1 Introduction

The Hele-Shaw model(flow)(see[5])describes the laminar motion of a(Newtonian)viscous fluid in a Hele-Shaw cell—also known as the Hele-Shaw analog or viscous flow analog—an experimental device composed by two parallel plates of size L kept at the distance b from each other,with b≪L(see Figure 1).The vertical,horizontal or angled Hele-Shaw cells can be considered to possibly account for gravity effect.The device was firstly proposed and used by Hele-Shaw[32]to experimentally investigate the viscous(potential)flows around bodies;later,vertical Hele-Shaw cells have been extensively used to study groundwater flows,oil production,drainage,etc.A wide literature is available on the topic,see e.g.[5,43,49,56].Indeed,the importance of the Hele-Shaw model lays in the fact that the potential flow occurring in the Hele-Shaw cell corresponds to a Darcy flow in a porous media(see[5])with permeability equivalent to(see[49]).We observe that,even if the Hele-Shaw flow occurs only for two-dimensional laminar flows,it represents a valid simplified model with respect to the full Navier-Stokes equations to study the effect of flows in porous media provided that the three-dimensional effects are negligible;moreover,it provides a framework for experimental comparisons:As reported in[5]and the references herein indicated,the Hele-Shaw cell is suitable for flows with Reynoldsnumbers smaller than 500-1′000,these being defined by means of the distance between the plates b.

The Cahn-Hilliard equation is a time dependent,mass conservative phase field model which describes the segregation of the phasesfrom a mixed configuration to a fully separated one,with the pure phases separated by smooth,but sharp interfaces(see[10–11,23]).The Cahn-Hilliard equation is a nonlinear,parabolic PDE with fourth-order spatial derivatives,which during the phasetransition exhibitsa fast and intermittent dynamics.It wasoriginally introduced by Cahn[10]to describe the separation of binary alloys systems in metallurgy and it later extensively studied both from a theoretical and numerical point of view(see e.g.[3,8,18,27–28,31,45,55]).

The ideal combination of the Hele-Shaw model with the Cahn-Hilliard phase field model yields the so-called Hele-Shaw-Cahn-Hilliard equations.These represent indeed a phase field model for Hele-Shaw flows with physical data varying through the smooth but sharp interfaces;the physical data depend on the phase field variable whose distribution in the computational domain depends on the phase transition according to a generalized Cahn-Hilliard equation.In the limit of interface thickness tending to zero(see[45]),these physical data are in fact discontinuous across the interface.The model was developed in[39]as a special case of the Navier-Stokes-Cahn-Hilliard model for immiscible fluids proposed in[42]of which it represents a simplified version under the hypothesis of Hele-Shaw flow(the inertial forces are negligible with respect to the viscous ones).In particular,the model is only two-dimensional and valid for isothermal laminar flows among the plates of the Hele-Shaw cell provided that there are no variation of the phase variable through the thickness among the plates.Nevertheless,the phase field model(see also[14,26,37])has the clear advantage of embedding into the formulation modifications in the shape of the interface and its evolution in time,as well as to naturally allow topological changes without resorting to interface capturing or tracking methods like the volume of fluid or level set methods(see[30,33,44,50]).Moreover,the asymptotic,sharp interface limit of the phase field model(see[9])yields the Darcy flow problem with internally discontinuous data across the interface(see[42]).

The analysis of Hele-Shaw-Cahn-Hilliard equations and related models is particularly challenging.For example,the well-posedness of Navier-Stokes-Cahn-Hilliard model was studied in[53]and its long time behavior described in[52].The convergence of weak solutions of the Cahn-Hilliard-Brinkman model to the Hele-Shaw-Cahn-Hilliard equations was proved in[7]and a non-local solution of the previous result determined in[47].The global,sharp interface limit of these equations was recently studied in[19].More recently,an alternative Hele-Shaw-Cahn-Hilliard model was derived in[16](starting from[1])and therein analyzed using a volume-averaged definition of the velocity field instead of the mass-averaged one of[39];see also[24]for the application of the model to tumor modeling.In this paper,however we refer to the original formulation of the Hele-Shaw-Cahn-Hilliard equations proposed in[39].

