Thermodynamical Modeling of Multiphase Flow System with Surface Tension and Flow

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. We make mathematical models for multiphase flow with surface flow by our energetic variational and thermodynamic approaches. 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. We study the conservative forms and conservation laws of our system by using the surface transport theorem and integration by parts. Moreover, we investigate the enthalpy, the entropy, the Helmholtz free energy, and the Gibbs free energy of our model by applying the thermodynamic identity. The key idea of deriving surface tension and viscosities is to make use of both the first law of thermodynamics and our energy densities.


Introduction
We are interested in a mathematical modeling of a soap bubble floating in the air.When we focus on a soap bubble, we can see the fluid flow in the bubble.We call the fluid flow in the bubble a surface flow.We can consider a surface flow as a fluid-flow on an evolving surface.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.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.More precisely, we apply the first law of thermodynamics and our energy densities to derive our multiphase flow system with surface tension and flow.
Remark 1.1.We call v S a total velocity, and π S a total pressure.Total velocity means that v S can be divided into surface velocity u S and motion velocity w S , that is, v S = u S + w S .Total pressure means one that includes surface pressure and tension.In this paper, we focus on the total velocity and the total pressure.

Let us introduce several operators and notations. For each
, and . Define the orthogonal projection P Γ to a tangent space by and the mean curvature where I 3×3 is the 3 × 3 identity matrix, and ⊗ denotes the tensor product.It is easy to check that P Γ n Γ = t (0, 0, 0) and Let us explain the key restrictions on the boundaries ∂Ω T and Γ T .We assume that (1.1) where r ∈ {0, 1}.We call and a no-slip boundary condition if r = 0. Note that we do not consider phase transition in this paper.This paper has three purposes.The first purpose is to derive the following multiphase flow system with surface tension and flow: where Here . The symbol : denotes the Frobenius inner product, that is, , where M, N are two 3×3 matrices, and [M] ij denotes the (i, j)-component of the matrix M. We call q A , q B , q S the heat fluxes, e D A , e D B , e D S the energy densities for the energy dissipation due to the viscosities, D(v A ), D(v B ) strain rate tensors, D Γ (v S ) a surface strain tensor, T A , T B stress tensors, and T S a surface stress tensor.We often call T S the surface stress tensor determined by the Boussinesq-Scriven law.More precisely, under the restrictions ( 1.1) we apply our energy densities and thermodynamic approaches to derive ( 1.2)-( 1.4).See Section 4 for details.
We often call π S H Γ n Γ surface tension.
(ii) If the fluids in Ω A,T , Ω B,T , Γ T are barotropic fluids, then we can write Here p A , p B , p S are three C 1 -functions, p = p (r) = dp/dr(r).See Theorem 2.4, Remark 2.5, and Section 4 for details.
The second purpose is to study the conservative forms and conservation laws of system ( 1.2)-( 1.4).In fact, if we set , 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: (1.9) Moreover, any solution to system ( 1.1)-( 1.4) satisfies that for t 1 < t 2 , (1.12) ) ) Here dH 2 x denotes the 2-dimensional Hausdorff measure.Under some assumptions (see Theorem 2.8), any solution to system ( 1.1)-( 1.3) satisfies that for t 1 < t 2 (1.15) x .
