a fractional phase-field model for two-phase flows with ... · viscosity and density in simulations...

$: A fractional phase-field model for two-phase flows with ... · viscosity and density in simulations of two-phase flow in a pipe and of a rising bubble. We also demonstrate that using$
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Comput. Methods Appl. Mech. Engrg. 305 (2016) 376–404www.elsevier.com/locate/cma

A fractional phase-field model for two-phase flows with tunablesharpness: Algorithms and simulations

Fangying Songa, Chuanju Xub, George Em Karniadakisa,∗

a Division of Applied Mathematics, Brown University, 182 George St, Providence, RI 02912, USAb School of Mathematical Sciences and Fujian Provincial Key Laboratory of Mathematical Modeling and High Performance Scientific Computing,

Xiamen University, 361005 Xiamen, China

Received 2 July 2015; received in revised form 9 March 2016; accepted 13 March 2016Available online 21 March 2016

Highlights

• A new fractional mass-conserving Allen–Cahn model.• A second-order (time) spectral (space) method for the coupled system of fractional equations.• A variable-fractional order model to control multi-rate diffusion.• First numerical solution of the fractional Navier–Stokes equations.

Abstract

We develop a fractional extension of a mass-conserving Allen–Cahn phase field model that describes the mixture of twoincompressible fluids. The fractional order controls the sharpness of the interface, which is typically diffusive in integer-orderphase-field models. The model is derived based on an energy variational formulation. An additional constraint is employed tomake the Allen–Cahn formulation mass-conserving and comparable to the Cahn–Hilliard formulation but at reduced cost. Thespatial discretization is based on a Petrov–Galerkin spectral method whereas the temporal discretization is based on a stabilizedADI scheme both for the phase-field equation and for the Navier–Stokes equation. We demonstrate the spectral accuracy of themethod with fabricated smooth solutions and also the ability to control the interface thickness between two fluids with differentviscosity and density in simulations of two-phase flow in a pipe and of a rising bubble. We also demonstrate that using a formulationwith variable fractional order we can deal simultaneously with both erroneous boundary effects and sharpening of the interface atno extra resolution.

c⃝ 2016 Elsevier B.V. All rights reserved.

Keywords: Fractional Allen–Cahn equation; Sharp interface; Eulerian method; ADI; Spectral method; Variable fractional order

This work was supported by the OSD/ARO/MURI on “Fractional PDEs for Conservation Laws and Beyond: Theory, Numerics andApplications (W911NF-15-1-0562)”. The second author was partially supported by NSF of China (Grant number 11471274).

∗ Corresponding author.E-mail addresses: fangying [email protected] (F. Song), [email protected] (C. Xu), george [email protected] (G.E. Karniadakis).

http://dx.doi.org/10.1016/j.cma.2016.03.0180045-7825/ c⃝ 2016 Elsevier B.V. All rights reserved.

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F. Song et al. / Comput. Methods Appl. Mech. Engrg. 305 (2016) 376–404 377

1. Introduction

The interfacial dynamics of immiscible fluids is of great interest in multi-phase flow systems but computationallynot easy to resolve. The common modeling approaches can be broadly divided into two types of sharp- and diffuse-interface models. The sharp-interface models assume a zero-thickness layer that separates the two fluids. The layeris endowed with properties such as surface tension, and matching boundary conditions are imposed on either side ofthe dividing surface. For numerical simulation of two-phase flow systems with sharp-interface models, moving-gridmethods are commonly used with conforming elements around the interface [1,2]. However, the possibility of meshentanglement restricts the moving-grid approaches to cases with mild deformation of the interface. Even for suchcases, careful remeshing of the domain is required to resolve the interface. In contrast, the diffuse-interface modelsassume a layer of finite thickness between the two phases. Instead of formulating the flow governing equations intwo domains separated by an interface, these methods represent the interfacial tension as a body-force or bulk-stressspreading over a narrow layer covering the interface. This approach yields a unified set of governing equations for thetwo phases, solved on a fixed grid in a purely Eulerian framework. Numerical methods such as Volume-of-Fluid [3,4]and level-set [4,5] have been successfully employed to simulate two-phase flows.