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The phase field model based on the Hele-Shaw-Cahn-Hilliard equations inherits similar mathematical features of the Cahn-Hilliard equations.It is a nonlinear system of partial differential equations(PDEsfor short)in parabolic-elliptic formulation in thephasefield and pressure variables,respectively,which involves high order spatial derivatives(up to four)for the phase field variable,whose solution may exhibit fast and intermittent dynamics.For thesereasons,the numerical approximation of this problem is particularly challenging as an accurate and efficient numerical method should be able to cope with all these features.Spatial approximations using the Finite Element method with the standard Lagrange polynomial basis have been widely adopted for phase field models(see[20–22,54])even if the fourth order derivatives involved in the problem require to resort to mixed formulations.In this paper,we propose instead NURBS-based isogeometric analysis in the framework of the Galerkin method(see[13,35])for spatially approximating such high order PDEs.Isogeometric analysis is a discretization method based on the isogeometric paradigm,for which the same basis functions are used first for the representation of the domain and then for the approximation of the solution of the PDEs.Besides of the geometric advantages related to this choice,the employment of high order continuous NURBSor B-splinesbasis functions to build the trial and test function subspaces allows solving the phase field model in the framework of the standard Galerkin formulation(see[51]).In particular,globally C 1-continuous B-splines basis functions of degree p=2 allow the construction of trial and test function subspaces which are H 2-conformal;moreover,these basis functions facilitate the introduction of periodic boundary conditions.These features of B-splines and NURBS basis functions have been extensively exploited for solving phase field model together with their accuracy in representing sharp but smooth interfaces(see e.g.[4,15,27,29,41]).In this paper,we use isogeometric analysis for solving the Hele-Shaw-Cahn-Hilliard equations for fluids in porous media endowed with discontinuous data,in particular the density;in this respect,we efficiently solve by means of our formulation the “rising bubble”benchmark problem(see[36]).In addition,we use the proposed method to verify the sharp interface limit of the Hele-Shaw-Cahn-Hilliard equations and we show that this phase field model can be efficiently used to represent an interface discontinuity problem involving the Poisson equation(see[17])without resorting to interface tracking or interface capturing techniques.

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This paper is organized as follows.In Section 2 we recall the Hele-Shaw and Cahn-Hilliard equations and then the phase field model described by the Hele-Shaw-Chan-Hilliard equations;we also discuss their sharp interface limit.In Section 3 we present the numerical discretization of the problem based on isogeometric analysis.In Section 4 we numerically verify the sharp interface limit of the model and we solve the benchmark problem of the “rising bubble”.Conclusions follow.

2 Phase Field M odel

In this section,we recall the basic notions of the Hele-Shaw and Cahn-Hilliard models.We provide the formulation of the Hele-Shaw-Cahn-Hilliard equations based on[39–40],their dimensionless form,the sharp interface limit,and the weak formulation of the problem.

2.1 The Hele-Shaw and Darcy flows

In this work,we are principally interested in flows under the effect of the gravity and therefore,we consider the vertical Hele-Shaw flow for the vertical analog(see[5]).Still referring to Figure 1,we assume that the Hele-Shaw cell lays in the plane x–y with center in z=0 such that??we introduce the velocity variablethe pressure variable P,the constant densityρ,the constant dynamic viscosityηand the gravitywhere the unit vector indicates the direction of the gravity and g the modulus.To derive the Hele-Shaw flow starting from the Navier-Stokes equations,we assume that V=0 at the walls of the plates,w=0 and the velocity gradients in the x and y directions are negligible with respect to those in the z direction.The Hele-Shaw flow velocity reads(see[5]):

with the gravitational potentialϕp:

defined in terms of a reference coordinateandin this manner,we have that V=V(x,y,z),while P=P(x,y).By introducing the specific discharge Q in the inters pace between the two plates,i.e.,the averaged velocity through the thicknesswe have

with Q=Q(x,y)as for the pressure P.It is straightforward to deduce the analogy of the Hele-Shaw flow with the two-dimensional Darcy law,since mass conservation leads to the requirement that∇·Q=0 in the computational domain,with representing the permeability of the porous medium.

