d  dε  ����  ε=0  ED+W [vε  A, vε  B, vε  S] =  �  ΩA(t)  divTA(vA, πA) · ϕA dx  +  �  ΩB(t)  divTB(vB, πB) · ϕB dx +  �  Γ(t)  divΓTS(vS, πS) · ϕS dH2  x  −  �  Γ(t)  {TA(vA, πA)nΓ} · ϕA dH2  x +  �  Γ(t)  {TB(vB, πB)nΓ} · ϕB dH2  x  −  �  ∂Ω  {TB(vB, πB)nΩ} · ϕB dH2  x,  where (TA, TB, TS) and (�TA, �TB) are defined by ( 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='7) and ( 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content=' Applying ( 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='8),  we check that  (R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='S) of ( 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='4) =  �  ΩA(t)  divTA(vA, πA) · ϕA dx +  �  ΩB(t)  divTB(vB, πB) · ϕB dx  +  �  Γ(t)  {divΓTS(vS, πS) + �TB(vB, πB)nΓ − �TA(vA, πA)nΓ} · ϕS dH2  x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='  14  HAJIME KOBA  Here we used the facts that  D(vA)nΓ · ϕA = (nΓ · {(nΓ · ∇)vA})(nΓ · ϕS) if r = 0,  D(vB)nΓ · ϕB = (nΓ · {(nΓ · ∇)vB})(nΓ · ϕS) if r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='  From fundamental lemma of calculation of variations, we have ( 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content=' Therefore,  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='6 is proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='  □  Proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content=' Fix 0 < t < T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content=' Let ψA, ψB, ψS ∈ C∞(R3) satisfying ( 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='  Since  d  dε  ����  ε=0  ET D[θε  A, θε  B, θε  S]  = −  �  ΩA(t)  κA∇θA·∇ψA dx−  �  ΩB(t)  κB∇θB·∇ψB dx−  �  Γ(t)  κS∇ΓθS·∇ΓψS dH2  x,  we apply the integration by parts (Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='1), ( 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='9), and ( 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='1) to see that  d  dε  ����  ε=0  ET D[θε  A, θε  B, θε  S] =  �  ΩA(t)  div(κA∇θA)ψA dx +  �  ΩB(t)  div(κB∇θB)ψB dx  +  �  Γ(t)  {divΓ(κS∇ΓθS) + κB(nΓ · ∇)θB − κA(nΓ · ∇)θA}ψS dH2  x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='  From fundamental lemma of calculation of variations, we have ( 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content=' Therefore,  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='7 is proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='  □  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content=' Thermodynamical Modeling  In this section we make mathematical models for multiphase flow with surface  tension and flow by our thermodynamic approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content=' Under the restrictions ( 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='1),  we apply Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='3, the first law of thermodynamics (Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='4), and our  energy densities (Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='2) to derive equations ( 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='2)-( 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content=' In this section we  consider the case when the fluids in ΩA,T , ΩB,T , ΓT are barotropic fluids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='  Let r ∈ {0, 1}, and pA, pB, pB ∈ C1(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content=' We assume that (vA, vB, vS, θA, θB, θS)  satisfies ( 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content=' We consider the energy densities defined in Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='2 as the  energy densities for multiphase flow with surface flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='  From Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='3, we admit that system ( 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='2) is the continuity equations of  our system, that is, we assume that (ρA, ρB, ρS) satisfies ( 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='  Let (QA, QB, QS) be the endothermic energies derived from energies dissipation  due to thermal diffusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content=' From Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='7, we set  �  �  �  �  �  QA = div(κA∇θA)  in ΩA,T ,  QB = div(κB∇θB)  in ΩB,T ,  QS = divΓ(κS∇ΓθS) + κB(nΓ · ∇)θB − κA(nΓ · ∇)θA  on ΓT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='  Let �Q♯ = �Q♯(x, t) be the quantity of heat supplied the fluid in Ω♯(t), where ♯ =  A, B, S, and ΩS(t) := Γ(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content=' Since eD♯ is the energy density for energy dissipation  due to the viscosities (µ♯, λ♯), we set �Q♯ = Q♯ + eD♯.