A relatively recent diffuse-interface model for multi-phase flow systems is based on phase fields, long usedin materials science problems. The energy-based variational framework of phase-field formulations makes them athermodynamically-consistent and physically attractive approach to model multi-phase flow systems (cf. [6–8]). Thephase-field model can be viewed as a physically motivated level-set method; however, unlike the level-set model,where an artificial smoothing function is prescribed for the interface, the Cahn–Hilliard model and Allen–Cahn modeldescribe the interface based on the concept of mixing energy. This energy-based description of phase-field modelsalso allows the modeling of complex rheology of non-Newtonian fluids or complex interfacial dynamics to be easilyincorporated into the formulation [9,10]. In this paper we will focus on a new fractional mass-conserving Allen–Cahnmodel, which we derive from a fractional order mixing energy formulation.

The Allen–Cahn equation was originally introduced to describe the motion of anti-phase boundaries in crystallinesolids [11]. If the phases are immiscible, one can introduce a non-local Lagrange multiplier ξ(t) in the Allen–Cahnequation to enforce the volume fraction conservation (cf. [12]), namely,

φt − γ (∆φ − f (φ)+ ξ(t)) = 0, (x, t) ∈ Ω × (0, T ],

d

dt

Ωφdx = 0,

∂φ

∂n

∂Ω

= 0, φ(x, 0) = φ0,

(1.1)

where Ω ∈ Rn, n = 2 (here) is a bounded domain with a convex polygonal domain, n is the outward normal,f (φ) = F ′(φ), F(φ) =

14η2 (φ

2− 1)2 is a double-well potential which achieves the global minimum value at

φ = ±1, and the Lagrange multiplier is ξ(t) =Ω f (φ)dx. Originally, φ represented the concentration of one of

the two metallic components of the alloy and the parameter η represented the interfacial length, which is extremelysmall compared to the characteristic (large) scale of the system. The homogeneous Neumann boundary conditionimplies that no mass loss occurs across the boundary walls. In order to demonstrate the mass-conserving propertyof the constrained Allen–Cahn equation we simulated two-phase flow in a tube with two fluids of different viscositycorresponding to Reynolds numbers Re1 = 66.6 (fluid 1) and Re2 = 200 (fluid 2). The initial phase-field profile isgiven by the following hyperbolic tangent function:

φ(x; 0) = −tanh((

x2 + y2 − r1)/(√

2η)), x ∈ Ω = (r, z) : 0 < r < L/2, 0 ≤ z ≤ 10L,

where r1 = 0.4 is the radius for the initial position of the interface, η is the interface thickness between the two fluidsand L = 1 is the characteristic length. The analytic solution and initial conditions for the velocity are exactly the sameas in [13,14]. In Fig. 1 we plot the error of the solution in the streamwise velocity obtained based on the constrainedAllen–Cahn model but also on the Cahn–Hilliard model; the latter conserves mass, see [15,16]. We see that theAllen–Cahn model is even more accurate than the Cahn–Hilliard as it employs a second-order operator unlike theCahn–Hilliard that requires numerical resolution of a fourth-order operator. Attempting to simulate the same problemwith the unconstrained Allen–Cahn model would lead to an instability due to the violation of mass conservation. The

378 F. Song et al. / Comput. Methods Appl. Mech. Engrg. 305 (2016) 376–404

Fig. 1. Comparison of velocity errors along the pipe radius for two-phase flow with the inside fluid at Re1 = 66.6 and the outside fluid atRe2 = 200. Interface thickness η = 0.01. Blue solid line: Cahn–Hilliard model; black dash–dot line: mass-conserving Allen–Cahn model. (Forinterpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Allen–Cahn equation can also be viewed as a gradient flow with Lyapunov energy functionalΩ

12 |∇φ|

2+

1η2 F(φ)dx.

Recently, the Allen–Cahn equation has been widely applied to many complicated moving interface problems, forexample, vesicle membranes, nucleation of solids, and mixture of two incompressible fluids, etc. (cf. [12,17–22]).

Fractional partial differential equations (FPDEs) are generalizations of the integer-order models, based on fractionalcalculus. They are becoming increasingly popular as a flexible modeling tool for diffusive processes associated withsub-diffusion (fractional in time), super-diffusion (fractional in space) or both. Recently, FPDEs have been applied todiverse fields, including control theory, biology, electro-chemical processes, viscoelastic materials, polymers, finance,etc.; see, e.g., [23–34] and the references therein. In particular, fractional diffusion equations have been used todescribe the so-called anomalous diffusion phenomenon. In fact, anomalous diffusion is ubiquitous in physical andbiological systems where trapping and binding of particles may occur; see, e.g., [35–39]. In this paper, we will considersub-diffusion effects for the phase-field model.