with the average density

2.2 The Cahn-Hilliard equation

In this manner,the first two dimensionless equations in the system(2.21)read

In addition,we introduce a total free energy,say E CH(c).For the problem under consideration,in which c is the unique dependent variable and the densityρconstant,we assume

where the total free energy function(Helmholtz potential)is defined as

with f 0(c)the bulk energy density function,andσandλsuitable positive,dimensional parameters.We remark that the thickness of the interfaces among the pure phases is proportional toFollowing[42],we define the chemical potentialµCH(c)in the case of the isothermal Cahn-Hilliard equation with constant density as

which assumes the explicit form

We remark that the choice of the bulk energy density function f 0(c)determines the separation of the phases typical in immiscible fluids,provided that it is in general non convex and in the form of a double-well function in the variable c,with the pure phases in c1 and c2;in particular,we choose the following quartic form for f 0(c):

which is polynomial,differentiable in c and with the pure phases in c1=1 and c2=0.2 A typical choice for f 0(c)is represented by the logarithm function with singular values(see e.g.[11]);for an alternative choice see e.g.[15].

Finally,the isothermal Cahn-Hilliard equation with constant density and periodic boundary conditions reads:

where D 0>0 is the mobility which we assume constant(with the dimension of a diffusivity); are abridged notations to indicate periodic boundary conditions.For example,ifΩ=(0,L 0)2 with

and

these conditions are being understood as c(0,y)=c(L 0,y)andfor y∈(0,L 0).

We observe that the total free energyassociated to the Cahn-Hilliard equation represents a Liapunov functional,i.e., for all t∈[0,T),and the system is mass conservative in the sense thatwith

2.3 The Hele-Shaw-Cahn-Hilliard equations

In this section,we briefly recall the Hele-Shaw-Cahn-Hilliard equations in the pressure and phase field variables and we highlight their properties.For their detailed derivation and thermodynamical aspects we refer the interested reader to[39,40]and also[42].

We start by assuming that the pure phases of the phase variable c are c1=1 and c2=0.We define the phase dependent densityρ(c)for a mixture as

and

and the phase dependent dynamic viscosityη(c)as

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for some constant densities ρi and dynamic viscosities ηi for i=1,2;we observe that when c=c i,we haveρ= ρi andη=ηi for i=1,2.In addition,we introduce the following coefficient α,which possesses the dimension of an inverse density,as

for which we have

where from(2.2)the average pressure p avg=p avg(x)is

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We introduce the surface tension vector s(c)which reads

In this manner,the specific discharge Q introduced in Subsection 2.1 for the Hele-Shaw flow is modified in q=q(p,c)to take into account for the dependency of the viscosity and density on c and the surface tension s(c):

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q(p,c)is also referred to as mass-averaged velocity.Furthermore,for the sake of simplicity,we define the fluxθ(p,c,˙c),for which we highlight the explicit dependency on the time derivative on the phase variable c,as

We introduce the total free energy E(p,c)in a similar manner for the Cahn-Hilliard equation of Subsection 2.2 to include the potential energy associated to the gravitational effect,which reads

with the free energy function f(c)given in(2.5)and the gravitational potentialϕP in(2.2);we recall thatΩrepresents a two-dimensional computational domain and b is the distance between the plates of the Hele-Shaw cell.The chemical potential,let sayµ(p,c),is defined in terms of the Gibbs energy g(p,c)for which we have the relationin the isothermal case(see[42]).Specifically,we have

with the explicit expression for the chemical potentialµ(p,c)from(2.5)and(2.11):

