Top Banner
See discussions, stats, and author profiles for this publication at: https://www.researchgate.net/publication/322959186 Three-Phase Model of Visco-Elastic Incompressible Fluid Flow and its Computational Implementation Article in Communications in Computational Physics · February 2019 DOI: 10.4208/cicp.OA-2017-0167 CITATION 1 READS 378 3 authors: Some of the authors of this publication are also working on these related projects: ion channel View project Multiscale Modeling View project Shixin Xu The Fields Institute for Research in Mathematical Sciences 24 PUBLICATIONS 105 CITATIONS SEE PROFILE Mark Alber University of California, Riverside 225 PUBLICATIONS 3,260 CITATIONS SEE PROFILE Zhiliang Xu University of Notre Dame 72 PUBLICATIONS 966 CITATIONS SEE PROFILE All content following this page was uploaded by Shixin Xu on 06 February 2018. The user has requested enhancement of the downloaded file.
50

Three-Phase Model of Visco-Elastic Incompressible …zxu2/blood_clot/three-phase-complex...Three-phase Model of Visco-elastic Incompressible Fluid Flow and its Computational Implementation

Jul 07, 2020

Download

Documents

dariahiddleston
Welcome message from author
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
Page 1: Three-Phase Model of Visco-Elastic Incompressible …zxu2/blood_clot/three-phase-complex...Three-phase Model of Visco-elastic Incompressible Fluid Flow and its Computational Implementation

See discussions, stats, and author profiles for this publication at: https://www.researchgate.net/publication/322959186

Three-Phase Model of Visco-Elastic Incompressible Fluid Flow and its

Computational Implementation

Article in Communications in Computational Physics · February 2019

DOI: 10.4208/cicp.OA-2017-0167

CITATION

1READS

378

3 authors:

Some of the authors of this publication are also working on these related projects:

ion channel View project

Multiscale Modeling View project

Shixin Xu

The Fields Institute for Research in Mathematical Sciences

24 PUBLICATIONS 105 CITATIONS

SEE PROFILE

Mark Alber

University of California, Riverside

225 PUBLICATIONS 3,260 CITATIONS

SEE PROFILE

Zhiliang Xu

University of Notre Dame

72 PUBLICATIONS 966 CITATIONS

SEE PROFILE

All content following this page was uploaded by Shixin Xu on 06 February 2018.

The user has requested enhancement of the downloaded file.

Page 2: Three-Phase Model of Visco-Elastic Incompressible …zxu2/blood_clot/three-phase-complex...Three-phase Model of Visco-elastic Incompressible Fluid Flow and its Computational Implementation

Three-phase Model of Visco-elastic Incompressible Fluid

Flow and its Computational Implementation

Shixin Xu1, Mark Alber1,2,∗, Zhiliang Xu2,∗

Abstract

Energetic Variational Approach is used to derive a novel thermodynamicallyconsistent three-phase model of a mixture of Newtonian and visco-elastic flu-ids. The model which automatically satisfies the energy dissipation law andis Galilean invariant, consists of coupled Navier-Stokes and Cahn-Hilliard e-quations. Modified General Navier Boundary Condition with fluid elasticitytaken into account is also introduced for using the model to study movingcontact line problems. Energy stable numerical scheme is developed to solvesystem of model equations efficiently. Convergence of the numerical schemeis verified by simulating a droplet sliding on an inclined plane under gravi-ty. The model can be applied for studying various biological or biophysicalproblems. Predictive abilities of the model are demonstrated by simulat-ing deformation of venous blood clots with different visco-elastic propertiesand experimentally observed internal structures under different biologicallyrelevant shear blood flow conditions.

Keywords:Phase field method; Energetic Variational Approach; multi-phase flow;visco-elasticity; variable density; slip boundary condition; deformation ofblood clot; thrombus.

∗Authors for correspondenceEmail addresses: [email protected] (Mark Alber ), [email protected] (Zhiliang Xu )

1Department of Mathematics, University of California, Riverside, Riverside, CA, 92521,USA

2Department of Applied and Computational Mathematics and Statistics, University ofNotre Dame, Notre Dame, IN, 46556, USA

February 6, 2018

Page 3: Three-Phase Model of Visco-Elastic Incompressible …zxu2/blood_clot/three-phase-complex...Three-phase Model of Visco-elastic Incompressible Fluid Flow and its Computational Implementation

1. Introduction

Phase field models [3, 5, 19, 28, 29, 40, 44, 46, 84, 86, 87] derived usingthe energy-based variational formulation, are widely used for studying multi-phase fluid flow problems. Labeling function or phase function is used in aphase field model to represent each of the phases. The sharp interface sepa-rating different species is replaced by narrow transition layer in which speciesmix. Free energy density functional of the labeling functions is constructedfor coupling different phases. (See, among others, [3, 47, 41] for reviews ofphase field approach.) A careless design of the free energy density function-al may lead to meta stable states [11]. For instance, traditional pairwisecombinations of double-well free energy functionals for coupling multiple flu-id components may give rise to non-physical results, such as growth of onephase due to the presence of saddle points inside the Gibbs triangle [88].

Additional problems with deriving a phase field model arise when somefluid components are non-Newtonian. Many existing non-Newtonian flowphase field models [2, 7, 10] do not satisfy the energy dissipation law. Thisimplies that numerical schemes designed for solving system of equations ofthese models likely do not to satisfy the discrete energy dissipation law ei-ther, and can result in large numerical errors [50]. These numerical errorssignificantly undermine accuracy of numerical model solutions over long timeperiods.

While most of the existing models [1, 3, 5, 17, 18, 37, 47, 52, 86, 89]focus on two-phase or Newtonian fluids, many biological and biophysical ap-plications require multi-phase or non-Newtonian fluid flow models. Thereare only few existing three (or more)-phase field models [20, 45, 46, 79].In particular, Wu and Xu [79] established the unisolvent property of coeffi-cient matrix involved in N-phase models based on pairwise surface tensions.By using obtained matrix, authors derived an N-phase inherently invariantCahn-Hillard model from the free energy functional. Important properties ofWu and Xu models are that the dynamics of concentrations are independentof the choice of phase variable, and the symmetric positive-definite propertyof the coefficient matrix can be proved equivalent to some physical conditionfor pairwise surface tensions. Among other multi-phase models, the model in[45] does not include components representing hydrodynamics, and modelsin [20, 46, 79] describe only Newtonian fluids. We use the Energy VariationalApproach (EnVarA) [21, 83] to derive in this paper a novel thermodynami-cally consistent phase field model of three-phase incompressible fluid system

2

Page 4: Three-Phase Model of Visco-Elastic Incompressible …zxu2/blood_clot/three-phase-complex...Three-phase Model of Visco-elastic Incompressible Fluid Flow and its Computational Implementation

with visco-elastic fluid components. Main novel modeling and numerical con-tributions of the paper in comparison with existing models, are as follows.

• A systematic approach is introduced to derive phase field model cou-pling Newtonian and Non-Newtonian fluids with large variations indensities or viscosities of individual fluid components. The Boussinesqapproximation under the assumption that density ratio between twofluids is relatively small [51, 50] is not needed in our model. Com-ponents of the fluid mixture are combined in a binary tree manner[12, 73]. The feasibility of this approach is demonstrated by derivinga three-phase fluid flow model, in which two of the fluid componentsare visco-elastic. The resulting model can be reduced in a physicallyconsistent manner to the two-phase model [51].

• The derived model is Galilean invariant and automatically satisfies theenergy dissipation law resulting in straightforward development of anefficient and energy stable numerical scheme. All model equations aredescribed in the Eulerian coordinate system which makes computation-al implementation of the model convenient. This is in contrast withmany computational models coupling Navier-Stokes equations and e-lastic equations for simulating fluid-structure interaction problems, inwhich Navier-Stokes equations are solved on a fixed mesh while elasticequations are solved on the Lagrangian mesh. Computational imple-mentation of interpolation between meshes to impose boundary condi-tion at the fluid-structure interface is not trivial.

• Modified General Navier Boundary Condition (GNBC) [58, 59, 60] withfluid elasticity taken into account is used for solving the moving contactline problem [40, 58, 63, 77] which describes movement of an interfaceseparating visco-elastic and pure Newtonian fluids on the solid wall.

• Efficient and energy stable numerical scheme is developed for solvingthe obtained model system with large variations in densities or viscosi-ties. The model system which couples Navier-Stokes and Cahn-Hilliardequations, is solved by using combination of the energy splitting method[22, 23] and the pressure stabilization method [68].

Convergence study of the new energy stable scheme is accomplished bysimulating deformation and motion of visco-elastic droplets on solid surface.

3

Page 5: Three-Phase Model of Visco-Elastic Incompressible …zxu2/blood_clot/three-phase-complex...Three-phase Model of Visco-elastic Incompressible Fluid Flow and its Computational Implementation

Creep-relaxation test of complex fluid is used to validate the approach adopt-ed by the model for representing visco-elasticity of the fluid. Additionally,simulations of a droplet wetting process are used to demonstrate differencesbetween fluid visco-elasticity models which give fluid-like and solid-like be-haviors, respectively.

To demonstrate the feasibility of the new model for studying biologi-cal and biophysical problems involving non-Newtonian fluids, it is appliedfor studying stability of venous blood clots with specific multi-componentstructures observed in experiments [43, 82]. Simulations of deformation ofhemophilic and normal blood clots, which consist of platelet aggregates andfibrin network, under physiologically relevant shear blood flows, are shownto be in good agreement with experimental observations.

The paper is organized as follows. Section 2 describes derivation of thethree-phase field model with variable densities and viscosities of fluid compo-nents. Moreover, two of the fluid components in the model are visco-elastic.An energy stable numerical scheme is introduced in Section 3 for solving mod-el equations described in Section 2. Section 4 describes simulation results.Conclusions are provided in Section 5.

2. Derivation of the Three-phase Model using EnVarA

A three-phase model describing mixture of Newtonian and non-Newtonianfluid components is derived in this section by using binary tree approach andapplying the EnVarA to ensure that the derived model satisfies the energydissipation law. We first outline below the general idea of the EnVarA andthen describe in detail steps employed to derive the three-phase model.

The EnVarA is based on the energy dissipation law [21, 26, 60, 66, 83], theLeast Action Principle (LAP), the Maximum Dissipation Principle (MDP)[25, 30, 39, 53, 54, 76], and Newton’s force balance law.

Under the assumption that the system is isothermal, the model derivedusing the EnVarA should obey the energy dissipation law, which states thatthe entropy change balances with the energy dissipation

d

dtEtotal + ∆ = 0 ⇔ d

dtEtotal = −∆ . (2.1)

Here Etotal = K + U − TS = K +H is the total energy of the system. K isthe kinetic energy, U is the internal energy, T is the temperature, S is theentropy, and H is the Helmholtz free energy. ∆ is the dissipation functional

4

Page 6: Three-Phase Model of Visco-Elastic Incompressible …zxu2/blood_clot/three-phase-complex...Three-phase Model of Visco-elastic Incompressible Fluid Flow and its Computational Implementation

which is usually represented as a quadratic function of certain rates such asthe fluid velocity u. (Other notations used in the paper are explained inAppendix A.)

The action functional for a Hamiltonian (or conservative) system is de-

fined as follows A =∫ t∗

0

∫Ω

(K − H)dxdt. The LAP states that the actionfunctional can be optimized with respect to the flow map x(t) = x(X, t)(with x(X, 0) = X(t = 0)) by taking the variation with respect to x. HereX stands for the Lagrangian coordinate system, which is called the referenceconfiguration, and x is the Eulerian coordinate, which is called deformedconfiguration. This gives rise to the variational derivative δA of the actionfunctional δA =

∫ t∗0

∫Ω0

[Fcon] · δxdXdt, where Fcon is the conservative force,Ω0 is the Lagrangian reference domain of Ω, and the trajectory x(t) is thepath that particle X moves from position x(X, 0) at time t = 0 to positionx(X, t∗) at time t = t∗ [4].

The MDP states that variation of ∆ with respect to certain rate (e.g.,velocity u) in the Eulerian coordinate system results in the dissipative forceFdis, which satisfies δ(1

2∆) =

∫Ω

[Fdis] · δudx. Note that the factor 12

is dueto the underlying assumption that ∆ is a quadratic function of u. In theend, the equation of motion is obtained by using the force balance law, i.e.,Fcon = Fdis.

The rest of this section is devoted to derivation of the three-phase modeldescribing Newtonian and non-Newtonian fluids mixture by using the En-VarA. A novel general Navier boundary condition is also introduced for im-posing the wall boundary condition for studying moving contact line probleminvolving visco-elastic fluid. This boundary condition includes contributionof the elasticity of the non-Newtonian fluid to the contact line slip velocity.

2.1. Three-phase model derivation

We consider in this section a complex fluid mixture consisting of visco-elastic fluids A and B, and Newtonian fluid C. These three fluid componentsof the mixture are separated in two groups: the visco-elastic fluids mixtureAB composed of fluids A and B, and the Newtonian fluid C. The volumefraction of the visco-elastic fluids mixture AB is denoted by φ2(φ2 ∈ [0, 1]),while the volume fraction of the fluid C is 1−φ2. Furthermore, φ1(φ1 ∈ [0, 1])is introduced to represent the volume fraction of fluid A in the mixture AB,and 1− φ1 is the volume fraction of fluid B in the mixture AB.