We often call ( 1.12), ( 1.13), ( 1.14), and ( 1.15), 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.See Theorem 2.8 and Section 5 for details.
) in this paper, we can define D N t for ρ S , ρ S v S .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. Assume that (ρ , θ ) are positive functions.Set the enthalpy h by h = e + π /ρ .Then Suppose that the thermodynamic identity (Gibbs [9]): Set the Helmholtz free energy F H by F H = e − θ ς .Then (1.18) Set the Gibbs free energy See Theorem 2.9 and Section 6 for details.
Note that We often call 1/ρ a specific volume.
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.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.To overcome these difficulties, we apply the first law of thermodynamics (Theorem 2.4), our energy densities (Definition 2.2), and the conservation law of total energy to derive equations ( 1.3) and ( 1.4).See Section 4 for details.
Let us mention the study of surface flow (interfacial flow).Boussinesq [5] first discovered the existence of surface flow.Scriven [22] considered their surface stress tensor.Slattery [23] investigated some properties of the surface stress tensor determined by the Boussinesq-Scriven law (see T S in ( 1.7)).Then many researchers have studied surface flow (see Slattery-Sagis-Oh [24] and Gatignol-Prud'homme [8] for the study of interfacial phenomena).
Let us state derivations of the governing equations for the motion of the viscous fluid on manifolds and surfaces.Taylor [26] introduced their surface stress tensor to make their incompressible viscous fluid system on a manifold.Mitsumatsu-Yano [20] applied their energetic variational approach to derive their incompressible viscous fluid system on a manifold.Arnaudon-Cruzeiro [2] made use of their stochastic variational approach to derive their incompressible viscous fluid system on a manifold.Koba-Liu-Giga [18] employed their energetic variational approach and the generalized Helmholtz-Weyl decomposition on a closed surface to derive their incompressible fluid systems on an evolving closed surface.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 surface with a boundary.This paper modifies and improves the methods in [14,15] to derive our multiphase flow system.Now we mention results for modeling of multiphase flow system with surface flow.Bothe-Prüss [4] made their multiphase flow system with surface flow by using the surface stress tensor determined by the Boussinesq-Scriven law.Koba [17] derived the inviscid multiphase flow system with surface flow by applying a geometric variational approach.This paper derives our multiphase flow system from a thermodynamic point of view.Therefore, our modeling methods are different from ones in [4] and [17].
Finally, we introduce some results and textbooks related to this paper.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.
Feireisl [7] studied the motion of the viscous fluid in a domain from a thermodynamic point of view.We refer the readers to Gyarmati [12] and Gurtin-Fried-Anand [11] for the theory of thermodynamics, Chapter XIII in Angel [1] for thermodynamical potential such as internal energy, enthalpy, entropy, and free energies, and Prüss-Simonett [21] for several elliptic and parabolic equations on hypersurfaces.
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.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.In Section 4, we apply our thermodynamic approaches to make mathematical models for multiphase flow with surface tension and flow.In Section 5, we study the conservation and energy laws of our system.In Section 6, we investigate the thermodynamic potential for our system.