Finally,we provide the strong form of the Hele-Shaw-Cahn-Hilliard equations(see[39])endowed with periodic boundary conditions:

where the constant mobility M 0>0 assumes a role similar to the diffusivity D 0 introduced in(2.8).3 The productdimensionally represents a diffusivity or kinematic viscosity(square length over time). The first equation represents the modified Hele-Shaw equation,while the second is the modified Cahn-Hilliard equation which accounts for the pressure p.The system of PDEs(2.21)is parabolic with second-order spatial derivatives in the pressure variable p and fourth-order derivatives in the phase variable c;the time derivative appears explicitly only for the phase variablewhereas the pressure variable p only depending implicitly on t through the phase transition.The requirement that the average pressure p is zero for all t∈(0,T)is introduced to ensure that the pressure problem is well posed and it is compatible with the change of variables in(2.12).The Hele-Shaw-Cahn-Hilliard equation is referred as quasi-incompressible(see[1,42])in the sense that,at the steady state,the model is incompressible except at the interfaces between the pure phases,provided that a full separation among them occurred;indeed,it is easy to see that in the pure phases,for which∇c=0,we have∇·q(p,c)=0 at the steady state;also,we observe that when the density is constant,i.e.,ρ(c)= ρandα=0,the model is fully incompressible a.e.for all t∈(0,T).Similarly to the Cahn-Hilliard equation,the total free energy E(c)(see 2.18)is a Liapunov functional,i.e., for all t∈(0,T);and the system is mass conservativesince for all t∈(0,T)for the periodic boundary conditions under consideration.

2.4 The dimensionless Hele-Shaw-Cahn-Hilliard equations

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The system(2.21)depends on the three following dimensionless parameters:

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the first being the Cahn number,the second being an analogue of the Mach number and the latter the Péclet number,with definitions similar to those for the Navier-Stokes-Cahn-Hilliard equations in[42].We remark that the Cahn number is related to the thickness of the interfaces between the pure phases,while the Mach number is related to the ratio between the pressure and chemical(surface tension)forces.We define the following dimensionless quantities:

Also,in place of the pressure variable P introduced in Subsection 2.1,we consider an averaged pressure which filters the gravitational effect and it is particularly suitable for the imposition of periodic boundary conditions.Specifically,we define

and

We rewrite the system of PDEs(2.21)in terms of dimensionless variables and parameters;we will denote with the superscript∗the dimensionless quantities.We start by introducing the scaling for the space and time variables and being L 0 and T0 the representative length and time scales,respectively.In addition we introduce the scalings p0,ρ0,η0 and q0 for the pressure,density,dynamic viscosity and velocity(specific discharge),respectively.In over,we define the parameterandthis manner,weset and c∗:=c sincethe phasevariableisalready dimensionless;morewe observe that and for i=1,2.In addition,we have that By assuming that the representative quantities L 0,ρ0 and η0 are chosen a priori,we obtain through dimensional analysis that the representative time,pressure and velocity are andrespectively.

In order to recall the isothermal Cahn-Hilliard equation for a binary fluid with constant densityρ,let us denote with c the dimensionless concentration of one of the phase variables defined in a computational domainΩ⊂R2 as the one represented in Figure 2.We specifically consider for example the case of periodic boundary conditions(see e.g.[27,41]),even if other cases can be considered as well 1 We remark that pairs of compatible boundary conditions need to be specified on each subset of the boundary∂Ωdue to the fourth-order spatial operator characterizing the Cahn-Hilliard equation..Albeit flow problems in the Hele-Shaw vertical cell—as that depicted in Figure 1—appear,at first glance,far from carrying periodic boundary conditions.We will see later that these conditions can instead be conveniently applied under certain circumstances.With this aim,let us introduce the notationΓin andΓout to indicate the subsets of the boundary∂Ω on which the pairs of periodic boundary conditions will be imposed,whereandthe vector b n represents the outward directed unit vector normal to ∂Ω (Γin orΓout).

with a straightforward scaling for the boundary and initial conditions.By observing that the dimensionless free energy function f∗(c∗)is

the dimensionless total free energy E∗ =E∗(p∗,c∗)associated to the system(2.21)reads

where

For the sake of simplicity,in the rest of the paper,we will omit the superscript∗to indicate dimensionless quantities and,otherwise else specified,we will refer only to dimensionless variables and quantities.