5

Page 7: Three-Phase Model of Visco-Elastic Incompressible …zxu2/blood_clot/three-phase-complex...Three-phase Model of Visco-elastic Incompressible Fluid Flow and its Computational Implementation

2.1.1. Definition of total energy and dissipation functionals

The total energy functional of the modeled complex fluid is defined as

Etotal = Ekin︸︷︷︸Macroscale

+Ecoh + Eela + Ew︸ ︷︷ ︸Microscale

, (2.2)

where Ekin is the kinetic energy, Ecoh is the mixing energy, Eela is the elasticenergy, and Ew is the specific wall energy.

The kinetic energy accounts for the transport of the trinary fluid mixtureand is defined as:

Ekin =

∫Ω

(1

2ρ|u|2

)dx , (2.3)

where ρ = ρ(x, t) = ρ(φ1, φ2, ρA, ρB, ρC , t) is the mixture density with ρibeing the density of phase i, i = A, B, C, and u the velocity of the fluidmixture, respectively.

According to the Cahn and Hilliard approach [13], the mixing energy Ecohrepresents competition between a homogeneous bulk mixing energy densityterm G(φ) (‘hydrophobic’ part) that establishes total separation of the phases

into pure components, and a gradient distortional term |∇φ|22

(‘hydrophilic’part) that represents the nonlocal interactions between different componentsand penalizes spatial heterogeneity. Therefore, the mixing energy is definedas follows:

Ecoh = Ecoh1 + Ecoh2

=

∫Ω

λ1φ22

(G1(φ1) +

γ21

2|∇φ1|2

)dx

+

∫Ω

λ2

(G2(φ2) +

γ22

2|∇φ2|2

)dx ,

(2.4)

where λi is the mixing energy density, γi is the capillary width of the in-terface, G1(φ1) = αφ3

1(φ14− β) [73], which has a nonzero minimum, is the

hydrophobic energy of the visco-elastic mixture AB. The choice of this cohe-sion energy G1(φ1) is for the purpose of using this model to describe complexfluids such as hydro-gel in which polymer network forms physical links andentanglements. The double well potential G2(φ2) = 1

4φ2

2(1 − φ2)2 [86] is thehydrophobic energy of the Newtonian and visco-elastic fluid mixture. In themixing energy Ecoh1, a factor φ2

2 is included because this energy makes senseonly when the volume fraction of the visco-elastic mixture AB is not zero.

6

Page 8: Three-Phase Model of Visco-Elastic Incompressible …zxu2/blood_clot/three-phase-complex...Three-phase Model of Visco-elastic Incompressible Fluid Flow and its Computational Implementation

To account for the visco-elastic property of the fluid mixture AB, we in-troduce an elastic free energy Eela. In the present paper, the Kelvin-Voigtmodel [56] is used to describe the fluid visco-elasticity. Following the re-sults in [49], the deformation gradient tensor F (X, t) defined by Fij = ∂xi

∂Xj,

in which x is the current (Eulerian) coordinate and X is the reference (La-grangian) coordinate, is introduced to write the elastic energy in the Eulerianframework,

Eela =

∫Ω

λe1φ22

2|F |2dx =

∫Ω

λe1φ22

2tr(FF T )dx , (2.5)

where λe1 = λe1(φ1, λA, λB) is the elastic energy density of non-Newtonianfluid mixture AB. λA and λB are the elastic energy density of fluids A andB, respectively. φ2

2 is used to ensure that only the elasticity of the mixtureAB is considered. tr(FF T ) is the trace of FF T .

If ∇ · F (X, 0) = 0 is satisfied at t = 0, ∇ · F = 0 for t ≥ 0 by thetransport equation of F [49]. Moreover, there exists a vector Ψ = (Ψ1,Ψ2)T

in the two-dimensional space [49], such that

F =

(−∂x2Ψ1 −∂x2Ψ2

∂x1Ψ1 ∂x1Ψ2

).

In the end, the elastic energy can be represented by using Ψ as

Eela =

∫Ω

λe2|F |2dx =

∫Ω

λe2

tr(FF T )dx

=

∫Ω

λe2

((∂x1Ψ1)2 + (∂x2Ψ1)2 + (∂x1Ψ2)2 + (∂x2Ψ2)2)dx

=

∫Ω

λe2|∇Ψ|2dx ,

where λe = φ22λe1.

For numerical study of the moving contact line problem [29, 40, 58, 59,63, 64] involving the interface of fluids intersecting with the wall, a wallfree energy Ew is introduced into the total energy functional to mimic theinteraction between the fluid interface and the wall. The moving contact lineproblem studied in this paper has an interface separating the non-Newtonianfluids mixture AB from the Newtonian fluid C. The wall free energy Ew inthis case is defined on the wall w and adopts the following form [84]

Ew = σ2

∫w

fw(φ2)ds , (2.6)

7

Page 9: Three-Phase Model of Visco-Elastic Incompressible …zxu2/blood_clot/three-phase-complex...Three-phase Model of Visco-elastic Incompressible Fluid Flow and its Computational Implementation

where fw is as follows:

fw(φ2) = −(2φ2 − 1) (3− (2φ2 − 1)2)

4cos(θs) . (2.7)

Here σ2 is the surface tension of the visco-elastic mixture and θs is the staticcontact angle [63, 64].

For the purpose of using Cahn-Hilliard equations to describe evolution ofφ1 and φ2, the chemical potentials µ1 and µ2 are defined as the variationalderivative of the Helmholtz free energy functional H = Ecoh + Eela and areas follows:

µ1 =δHδφ1

= λ1

(φ2

2G′1(φ1)− γ2

1∇ · (φ22∇φ1)

)+

1

2(∂1λe)|∇Ψ|2 , (2.8)

and

µ2 =δHδφ2

= λ2

(G′2(φ2)− γ2

24φ2

)+2λ1φ2

(G1(φ1) +

γ21

2|∇φ1|2

)+

1

2(∂2λe)|∇Ψ|2 . (2.9)

For the sake of simplicity, here and in the rest of the paper, ∂i denotes ∂φifor i = 1, 2.

Remark 2.1. There exist different definitions of the chemical potential. Inpapers [41, 50], the chemical potential is defined as the variational derivativeof the total energy. When the mixed fluids have variable densities, there

is a term ρ′|u|22

in the chemical potential, which is not Galilean invariant.However, as values of the mobility parameters in the Cahn-Hilliard systemapproach zero, the whole system converges to a Galilean invariant system.The chemical potential in [1, 34] is defined as the variational derivative of

the mixing energy, which eliminates the ρ′|u|22

term. In our work, we definethe chemical potential as the variational derivative of the sum of the mixingenergy and elastic energy. This introduces the 1

2(∂iλe)|∇Ψ|2 term in the

chemical potentials. The reason to include the elastic energy is to ensurethat the obtained system satisfies the energy dissipation law. When a complexfluid with variable elasticities is considered, it is difficult, if not impossible, toprove that the obtained system satisfies the energy dissipation law in case theelastic energy is not included in the derivation of the chemical potential. We

8

Page 10: Three-Phase Model of Visco-Elastic Incompressible …zxu2/blood_clot/three-phase-complex...Three-phase Model of Visco-elastic Incompressible Fluid Flow and its Computational Implementation

note that, in fact, the Cahn-Hilliard type of dynamics should not be viewedstrictly as a physics law. Rather, it is just a relaxation of the pure transportequation [41].

The dissipation functional is defined as

∆=

∫Ω

(η2|D|2 +M1|∇µ1|2 +M2|∇µ2|2

)dx+

∫w

(κ|φ2|2+βs|us|2)ds ,(2.10)

where η = η(φ1, φ2, ηA, ηB, ηc, t) is the viscosity of the mixture,with ηi beingthe viscosity of phase i, i = A,B, and C. Mi is the phenomenological mobil-ity coefficient of the phase i. D = ∇u + (∇u)T . κ is the phenomenologicalrelaxation time of φ on the wall. βs is the slip friction coefficient, and us isthe slip speed on the wall.

2.1.2. Microscale transport of φ1 and φ2

We assume that φ1 and φ2 satisfy the following conservation laws:

∂tφ1 +∇ · (u∇φ1) = 0 , (2.11)

∂tφ2 +∇ · (u∇φ2) = 0 . (2.12)

Equations (2.11) and (2.12) are approximated in the phase field methodby the following Cahn-Hilliard equations

∂tφ1 +∇ · (u∇φ1) = ∇ · (M1∇µ1) , (2.13)

∂tφ2 +∇ · (u∇φ2) = ∇ · (M2∇µ2) . (2.14)

In addition, φ2 satisfies the following relaxation boundary condition on thesolid wall boundary w:

κφ2 + L(φ2) = 0 , (2.15)

where L(φ2) = λ2γ22∇nφ2 + f ′w, and φ2 = ∂tφ2 + u · ∇φ2 is the material

derivative of φ2 on the wall.

2.1.3. Macroscale momentum equation

The conservative and dissipative forces are obtained by applying the LAPto the Hamiltonian part of the system and the MDP to the dissipative partof the system, respectively.

9

Page 11: Three-Phase Model of Visco-Elastic Incompressible …zxu2/blood_clot/three-phase-complex...Three-phase Model of Visco-elastic Incompressible Fluid Flow and its Computational Implementation

Application of the LAP yields that

Fcon = −(

1

2[ρ(∂tu+ u · ∇u) + (∂t(ρu) +∇ · (ρu⊗ u))] +

+λ2γ22∇ · (∇φ2 ⊗∇φ2) + λ1γ

21∇ · (φ2∇φ1 ⊗∇φ1)

+∇ · (λe(∇Ψ)T∇Ψ) +∇P1

). (2.16)

By using the MDP and the flow incompressibility constraint, we obtainthe following dissipative force for deriving the equation of motion in the bulkflow region

Fdis = −∇ · (ηD) +∇P2 , (2.17)

and the dissipative force on the wall w

Fdis,w = τ · (ηD) · n+ κφ2∂τφ2 + βsus . (2.18)

Finally, the Navier-Stokes type of equation of motion for the macroscopictrinary fluid mixture is obtained as a result of the macroscopic force balance,i.e., Fcon = Fdis:

12

[ρ(∂tu+ u · ∇u) + (∂t(ρu) +∇ · (ρu⊗ u))] = ∇ · (ηD)−∇P−λ2γ

22∇ · (∇φ2 ⊗∇φ2)− λ1γ

21∇ · (φ2∇φ1 ⊗∇φ1)

−∇ · (λe(∇Ψ)T∇Ψ) (2.19)

where P = P1 + P2.The following slip boundary condition is used for the equation (2.19),

βsus = −τ · (ηD− λe(∇Ψ)T∇Ψ) · n+ L(φ2)∂τφ2 . (2.20)

If the elastic property of fluid is not considered, i.e., λe = 0, then theabove slip boundary condition is reduced to the General Navier BoundaryCondition (GNBC) [58, 59, 60]. In other words, the boundary condition(2.20) is a generalized form of the GNBC for the moving contact line probleminvolving visco-elastic fluid.

Remark 2.2. Details of using the LAP and MDP to derive the three-phasefluid model are described in Appendix B.

10

Page 12: Three-Phase Model of Visco-Elastic Incompressible …zxu2/blood_clot/three-phase-complex...Three-phase Model of Visco-elastic Incompressible Fluid Flow and its Computational Implementation

After the right hand side of the Navier-Stokes equation (2.19) is simplifiedby using the method described in Appendix B, we obtain the following three-phase Navier-Stokes Cahn-Hilliard model

12

[ρ(∂tu+ u · ∇u) + (∂t(ρu) +∇ · (ρu⊗ u))] +∇P = ∇ · (ηD)

−∇µ1φ1 −∇µ2φ2 − (∇Ψ)T∇ · (λe∇Ψ) ,

∇ · u = 0 ,

∂tΨ + u · ∇Ψ = 0 ,

∂tφ1 +∇ · (uφ1) = ∇ · (M1∇µ1) ,

∂tφ2 +∇ · (uφ2) = ∇ · (M2∇µ2) ,

µ1 = λ1 (φ22G′1(φ1)− γ2

1∇ · (φ22∇φ1)) + 1

2(∂1λe)|∇Ψ|2 ,

µ2 = λ2 (G′2(φ2)− γ224φ2) + 2λ1φ2

(G1(φ1) +

γ212|∇φ1|2

)+ 1

2(∂2λe)|∇Ψ|2 .

(2.21)

The initial and the wall boundary conditions are given as follows:

u · n = 0 , ∇nµ1 = ∇nµ2 = 0 , ∂nφ1 = 0 ,

κφ2 = −L(φ2) = −(ε∇nφ2 + f ′w) ,

βsus = −τ ·[ηD− λe(∇Ψ)T∇Ψ

]· n+ L(φ2)∂τφ2 ,

φ1(·, 0) = φ10, φ2(·, 0) = φ20, Ψ(·, 0) = Ψ0 .

(2.22)

Remark 2.3. This three-phase model satisfies the following conditions pro-posed in [11, 88]:

• When a phase does not present in the mixture at the initial time, thisphase should not appear during the time evolution of the system. E.g.,if φi(·, t = 0) = 0, then φi(·, t) ≡ 0, ∀t > 0, i = 1 or 2. This is to makesure that each phase does not appear without basis.

• The three-phase model should be reduced to the two-phase model bysetting one of the phase to be equal to zero. For example, if let φ1 ≡1 and λe = 0, the system (2.21) is reduced to the two-phase modelproposed in [50].