One of the shortcomings of FPDEs is that they are computationally expensive since even low-order discretizationslead to dense matrices due to their non-local character. To this end, direction splitting methods (alternating directionimplicit or ADI methods) are powerful techniques, which allow us to split the underlying high dimensional probleminto a set of one-dimensional sub-problems, thus reducing considerably the computational complexity for someclassical problems (Navier–Stokes equation, diffusion equations, and so on.); see, e.g., [40,41]. The ADI scheme wedevelop below for the fractional phase field model is a straight-forward extension of its counterpart for the traditionaldiffusion equation [42,43].

The objective of this paper is to re-formulate the constrained Allen–Cahn equation coupled to Navier–Stokes equa-tions within the fractional calculus framework in order to simulate multi-phase flow systems with tunable sharpnessof the interface. The paper is organized as follows. In Section 2 we develop the fractional governing equations. InSection 3 we present the numerical discretization schemes and corresponding convergence results. In Section 4 wepresent numerical simulations of two-phase flows, including simulations with variable-fractional order. In Section 5we provide a short summary. In the Appendices we derive an energy law for the fractional Navier–Stokes/phase-fieldmodel and also present discretization details along with some convergence results.

2. A new fractional phase-field model for incompressible two-phase flows

We consider a mixture of two immiscible and incompressible fluids with densities ρ1, ρ2 and viscosities µ1, µ2.In order to identify the regions occupied by the two fluids, we introduce a phase-field function φ such that

φ(x, t) =

1, fluid 1,−1, fluid 2,

(2.1)

with the smooth transition layer of thickness η connecting the two fluids so the interface of the mixture can bedescribed by Γt = x : φ(x, t) = 0. Let F(φ) =

14η2 (φ

2− 1)2 be the Ginzburg–Landau double-well potential, and


define the fractional mixing energy functional

E(φ,∇sφ) =

Ω

12|∇

sφ|2+ F(φ)

dx, (2.2)

where the fractional gradient is defined as ∇s

= ( ∂s

∂xs ,∂s

∂ys ),12 < s ≤ 1, and ∂s

∂xs and ∂s

∂ys are fractional operatorswhich will be defined in the following. Also, E(φ,∇sφ) represents the competition between the hydrophilic andhydrophobic properties of the two-phase flow. Then, the dynamics of the phase-field φ is determined by the gradientflow

φt + (u · ∇)φ = −γδE

δφ, (2.3)

where γ is a relaxation time and δEδφ

is the variational derivative. Specifically, δEδφ

is a generalization of the

Euler–Lagrange equation, and it can be taken in the L2 space, hence leading to the (non-conserving) fractionalAllen–Cahn equation

φt + (u · ∇)φ = γ∆sφ − f (φ)

, (2.4)

where f (φ) = F ′(φ), and ∆s is the fractional Laplace operator to be defined. For immiscible two-phase flows,we can introduce a non-local Lagrange multiplier ξ(t) in the Allen–Cahn equation to enforce the volume fractionconservation,

φt + (u · ∇)φ = γ∆sφ − f (φ)+ ξ(t)

,

d

dt

Ωφdx = 0.

(2.5)

Next, we introduce the fractional operators. The fractional derivative ∂s

∂xs , with 0 < s ≤ 1, Λ = [a, b], can be definedin the Riemann–Liouville sense as follows:

RLDsxφ(x) =

1Γ (1 − s)

d

dx

x

a

φ(ξ)dξ

(x − ξ)s, ∀x ∈ Λ, (2.6)

RLxDsφ(x) = −

1Γ (1 − s)

d

dx

b

x

φ(ξ)dξ

(ξ − x)s, ∀x ∈ Λ. (2.7)

We will also make use of the Caputo definition for the fractional derivative:

Dsxφ(x) =

1Γ (1 − s)

x

a

φ′(ξ)dξ

(x − ξ)s, ∀x ∈ Λ, (2.8)

x Dsφ(x) = −1

Γ (1 − s)

b

x

φ′(ξ)dξ

(ξ − x)s, ∀x ∈ Λ. (2.9)