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Remark 2.1 The specific discharge q of(2.24)stands for a mass—averaged velocity and is not divergence—free as seen in(2.27);in addition,the chemical potentialµ of(2.26)directly depends on the pressure variable p.These make the analysis of such Hele-Shaw-Chan-Hilliard formulation quite involved.Conversely,the phase field model of[16]yields a straightforward analysis as it considers a volume–averaged velocity and a modified pressure instead of the physical pressure p which,in our model,is unbounded with respect to the interface thickness

2.5 The sharp interface limit:Darcy flows with discontinuous physical data

By recalling the results of[39–40],we briefly discuss the sharp interface limit of the Hele-Shaw-Cahn-Hilliard equations.In particular,we are interested in determining the so-called sharp interface model(PDEs)which is obtained from the phase field model if the thickness of the interfaces between the phases tends to zero.The procedure for the derivation of the sharp interface limit is based on the approach of[45]for which asymptotic expansions of the variables p and c in terms of the thickness of the interface are used both far away from the interface(outer expansion)and within the interface(inner expansion);see also[1,9,26,42].In fact,the investigation of the sharp interface limit constitutes a validation of the phase field model by means of consistency with the interface problem with discontinuous physical data which the model aims at representing through smooth by sharp interfaces.Specifically,the sharp interface model obtained as limit of the Hele-Shaw-Cahn-Hilliard equations is represented by Darcy flows(see[5])with discontinuous data across an internal interface.

We recall the Darcy equations to model the motion of two immiscible fluids contained in two separated regions(subdomains)Ω1 andΩ2 such thatanda.e.with an interface(see Figure 3).Let us observe that when a prescribed phase field variable c is provided and the phases are fully decomposed,the variable c can be used to track the subdomainsΩi,i=1,2,and the interfaceΓby identifying the locations for which c=c1 or c=c2,respectively.For the specific choice of the bulk energy functional(2.7)for which the pure phases corresponds to c1=1 and c2=0,we can define

As consequence,for this choice,we have that where c=c1=1 we are referring to the subdomain Ω1 filled with a fluid with density ρ1 and viscosity η1;conversely,when c=c2=0,we indicate the subdomain Ω2 with the physical propertiesρ2 and η2.In each subdomain Ωi,i=1,2,we define two variables p i for the pressure and the fluxessuch that

Then,the Darcy model in each subdomainΩi,i=1,2,reads

with suitable boundary conditions(Dirichlet,Neumann or Robin)imposed on the external boundary ifIf we denote with p the pressure variable in Ω,we have that and similarlyfor i=1,2.By denoting with the jump of a generic variable across the interfaceΓand withthe unit vector normal to Γ which,by convention,we assume as outward directed with respect to the subdomainΩ2(see Figure 3),we define the following interface conditions for two immiscible fluids:

whereτis a dimensionless surface tension coefficient andκindicates the signed dimensionless curvature of the interface.Due to the convention chosen for the normal to the interfaceΓ,we have that

Remark 2.2 The interface conditions(2.32)corresponding to the Darcy equations(2.31)represent a particular,but physically consistent case of an internal discontinuity interface problem associated to the Laplace equations defined on the separated sub domains.The latter problem is afforded for example in[17]where the jumps of the variable p and the flux q(p)are set equal to some prescribed generic functions g p and g q defined on Γ,i.e.,[[p]]Γ =g p and[[q(p)]]Γ =g q;for the specific case of Darcy flows,we have g p= τκ and g q=0.For further detailson internal discontinuity interface problems and their numerical approximation,we refer the interested reader e.g.to[6,17].

Let us indicate withεthe dimensionless characteristic size of the interfaces between the pure phases;specifically we setmoreover,we introduce the parameterΣref defined as:4 As discussed in[42]the value ofΣref obtained for the cylindrical coordinates tends in the limit to the value obtained for the planar case;for this reason,we consider an unique value ofΣref for both the cases.

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an example of isreported in Figure4.Following[39,42]it ispossibleto show that under the assumption thatand with the choice of the following scalings inεfor the dimensionless parameters:5 The Péclet number can be also set as independent of ε,specifically we can take P e=1(see[39]).The scaling for the parameter M a corresponds to the choice of the dimensional surface tension coefficient

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the Hele-Shaw-Cahn-Hilliard equations(2.27)tend,in the sharp interface limitto the sharp interface model represented by the Darcy equations(2.31)defined in the subdomainsΩi with the interface conditions(2.32)defined onΓ.In this manner,the Darcy flow problem of two immiscible fluids with discontinuous physical properties can be obtained as the limit of the Hele-Shaw phase field model for the thickness of the interfaces between the pure phases tending to zero.Asa consequence,the numerical approximation of the Hele-Shaw-Cahn-Hilliard equations represents a viable,mathematically and thermodynamically consistent approach for the solution of the Darcy flow problem(2.31)with the interface conditions(2.32)without resorting to interface capturing or interface tracking methods,as e.g.the volume of fluid or level set methods(see[33,44,50]).