11

Page 13: Three-Phase Model of Visco-Elastic Incompressible …zxu2/blood_clot/three-phase-complex...Three-phase Model of Visco-elastic Incompressible Fluid Flow and its Computational Implementation

2.2. Energy dissipation law

The following dimensionless form of the system (2.21)-(2.22) for conve-nience of discussion is obtained by scaling the density, viscosity, elasticity,length and velocity by ρA, ηA, λA, L and U , respectively,

Re12

[ρ(∂tu+ u · ∇u) + (∂t(ρu) +∇ · (ρu⊗ u))] +∇P

= ∇ · (ηD)− φ1∇µ1 − φ2∇µ2 − αe(∇Ψ)T∇ · (λe∇Ψ) ,

∇ · u = 0 ,

∂tΨ + u · ∇Ψ = 0 ,

∂tφ1 + u · ∇φ1 = ∇ · (M1∇µ1) ,

∂tφ2 + u · ∇φ2 = ∇ · (M2∇µ2) ,

(2.23)

where

µ1 = α1

(1

ε1

G′1φ22 − ε1∇ · (φ2

2∇φ1)

)+ αe

|∇Ψ|2

2∂1λe , (2.24)

and

µ2 = α2

(1

ε2

G′2 − ε2∆φ2

)+ 2α1φ2

(1

ε1

G1 +ε1

2|∇φ1|2

)+αe|∇Ψ|2

2∂2λe . (2.25)

The initial and the wall boundary conditions for the system (2.23) are givenby

u · n = 0 , ∇nµ1 = ∇nµ2 = 0 , ∂nφ1 = 0 ,

κφ2 = −L(φ2) ,

l−1s us = −τ ·

[ηD− αeλe(∇Ψ)T∇Ψ

]· n+ α2L(φ2)∂τφ2 ,

φ1(·, 0) = φ10, φ2(·, 0) = φ20, Ψ(·, 0) = Ψ0 ,

(2.26)

where the dimensionless constants are Re = ρALUηA

, ε1 = γ1L

, ε2 = γ2L

, α1 =λ1γ1ηAU

, α2 = λ2γ2ηAU

, αe = λALηAU

, M1 = M1ηAL2 , M2 = M2ηA

L2 , and κ = κLλ2γ2/U

.One advantage of using the EnVarA is that the obtained system automat-

ically satisfies the energy dissipation law. Theorem 2.4 states the energydissipation law that the system (2.23)-(2.26) satisfies.

12

Page 14: Three-Phase Model of Visco-Elastic Incompressible …zxu2/blood_clot/three-phase-complex...Three-phase Model of Visco-elastic Incompressible Fluid Flow and its Computational Implementation

Theorem 2.4. If φ1, φ2, Ψ, u and P are smooth solutions of the abovesystem (2.23)-(2.26), then the following energy law is satisfied:

d

dtEtotal =

d

dt(Ekin + Ecoh + Eela + Ew)

= −‖η1/2D‖2

2−M1‖∇µ1‖2 −M2‖∇µ2‖2

−κα2‖φ2‖2w − ‖l1/2s us‖2

w , (2.27)

where ζ =√ρ, Ekin = Re

2‖ζu‖2,

Ecoh =

∫Ω

α1φ22

(G(φ1)

ε1

+ε1

2|∇φ1|2

)dx+

∫Ω

α2

(G(φ2)

ε2

+ε2

2|∇φ2|2

)dx,

Eela = αe

∫Ω

1

2λe|∇Ψ|2dx, and Ew=α2

∫w

fwds.

Proof. The main idea of the proof is to show how the left hand side of theequation (2.27) can be obtained by multiplying the Navier-Stokes equationby u, the phase transport equations by µi, the chemical potentials by φi,i = 1, 2, and the gradient of the equation for Ψi by αeλe∇Ψi, and summingthem up. The dissipation terms on the right hand side of the equation(2.27) are obtained by using integration by parts and the boundary conditionsspecified in equations (2.26).

By using the fact that∫

Ω(∇ · (ρu⊗ u) + ρu∇u,u) dx = 0, if we multiply

the first equation of the system (2.23) by u and use integration by parts, therate of change of kinetic energy d

dtEkin is calculated

d

dtEkin =

d

dt

Re

2‖ζu‖2 = −1

2‖η1/2D‖2 − (φ1∇µ1,u)− (φ2∇µ2,u)

−αe((∇Ψ)Tω,u) + (ητ ·D · n, us)w , (2.28)

where ω = ∇ · (λe∇Ψ).By taking the gradient of each component of the third equation of (2.23),

the following equation is obtained

∂k(∂tΨi) + ∂k(uj∂jΨi) = 0 . (2.29)

13

Page 15: Three-Phase Model of Visco-Elastic Incompressible …zxu2/blood_clot/three-phase-complex...Three-phase Model of Visco-elastic Incompressible Fluid Flow and its Computational Implementation

Inner product of above equation with αeλe∂kΨi has the form

αe(λe∇Ψ : ∇(∂tΨ))

= αe(λe∂kΨi, ∂k(∂tΨi))

= −αe(λe∂kΨi, ∂k(uj∂jΨi))

= αe(∂k(λe∂kΨi), uj∂jΨi)− αe(λe(∂kΨi∂jΨi)nk, uj)w

= αe((∇Ψ)Tω,u)− αe(τ · (λe(∇Ψ)T∇Ψ) · n, us)w . (2.30)

Adding equation (2.28) to (2.30) and using the third boundary conditionin (2.26) result in the following equation

d

dtEkin

= −1

2‖η1/2D‖2 − (φ1∇µ1,u)− (φ2∇µ2,u)

−αe(λe∇Ψ,∇(∂tΨ))− ‖l−1/2s us‖2

w + α2(L(φ2)∂τφ2, us)w

= −1

2‖η1/2D‖2 − (φ1∇µ1,u)− (φ2∇µ2,u) (2.31)

−αe(λe∇Ψ,∇(∂tΨ))− ‖l−1/2s us‖2

w − α2(κφ2, us∂τφ2)w .

Taking inner product of the fourth and fifth equations in system (2.23) withµ1 and µ2, respectively, results in the following system

(∂tφ1, µ1)− (uφ1,∇µ1) +M1‖∇µ1‖2 = 0 , (2.32)

(∂tφ2, µ2)− (uφ2,∇µ2) +M2‖∇µ2‖2 = 0 . (2.33)

Inner product of the chemical potential (2.24) with −∂tφ1 yields

−(∂tφ1, µ1) = −α1

(φ2

2

G′1ε1

, ∂tφ1

)− α1

(ε1φ

22∇φ1,∇(∂tφ1)

)−αe

(|∇Ψ|2

2∂1λe, ∂tφ1

). (2.34)

Inner product of the chemical potential (2.25) with −∂tφ2 and integrationby parts, together with the dynamics boundary condition of φ2 on the wall,result in the following equation

− (∂tφ2, µ2) = − d

dtEcoh2 − α1

(G1

ε1

2|∇φ1|2, 2φ2∂tφ2

)−αe

(∂2λe|∇Ψ|2

2, ∂tφ2

)− α2(κφ2 + f ′w, ∂tφ2)w .(2.35)

14

Page 16: Three-Phase Model of Visco-Elastic Incompressible …zxu2/blood_clot/three-phase-complex...Three-phase Model of Visco-elastic Incompressible Fluid Flow and its Computational Implementation

Summing up the equations (2.31)-(2.35) gives rise to

d

dtEtotal =

d

dt(Ekin + Ecoh + Eela + Ew)

= −‖η1/2D‖2

2−M1‖∇µ1‖2 −M2‖∇µ2‖2

−α2(κφ2, ∂tφ2)w − α2(κφ2, us∂τφ2)w − ‖l1/2s us‖2w ,

= −‖η1/2D‖2

2−M1‖∇µ1‖2 −M2‖∇µ2‖2

−κα2‖φ2‖2w − ‖l1/2s us‖2

w .

3. Numerical Scheme for Solving Model Equations

Many techniques were proposed to improve stability and efficiency ofnumerical schemes for solving the Cahn-Hilliard equation [24, 35, 44, 73].Here we use the energy convex splitting method [22, 23, 28, 29, 67, 69], whichdiscretizes the chemical potentials related to the convex energy implicitly andthe rest explicitly. Traditional projection-like methods [8, 15, 33, 72] for thevariable density Navier-Stokes equations require solving an elliptic equationwith variable coefficient to obtain the pressure or related scalar quantity. Thisis time consuming, especially when there is a large variation in fluid density.To overcome this difficulty, we choose the pressure stabilization method [29,32, 50, 68] to solve the Navier-Stokes equation, which only involves solvingpressure Poisson equation with constant coefficient and treats the divergencefree condition as a penalty.

In [50], the authors proposed a decoupled scheme by introducing a half-step velocity when solving the Navier-Stokes Cahn-Hilliard system numeri-cally. If we ignore the elastic terms in the system (2.23)-(2.26), the decoupledscheme can also be used for solving the Cahn-Hilliard equations in our modelby setting the half-step velocity u∗ = un− 4t

Reρn+1 (φn1∇µn+11 +φn2∇µn+1

2 ). For

solving the system (2.23)-(2.26), the half-step velocity should be set to beu∗ = un− 4t

Reρn+1 (φn1∇µn+11 +φn2∇µn+1

2 −αe(∇Ψn)T∇·(λe∇Ψn+1)). However,

according to the boundary conditions (2.26), it can be found that u∗ · n isnot zero. This means the decoupled scheme developed in [50] might not workfor the system (2.23)-(2.26).

15

Page 17: Three-Phase Model of Visco-Elastic Incompressible …zxu2/blood_clot/three-phase-complex...Three-phase Model of Visco-elastic Incompressible Fluid Flow and its Computational Implementation

We propose in this section an efficient and energy stable scheme based onthe convex splitting method [28, 29] for solving the coupled system (2.23)-(2.26) without using the half-step velocity. The first-order accurate versionof the scheme is described here. Stability analysis of the scheme is describedin Appendix C.

The first-order accurate energy stable scheme is constructed as follows.Given initial condition (φ0

1, φ02, P

0, u0, Ψ0), numerical solution (φn+11 , φn+1

2 ,un+1, P n+1, Ψn+1) is updated for n ≥ 1 by

φn+11 −φn14t +∇ · (un+1φn+1

1 ) = ∇ · (M1∇µn+11 ) ,

φn+12 −φn24t +∇ · (un+1φn+1

2 ) = ∇ · (M2∇µn+12 ) ,

∂nφn+11 = 0 ,

κφ2n+1

= κ(φn+12 −φn24t + un+1

s ∂τφn+12 ) = −L(φn+1

2 ) ,

(3.1a)

Re(ρn+1un+1−ρnun

24t + 12∇ · (ρn+1un+1 ⊗ un)

)+Re

(ρnu

n+1−un

24t + ρn+1

2un · ∇un+1

)= −∇(2P n − P n−1) +∇ · (ηn+1D(un+1))−φn+1

1 ∇µn+11

−φn+12 ∇µn+1

2 − αe(∇Ψn)T∇ · (λn+1e ∇Ψn+1) ,

l−1s u

n+1s = −ηn+1τ · [D(un+1)− αeλn+1

e (∇Ψn)T∇Ψn+1] · n

+α2L(φn+12 )∂τφ

n+12 ,

Ψn+1−Ψn

4t + un+1 · ∇Ψn = 0 ,

(3.1b)

∆(P n+1 − P n) = ρ

4tRe∇ · un+1 ,

∂nPn+1 = 0 ,

(3.1c)

where

µn+11 = α1µ

n+11 − α1ε1∇ · ((φn+1

2 )2∇φn+11 )

+αe1

2((φn+1

2 )2(1− α12))|∇Ψn|2 ,

µn+12 = α2µ

n+12 − α2ε2∆φn+1

2 + α1φn+12 ε1|∇φn1 |2

+αe(φn+12 (φn1 + (1− φn1 )α12))|∇Ψn|2 ,

16

Page 18: Three-Phase Model of Visco-Elastic Incompressible …zxu2/blood_clot/three-phase-complex...Three-phase Model of Visco-elastic Incompressible Fluid Flow and its Computational Implementation

µn+11 =

s1

ε1

φn+11 − (

s1

ε1

φn1 − (φn2 )2 1

ε1

G′1(φn1 )) ,

µn+12 =

s2

ε2

φn+12 − (

s2

ε2

φn2 −1

ε2

G′2(φn2 )− 2φn2ε1

G1(φn1 )) ,

ρn+1 = ρ13(1− φn+12 ) + φn+1

2 (φn+11 + (1− φn+1

1 )ρ12) ,

λn+1e = (φn+1

2 )2(φn+11 + (1− φn+1

1 )α12) ,

L(φn+12 ) = ε2∂nφ

n+12 + f ′w(φn2 ) + αw(φn+1

2 − φn2 ) ,

with ρ12 = ρ2ρ1

, ρ13 = ρ3ρ1

, ρ = min(1, ρ12, ρ13) and α12 = λBλA

.The following theorem with proof provided in Appendix C shows that

the above discrete system satisfies discrete energy law.