Usually, RLDsx and Ds

x are called the left-sided fractional derivatives, and RLxDs and xDs the right-sided fractional

derivatives of order s. It can be proved [44] lims→1RLDs

x = − lims→1RL

xDs= lims→1 Ds

x = − lims→1 xDs=

∂∂x ,∀x ∈ (a, b). The fractional gradient operator can be defined:

∇s

= (Dsx , Ds

y), Left-type, (2.10)

∇s

= (x Ds ,y Ds), Right-type, (2.11)

∇s

=12(Ds

x + x Ds, Dsy + y Ds), Symmetry-type. (2.12)

We can also define the fractional Laplace operator in different ways, but this is still an open issue for bounded domains(cf. [45–50]). In these papers the fractional Laplacian is often defined through a fractional power of the spectraldecomposition of the standard Laplacian. Another possible definition is ∆s

:= ∇s· ∇

s . Here we choose to use (2.12)as the definition of ∇

s and define the fractional Laplace operator ∆s of the above system (2.28)–(2.31) as follows (see


comments in Remark 2.3 for a motivation):

∆s:=

12

RLDsx Ds

x +RL

xDsx Ds

+RLDs

y Dsy +

RLyDs

y Ds. (2.13)

The fractional Allen–Cahn equation can describe spinodal decomposition [46,50–52], and chemical and contami-nant transport in heterogeneous aquifers [26,53,54].

The fractional momentum equation for the two-phase system takes the form:

ρ(ut + (u · ∇)u) = ∇ · µD(u)− ∇ p − λ∆sφ∇φ +

12∇|∇

sφ|2, (2.14)

where D(u) = ∇u + ∇uT , and λ is the mixing energy density.

Remark 2.1. The last term on the right-hand-side of the above equation can be described as follows in the case ofinteger-order (i.e., s = 1):

∆φ∇φ +12∇|∇φ|

2= ∇ · (∇φ ⊗ ∇φ), (2.15)

where ∇φ ⊗ ∇φ is the extra elastic stress induced by the interfacial surface tension.

In Eq. (2.14), ρ and µ are dependent variables defined by the linear average

ρ(φ) =ρ1 − ρ2

2φ +

ρ1 + ρ2

2, µ(φ) =

µ1 − µ2

2φ +

µ1 + µ2

2. (2.16)

Hence, the Allen–Cahn phase equation (2.5) and the momentum equation (2.14), together with the incompressibilityconstraint

∇ · u = 0, (2.17)

and suitable boundary and initial condition, form a complete system for (u, p, φ) with ρ and µ given by (2.16).To be specific, we consider the homogeneous Dirichlet boundary conditions for u:

u|∂Ω = 0, (2.18)

and homogeneous Neumann boundary conditions for φ:

−12

I 1−sx Ds

xφ(x, y)+ x I 1−sx Dsφ(x, y)

x=a = 0,

12

I 1−sx Ds

xφ(x, y)+ x I 1−sx Dsφ(x, y)

x=b = 0,

(2.19)

−12

I 1−s

y Dsyφ(x, y)+ y I 1−s

y Dsφ(x, y)

y=a = 0,12

I 1−s

y Dsyφ(x, y)+ y I 1−s

y Dsφ(x, y)

y=b = 0,

(2.20)

where I γz denotes the fractional integral of order γ , defined in the Riemann–Liouville sense as follows:

I γz ϕ(z) =1

Γ (γ )

z

aϕ(ξ)(z − ξ)γ−1dξ, ∀z ∈ Λ, (2.21)

z I γ ϕ(z) =1

Γ (γ )

b

zϕ(ξ)(ξ − z)γ−1dξ, ∀x ∈ Λ. (2.22)

Remark 2.2. We can rewrite the boundary conditions as follows:

−12

D2s−1

x φ(x, y)+ x D2s−1φ(x, y)

x=a = 0,12

D2s−1

x φ(x, y)+ x D2s−1φ(x, y)

x=b = 0, (2.23)

−12

D2s−1

y φ(x, y)+ y D2s−1φ(x, y)

y=a = 0,12

D2s−1

y φ(x, y)+ y D2s−1φ(x, y)

y=b = 0. (2.24)


If s = 1, the boundary conditions in (2.19) and (2.20) can be replaced by the standard homogeneous Neumannboundary condition, i.e.,

∇φ · n∂Ω

= 0.