Main Results
We first introduce the transport theorems and the energy densities for our multiphase flow system.Then we state the main results.Definition 2.1 (Ω T is flowed by the velocity fields (v A , v B , v S )).We say that Ω T is flowed by the velocity fields , where = A, B, S, and j = 1, 2, 3.
We often call ( 2.1), ( 2.2) the transport theorems, and ( 2.2) the surface transport theorem.The derivation of the surface transport theorem can be founded in [3,10,6,18].Throughout this paper we assume that Ω T is flowed by the velocity fields (v A , v B , v S ).

Definition 2.2 (Energy densities). Set
We call e K the kinetic energy, e D the energy density for the energy dissipation due to the viscosities (µ , λ ), e W the power density for the work done by the pressure π , and e Q the energy density for the energy dissipation due to thermal diffusion.
We now state the main results of this paper.From Definition 2.1, we have Proposition 2.3 (Continuity equations).Assume that for each 0 < t < T and Λ ⊂ Ω, The proof of Proposition 2.3 is left for the readers.

Then
(2.4) (ii) Let p A , p B , p S ∈ C 1 (R).Suppose that for every 0 < t < T and Λ ⊂ Ω, Then Remark 2.5.(i) We can write system ( 2.4) as follows: where D f = ρ D t f .Therefore, we call Theorem 2.4 the first law of thermodynamics in this paper.See also (ii) in Remark 1.4.
(ii) The pressures (Π A , Π B , Π S ) derived from the assertion (ii) of Theorem 2.4 correspond to the pressures derived from an energetic variational approach (see [17]).
Next we consider the variation of our dissipation energies.Let r ∈ {0, 1} and 0 < Then we have

and
(2.9) For each variation ( We call E D the energy dissipation due to viscosities, E W the work done by pressures, and E T D the energy dissipation due to thermal diffusion. Theorem 2.6 (Forces derived from variation of dissipation energies).Let r ∈ {0, 1}, 0 < t < T , and Then where (T A , T B , T S ) and ( T A , T B ) are defined by ( 1.7) and ( 1.8), respectively.Theorem 2.7 (Endothermic energies derived from thermal diffusion).Let 0 < t < T , and

Application of Transport Theorems and Integration by Parts
We apply the transport theorems (Definition 2.1) and several formulas for integration by parts (Lemma 3.1) to prove Theorems 2.4, 2.6, and 2.7.Lemma 3.1 (Formulas for integration by parts).Fix 0 ≤ t < T and j = 1, 2, 3. Then for every f, g ∈ C 1 (R 3 ), where ) denotes the unit outer normal vector at x ∈ Γ(t) and ) the unit outer normal vector at x ∈ ∂Ω (see Figure 1).
Applying the Gauss divergence theorem and the surface divergence theorem (Section 9 in [25], Theorem 2.3 in [16]), we can prove Lemma 3.1.We now make use of the transport theorems to prove Theorem 2.4.
Let us apply integration by parts to prove Theorems 2.6 and 2.7.
Proof of Theorem 2.6.Let r ∈ {0, 1} and 0 Using integration by parts (Lemma 3.1), we find that where (T A , T B , T S ) and ( T A , T B ) are defined by ( 1.7) and ( 1.8).Applying ( 2.8), we check that (R.H.S) of ( 3.4) = thermodynamics, that is, suppose that for every 0 < t < T and Λ ⊂ Ω, From Theorem 2.4, we have where (q A , q B , q S ) and (Π A , Π B , Π S ) are defined by ( 1.5) and ( 2.6).Let F A , F B , F S ∈ [C(R 4 )] 3 .We assume that the momentum equations of our system are written by From a thermodynamic point of view we assume that our system satisfies the conservation law of total energy, that is, (F A , F B , F S ) satisfies that for each 0 < t < T , Using the transport theorems with ( 1.2), ( 4.1), ( 4.2), we see that Applying the integration by parts with ( 1.1), we observe that (L.H.S) of ( 4.3) = where (T A , T B , T S ) and ( T A , T B ) are defined by ( 1.7) and ( 1.8).Here we used the facts that Thus, we set (4.4) to see that (L.H.S) of ( 4.3) equals to zero.Combining ( 4.1), ( 4.2), ( 4.4), we have ( 1.3) and ( 1.4).Therefore, we derive ( 1.2)-( 1.4) by our thermodynamic approach.Finally, we introduce another approach to derive the momentum equations ( 1.4).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, where (F A , F B , F S ) is defined by ( 2.10).Using the transport theorems with ( 1.2), we have ( 1.4).

Conservative Forms and Conservation Laws
We study the conservation laws of our model to prove Theorem 2.8.

Proof of
Applying integration by parts (Lemma 3. where (e D A , e D B , e D S ) is defined by ( 1.6).Integrating with respect to t, we have ( 1.14).Finally, we show (iii).Assume that r = 1.Using the transport and divergence theorems (Definition 2.1 and Lemma 3.1) with ( 2.12), we see that x = t (0, 0, 0).
Integrating with respect to t, we have ( 1.15).Therefore, Theorem 2.8 is proved.

Thermodynamic Potential
We investigate thermodynamic potential for our model to prove Theorem 2.9.
Proof of Theorem 2.9.We only prove the case when = S. = div Γ q S + e D S + q B • n Γ − q A • n Γ + D S t π S .