Theorem 3.1. Let N = maxφn2 (|√

22

(2φn2 − 1) cos(θs)|). If αw ≥ N , and s1

and s2 satisfy the condition in the lemma described in Appendix C.1, thenthe solution (φn+1

1 , φn+12 , un+1, P n+1, Ψn+1) of the scheme (3.1) satisfies the

following discrete energy law for any 4t > 0:

En+1 +(4t)2

2ρRe‖∇P n+1‖2 +4t

(1

2‖η1/2D(un+1)‖2

)+4t

(‖M1/2

1 ∇µ1‖2 + ‖M1/22 ∇µ2‖2

)+4t

(‖l−1/2s un+1

s ‖2w + κα2‖φn+1

2 ‖2w

)≤ En +

(4t)2

2ρRe‖∇P n‖2 (3.2)

Remark 3.2. In the actual numerical implementation, we use finite elementmethod to discretize the space. The nonlinear terms ∇ · (uφi) are discretizedin time as ∇ · (unφn+1

i ) to make the resulting numerical equations easy tosolve [29]. Even though this treatment introduces a CFL-like constraint forchoosing the time step size 4t, it decouples the system (3.1) into three inde-pendent subsystems. This makes the numerical implementation much easierthan implementation which involves solving a large nonlinear system by iter-ation method. Moreover, (φn+1

2 , µn+12 ) is updated by using the nth step infor-

mation on the numerical implementation before computing other unknowns.Then they are used to update (φn+1

1 , µn+11 ). (Ψn+1,un+1, pn+1) are calculated

in the end.

17

Page 19: Three-Phase Model of Visco-Elastic Incompressible …zxu2/blood_clot/three-phase-complex...Three-phase Model of Visco-elastic Incompressible Fluid Flow and its Computational Implementation

4. Simulation Results

4.1. Droplet sliding on an inclined plane under gravity

Dynamics of a droplet sliding on an inclined plane under gravity [64,65, 80] is used in this subsection to demonstrate convergence of the numer-ical scheme proposed in the previous section. The gravitational force ρG isadded to the right hand side of the Navier-Stokes equation. Initial profile ofthe droplet is chosen in the form of a circular cap with contact angle 90.Computational domain is chosen to be [0, 1.5]× [0, 0.5]. (See also Fig. 1.)

Droplet in this study is treated as a two-phase fluid. The droplet and theambient fluid surrounding the droplet make the three-phase system. Densi-ties of the two fluid components of the droplet are ρA = ρB = 103kg/m3, theirviscosities are ηA = ηB = 100cP , and elasticities are λA = 1Pa, λB = 0.5Pa,respectively. The density ratio of the droplet to the ambient fluid is 1000 andthe viscosity ratio is 10. Values of non-dimensional parameters correspondingto the characteristic length L = 1 × 10−3m and velocity U = 1 × 10−2m/s,are listed in Table 1. The static contact angle of the droplet is 90, and the

Re ls ε α βg0.1 0.005 0.01 10 20

Table 1: The parameters used in convergence study. Here Re is the Reynolds number; ls isthe slip length; ε = 0.01 is the capillary width; α = λAL

ηAUis the mixture energy coefficient;

βg = ρAgL2

ηAU= 20 is the gravitational force.

inclination angle of the wall is α = 45. Evolution of the advancing contactpoint xa and the receding contact point xr of the droplet from the initialtime t = 0 to the time t = 5 was computed using three different meshes withmesh sizes h = 1/64, 1/128 and 1/256, respectively. Fig. 2 demonstratesconvergence of the numerical solution computed by the proposed numericalscheme.

4.2. Creep-recovery test

The Kelvin-Voigt model [42, 56] is used to represent behavior of a solid-like material undergoing reversible, visco-elastic deformation. Namely, thematerial described by the Kelvin-Voigt model deforms at a decreasing rate,and approaches asymptotically the steady-state strain under a constant stress.When the stress is released, the material gradually relaxes towards it initial

18

Page 20: Three-Phase Model of Visco-Elastic Incompressible …zxu2/blood_clot/three-phase-complex...Three-phase Model of Visco-elastic Incompressible Fluid Flow and its Computational Implementation

Figure 1: Diagram of the droplet sliding on an inclined plane under gravitational force.

0 0.5 1 1.5 2

Time

0.25

0.3

0.35

Po

sitio

n

h=1/64

h=1/128

h=1/256

(a)

0 0.5 1 1.5 2Time

0.74

0.76

0.78

0.8

0.82

0.84

0.86

Po

sitio

n

h=1/64

h=1/128

h=1/256

(b)

0.5 1 1.5 2Time0.015

0.02

0.025

Ve

locity

h=1/64

h=1/128

h=1/256

(c)

Figure 2: Convergence study of the numerical scheme by simulating droplet sliding onan inclined plane. (a) Motion of the receding contact point xr by using different meshes.(b) Motion of the advancing contact point xa with different meshes. (c) Velocity of thereceding contact point xr by using different meshes.

un-deformed configuration. However, complete recovery to the initial config-uration is never achieved in finite time. This is called creep-recovery.

In this section, we use a half circular-shaped droplet on a plane to dothe creep-recovery test numerically. The droplet is surrounded by a constantshear Newtonian flow. Values of non-dimensional parameters correspondingto the characteristic length L = 1×10−3m and velocity U = 1×10−2m/s arethe same as ones listed in Table 1. Computational domain is chosen to be[0, 1.5]× [0, 0.5]. From the time t = 0 to t = 1, a constant inlet flow conditionwith velocity v = 20(0.5 − y)y is added on the left of the boundary. Aftert = 1, the inlet flow is stopped and the droplet gradually recovers.

In Fig. 3, we show the creep-recovery test result. It shows that beforet = 1, the droplet strain increases monotonically, i.e., the droplet undergoescreep process. After t = 1, the droplet strain decreases with time to aconstant value, which is called permanent deformation due to dissipationof the system. Snapshots of the droplet profiles are presented in Fig. 4.The largest permanent deformation is around the left corner of the droplet.

19

Page 21: Three-Phase Model of Visco-Elastic Incompressible …zxu2/blood_clot/three-phase-complex...Three-phase Model of Visco-elastic Incompressible Fluid Flow and its Computational Implementation

This is caused by the dissipation on the boundary with rate κα2|φ2|2. Seealso Theorem 2.4. This numerical study confirms that our model producesvisco-elastic behavior of the fluid described by the Kelvin-Voigt model.

In the next section, we compare simulations by using the Oldroyd-B andKelvin-Voigt models for describing fluid visco-elasticity to reveal their differ-ences.

0 0.5 1 1.5 2 2.5 3Time

0

0.05

0.1

0.15

Str

ain

Recovery

Creep

Figure 3: Creep and Recovery. The inlet flow velocity specified by u = 20(0.5 − y)y isadded until t = 1 on the left boundary of the domain. Then the inlet flow velocity is setto be zero.

4.3. Droplet spreading test for Oldroyd-B and Kelvin-Voigt models

As we mentioned in the previous section, the Kelvin-Voigt model is usedfor describing behavior of solid-like visco-elastic materials. For fluid-likevisco-elastic materials, one of the most popular model is the Oldroyd-B model[9, 27, 77, 85, 90]. Conceptually, the Oldroy-B model is constructed by con-necting a spring and a dashpot sequentially. The deformation of the springis finite, while the dashpot retains deformation when the load is removed.Therefore, a material described by the Oldroyd-B model is more like a fluidthan a solid.

In this section, we describe simulations of a droplet spreading on a plane,with its visco-elastic property described by the Kelvin-Voigt model and theOldroy-B model, respectively. We also simulate a pure Newtonian dropletspreading for comparison. Initial shapes of these droplets are all chosen

20

Page 22: Three-Phase Model of Visco-Elastic Incompressible …zxu2/blood_clot/three-phase-complex...Three-phase Model of Visco-elastic Incompressible Fluid Flow and its Computational Implementation

0.4 0.6 0.8 1 1.2 1.4

x

0

0.1

0.2

0.3

0.4y

(a)

0.5 1 1.5x

0

0.1

0.2

0.3

0.4

y

(b)

0.4 0.6 0.8 1 1.2 1.4

x

0

0.1

0.2

0.3

0.4

y

(c)

0.4 0.6 0.8 1 1.2 1.4

x

0

0.1

0.2

0.3

0.4

y

(d)

Figure 4: Interface of the droplet at time (a) t = 0.1, (b) t = 1, (c) t = 2, and (d) t = 3 forcreep-recovery test. The Kelvin-Voigt model is used for describing droplet visco-elasticity.

to be a half circle with radius 0.2 and center (0.75, 0). Other parametersvalues are the same as those in [77]. Computational domain is chosen to be[0, 1.5] × [0, 0.5]. For the Oldroyd-B model simulation, we use the equation(6) in [77] to describe evolution of the visco-elastic tensor, and couple it withthe Cahn-Hilliard Navier-Stokes equations in our model.

Fig. 5 shows the interface profiles of these droplets at different times. Itcan be seen that the Oldroyd-B droplet (blue dash line) spreads much fasterthan the pure Newtonian droplet (black line) before t = 1.5. After that, thespreading speed of the pure Newtonian droplet is greater than the Oldroyd-B droplet as observed in [77]. The Kevin-Voigt droplet (red dash dot line)spreads slower than the pure Newtonian droplet as expected.

The dynamics of contact angles of different type droplets are shown inFig. 6(a). The results shows the contact angles quickly decay from initial 150

to 60 and then slowly approach equilibrium angle 45. Fig. 6(b) describesevolution of spreading radius, which is defined as the distance between two

21

Page 23: Three-Phase Model of Visco-Elastic Incompressible …zxu2/blood_clot/three-phase-complex...Three-phase Model of Visco-elastic Incompressible Fluid Flow and its Computational Implementation

(a) (b)

(c) (d)

Figure 5: Spontaneous plot of interface profiles of the simulated droplets when they spreadon the plane. Due to symmetry, only parts of the interfaces in x > 0 plane are plotted.(a) t = 0.5; (b) t = 1; (c) t = 1.5; and (d) t = 5. Black line: Pure Newtonian droplet.Blue dash line: Droplet with the Oldroyd-B model. Red dash dot line: Droplet with theKelvin-Voigt model.

contact points. The Oldroyd-B droplet and the pure Newtonian dropletachieve the same spreading radius (d = 0.8146) when they reach steadystate. While the spreading radius of the Kelvin-Voigt droplet (d = 0.776)is 5% smaller than the pure Newtonian droplet. This result is consistentwith the findings in [77]. Thus our simulations also showed importance ofincluding physical properties of fluids when studying its dynamics. Moreover,when the Weissenberg number Wi = λoU/L , which compares elastic forceto viscous force, where λ0 is the relaxation time in the Oldroyd-B model,increases from 2 to 5, the Oldroyd-B droplet spreading speed also increases.For the Kevin-Voigt droplets, the spreading speeds decrease with increasingshear modulus.

22

Page 24: Three-Phase Model of Visco-Elastic Incompressible …zxu2/blood_clot/three-phase-complex...Three-phase Model of Visco-elastic Incompressible Fluid Flow and its Computational Implementation

0 0.5 1 1.5 2 2.5Time 40

60

80

100

120

140

Con

tact

ang

le

Pure NewtonianKelvin-Voigt 1Oldroyd-B1

(a)

0 1 2 3 4 5Time0.2

0.4

0.6

0.8

1

Spr

ead

radi

us

Oldroyd-B 1Oldroyd-B 2Pure NewtonianKelvin-Voigt 1Kelvin-Voigt 2Kelvin-Voigt 3

(b)

Figure 6: (a) Contact angles and (b) radius of simulated spreading droplets. Black line cor-responds to the pure Newtonian droplet. Red, brown and purple dash dot lines correspondto the Kelvin-Voigt type visco-elastic droplet with different shear modulus, αe = 2, 5, 10,respectively. Blue and green dash lines indicate data for the Oldroyd-B visco-elasticdroplets with different relaxation times Wi = 2, 5, respectively.

4.4. Deformation of Venous Blood Clot under Shear Flow

To demonstrate applicability and relevance of the novel three-phase mod-el introduced in this paper for studying variety of problems in science andengineering, the model has been applied for studying the role of mechanicalproperties of a blood clot formed in a vein [82] in determining its stabilityunder biologically relevant flow conditions. This is an important biomedicalproblem for many reasons. For example, fragile blood clot may break to formseveral large pieces, or emboli, which can end up in lungs and subsequentlycause fatal outcomes for patients [14]. Also, hemophilia patients suffer frombleeding disorder, which is partially attributed to the mechanical propertiesof the clots. Fibrin networks in a hemophilic clot are more sparse than in aclot formed in normal blood, and they are less resistent to the shear stressgenerated by the blood flow [43]. The three-phase model simulations present-ed here reveal how changes in bulk properties of blood clots result in differentresponses of normal and hemophilic blood clots to the blood flow. Parame-ter values of elasticities of blood clot components in our simulations used theexperimental data provided in Tables 4 and 5 of Section 3 of reference [43].

For simplicity, we consider stability of small blood clots formed in micronsize blood venules. We assume that a blood clot, which is a porous andvisco-elastic gel type substance, consists of three major components: plasma,fibrin network and platelet aggregates. (See Fig. 7(a) for an example ofits structure.) Fibrin network is composed of thin fibers [31, 75]. Plateletaggregates are formed by the activated platelets, which change their shapes

23

Page 25: Three-Phase Model of Visco-Elastic Incompressible …zxu2/blood_clot/three-phase-complex...Three-phase Model of Visco-elastic Incompressible Fluid Flow and its Computational Implementation

after activation and tightly adhere with each other [36, 91]. Experimentalimage Fig. 7(b) shows that stabilized non-occluding blood clot formed in veinhas a dense core (in yellow color) consisting mainly of aggregates of activatedplatelets and fibrin network. A porous shell (in green color) which coversthe core, has high concentration of fibrin network and low concentrationof platelets. This clot structure was used as the initial structure for clotsimulations presented in this section.