When the density ratio is small, a usual approach is to use the Boussinesq approximation, i.e., replacing (2.14) by

ρ0(ut + (u · ∇)u) = ∇ · µD(u)− ∇ p − λ∆sφ∇φ +

12∇|∇

sφ|2

+ g(ρ1, ρ2), (2.25)

where ρ0 =ρ1+ρ2

2 and g(ρ1, ρ2) is an additional gravitational force to account for the density difference.

Remark 2.3. We emphasize that there exist several ways to define the fractional gradient and fractional Laplacian.These definitions are not necessarily equivalent to each other, and an energy law is only available for those definitionswhich possess certain self-adjoint property. An energy law analysis of the new system (2.5)–(2.17)–(2.25), subjectto the boundary conditions (2.18)–(2.20), is given in Appendix A Theorem A.1. When the density ratio is large, theBoussinesq approximation is no longer valid, and thus we have to face a fractional momentum equation with variabledensity. In this case the momentum equations have to be modified in a physically consistent way to obtain an energylaw; see also Appendix A for more details.

Next, we reformulate the system (2.4)–(2.14)–(2.17) into an equivalent form, which is more convenient fornumerical approximation. Here, F(φ) is given by

F(φ) =1

4η2 (φ2− 1)2. (2.26)

Using the phase-field equation (2.4), and the fact that f (φ)∇φ = ∇F(φ), we have

∇ p + λ∆sφ∇φ +

12∇|∇

sφ|2

= ∇

p +

λ

2|∇

sφ|2+ λF(φ)

+λ

γ

φt + u · ∇φ

∇φ. (2.27)

Therefore, if we define the modified pressure as Π = p +λ2 |∇

sφ|2

+ λF(φ), we can rewrite the system(2.4)–(2.14)–(2.17) as

φt + (u · ∇)φ = γ∆sφ − f (φ)

, (2.28)

ρ(φ) =ρ1 − ρ2

2φ +

ρ1 + ρ2

2, µ(φ) =

µ1 − µ2

2φ +

µ1 + µ2

2, (2.29)

ρut + (u · ∇)u

= ∇ · µD(u)− ∇Π − λ(φt + u · ∇φ)∇φ, (2.30)

∇ · u = 0. (2.31)

We will solve the system of equations (2.28)–(2.31) by replacing f (φ) by f (φ)− ξ(t) in the phase-field equation(2.28) to ensure mass conservation.

3. Numerical schemes and convergence tests

In this section we present the direction splitting scheme (ADI) for the phase-field equation, which is second-order intime. In Appendix B, we also provide a similar second-order scheme for the incompressible Navier–Stokes equations.We employ the Legendre spectral method for spatial discretization for constant fractional order, and Petrov–Galerkinmethod for variable fractional model. We then verify the accuracy of the coupled scheme and quantify its convergencerates for different state variables through a fabricated smooth solution.

3.1. ADI scheme for phase-field in two dimensions

Let L be the number of the time steps to integrate up to final time T , then ∆t = T/L . We denote by superscripts

the time levels and set initial conditions φ0= φ(x, 0), φ−1

= φ0, u0= u(x, 0), u−1

= u0, Π −12 = p0, ψ−

12 = 0,


and ρ = min(ρ1, ρ2). (Here, we simulate with a first-order scheme in the first time step.) We look for solutions of

(φn+1,un+1,Π n+12 , ψn+

12 ) for n = 0, . . . , L − 1. We introduce the following notation for convenience:

φn+12 =

12(φn+1

+ φn), φ∗,n+12 =

12(3φn

− φn−1), φn+1=

1, φn+1

≥ 1,φn+1, |φn+1

| ≤ 1,−1, φn+1

≤ −1,

ρn+12 =

ρn+1+ ρn

2, ρ∗,n+

12 =

12(3ρn

− ρn−1), µn+12 =

µn+1+ µn

2,

un+12 =

12(un+1

+ un), u∗,n+12 =

12(3un

− un−1),

p⋆,n+12 = Π n−

12 + ψn−

12 .