Remark 2.3 Following Remark 2.2 and the previous considerations on the sharp interface limit,suitable phase field models coupled with the Cahn-Hilliard equations can be eventually developed to represent generic internal discontinuity interface problems when the interface thickness among the pure phases tends in the limit to zero(see[1,42]).

2.6 The weak formulation

By using a standard notation to denote the Sobolev spaces of functions with Lebesgue measurable derivatives and norms(see[2]),we define the function spaces:

with V and H accounting for the periodic boundary conditions.

We define,for all t∈ (0,T),the residuals R p(p,c)(·): and R c(p,c)(·):H ∈ R for given p∈V and c∈H:

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where,by integration by parts and using(2.26),we have introduced the real-valued form D(p,c)(ψ)for any ψ ∈ H:

Finally,the weak formulation of the Hele-Shaw-Cahn-Hilliard equations reads:

We notice that the choice of the function space H for the weight functionφin(2.36)and the first of the equations(2.39)is due to the second order derivatives in the form D(p,c)(φ)of(2.38).

3 Numerical Approximation

We discuss the numerical solution of problem(2.39)both in terms of the spatial and time approximations which are based on isogeometric analysis and the generalized-αmethod,respectively.For alternative formulations,we refer the reader to[21–22,40,54].

3.1 The spatial approximation:Isogeometric analysis

For the spatial approximation of problem(2.39)we consider isogeometric analysis(see[13,35]).Since the solution field c∈ H ⊂ H 2(Ω)for all t∈(0,T),it is necessary to approximate it with at least globally C1-continuous basis functions;see[51]and e.g.[4,15,27,29,41].For the sake of simplicity,despite the fact that only globally C0-continuous basis functions are necessary for the approximation of the pressure field p ∈ V ⊂ H 1(Ω),we use the same basis functions for both the pressure and phase fields.

植酸的抗氧化特性在于它能与金属离子发生极强的螯合作用,使促进氧化作用的金属离子因螯合失去活性,同时释放出氢,破坏氧化过程中产生的过氧化物,使之不能继续形成醛、酮等产物,由此产生良好的抗氧化性[9]。

We introduce the multivariate B-splines(NURBS for short)basiswhich is composed by n bf basis functions(see[46]).Then,we write the approximate pressure p h=p h(t,x)and phase c h=c h(t,x)fields as

梨黑星病能够侵染梨树地上部分所有绿色幼嫩组织，包括花序、叶片、叶柄、新梢和果实，以叶片和果实受害最重。其症状的共同特点是病部形成明显的黑色霉层。

respectively,with the coefficients and being time dependent;similarly,for the test functions,we choose

and

More specifically,we introduce the discrete function spaces V h⊂V and H h⊂H composed of basis functions of local degree p=2 and such that the periodic boundary conditions are included 6 The requirement of a zero mean value for the pressure field is replaced by setting equal to zero a control variable of the approximate pressure field.;the reader interested in a discussion about the strong imposition of periodic boundary conditions in the field of isogeometric analysis is referred to e.g.[4,41].We indicate with n h,p the dimension of the space V h and with n h,c the dimension of H h;we observe that,due to the previous considerations and the definition of the space V,we have that n h,p=n h,c−1.

The semi-discrete version of the weak problem(2.39)reads:

where c0,h is the L 2 projection of the initial condition c0 onto the space H h;the total number of spatial degrees of freedom is n h:=n h,p+n h,c(or equivalently n h=2n h,c−1).