Fibrin network and platelet aggregates are treated in the model as visco-elastic fluids, and plasma is treated as a Newtonian fluid. The initial valueof the volume fraction of the simulated blood clot (φ2) is set to be close to1 and 0.7 in the core and shell regions, respectively. Initial values of thevolume fraction of platelets (φ1) are 0.7 and 0.5 in the core and shell regionsof the simulated blood clot, respectively. The maximum volume fraction ofthe fibrin network is assumed to occur near the surface of the clot (Fig. 7(f))to mimic the fiber cap observed in the experiments [43, 48]. Fig. 7(d-f)shows the initial distributions of the volume fractions of components of thesimulated clot, which correspond to the experimental observations describedin [48]. Small spike-like extensions on the surface of the clot, which aresimilar to the ones seen in experimental figures (8-9) from [81], are added tothe initial surface of the simulated blood clot to represent its surface in morerealistic way.

Computational domain is chosen to be [0, Lx] × [0, Ly], where Lx =800× 10−6m and Ly = 320× 10−6m are the length and width of the domain,respectively. The inlet flow velocity imposed on the left boundary of the do-

main is given by uin =(

4umaxy(Ly−y)

L2y

, 0)

, where umax = 3.2×10−2m/s. (See

also Fig. 7(c).) Based on the experimental results in [57, 74], we assume thatdensities of the plasma and the blood clot are both ρ = 1.025 × 103kg/m3.Adhesion between blood clot and vessel wall [70, 71, 78] prevents the bloodclot from moving on the vessel wall. Therefore, the no-slip boundary condi-tion is used for the Navier-Stokes equations in the simulations.

The viscosity of the fibrin network ηn in a hemophilia clot is varied insimulations between 4cP and 40cP [55]. The viscosity of the platelet aggre-gate ηp is chosen to be 40cP [38]. Also, the viscosity of the fibrin network ina normal clot is set to be 400cP , and the viscosity of the platelet aggregateis varied between 40cP and 400cP [38, 62]. The viscosity of the plasma isassumed to be ηf = 4cP [81]. The elastic modulus λn of the fibrin networkof hemophilia clots is about O(1Pa) [43]. The elastic modulus of the fib-

24

Page 26: Three-Phase Model of Visco-Elastic Incompressible …zxu2/blood_clot/three-phase-complex...Three-phase Model of Visco-elastic Incompressible Fluid Flow and its Computational Implementation

(a) (b) (c)

(d) (e)

(f)

Figure 7: Initial structure of the blood clot in a vein. (a) Reconstructed three-dimensionalimage of a venule blood clot from in vivo experiments in mice. (Original image waspublished in [43, 82].) Platelets are indicated in red, fibrin is in green, yellow indicatescombination of platelets and fibrin, and black is used for other blood cells. Images showthat platelet aggregate in the middle of the clot is covered by the fibrin network and thatthe surface of the blood clot consists mainly of the fiber network. (b) yz cross sectionof the reconstructed image of the blood clot; (c) Schematic diagram of the clot structureused in simulations of a blood clot deformation under shear flow. Blood flow enters on theleft side of the simulation domain with a parabolic profile and exits on the right side ofthe domain. (d) Initial distribution of the volume fraction of the blood clot representedby the phase function φ2(t = 0). (e) Initial distribution of the volume fraction of plateletsrepresented by φ1(t = 0). (f) Initial distribution of the volume fraction of the fibrinnetwork represented by (1− φ1(t = 0)).

25

Page 27: Three-Phase Model of Visco-Elastic Incompressible …zxu2/blood_clot/three-phase-complex...Three-phase Model of Visco-elastic Incompressible Fluid Flow and its Computational Implementation

rin network generated by using normal blood varies between O(10Pa) andO(100Pa) [43]. Simulations are stopped when no blood clot deformation isdetected.

Panels (a-b) of the Fig. 8 demonstrate that small spike-like extensions,which mainly consist of fibrin, on the surface of a hemophilia clot developinto extensively elongated thin structures (emboli) (breakup of the emboli isnot shown); while panels (c-d) of the Fig. 8 show that normal clots deformonly slightly. The simulations reveal a possible novel mechanism of desta-bilization of a hemophilia clot. Since surfaces of clots in general are notsmooth, emboli can develop by the fibrin network on hemophilia clot surfaceeven under normal blood flow conditions, and subsequently detach from theclot. This makes formation of a stable clot in hemophilia blood much harderthan in normal blood. Simulations also predict that size of hemophilia clotwas significantly smaller than normal clot. Volume changes of normal (redcircles) and hemophilia (blue triangles) clots with respect to time are shownin Fig. 8e. The volume of the hemophilia clot gradually decreases after 0.5sby flow removal of the emboli and reaches a constant value around 1s, whichis about 28.9% of its initial volume. On the other hand, the volume of thenormal clot almost does not change. This is consistent with the experimen-tally observed clots [55]. Note that our simulation did not consider blood clotgrowth. This is why size of the simulated hemophilia clot reaches a constantvalue around 1s.

Simulations were used to study effects of changes in elasticity of the fibrinnetwork on clot stability. The following values of the elasticity modulusof the fibrin network [43] are used in simulations: 0.1Pa, 1Pa, 10Pa and50Pa. Viscosities of the fibrin network and platelet aggregate are fixed at40cP . Fig. 9 shows that the clot with 0.1Pa elasticity modulus of the fibrinnetwork is stretched to form a long and thin tail. When value of the elasticitymodulus of the fibrin network increases, the clot becomes less deformable.This is consistent with the results in [43], and shows how elasticity of fibrinnetwork affects clot deformation. Note that viscosities of the fibrin networkand platelet aggregate used in these simulations are for hemophilia clots. Oursimulations predicted that compared with viscosities of the fibrin network andplatelet aggregate, the elasticity of the fibrin network played major role inresisting clot deformation induced by blood flow.

Fig. 10 shows that increase of fibrin network elasticity decreases the av-erage speed of the intrathrombous flow, which also indicates that clot is lessdeformable. Simulations described in this section suggest that clots forming

26

Page 28: Three-Phase Model of Visco-Elastic Incompressible …zxu2/blood_clot/three-phase-complex...Three-phase Model of Visco-elastic Incompressible Fluid Flow and its Computational Implementation

(a) (b)

(c) (d)

0.2 0.4 0.6 0.8 1

Time (s)

0

0.02

0.04

0.06

0.08

Vo

lum

e (

L2)

ηn=4cP,η

p=40cp

ηn=400cp,η

p=400cp

(e)

Figure 8: Shapes of clots with different values of viscosity described by steady statesolutions of the model system of equations with fixed elasticity modulus λn = 1Pa, λp =10Pa. (a) ηn = 4cP, ηp = 40cP , (b) ηn = 40cP, ηp = 40cP , (c) ηn = 400cP, ηp = 40cP ,(d) ηn = 400cP, ηp = 400cP . ηp and ηn are the viscosities of the platelet aggregate andfibrin network, respectively. (e) Dynamics of total volumes of the clot. Blue triangle:ηn = 4cP, ηp = 40cP . Red circle: ηn = 400cP, ηp = 400cP .

27

Page 29: Three-Phase Model of Visco-Elastic Incompressible …zxu2/blood_clot/three-phase-complex...Three-phase Model of Visco-elastic Incompressible Fluid Flow and its Computational Implementation

in hemophilia patients can develop emboli resulting in them being much lessstable then clots developing in healthy individuals.

5. Conclusions

Novel thermodynamically consistent three-phase Navier-Stokes Cahn-Hilliardmodel for simulating complex fluids is presented in the paper. The new mod-el which is derived by using the EnVarA, is shown to be capable of simu-lating fluids with large density and viscosity ratios, and satisfy the energydissipation law. Energy stable numerical scheme is also developed to solveobtained system of model equations. Convergence of the numerical schemeis demonstrated by simulating droplet sliding on an inclined plane. ModifiedGeneral Navier Boundary Condition with fluid elasticity taken into accountis introduced for purpose of simulating contact line problems.

Differences between outcomes obtained using Kelvin-Voigt and Oldroyd-B models representing visco-elasticity of complex fluids are studied by usingcreep-recovery test for fluids and droplet spreading. Simulations suggest thatthe Kelvin-Voigt model is suitable for modeling complex fluid with reversible,visco-elastic deformation. While the Oldroyd-B model is more suitable formodeling complex fluid with fluid-like behavior.

Obtained model was used for studying deformation and stability of mi-cron size blood clots under physiologically relevant blood flow conditions.Blood clot simulations showed that hemophilia clots are more deformableand unstable than blood clots obtained using normal blood [55]. Model sim-ulations revealed that different responses of hemophilia and normal clots toblood flow are partially due to different structures and densities of fibrinnetworks. Notice that the viscosity and elasticity of platelet aggregates werevaried in simulations as well.

The three-phase model can be generalized to study lysis (disintegration)of a blood clot due to activity of thrombolytic agents. It has been shown in[70, 71] that intra-thrombus molecular transport is affected by the structureof the blood clot. Therefore the model described in this paper can be coupledwith the anti-coagulation transport sub-models to predict conditions of thegradual resolution of a blood clot [6].

Our model includes three phases. It can be viewed as a special case ofthe models described in the reference [79] with additional modification. Thismodification was motivated by the fact that for N -phase (N > 3) system,the surface tension between two phases cannot be uniquely represented by

28

Page 30: Three-Phase Model of Visco-Elastic Incompressible …zxu2/blood_clot/three-phase-complex...Three-phase Model of Visco-elastic Incompressible Fluid Flow and its Computational Implementation

(a)

(b)

(c)

(d)

Figure 9: Shapes of clots with different values of elasticity modulus described by steadystate solutions of the model system of equations. (a) λn = 0.1Pa, λp = 1Pa. (b)λn = 1Pa, λp = 10Pa. (c) λn = 10Pa, λp = 10Pa. (d) λn = 50Pa, λp = 10Pa. λp andλn are elasticity moduli of the platelet aggregates and fibrin networks, respectively.

29

Page 31: Three-Phase Model of Visco-Elastic Incompressible …zxu2/blood_clot/three-phase-complex...Three-phase Model of Visco-elastic Incompressible Fluid Flow and its Computational Implementation

0 0.2 0.4 0.6 0.8 1 1.2Time

0

0.2

0.4

0.6

0.8

Avera

ge V

elo

city in C

lot λ

n=0.1Pa,λ

p=1Pa

λn=1Pa,λ

p=10Pa

λn=10Pa,λ

p=10Pa

λn=50Pa,λ

p=10Pa

(a)

Figure 10: Evolution of averaged velocity inside clot∫

ΩχClot|u|dx/

∫ΩχClotdx. λp and

λn are elasticity moduli of the platelet aggregates and fibrin networks, respectively.

phase specific surface tension. Many previous works let surface tensionsbe homogeneous in this situation. In order to include non-homogeneoussurface tensions and ensure no phase appears artificially, we couple phaseshierarchically. Namely, the phase function in our model is treated not ina pairwise way but by using the binary tree approach. The binary treeapproach is used to avoid deriving complicated algebraic relations betweenpairwise surface tension and phase specific surface tension for N ≥ 3 phases.Note that our model also satisfies Assumptions 2 and 3 in [79]. Moreover, the

mixing energy as Λ =

[φ2

2 00 1

]described in our paper is a generation of the

case Λ = I described in [79]. Therefore, our binary tree approach provides asimple alternative for coupling N-phase (N ≥ 3) fluids.

AcknowledgmentsThe authors would like to thank Professor Chun Liu from Penn State

University for discussion of the application of the Energetic Variational Ap-proach for the derivation of the model. This research was partially supportedby the NSF grant DMS-1517293 and NIH grant IU01HL116330.

References

[1] H. Abels, H. Garcke, G. Grun, Thermodynamically consistent, frameindifferent diffuse interface models for incompressible two-phase flows

30

Page 32: Three-Phase Model of Visco-Elastic Incompressible …zxu2/blood_clot/three-phase-complex...Three-phase Model of Visco-elastic Incompressible Fluid Flow and its Computational Implementation

with different densities, Math. Models Methods Appl. Sci. 22(3),1150013, 2012.

[2] A. Abedijaberi, G. Bhatara, E. S. G. Shaqfeh, B. Khomami, A com-putational study of the influence of visco-elasticity on the interfacialdynamics of dip coating flow, J. Non-Newtonian Fluid Mech. 166, 614-627, 2011.

[3] D. M. Anderson, G. B. McFadden, A. A. Wheeler, Diffusive-interfacemethods in fluid dynamics, Annu. Rev. Fluid. Mech. 30 139-165, 1998.

[4] V. I. Arnold, Mathematical methods of classical mechanics, secondedition, Springer-Verlag, New York, 1989.

[5] V. E. Badalassi, H. D. Ceniceros, S. Banerjee, Computation of multi-phase systems with phase field models, J. Comput. Phys. 190, 371-397,2003.

[6] F. Bajd, I. Sersa, Mathematical Modeling of Blood Clot FragmentationDuring Flow-Mediated, Thrombosis, Biophys. J. 104: 1181-90, 2013;

[7] M. Bajaj, J. R. Prakash, M. Pasquali, A computational study of theeffect of visco-elasticity on slot coating flow of dilute polymer solutions,J. Non-Newtonian Fluid Mech. 149, 104-123, 2008.

[8] J. B. Bell, D. L. Marcus, A second-order projection method for variable-density flows, J. Comput. Phys. 101, 334-348, 1992.

[9] R. B. Bird, R. C. Armstrong, O. Hassafer, Dynamics of polymericfluids, 1, New York, NY: Wiley.