(3.1)

• Phase field:

φn+1− φn

∆t− γ

∆sφn+

12 − f (φ∗,n+

12 )

+ (u∗,n+

12 · ∇)φ∗,n+

12 +

S∆t

η2 (φn+1− φn) = 0, (3.2)

with boundary conditions defined in (2.19) and (2.20). Here, S∆tη2 (φ

n+1− φn) is an extra stability term, which

was introduced in [55] for the integer-order case. If the parameter S satisfies S ≥l2

4η2 , then the above scheme is

unconditionally stable, where l is the upper bound of | f ′(φ)|, with F(φ) modified as Eq. (4.9) in [10].

Remark 3.1. We will solve the phase-field equation (3.2) by replacing f (φ∗,n+12 ) by f (φ∗,n+

12 )− ξ∗,n+

12 to ensure

mass conservation, where ξ∗,n+12 =

1|Ω |

Ω f (φ∗,n+

12 )dx.

• Velocity:

ρn+1=ρ1 − ρ2

2φn+1

+ρ1 + ρ2

2, µn+1

=µ1 − µ2

2φn+1

+µ1 + µ2

2, (3.3)

ρn+12 (un+1

− un)

∆t− ∇ · µn+

12 D(un+

12 )+ ρ∗,n+

12 (u∗,n+

12 · ∇)u∗,n+

12

+ ∇ p⋆,n+12 +

λ

γ

φn+1− φn

∆t+ u∗,n+

12 · ∇φ∗,n+

12∇φn+

12 = gn+

12 ,

un+1|∂Ω = 0.

(3.4)

• Pressure correction:

−∇2ψn+

12 = −

ρ

∆t∇ · un+1,

∇ψn+12 · n|∂Ω = 0,

Π n+12 = Π n−

12 + ψn+

12 − χµn+

12 ∇ · un+

12 ,

(3.5)

where χ ∈ [0, 1] is a user-defined coefficient; the choice χ = 0 yields the standard form of the algorithm, whereasχ = 1 yields the rotational form [12].

In order to solve the incompressible Navier–Stokes equations, we will employ the spectral direction splittingmethod [40] based on pressure-stabilization. In Appendix B, we summarize this scheme and present numerical resultsin order to obtain the convergence rate of the velocity and pressure fields before coupling them to the phase-field.

The ADI scheme for the fractional phase-field (3.2) is given by:

ζ n+1− φn

∆t− γ

∆sφn

− f (φ∗,n+12 )

+ (u∗,n+

12 · ∇)φ∗,n+

12 +

S∆t

η2 (ζ n+1− φn) = 0,

ϕn+1− ζ n+1

12∆t

−14γ (RLDs

x Dsx +

RLxDs

x Ds)(ϕn+1− φn)+

S∆t

η2 (ϕn+1− ζ n+1) = 0,

φn+1− ϕn+1

12∆t

−14γ (RLDs

y Dsy +

RLyDs

y Ds)(φn+1− φn)+

S∆t

η2 (φn+1− ϕn+1) = 0,

(3.6)


Fig. 2. Density (phase-field) (a), velocity (b), and pressure (c) errors in L2-norm as a function of ∆t in log–log scale. The space polynomial degreeis N = 64.

with the homogeneous fractional Neumann boundary conditions (2.19) and (2.20). The velocity and phase-field arerepresented as Legendre expansions of order N whereas the pressure is represented as a Legendre expansion of orderN − 2, see Appendices B and C.

3.2. Verification of accuracy and convergence results

In order to test the convergence rates of all state variables, we consider the system in Ω = [−1, 1]2 with µ = 1

and suitable forcing functions such that the exact solutions of phase-field φ, density ρ, velocity u and pressure p aregiven by:

φ(x, t) = sin(t) cos(πx) cos(πy),

ρ(x, t) = φ + 2,

u(x, t) = πsin(t)sin(2πy) sin2(πx),− sin(2πx) sin2(πy)

,

p(x, t) = sin(t) cos(πx) sin(πy),

and the densities of the two fluids are ρ1 = 3, ρ2 = 1, while the viscosities are µ1 = µ2 = 1. Here we employ thefixed parameter χ = 0.5.

We used 652 Legendre–Gauss–Lobatto points so the spatial discretization errors are negligible compared with thetime discretization error. In Fig. 2, we plot the L2-errors of the velocity, pressure, and density between the numericalsolution and the exact solution at t = 1 with different time step sizes and different fractional orders s = 0.85 ands = 1.00, respectively. It is clear from Fig. 2 that the scheme is second-order accurate in time for density (i.e., phase-field) and velocity, while the pressure approximation has 1.7-order accuracy in time; the latter is consistent with theresults for the Navier–Stokes equation presented in the Appendix B.