In view of the time approximation it is convenient to express also the residuals in terms of time derivative of the approximate phase field,say for the sake of simplicity.We define

and

and,from(2.36)–(2.37),the residuals

Since in this work we consider B-splines(NURBS)basis of degree p=2 which are globally C 1-continuous,a 3×3 Gauss-Legendre quadrature rule is used in each element.

3.2 The time approximation:Generalized-αmethod

For the time approximation of the semi-discrete problem(3.1)we consider the generalized-α method(see[12,38]);we outline the numerical approach and,for a more detailed description,we refer the reader to[15,27].

We partition the whole time interval[0,T]into n ts time steps of size,with the discrete time vector,for which the discrete variables readandGiven the variables and at the time t n,the generalized-αmethod consists in solving the following problem at the time step t n+1:

with the residuals given in(3.2).For the parametersαm,αf and δ∈ R we chooseαm=with ρ∞ ∈ [0,1]the spectral radius of the amplification matrix at (see[38]);specifically,we chooseρ∞ =0.5.We solve the problem(3.3)for n=0,···,n ts − 1 with a predictor-multicorrector scheme,similarly to[15,41].The associated linear system is solved by means of the GMRES method(see[48])preconditioned by an algebraic multigrid algorithm with smoothed aggregation(see[25]);the stopping criterion is based on the relative residual with the tolerance 10−6.

4 Numerical Results

We provide and discuss numerical results for the Hele-Shaw-Cahn-Hilliard equations.First,we numerically analyze the sharp interface limit with discontinuous physical data of Subsection 2.5 in the one-dimension.Then,we provide an example of the numerical solution of the Hele-Shaw-Cahn-Hilliard problem in the two-dimensional setting,specifically the so-called“rising bubble”problem driven by two phases with different densities.

4.1 The sharp interface limit under mesh refinement

We consider the numerical approximation of the steady pressure equation(2.39)in onedimension,both in Cartesian and radial coordinates,for a prescribed distribution of the phase variable,say c,inΩ=(0,1).Specifically,the problem reads:

where

and the spatial differential operators are expressed in Cartesian or radial coordinates;consequently,the unit vector is identified as orThe prescribed phase variable assumes the form:

where the independent variable s represents the Cartesian(x)or radial(r)coordinate,s0∈Ωand C a is the Cahn number introduced in(2.22)7 The prescribed phase distribution c of(4.2)solves the steady Cahn-Hilliard equation with suitable boundary conditions;its expression is compatible with the bulk energy function f 0(c)given in(2.7)..

By noticing that(4.1)is linear in the pressure variable,we solve the steady problem by using the isogeometric spatial approximation outlined in Subsection 3.1.Moreover,by following the paradigm introduced in[27]and used in[29,15,41],we relate the interface thicknessεto the dimensionless characteristic mesh size,say h,through a safety coefficientγs>0;specifically,we assume

a typical and effective choice of the safety parameter isγs=2.This choice leads to interfaces between the pure phases to become sharper and shaper as the mesh size reduces,thus allowing to explore the sharp interface limit of the pressure equation.

In Figure 5(a)we report the prescribed phase variable both in Cartesian and radial coordinates for uniform meshes of size,and with the interfaceΓlocated inIn Figure 5(b,c,d)we highlight the pressure variable p h around the interfaces for different data.In Figure 5(a)we consider the Cartesian coordinate withτ=0.5,ρ1=1 and ρ2=0.1;as expected the jump of the pressure across the interface tends for to[[p]]Γ=0,being the curvatureκ=0,while the jump of the derivative of the pressureIn Figure 5(c)we report the result for the same data,but in radial coordinates;in this case,being the curvature of the interfacethe jump of the pressure tends in the limit to[[p]]Γ=1.0,while

In order to evaluate the convergence orders of the pressure and pressure gradients jumps across the interfaces,we introduce the following notions of jumps,dependent onε:

whereandfor c21 and some tolerance tolε>0;specifically,for the one-dimensional case,we set c1=1,c2=0,and tol=10−10.We compute the errors associated to the jumpsand of(4.4),say e p and e dp,as

respectively.