[10] A. V. Borkar, J. A. Tsamopoulos, S. A. Gupta, R. K. Gupta, Spincoating of visco-elastic and nonvolatile fluids over a planar disk, Phys.Fluids 6, 3539-3553, 2007.

[11] F. Boyer, C. Lapuerta, Study of a three component Cahn-Hilliard flowmodel, Math. Model. Numer. Anal. 40, 653-687,2006.

[12] J. Brannick, C. Liu, T. Qian, H. Sun, Diffuse Interface Methods forMultiple Phase Materials: An Energetic Variational Approach, Numer.Math. Theor. Meth. Appl. 8, 220-236,2014.

31

Page 33: Three-Phase Model of Visco-Elastic Incompressible …zxu2/blood_clot/three-phase-complex...Three-phase Model of Visco-elastic Incompressible Fluid Flow and its Computational Implementation

[13] J. W. Cahn, J. E. Hilliard, Free energy of a nonuniform system. I.Interface free energy, J. Chem. Phys. 28, 258-267, 1958.

[14] R. A Campbell, K. A. Overmyer, C. R. Bagnell, A. S. Wolberg, Cel-lular procoagulant activity dictates clot structure and stability as afunction of distance from the cell surface, Arterioscler Thromb. Vasc.Biol. 28(12), 2247-2254, 2008.

[15] A. J. Chorin, Numerical solution of the Navier-Stokes equations, Math.Comput. 22, 745-762, 1968.

[16] B.S. Dandapat, S.K. Singh, Cell Movement Is Guided by the Rigidityof the Substrate, Biophys. J. 79, 1,Volume 79, 144-152, 2000.

[17] Y. Di, R. Li, T. Tang, A general moving mesh framework in 3D and itsapplication for simulating the mixture of multi-phase flows, Commun.Comput. Phys. 3, 582 - 602, 2008.

[18] H. Ding, P. D. M. Spelt, and C. Shu, Diffuse interface model for incom-pressible two-phase flows with large density ratios, J. Comput. Phys.226, 2078-2795, 2007.

[19] H. Ding, P. D. M. Spelt, Wetting condition in diffuse interface simula-tions of contact line motion, Phys. Rev. E. 75, 046708, 2007.

[20] S. Dong, An efficient algorithm for incompressible N-phase flows, J.Comput. Phys. 276, 691-728, 2014.

[21] B. Eisenberg, Y. Hyon, Chun Liu, Energy variational analysis of ionsin water and channels: Field theory for primitive models of complexionic fluids, J. Chem. Phys. 133(10), 104104, 2010.

[22] D. J. Eyre, An unconditionally stable one-step scheme for gradientsystems, Unpublished article, June 1998.

[23] D.J. Eyre, in J.W. Bullard et al. (Eds.), Computational and mathe-matical models of microstructural evolution, The Materials ResearchSociety, Warrendale, PA, 39-46, 1998.

[24] X. Feng, T. Tang, J. Yang, Long time numerical simulations for phase-field problems using p-adaptive spectral deferred correction methods,SIAM J. Sci. Comput. 37(1), 271-294, 2014.

32

Page 34: Three-Phase Model of Visco-Elastic Incompressible …zxu2/blood_clot/three-phase-complex...Three-phase Model of Visco-elastic Incompressible Fluid Flow and its Computational Implementation

[25] M. Fontelos, G. Grun, S. Jorres, On a Phase-Field Model for Elec-trowetting and Other Electrokinetic Phenomena, SIAM J. Math. Anal.43, 527-563, 2011

[26] Johannes Forster, Mathematical Modeling of Complex Fluids, mastertheis, University of Wuurzburg, 2013.

[27] G. P. Galdi, R. Rannacher, A. M. Robertson and S. Turek, Hemo-dynamical flows Modeling, Analysis and Simulation, 37, BirkhauserBasel, 2008.

[28] M. Gao, X.-P. Wang, A gradient stable scheme for a phase field modelfor the moving contact line problem, J. Comput. Phys. 231, 1372-1386,2012.

[29] M. Gao, X.-P. Wang, An efficient scheme for a phase field model forthe moving contact line problem with variable density and viscosity, J.Comput. Phys. 272, 704-718, 2014.

[30] H. Garcke, M. Hinze, C. Kahle, A stable and linear time discretizationfor a thermodynamically consistent model for two-phase incompressibleow, Applied Numerical Mathematics, 99, 151-171, 2016.

[31] K. C. Gersh, C. Nagaswami, J. W. Weisel, Fibrin network structure andclot mechanical properties are altered by incorporation of erythrocytes,Thromb Haemost. 102(6), 1169-75, 2009.

[32] J.-L. Guermond, A. Salgado, A splitting method for incompressibleflows with variable density based on a pressure Poisson equation, J.Comput. Phys. 228, 2834-2846, 2009.

[33] J.-L. Guermond, L. Quartapelle, A projection FEM for variable densityincompressible flows, J. Comput. Phys. 165(1), 167-188, 2000.

[34] Z. Guo, P. Lin, A thermodynamically consistent phase-field modelfor two-phase flows with thermocapillary effects, J. Fluid Mech. 766,226?71, 2015.

[35] Y. He, Y. Liu, T. Tang, On large time-stepping methods for the Cahn-Hilliard equation, Appl. Numer. Math. 57, 616-628, 2007.

33

Page 35: Three-Phase Model of Visco-Elastic Incompressible …zxu2/blood_clot/three-phase-complex...Three-phase Model of Visco-elastic Incompressible Fluid Flow and its Computational Implementation

[36] J. W. Heemskerk, E. M.Bevers, T. Lindhout, Platelet activation andblood coagulation, Thromb. Haemost. 88(2), 186-93, 2002.

[37] P. C. Hohenberg and B. I. Halperin, Theory of dynamic critical phe-nomena, Rev. Mod. Phys. 49, 435-479, 1977

[38] C.-C. Huang, P.-Y. Chen, C.-C. Shih, Estimating the viscoelastic mod-ulus of a thrombus using an ultrasonic shear-wave approach, Med.Phys. 40(4),042901, 2013

[39] Y. Hyon, D. Y. Kwak, C. Liu, Energetic variational approach in com-plex fluids: maximum dissipation principle, DCDS-A 24(4), 1291-1304,2010.

[40] D. Jacqmin, Contact-line dynamics of a diffuse fluid interface, J. FluidMech. 402, 57-88, 2000.

[41] J. Jiang, Y. Li, C. Liu, Two-phase Incompressible Flows with VariableDensity: An Energetic Variational Approach, DCDS-A 37(6), 2017.

[42] T. H. S. Van Kempen, A. C. B.Bogaerds, G. W. M. Peters , F. N.van de Vosse, A Constitutive Model for a Maturing Fibrin Network,Biophys. J. 107(2), 504-513, 2014.

[43] E. Kim, O. V. Kim, K. R. Machlus, X. Liu, T. Kupaev, J. Lioi, A.S. Wolberg, D. Z. Chen, E. D. Rosen, Z. Xu, M. Alber, Correlationbetween fibrin network structure and mechanical properties: an exper-imental and computational analysis, Soft Matter 7, 4983-4992, 2011.

[44] J. S. Kim, K. K. Kang, J. S. Lowengrub, Conservative multigrid meth-ods for Cahn- Hilliard fluids, J. Comput. Phys. 193, 511-543, 2004.

[45] J. S. Kim, K. K. Kang, J. S. Lowengrub, Conservative multigrid meth-ods for ternary Cahn- Hilliard systems, Comm. Math. Sci. 2, 53-77,2004.

[46] J. S. Kim, J. S. Lowengrub, Phase field modeling and simulation ofthree-phase flows, Int. Free Bound. 7, 435-466, 2005.

[47] J. S. Kim, Phase-field models for multi-component fluid flows, Com-mun. Comput. Phys. 12(3), 613-661, 2012.

34

Page 36: Three-Phase Model of Visco-Elastic Incompressible …zxu2/blood_clot/three-phase-complex...Three-phase Model of Visco-elastic Incompressible Fluid Flow and its Computational Implementation

[48] O. V. Kim, Z. Xu, E. D. Rosen, M. Albe, Fibrin networks regulateprotein transport during thrombus Development, PLoS Comput. Biol.9(6): e1003095, 2013.

[49] F. H. Lin, C. Liu, P. Zhang, On hydrodynamcis of visco-elastic fluids,Commum. Pure Appl. Math. 58, 1437-1471, 2005.

[50] C. Liu, J. Shen, X. F. Yang, Decoupled energy stable schemes fora phase-field model of two-phase incompressible flows with variabledensity, J. Sci. Comput. 62(2), 601-622, 2015.

[51] C. Liu, J. Shen, A phase field model for the mixture of two incompress-ible fluids and its approximation by a Fourier-spectral method, PhysicaD 179, 211-228, 2003.

[52] J. Lowengrub, L. Truskinovsky, Quasiincompressible CahnHilliard flu-ids and topological transitions, Proc. R. Soc. Lond. A 454, 2617-2654,1998 ;

[53] L. Onsager, Reciprocal relations in irreversible processes. I., Phys. Rev.II. Ser. 37, 405-426, 1931.

[54] L. Onsager, Reciprocal relations in irreversible processes. II., Phys.Rev. II. Ser. 38, 2265-2279, 1931.

[55] M. V. Ovanesov, J. V. Krasotkina, L. I. Ul’yanova et al., HemophiliaA and B are associated with abnormal spatial dynamics of clot growth.Biochim. Biophys. Acta 1572(1), 45-57, 2002.

[56] N. zkaya, Fundamentals of Biomechanics : Equilibrium, Motion, andDeformation. New York : Springer Verlag, 2012.

[57] R. Polanowska-Grabowska, S. Raha, A. R. Gear, Adhesion efficiency,platelet density and size, Br. J. Haematol. 82, 715-720, 1992.

[58] T. Qian, X.-P. Wang, P. Sheng, Molecular scale contact line hydrody-namics of immiscible flows, Phys. Rev. E 68, 016306, 2003.

[59] T. Qian, X.-P. Wang, P. Sheng, Molecular hydrodynamics of the mov-ing contact line in two-phase immiscible flows, Commun. Comput.Phys. 1, 1-52, 2006.

35

Page 37: Three-Phase Model of Visco-Elastic Incompressible …zxu2/blood_clot/three-phase-complex...Three-phase Model of Visco-elastic Incompressible Fluid Flow and its Computational Implementation

[60] T. Qian, X.-P. Wang, P. Sheng, A variational approach to the movingcontact line hydrodynamics, J. Fluid Mech. 564, 333-360, 2006.

[61] S. Rafaı, D. Bonn, and A. Boudaoud, Spreading of non-Newtonianfluids on hydrophilic surfaces, J. FluidMech. 513, 77, 2004.

[62] M. Ranucci, T. Laddomada, M. Ranucci, and E. Baryshnikova, Bloodviscosity during coagulation at different shear rates, Physiological Re-ports, 2(7), 2014.

[63] W. Ren, W. E, Boundary conditions for the moving contact line prob-lem, Phys. Fluids 10, 022101, 2007.

[64] W. Ren, D. Hu, W. E, Continuum models for the contact line problem,Phys. Fluids 22, 102103, 2010.

[65] G. Della Rocca, G. Blanquart, Level set reinitialization at a contactline, J. Comput. Phys. 265 (2014) 34-49.

[66] R. J. Ryham, An energetic variational approach to mathematical mod-eling of charged fluids: charge phases,simulation and well posednes,thesis Pennsylvania State University, 2006.

[67] J. Shen, C. Wang, X. M. Wang, S.M. Wise, Second-order convex split-ting schemes for gradient flows with Ehrlich-Schwoebel type energy:Application to thin film epitaxy, SIAM J. Numer. Anal. 50(1), 105-125, 2012.

[68] J. Shen, X. Yang, A phase-field model and its numerical approximationfor two-phase incompressible flows with different densities and viscosi-ties, SIAM, J. Sci. Comput. 32(3), 1159-1179, 2010.

[69] J. Shen, X. Yang, Numerical approximations of Allen-Cahn and Cahn-Hilliard equations, DCDS-A 28(4), 1669-1691, 2010.

[70] T. J. Stalker, J. D. Welsh, M. Tomaiuolo, J. Wu, T. V. Colace, S.L. Diamond, and L. F. Brass, A systems approach to hemostasis: 3.Thrombus consolidation regulates intrathrombus solute transport andlocal thrombin activity, Blood. 124(11), 1824-31, 2014.

36

Page 38: Three-Phase Model of Visco-Elastic Incompressible …zxu2/blood_clot/three-phase-complex...Three-phase Model of Visco-elastic Incompressible Fluid Flow and its Computational Implementation

[71] T. J. Stalker, E. A. Traxler, J. Wu, K. M. Wannemacher, S. L. Cer-mignano, R. Voronov, S. L. Diamond, and L. F. Brass, Hierarchicalorganization in the hemostatic response and its relationship to theplatelet-signaling network, Blood. 121(10), 1875-85, 2013.

[72] R. Temam, Sur l’approximation de la solution des equations de Navier-Stokes par la methode des pas fractionnaires II, Arch. Rat. Mech. Anal.33, 377-385, 1969.

[73] G. Tierra, J. P. Pavissich, R. Nerenberg, Z. Xu, M. Alber, Multicom-ponent model of deformation and detachment of a biofilm under fluidflow, J. R. Soc. Interface 12, 20150045, 2015.