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content=' Now we admit the first law of  MULTIPHASE FLOW SYSTEM WITH SURFACE TENSION AND FLOW  15  thermodynamics, that is, suppose that for every 0 < t < T and Λ ⊂ Ω,  d  dt  �  ΩA(t)∩Λ  {ρAeA − pA(ρA)} dx =  �  ΩA(t)∩Λ  (QA + eDA) dx,  d  dt  �  ΩB(t)∩Λ  {ρBeB − pB(ρB)} dx =  �  ΩB(t)∩Λ  (QB + eDB) dx,  d  dt  �  Γ(t)∩Λ  {ρSeS − pS(ρS)} dH2  x =  �  Γ(t)∩Λ  (QS + eDS) dH2  x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='  From Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='4, we have  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='1)  �  �  �  �  �  ρADA  t eA + (divvA)ΠA = divqA + eDA in ΩA,T ,  ρBDB  t eB + (divvB)ΠB = divqB + eDB in ΩB,T ,  ρSDS  t eS + (divΓvS)ΠS = divΓqS + eDS + qB · nΓ − qA · nΓ on ΓT ,  where (qA, qB, qS) and (ΠA, ΠB, ΠS) are defined by ( 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='5) and ( 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='  Let FA, FB, FS ∈ [C(R4)]3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content=' We assume that the momentum equations of our  system are written by  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='2)  �  �  �  �  �  ρADA  t vA = FA  in ΩA,T ,  ρBDB  t vB = FB  in ΩB,T ,  ρSDS  t vS = FS  on ΓT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='  From a thermodynamic point of view we assume that our system satisfies the con-  servation law of total energy, that is, (FA, FB, FS) satisfies that for each 0 < t < T,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='3)  d  dt  � �  ΩA(t)  �1  2ρA|vA|2 + ρAeA  �  dx +  �  ΩB(t)  �1  2ρB|vB|2 + ρBeB  �  dx  +  �  Γ(t)  �1  2ρS|vS|2 + ρSeS  �  dH2  x  �  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='  Using the transport theorems with ( 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='2), ( 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='1), ( 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='2), we see that  (L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='S) of ( 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='3) =  �  ΩA(t)  (ρADA  t vA · vA + ρADA  t eA) dx  +  �  ΩB(t)  (ρBDB  t vB · vB + ρBDB  t eB) dx +  �  Γ(t)  (ρSDS  t vS · vS + ρSDS  t eS) dH2  x  =  �  ΩA(t)  {FA · vA + divqA + eDA − (divvA)ΠA} dx  +  �  ΩB(t)  {FB · vB + divqB + eDB − (divvB)ΠB} dx  +  �  Γ(t)  {FS · vS + divΓqS + eDS − (divΓvS)ΠS + qB · nΓ − qA · nΓ} dH2  x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='  16  HAJIME KOBA  Applying the integration by parts with ( 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='1), we observe that  (L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='S) of ( 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='3) =  �  ΩA(t)  {FA − divTA(vA, ΠA)} · vA dx  +  �  ΩB(t)  {FB − divTB(vB, ΠB)} · vB dx +  �  Γ(t)  {FS − divΓTS(vS, ΠS)} · vS dH2  x  +  �  Γ(t)  �TA(vA, ΠA)nΓ · vS dH2  x −  �  Γ(t)  �TB(vB, ΠB)nΓ · vS dH2  x,  where (TA, TB, TS) and (�TA, �TB) are defined by ( 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='7) and ( 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content=' Here we used the  facts that  �  Γ(t)  {TA(vA, ΠA)nΓ} · vA dH2  x =  �  Γ(t)  �TA(vA, ΠA)nΓ · vS dH2  x,  �  Γ(t)  {TB(vB, ΠB)nΓ} · vB dH2  x =  �  Γ(t)  �TB(vB, ΠB)nΓ · vS dH2  x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='  Thus, we set  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='4)  �  �  �  �  �  FA = divTA(vA, ΠA),  FB = divTB(vB, ΠB),  FS = divΓTS(vS, ΠS) + �TB(vB, ΠB)nΓ − �TA(vA, ΠA)nΓ  to see that (L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='S) of ( 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='3) equals to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content=' Combining ( 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='1), ( 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='2), ( 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='4), we  have ( 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='3) and ( 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content=' Therefore, we derive ( 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='2)-( 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='4) by our thermodynamic  approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='  Finally, we introduce another approach to derive the momentum equations ( 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='  We assume that the time rate of change of the momentum equals to the forces  derived from the variation of energies dissipation due to the viscosities, that is,  suppose that for every 0 < t < T and Λ ⊂ Ω,  d  dt  �  ΩA(t)∩Λ  ρAvA dx =  �  ΩA(t)∩Λ  FA dx,  d  dt  �  ΩB(t)∩Λ  ρBvB dx =  �  ΩB(t)∩Λ  FB dx,  d  dt  �  Γ(t)∩Λ  ρSvS dH2  x =  �  Γ(t)∩Λ  FS dH2  x,  where (FA, FB, FS) is defined by ( 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content=' Using the transport theorems with ( 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='2),  we have ( 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content=' Conservative Forms and Conservation Laws  We study the conservation laws of our model to prove Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='  Proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content=' Let r ∈ {0, 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content=' Direct calculations give (i) (see Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='  We now prove (ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content=' From Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='3 and the arguments in Section 4, we see  ( 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='12) and ( 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content=' Using the transport theorems (Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='1) with ( 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='2) and  MULTIPHASE FLOW SYSTEM WITH SURFACE TENSION AND FLOW  17  ( 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='4), we see that  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='1)  d  dt  � �  ΩA(t)  1  2ρA|vA|2 dx +  �  ΩB(t)  1  2ρB|vB|2 dx +  �  Γ(t)  1  2ρS|vS|2 dH2  x  �  =  �  ΩA(t)  ρADA  t vA · vA dx +  �  ΩB(t)  ρBDB  t vB · vB dx +  �  Γ(t)  ρSDS  t vS · vS dH2  x  =  �  ΩA(t)  divTA(vA, πA) · vA dx +  �  ΩB(t)  divTB(vB, πB) · vB dx  +  �  Γ(t)  {divΓTS(vS, πS) + �TB(vB, πB)nΓ − �TA(vA, πA)nΓ} · vS dH2  x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='  Applying integration by parts (Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='1) and ( 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='1), we check that  (R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='S) of ( 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='1) =  �  ΩA(t)  {−eDA + (divvA)πA} dx  +  �  ΩB(t)  {−eDB + (divvB)πB} dx +  �  Γ(t)  {−eDS + (divΓvS)πS} dH2  x,  where (eDA, eDB, eDS) is defined by ( 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content=' Integrating with respect to t, we have  ( 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='  Finally, we show (iii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content=' Assume that r = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content=' Using the transport and divergence  theorems (Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='1 and Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='1) with ( 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='12), we see that  d  dt  � �  ΩA(t)  ρAvA dx +  �  ΩB(t)  ρBvB dx +  �  Γ(t)  ρSvS dH2  x  �  =  �  ΩA(t)  divTA(vA, πA) dx +  �  ΩB(t)  divTB(vB, πB) dx  +  �  Γ(t)  {divΓTS(vS, πS) + TB(vB, πB)nΓ − TA(vA, πA)nΓ} dH2  x = t(0, 0, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='  Integrating with respect to t, we have ( 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content=' Therefore, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='8 is proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='  □  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content=' Thermodynamic Potential  We investigate thermodynamic potential for our model to prove Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='  Proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content=' We only prove the case when ♯ = S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content=' Let r ∈ {0, 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content=' Assume  that ρS and θS are positive functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content=' Set hS = eS + πS/ρS, F H  S  = eS − θSςS,  F G  S = hS − θSςS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content=' Assume that eS satisfies the thermodynamic identity:  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='1)  DS  t eS = θSDS  t ςS − πSDS  t  � 1  ρS  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='  We first derive ( 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content=' By ( 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='2) and ( 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='3), we see that  ρSDS  t hS = ρSDS  t eS + ρSDS  t  �πS  ρS  �  = divΓqS + eDS + qB · nΓ − qA · nΓ + DS  t πS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='  18  HAJIME KOBA  This shows that  DN  t (ρShS) + divΓ(ρShSvS − qS) = ρSDS  t hS − divΓqS  = eDS + DS  t πS + qB · nΓ − qA · nΓ,  which is ( 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='  Next we show ( 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content=' From ( 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='3) and ( 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='1), we find that  θSρSDS  t ςS = ρSDS  t eS + ρSπSDS  t  � 1  ρS  �  = divΓqS + eDS + qB · nΓ − qA · nΓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='  Using the above equality and ( 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='16), we check that  DN  t (ρSςS) + divΓ  �  ρSςSvS − qS  θS  �  = ρSDS  t hS − divΓ  �qS  θS  �  = eDS  θS  + qS · gradΓθS  θ2  S  + qB · nΓ − qA · nΓ  θS  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='  Thus, we have ( 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='  Finally, we derive ( 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='18) and ( 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content=' Applying ( 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='1) and ( 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='2), we see that  ρSDS  t F H  S + ρSςSDS  t θS = ρSDS  t eS − ρSθSDS  t ςS  = −(divΓvS)πS,  and that  ρSDS  t F G  S + ρSςSDS  t θS = ρSDS  t hS − ρSθSDS  t ςS  = DS  t πS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='  Therefore, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='9 is proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='  □  Data Availability : The author declares that data sharing not applicable to this  article as no datasets were generated or analyzed during the current study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='  Conflict of interest : The author declares no conflict of interest associated with  this manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='  Acknowledgments : This work was partly supported by the Japan Society for  the Promotion of Science (JSPS) KAKENHI Grant Number JP21K03326.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