Next, we investigate the fractional phase-field equation converging to the integer-order model as the fractionalorder 2s is approaching 2. We set the forcing term of the phase-field equation equal to the one derived from theexact solutions corresponding to s = 1. In Fig. 3 we see that the solutions of the fractional phase-field model areapproaching the integer-order phase-field solutions with linear convergence with respect to the parameter (1 − s).

4. Numerical simulations

In the following, we test the new fractional model for two multi-phase flows: the first one is pipe flow and thesecond one is a rising bubble in the presence of gravity. We demonstrate that the fractional model recovers the integer-order model but additionally we can control the sharpness of the interface by the fractional order s. We also addresssome issues regarding boundary conditions, and to this end we introduce a variable-order fractional model, where theorder s varies in space.

4.1. Simulation of two-phase pipe flow

Here we set the domain Ω = [−10, 10] × [−1, 1], and define the boundary conditions on top and bottom asno-slip walls, with the left boundary being the inlet (Dirichlet) and the right boundary the outlet (zero Neumann).


Fig. 3. Density (phase-field) (a), velocity (b) and pressure (c) errors in L2-norm as a function of ∆t in log–log scale with different fractional orderof the phase-field equations 2s. The space polynomial degree is N = 64.

Fig. 4. (a) Streamwise profiles of the velocity u in steady state; (b) distribution of the phase-field for different fractional orders.

Specifically, the initial and boundary conditions for the two-phase laminar flow are as follows: for velocity, we use thefully-developed velocity profile at the inlet and also as the initial set up:

u(y) =

C[1 − δ2

+ µ(δ2− y2)]

δ3(µ+ 1), 0 ≤ |y| ≤ δ,

C(1 − y2)

δ3(µ+ 1), δ < |y| ≤ 1,

(4.1)

where µ =µ2µ1

, δ = 0.8, C = 2. The initial profile of the phase-field is φ = − tanh( |y|−0.8η

).

In Fig. 4 we plot: (a) streamwise profiles of the velocity u in steady state, and (b) the distribution of the phase-fieldfor different fractional orders. Here we use a grid with 2572 points (N = 256) on the domain Ω , and set η =

132 ,

∆t = 0.005, densities ρ1 = ρ2 = 1, and viscosities µ1 = 0.015, µ2 = µ1/3. The profile of the velocity withfractional order s = 0.75 is more accurate than the integer-order model; this difference is more pronounced in thephase-field that exhibits a much sharper interface than the case of s = 1.

4.2. Simulations of a rising bubble

Next, we add the gravity term in the Navier–Stokes equation and simulate the rising of a bubble under differentconditions. For the simulations shown in Fig. 5, we set η = 0.02, γ = 1, λ = 0, ρ1 = ρ2 = 1, µ1 = µ2 = 0.1, ∆t =

0.001, N = 200, and the initial condition φ(x, 0) = tanh( |x+y|+|x−y|−0.6η

). First, without any flow motion (λ = 0) wetest the mass-conserving property of the fractional Allen–Cahn model. We see from the plots of Fig. 5(a) and (b) thatthe mass of the phase-field is not conserved without the Lagrange multiple ξ(t). We notice that the rectangular bubblequickly deforms into a circular bubble due to the surface tension, while the size of the bubble shrinks and eventually


(a) s = 0.9. (b) s = 1.0. (c) s = 0.9. (d) s = 1.0.

Fig. 5. Time evolution of the phase-field without Lagrange multiplier ξ(t) (cases a and b) and with (cases c and d). From top to bottomt = 0.2, 1.0, 3.0.