[74] R. J. Trudnowski , R. C. Rico, Specific gravity of blood and plasma at4 and 37 degrees C, Clin. Chem. 20, 615-616, 1974.

[75] A. Undas, R. A. Ariens, Fibrin clot structure and function: a role inthe pathophysiology of arterial and venous thromboembolic diseases,Arterioscler Thromb. Vasc. Biol. 31(12), 88-99, 2011.

[76] S. W. Walker, A Mixed Formulation of A Sharp Interface Model OfStokes Flow With Moving Contact Lines ESAIM: Mathematical Mod-elling and Numerical Analysis, 48, 969-1009, 2014

[77] Y. Wang, D.-Q. Minh, G. Amberg, Dynamic wetting of viscoelasticdroplets, Phys. Rev. E, 92, 043002, 2015.

[78] J. D. Welsh, T. J. Stalker, R. Voronov, R. W. Muthard, M. Tomaiuolo,S. L. Diamond, L. F. Brass, A systems approach to hemostasis: 1. Theinterdependence of thrombus architecture and agonist movements inthe gaps between platelets, Blood 124(11), 1808-15, 2014.

[79] S. Wu, J. Xu, Multiphase AllenCahn and CahnHilliard models andtheir discretizations with the effect of pairwise surface tensions, Journalof Computational Physics 343 (15), 10-32 2017.

[80] S. Xu, W. Ren, Reinitialization of the level set function in 3d simulationof moving contact lines, Commun. Comput. Phys. 20(5), 1163-1182,2016.

37

Page 39: Three-Phase Model of Visco-Elastic Incompressible …zxu2/blood_clot/three-phase-complex...Three-phase Model of Visco-elastic Incompressible Fluid Flow and its Computational Implementation

[81] Z. Xu, N. Chen, M. M. Kamocka, E. D. Rosen, M. Alber, A multiscalemodel of thrombus development, J. R. Soc. Interface. 5(24), 705-22,2008.

[82] Z. L. Xu, J. Lioi, J. Mu, M. M. Kamocka, X. Liu, D. Z. Chen, E.D. Rosen, M. Alber, A multiscale model of venous thrombus formationwith surface-mediated control of blood coagulation cascade, BiophyicalJ. 98, 1723-1732, 2010.

[83] S. Xu, P. Sheng, C. Liu, An energetic variational approach for iontransport, Commun. Math. Sci. 12(4), 779-789, 2014.

[84] P. Yue, J. J. Feng, Wall energy relaxtion in the Cahn-Hilliard modelfor moving contact lines, Phys. Fluids 23, 012106, 2011.

[85] P. Yue, J. J. Feng, Phase-field simulations of dynamic wetting of visco-elastic fluids, J. Non-Newton. Fluid 189-190, 8-13, 2012.

[86] P. Yue, J. J. Feng, Chun Liu, J. Shen, A diffuse-interface method forsimulation two-phase flows of complex fluids, J. Fluid Mech. 515, 293-317, 2004.

[87] P. Yue, C. Zhou, J. J. Feng, Sharp-interface limit of the Cahn-Hilliardmodel for moving contact lines, J. Fluid Mech. 645, 279-294, 2010.

[88] Q. Zhang, X.-P. Wang, Phase field modeling and simulation of threephase flow on solid surface, J. Comput. Phys. 319, 79-107, 2016.

[89] T. Y. Zhang, N. Cogan, Q. Wang, Phase Field Models for Biofilms.I. Theory and 1-D simulations, SIAM J. Appl. Math. 69 (3), 641-669,2008.

[90] C. Zhou, P. Yue, J. J. Feng, C. F. Ollivier-Gooch, H. H. Hu, 3D phase-field simulations of interfacial dynamics in Newtonian and visco-elasticfluids, J. Comput. Phys. 229, 498-511, 2010.

[91] M. B. Zucker, V. T. Nachmias, Platelet activation, Arterioscler ThrombVasc Biol. 5, 2-18, 1985.

38

Page 40: Three-Phase Model of Visco-Elastic Incompressible …zxu2/blood_clot/three-phase-complex...Three-phase Model of Visco-elastic Incompressible Fluid Flow and its Computational Implementation

Appendix A. Mathematical notations used in the paper

Mathematical notations used in this paper are as follows. Suppose a 2-rank tensor is denoted as F , its l2 norm is |F |2 =

∑ij=1,2 F

2ij. If A and B are

two 2-rank tensors, then (AB)ij =∑

k AikBkj and the double dot product ofthese two tensors is A : B =

∑ij AijBij. If a and b are two vectors, the outer

product a ⊗ b means (a ⊗ b)ij = aibj. L2 norm of the smooth function fin the domain Ω, (

∫Ω|f |2dx)1/2, is denoted by ‖f‖Ω and the L2 norm on the

boundary w, (∫

Γ|f |2ds)1/2, is denoted by ‖f‖w. If f and g are two smooth

functions in Ω, (f, g) stands for the inner product of these two functions andit is defined by (f, g) =

∫Ωfgdx.

Appendix B. Derivation of the three-phase model

We first use LAP to derive the conservative force. The action functionalis defined as follows:

A =

∫ t∗

0

∫Ω

1

2ρ|u|2 −

∫ t∗

0

∫Ω

λ1φ22

(G1(φ1) +

γ21

2|∇φ1|2

)dx

−∫ t∗

0

∫Ω

λ2

(G2(φ2) +

γ22

2|∇φ2|2

)dx−

∫ t∗

0

∫Ω

λe2|∇Ψ|2dx .(B.1)

We use 1-parameter family of volume preserving diffeomorphisms to per-

form the variation xε, such that x0 = x and dxε

∣∣∣ε=0

= y, where y is smooth

function with compact support and satisfies y(X, 0) = y(X, t∗) = 0 for anyX ∈ Ω0. For any ε, xε is required to satisfy det ∂xε

∂X= 1. This leads to the

divergence free condition for y(X, t) = y(x(X, t), t), i.e. ∇x · y = 0. ForLAP, we use the variations xε of x as described above. The variation of

39

Page 41: Three-Phase Model of Visco-Elastic Incompressible …zxu2/blood_clot/three-phase-complex...Three-phase Model of Visco-elastic Incompressible Fluid Flow and its Computational Implementation

action functional A is calculated as follows:

d

∣∣∣ε=0

A(xε)

=d

∣∣∣ε=0

∫ t∗

0

∫Ω0

(ρ(φ10(X), φ20(X))

2|xt|2

)dXdt

− d

∣∣∣ε=0

∫ t∗

0

∫Ω0

(λ1φ20(X)(G1(φ10) +

γ21

2|F−T∇Xφ10(X)|2)

)dXdt

− d

∣∣∣ε=0

∫ t∗

0

∫Ω0

(λ2(G2(φ20(X)) +

γ22

2|F−T∇Xφ20(X)|2)

)dXdt

− d

∣∣∣ε=0

∫ t∗

0

∫Ω0

(1

2λe(φ0(X))|∇XΨ0F

−1|2)dXdt

= I1 + I2 + I3 + I4. (B.2)

Here φi(x) =φi,0

detF[83], for i = 1, 2, is used. detF = 1 for the incompressible

fluid.This yields the following form of the first term of the right hand side of

equation (B.2)

I1 =

∫ t∗

0

∫Ω0

ρ(φ10, φ20)xtytdXdt

= −∫ t∗

0

∫Ω0

ρ(φ10, φ20)(xtty)dXdt

= −∫ t∗

0

∫Ω

ρ(φ1, φ2)(∂tu+ u · ∇u, y)dxdt .

(B.3)

At the same time,if we draw back from Lagrangian to Eulerian and then dothe integration by parts, we have

I1 =

∫ t∗

0

∫Ω0

ρ(φ10, φ20)xtytdXdt

=

∫ t∗

0

∫Ω

ρ(φ1, φ2)(u, yt + u · yt)dxdt

= −∫ t∗

0

∫Ω

(∂t(ρu) +∇ · (ρu⊗ u), y)dxdt .

(B.4)

40

Page 42: Three-Phase Model of Visco-Elastic Incompressible …zxu2/blood_clot/three-phase-complex...Three-phase Model of Visco-elastic Incompressible Fluid Flow and its Computational Implementation

In this work, we combine above two formula as shown in [41]

I1 = −∫ t∗

0

∫Ω

(1

2ρ(∂tu+ u · ∇u), y

)dxdt

−∫ t∗

0

∫Ω

(1

2(∂t(ρu) +∇ · (ρu⊗ u)), y

)dx (B.5)

dF ε

∣∣∣ε=0

= −F−1(∇Xy)F−1 [26] results in the following form of the second

and third terms

I2 = −∫ t∗

0

∫Ω0

λ1γ21φ20

(F−T∇Xφ10(X), (−F−T (∇Xy)TF−T∇Xφ10)

)dXdt

= −∫ t∗

0

∫Ω

λ1γ21 (φ2∇φ1, (−∇y∇φ1)) dxdt

= −∫ t∗

0

∫Ω

(λ1γ21∇ · (φ2∇φ1 ⊗∇φ1))ydxdt . (B.6)

I3 = −∫ t∗

0

∫Ω

(λ2γ22∇ · (∇φ2 ⊗∇φ2))ydxdt . (B.7)

The fourth term are transformed in a similar way as follows

I4 =−∫ t∗

0

∫Ω0

λe(φ10, φ20)((∇XΨ0)F−1 : (−(∇XΨ0)F−1(∇Xy)F−1)

)dXdt

= −∫ t∗

0

∫Ω

λe(φ1, φ2) (∇Ψ : (−∇Ψ∇y)) dxdt

= −∫ t∗

0

∫Ω

(∇ · (λe(φ)(∇Ψ)T∇Ψ), y)dxdt . (B.8)

Combining formula from (B.5) to (B.8), and taking into account of the in-compressibility and the Weyl’s decomposition or Helmholtz’s decomposition,for some P1 ∈ W 1,2(Ω) yields

Fcon = −([

1

2ρ(∂tu+ u · ∇u) +

1

2(∂t(ρu) +∇ · (ρu⊗ u))

]+

+λ2γ22∇ · (∇φ2 ⊗∇φ2) + λ1γ

21∇ · (φ2∇φ1 ⊗∇φ1)

+∇ · (λe(∇Ψ)T∇Ψ) +∇P1

).

41

Page 43: Three-Phase Model of Visco-Elastic Incompressible …zxu2/blood_clot/three-phase-complex...Three-phase Model of Visco-elastic Incompressible Fluid Flow and its Computational Implementation

Variation of the dissipation functional with respect to uε = u+ εv with∇ · v = 0 in Ω and v · n = 0 on the wall w, where n is an outer normalvector of the wall, in the Eulerian coordinate system is as follows

1

2

δ∆

δu=

∫Ω

(−∇ · (ηD))vdx+

∫w

(τ · (ηD) · n+ κφ2∇τφ2 + βsus)vτds ,(B.9)

where vτ = v · τ and τ is a tangential vector to the wall. The followingexpressions are also taken into account u · τ = us and φ = ∂tφ2 + us∂τφ2.The following expression for the dissipative force in the equation of motionin the bulk region is obtained using MDP and the incompressible constraint

Fdis = −∇ · (ηD) +∇P2 . (B.10)

Finally, after using the force balance in the bulk region, i.e., Fcon = Fdis,we obtain the equation of motion for the macroscopic fluid mixture

1

2[ρ(∂tu+ u · ∇u) + (∂t(ρu) +∇ · (ρu⊗ u))]

= ∇ · (ηD)−∇P − λ2γ22∇ · (∇φ2 ⊗∇φ2)

−λ1γ21∇ · (φ2∇φ1 ⊗∇φ1)−∇ · (λe(∇Ψ)T∇Ψ) , (B.11)

where P = P1 − P2. The right hand side terms of the previous equation canbe written

λ1γ21∇ · (φ2(∇φ1 ⊗∇φ1))

=λ1γ21∇ · (φ2∇φ1)∇φ1 +

γ21

2λ1φ2∇|∇φ1|2

=− λ1

(−γ2

1∇ · (φ2∇φ1) + φ2G′1(φ1)

)∇φ1 + λ1φ2∇

(G1(φ1) +

γ21

2|∇φ1|2

)=− µ1∇φ1 + λ1φ2∇

(G1(φ1) +

γ21

2|∇φ1|2

)+

(∂1λe

1

2|∇Ψ|2

)∇φ1,

42

Page 44: Three-Phase Model of Visco-Elastic Incompressible …zxu2/blood_clot/three-phase-complex...Three-phase Model of Visco-elastic Incompressible Fluid Flow and its Computational Implementation

λ2γ22∇ · (∇φ2 ⊗∇φ2)

=λ2γ224φ2∇φ2 +

γ22λ2

2∇|∇φ2|2

=− λ2(−γ224φ2 +G′2(φ2))∇φ2 + λ2∇(

γ22

2|∇φ2|2 +G2(φ2))

=− µ2∇φ2 + λ2∇(γ2

2

2|∇φ2|2 +G2(φ2)

)+ λ1

(G1 +

γ21

2|∇φ1|2

)∇φ2

+

(∂2λe

1

2|∇Ψ|2

)∇φ2

with the form of the elastic force term

∇ · (λe(∇Ψ)T∇Ψ) = (∇Ψ)Tω +λe2∇|∇Ψ|2

where ω = ∇ · (λe∇Ψ). This results in the following form of the equation(B.11)

∇ · (λe(∇Ψ)T∇Ψ) + λ2γ22∇ · (∇φ2 ⊗∇φ2) + λ1γ

21∇ · (φ2∇φ1 ⊗∇φ1)

=− µ1∇φ1 − µ2∇φ2 + (∇Ψ)Tω +∇P ,

with

P =

(λ2γ2

2

2|∇φ2|2 + λ2G2(φ2) + (λ1G1 +

λ1γ21

2|∇φ1|2)φ2 +

λe2|∇Ψ|2

).