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+page_content='  [15] Hajime Koba, On Generalized Compressible Fluid Systems on an Evolving Surface with a  Boundary, preprint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content=' arXiv:1810.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='07909.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content=' to appear in Quart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='  [16] Hajime Koba, On Generalized Diffusion and Heat Systems on an Evolving Surface with a  Boundary, Quart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
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+page_content=' arXiv:2211.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
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+page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
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+page_content='  MR3614501.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content=' Errata to Energetic variational approaches for incompressible fluid systems on  an evolving surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content=' Quart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
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+page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content=' Angew.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
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+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
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+page_content=' (Japanese) Geometric mechanics (Japanese) (Kyoto,  2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='  Surikaisekikenkyusho Kokyuroku No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
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+page_content=' Monographs in Mathematics, 105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content=' Birkh¨auser/Springer, [Cham], 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content=' xix+609 pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='  ISBN: 978-3-319-27697-7;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content=' 978-3-319-27698-4 MR3524106  [22] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content=' Scriven, Dynamics of a fluid interface Equation of motion for Newtonian surface fluids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='  Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content=' Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
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+page_content='  [23] John  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='  Slattery,  Momentum  and  moment-of-momentum  balances  for  moving  sur-  facesChemical Engineering Science, Volume 19, 1964, Pages 379–385.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='  [24] John C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content=' Slattery, Leonard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content=' Sagis, and Eun-Suok Oh, Interfacial transport phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content=' Second  edition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content=' Springer, New York, 2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content=' xviii+827 pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content=' ISBN: 978-0-387-38438-2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content=' 0-387-38438-3  MR2284654.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='  [25] Leon Simon, Lectures on geometric measure theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content=' Proceedings of the Centre for Mathe-  matical Analysis, Australian National University, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content=' Australian National University, Centre  for Mathematical Analysis, Canberra, 1983.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content=' vii+272 pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content=' ISBN: 0-86784-429-9 MR0756417.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='  [26] Michael E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content=' Taylor, Analysis on Morrey spaces and applications to Navier-Stokes and other  evolution equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content=' Comm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content=' Partial Differential Equations 17 (1992), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content=' 9-10, 1407–1456.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='  MR1187618.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='  Graduate School of Engineering Science, Osaka University,, 1-3 Machikaneyamacho,  Toyonaka, Osaka, 560-8531, Japan  Email address: iti@sigmath.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='es.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='osaka-u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfCALf/content/2301.02860v1.pdf'}
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