In Figure 6,we report the errors e p and e dp on the pressure and pressure gradient jumps across the interface vs.the parameterεfor two test problems;in the first case,the Cartesian coordinate is used with τ=0.5,ρ1=1.0 and ρ2=0.1,for which[[p]]Γ =0 andwhile in the second case,which is in radial coordinate,τ=1.0,ρ1=1.0 and ρ2=0.05,yielding[[p]]Γ=1.0 andWe remark that the error e p on the pressure jump converges to zero with order 1 inε,while the error e dp on the jump of the pressure gradient converges to zero with order 2 for both the test problems;the same convergence orders are obtained for spatial discretizations with B-splines basis functions of degree r>2,which are at least globally C 1-continuous.Indeed,such results correspond to an intrinsic property of the sharp interface limit(see Subsection 2.5)obtained for the specific Hele-Shaw-Cahn-Hilliard model at hand(see[39]);for more details,we refer the reader to[16]and the analysis therein,which exploits the sharp asymptotic limit for high order terms inε.From this result,extendable to the two-and three-dimensional cases,we deduce that the pressureequation(4.1),endowed with a prescribed phasefield variable c,solves a Poisson problem in p with conditions on the jumps of the pressure and gradient of the pressure across an internal interfaceΓinΩ.

4.2 Discontinuous d ensity:The “rising bubble” p roblem

We consider the case of a “rising bubble”,i.e.,a phase comprised of a “light” fluid(phase 1)embedded in an heavier one(phase 2),which rises in presence of the gravitational force.The numerical solution of this problem was addressed in[36]by comparing different numerical methods;more recently,isogeometric analysis was used in[34]to solve the “rising bubble”problem even if modeled by the Navier-Stokes-Cahn-Hilliard equations.

For this test,we select the computational domainΩ=(0,1)2 and the following parameters introduced in Section 2: ρ1=5.8507,ρ2=14.8507,η1= η2=0.5,σ =73.186,τ=0.235702,g=9.81 andwe deduce that from(2.33), from(2.34).The initial condition for the phase field is c0(x,y)=where

being

this choice determines the distribution of the “light” and “heavy” fluids shown in Figure 7(topleft)which are depicted in red and blue,respectively.We remark that periodic type boundary conditions are used in the sense outlined in Subsection 2.2.

We solve the problem by means of IGA with B-splines basis functions of degree 2 and globally C 1-continuous inΩ;specifically,we select a uniform mesh of size yielding a number of basis functions n bf=16′384.For the time discretization,we use the generalized-α method as described in Subsection 3.2 with ∆t=5.0·10−4.

We report in Figures 7–9 the phase field variable c at different times and,correspondingly in Figures 10–12,the pressure field p.We highlight that the “light” fluid raises in the heavier one and the initial condition c0 determines the pinch-off of the stratified fluid distribution up to the generation of a bubble;the solution exhibits a topological change that is taken into account from the phase field model in a straightforward fashion.Similarly,the pressure p highlights the mechanism driving this topological change and the consequent rise of the “light” bubble into the “heavy” fluid.

5 Conclusions

In this paper,we have considered isogeometric analysis for the spatial approximation of the Hele-Shaw-Cahn-Hilliard equations,a phase field model particularly useful to represent Darcy flows in a two-dimensional cell in presence of discontinuous data through interfaces.We have showed that our numerical formulation is suitable and efficient for approximating this model by solving the benchmark “rising bubble” problem of fluids with different densities in presence of the gravitational force.In addition,we numerically verified the sharp interface limit of the Hele-Shaw-Cahn-Hilliard equations which is represented by Poisson problems in the pressure variable across interfaces and endowed with interface conditions to match discontinuities in the data.Specifically,these interface conditions are jump conditions on the pressure and its gradient and are proportional to the surface tension and the density gap across the interface,respectively.Our numerical tests have showed that the pressure equation of the Hele-Shaw-Cahn-Hilliard equations can also be used to model internal discontinuity interface Poisson problems by prescribing a suitable phase field variable in the computational domain to represent interfaces.

Acknowledgements The authors acknowledge Prof.H.Garcke,Dr.K.F.Lam(University of Regensburg),and Prof.S.Salsa(Politecnico di Milano)for the fruitful discussions and insights into the topic.

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