Finally, this yields the macroscale momentum equation of the three-phasemodel

12

[ρ(∂tu+ u · ∇u) + (∂t(ρu) +∇ · (ρu⊗ u))]

= ∇ · (ηD)−∇P −∇µ1φ1 −∇µ2φ2 − (∇Ψ)Tω ,

where P = P − µ1φ1 − µ2φ2.

Remark Appendix B.1. Notice that w the variation is taken in the La-grangian coordinate system when using LAP approach and it is taken in theEulerian coordinate system in the MDP method. This is done because thevariation of the action functional is taken with respect to the flow map (ortrajectory x(X, t) ) and it is more convenient to use the LAP in the La-grangian coordinates.

43

Page 45: Three-Phase Model of Visco-Elastic Incompressible …zxu2/blood_clot/three-phase-complex...Three-phase Model of Visco-elastic Incompressible Fluid Flow and its Computational Implementation

Appendix C. Stability analysis of the numerical scheme

We present in this section the stability analysis of the numerical scheme(3.1). We start by proving the following lemma similar to the one in [29].This lemma will be used in proving Theorem 3.1.

Lemma Appendix C.1. Let E = Ec − Ee, where Ec =∫

Ω( s1

2ε1|φ1|2 +

s22ε2|φ2|2)dx, Ee =

∫Ω

( s12ε1|φ1|2 + s2

2ε2|φ2|2 − 1

ε2G2(φ2) − φ22

ε1G1(φ1))dx. If s1 ≥

max(G′′1(φn1 )(φn2 )2, G

′′1(φn1 )(φn2 )2 − 2φ2G

′1(φn1 )), s2 ≥ max(G

′′2(φn2 ) + 2ε2

ε1G1,

G′′2(φn2 ) + 2ε2

ε1(G1 − φ2G

′1(φn1 ))), and supx∈Ω|φn1 |, |φn2 | ≤ C with a constant

C > 0 then for given φn1 and φn2 , we have

E(φn+11 , φn+1

2 )− E(φn1 , φn2 ) ≤ (µn+1

1 , φn+11 − φn1 ) + (µn+1

2 , φn+12 − φn2 ) ,(C.1)

where µ1 = s1ε1φn+1

1 − (s1φn1ε1− (φn2 )2G

′1(φn1 )

ε1), µ2 = s2

ε2φn+1

2 − (s2φn2ε2− G′2(φn2 )

ε2−

(2φn2 )G1(φn1 )

ε1).

Proof. By mean value theorem, we have

Ec(φn1 , φ

n2 )− Ec(φn+1

1 , φn+12 ) ≥ (

δEcδφ1

, φn1 − φn+11 ) + (

δEcδφ2

, φn2 − φn+12 )

+s1

ε1

|φn1 − φn+11 |2 +

s2

ε2

|φn2 − φn+12 |2.(C.2)

Similarly, we can get

Ee(φn+11 , φn+1

2 )− Ee(φn1 , φn2 )

= (δEe(φ

n1 , φ

n2 )

δφ1

, φn+11 − φn1 ) + (

δEe(φn1 , φ

n2 )

δφ2

, φn+12 − φn2 )

+(H11, (φn+11 − φn1 )2) + (2H12, (φ

n+11 − φn1 )(φn+1

2 − φn2 ))

+(H22, (φn+12 − φn2 )2) , (C.3)

where

H =

(s1ε1− φ21

ε1G′′1(φ1) −2φ2

ε1G′1(φ1)

−2φ2ε1G′1(φ1) s2

ε2− G′′2 (φ2)

ε2− 2G1(φ1)

ε1

). (C.4)

If s1 ≥ max(G′′1(φn1 )(φn2 )2, G

′′1(φn1 )(φn2 )2−2φ2G

′1(φn1 )), s2 ≥ max(G

′′2(φn2 )+

2ε2ε1G1, G

′′2(φn2 ) + 2ε2

ε1(G1 − φ2G

′1(φn1 ))), then matrix H is a positive defined

matrix.

44

Page 46: Three-Phase Model of Visco-Elastic Incompressible …zxu2/blood_clot/three-phase-complex...Three-phase Model of Visco-elastic Incompressible Fluid Flow and its Computational Implementation

Then, there exist two constants C1 and C2, such that

Ee(φn+11 , φn+1

2 )− Ee(φn1 , φn2 )

≥ (δEe(φ

n1 , φ

n2 )

δφ1

, φn+11 − φn1 ) + (

δEe(φn1 , φ

n2 )

δφ2

, φn+12 − φn2 )

+C1|φn1 − φn+11 |2 + C2|φn2 − φn+1

2 |2. (C.5)

Adding (C.2) with (C.5) together gives

E(φn+11 , φn+1

2 )− E(φn1 , φn2 ) ≤ (µn+1

1 , φn+11 − φn1 ) + (µn+1

2 , φn+12 − φn2 ) .

By using above Lemma and multiplying each equation in system (3.1)with proper function, we can prove the energy stable Theorem (3.1) in Section3.

Theorem 3.1: Let N = maxφn2 (|√

22

(2φn2 − 1) cos(θs)|). If s1 and s2

satisfy the condition in Lemma Appendix C.1 and αw ≥ N , then the solution(φn+1

1 , φn+12 , un+1, P n+1, Ψn+1) of the scheme (3.1) satisfies the following

discrete energy law for any 4t > 0:

En+1 +(4t)2

2ρRe‖∇P n+1‖2 +4t

(1

2‖η1/2D(un+1)‖2

)+4t

(‖M1/2

1 ∇µ1‖2 + ‖M1/22 ∇µ2‖2

)+4t

(‖l−1/2s un+1

s ‖2w + κα2‖φn+1

2 ‖2w

)≤ En +

(4t)2

2ρRe‖∇P n‖2 (C.6)

Proof of Theorem 3.1. By the definition of λe in Section 3, inner product

of ∂iλn+1e

|∇Ψn|22

and φn+1i − φni , i = 1, 2, respectively, and summing them up,

45

Page 47: Three-Phase Model of Visco-Elastic Incompressible …zxu2/blood_clot/three-phase-complex...Three-phase Model of Visco-elastic Incompressible Fluid Flow and its Computational Implementation

result in the following

1

2((φn+1

2 )2(1− α12)φn+11 − φn1 , |∇Ψn|2)

+(φn+12 (φn1 + (1− φn1 )α12))φn+1

2 − φn2 , |∇Ψn|2|)

=1

2((φn+1

2 )2(φn+11 + (1− φn+1

1 )α12), |∇Ψn|2)

−1

2((φn+1

2 )2(φn1 + (1− φn1 )α12), |∇Ψn|2)

+1

2((φn+1

2 )2(φn1 + (1− φn1 )α12), |∇Ψn|2)

−1

2((φn2 )2(φn1 + (1− φn1 )α12), |∇Ψn|2)

+1

2((φn+1

2 − φn2 )2(φn1 + (1− φn1 )α12), |∇Ψn|2)

=1

2‖(λn+1

e )1/2∇Ψn‖2 − 1

2‖(λne )1/2∇Ψn‖2

+1

2((φn+1

2 − φn2 )2(φn1 + (1− φn1 )α12), |∇Ψn|2) . (C.7)

And for the hydrophilic term, inner product of ∇ · ((φn+12 )2∇φn+1

1 ) andφn+1

2 |∇φn1 |2 by φn+11 − φn1 and φn+1

2 − φn2 , respectively, and summing themup have the form

−ε1

(∇ · ((φn+1

2 )2∇φn+11 ), φn+1

1 − φn1)

+ ε1(φn+12 |∇φn1 |2, φn+1

2 − φn2 )

= ε1

((φn+1

2 )2∇φn+11 ,∇φn+1

1 −∇φn1)

+ ε1(|∇φn1 |2φn+12 , φn+1

2 − φn2 )

=ε1

2((φn+1

2 )2, |∇φn+11 |2 − |∇φn1 |2 + |∇(φn+1

1 −∇φn1 )|2)

+ε1

2(|∇φn1 |2, (φn+1

2 )2 − (φn2 )2 + (φn+12 − φn2 )2)

=ε1

2‖φn+1

2 ∇φn+11 ‖2 − ε1

2‖φn2∇φn1‖2

+ε1

2‖(φn+1

2 )∇(φn+11 − φn1 )‖2 +

ε1

2‖(φn+1

2 − φn2 )∇φn1‖2 . (C.8)

Combining equations (C.7)-(C.8) and Lemma Appendix C.1, results in

46

Page 48: Three-Phase Model of Visco-Elastic Incompressible …zxu2/blood_clot/three-phase-complex...Three-phase Model of Visco-elastic Incompressible Fluid Flow and its Computational Implementation

the following inequality

En+1coh + En+1

w − (Encoh + Enw)

+αe2

(‖(λn+1e )1/2∇Ψn‖2 − ‖(λne )1/2∇Ψn‖2)

≤ (µn+11 , φn+1

1 − φn1 ) + (µn+12 , φn+1

2 − φn2 )

+α2(L(φn+12 ), φn+1

2 − φn2 )w . (C.9)

After taking the inner product of the first and second equations in (3.1a)with 4tµn+1

1 and 4tµn+12 , respectively, we have

(φn+11 − φn1 , µn+1

1 )−4t(un+1φn+11 ,∇µn+1

1 ) +4tM1‖∇µn+11 ‖2 = 0 , (C.10)

(φn+12 − φn2 , µn+1

2 )−4t(un+1φn+12 ,∇µn+1

2 ) +4tM2‖∇µn+12 ‖2 = 0 . (C.11)

Taking the inner product of the first equation in (3.1b) with 4tun+1

yields

Re(ρn+1un+1 − ρnun

24t+ρn(un+1 − un)

24t,un+1) (C.12)

= Re(1

2(ρn+1 + ρn)un+1 − ρnun,un+1) (C.13)

=Re

2

(‖ζn+1un+1‖2 − ‖ζnun‖2 + ‖ζn(un+1 − un)‖2

)(C.14)

= −4t2‖(η(φn+1))1/2D(un+1)‖2 +4t(∇(−2P n + P n−1),un+1)

−4t(φn+11 ∇µn+1

1 ,un+1)−4t(φn+12 ∇µn+1

2 ,un+1)

−αe4t((∇Ψn)Tωn+1,un+1) +4t(η(φn+1)τ ·D(φn+1) · n,un+1s )w ,

where ωn+1 = ∇·(λn+1e ∇Ψn+1). Here we use the fact that

∫Ω

(∇·(ρn+1un+1⊗un) + ρn+1un · ∇un+1,un+1)dx = 0.

By using the same argument as in [29], the pressure can be estimated asfollows

4t(un+1,∇(−2P n+1 + P n)) (C.15)

≤ (4t)2

2ρRe(−‖∇(P n − P n−1)‖2 − ‖∇P n+1‖2 + ‖∇P n‖2)

+Re

2‖ζn(un+1 − un)‖2 .

47

Page 49: Three-Phase Model of Visco-Elastic Incompressible …zxu2/blood_clot/three-phase-complex...Three-phase Model of Visco-elastic Incompressible Fluid Flow and its Computational Implementation

Taking gradient of each component of the third equation in (3.1b), yields

∂jΨn+1i − ∂jΨn

i

4t+ ∂j(u

n+1k ∂kΨ

ni ) = 0 . (C.16)

The inner product of above equation with αe4tλn+1e ∂kΨ

n+1i results in

αe2

(‖(λn+1e )1/2Ψn+1‖2 − ‖(λn+1

e )1/2Ψn‖2)

≤ αe4t((∇Ψn)Tωn+1 ,un+1)

−αe(τ · (φn+12 )2(∇Ψn)T∇Ψn+1 · n,un+1

s )w . (C.17)

Adding equation (C.12) to the equation (C.17) yields

Re

2

(‖ζn+1un+1‖2 − ‖ζnun‖2 + ‖ζn(un+1 − un)‖2

)+αe2

(‖(λn+1e )1/2Ψn+1‖2 − ‖(λn+1

e )1/2Ψn‖2

≤ −4t2‖(η(φn+1))1/2D(un+1)‖2 +4t(∇(−2P n + P n−1),un+1)

−4t(φn+11 ∇µn+1

1 ,un+1)−4t(φn+12 ∇µn+1

2 ,un+1)

−‖l1/2s un+1s ‖2

w + α2(L(φn+12 )∂τφ

n+12 ,un+1

s )w . (C.18)

Combing equations (C.9)-(C.11), (C.18) with pressure estimation (C.15)results in the equation (3.2).

Appendix D. Additional Simulation Figures

48

Page 50: Three-Phase Model of Visco-Elastic Incompressible …zxu2/blood_clot/three-phase-complex...Three-phase Model of Visco-elastic Incompressible Fluid Flow and its Computational Implementation

(a) (b)

(c) (d)

Figure D.11: Profiles of velocity norm at steady state with different values of elasticity forfibrin network and platelet aggregate. (a) λn = 0.1Pa, λp = 1Pa. (b) λn = 1Pa, λp =10Pa. (c) λn = 10Pa, λp = 10Pa. (d) λn = 50Pa, λp = 10Pa.

49

View publication statsView publication stats