Table 1Time evolution of the discrete mass (φ = 1) for each different simulation corresponding to Fig. 5.

t (a) (b) (c) (d)

0 0.3600 0.3600 0.3600 0.36000.5 0.3085 0.2555 0.3414 0.33111.0 0.2593 0.1509 0.3348 0.33111.5 0.2087 0.0494 0.3405 0.33113.0 0.0632 0 0.3386 0.33115.0 0 0 0.3386 0.3311

approaches zero size. The decaying rate of the size of the bubble is smaller for the fractional order (s = 0.9) comparedto the integer model (s = 1.0). When we employ the constrained Allen–Cahn model mass conservation is valid, andas shown in Fig. 5(c) and (d), the initially rectangular bubble approaches a circular bubble in steady state. Table 1shows that discrete mass is conserved in steady state using this model. Fig. 6 shows the profiles of the phase fieldalong the horizontal centerline; the interface corresponding to the fractional order (s = 0.9) model is sharper than theinteger-order model (s = 1.0). Next, we perform a parametric study to investigate the effect of grid resolution andof the parameter η governing the interface thickness. Figs. 7 and 8 summarize these simulations. It is interesting thatfor low resolution and small η the initial square bubble does not achieve a circular shape, the latter being the correctanswer obtained for higher resolution and small η. The smaller fractional order s = 0.75 leads to a sharper interfacefor all resolutions and for all values of η.

Next, we present numerical simulations of the bubble rising for a low density ratio. In this case all four boundariesare walls with a no-slip flow velocity as boundary condition. The densities are ρ1 = 0.5, ρ2 = 3ρ1, and the viscositiesareµ1 = µ2 = 0.1, g = 10, λ = 0.1, γ = 1, ∆t = 10−3, T = 1.8, η = 0.04, N = 256, and the external body force

g is the buoyancy force. Initially the bubble is described by φ(x, 0) = tanh(√

x2+(y+0.4)2−0.3η

). It starts as a circularbubble near the bottom of the domain and then it rises, see Fig. 9. Specifically, Fig. 9 shows the bubble rising forfractional order s = 0.85, 0.90, 1.00. The interface between the bubble and the background fluid is sharper for smallerfractional order, however the rising speed also depends on the fractional order. Moreover, for even smaller values of thefractional order s the flow physics we simulate changes substantially corresponding to anomalous diffusion, which hasan effect not only on the rising speed but also on the shape of the bubble. This is shown in simulations correspondingto s = 0.75 in Fig. 10. All parameters have the same values as before except for the fractional order. In addition,Fig. 11 shows the profiles of the phase field at steady state T = 1.8. We can see that the phase-field profiles along


Fig. 6. Profiles of the phase-field with the mass-conserving Allen–Cahn model at time t = 3 (steady state).

(a) s = 0.75, N = 64. (b) s = 1.00, N = 64. (c) s = 0.75, N = 256. (d) s = 1.00, N = 256.

Fig. 7. Phase-field plots at time t = 3. From top to bottom η = 0.02, 0.04, 0.06, for different fractional order and different resolutions N .

the horizontal direction at y = 1 (upper wall) are quite different for different fractional orders. Fig. 11(a) shows thatin terms of the height φ0.75(0, 1) < φ0.85(0, 1) < φ0.90(0, 1) < φ1.00(0, 1), where φs(0, 1) denotes the value of thephase-field with fractional order s at the point (0, 1). The fractional phase-field model causes a boundary layer, dueto its non-local nature, see Fig. 11(b). In the next subsection, we will present a variable-fractional order phase-fieldmodel that can eliminate this boundary layer artifact.

4.3. Variable-fractional order phase-field model

Fractional calculus allows us to define derivatives of variable order that may depend on time or space or both[56,57]. Hence, we can exploit this new freedom to deal with boundary conditions, discontinuities but also with multi-rate physics that may change locally or in time; see several examples in [56]. Here, we employ variable-fractionalmodels to remove the artificial boundary layers at the walls in the rising bubble simulations, which were created dueto the non-local nature of the fractional operators with fractional order s smaller than 1; see examples in Fig. 4(b) andFig. 11. Hence, we introduce a variable-fractional order s(x, y), i.e., depending on the position, and we choose s →1close to the walls but maintain s < 1 away from the walls where the bubble is located.


(a) η = 0.02. (b) η = 0.04.

(c) η = 0.06.

Fig. 8. Phase-field profiles along the horizontal centerline at t = 3.

We have repeated the bubble rising simulations with the variable-fractional order along the x-direction (horizontal)

given by 2s(x) =r + 2s(0)

−

√r2 − x2, r =

1+(2−2s(0))2

2(2−2s(0)) , where s(0) < 1 is the fixed order at x = 0. Thevariation of the fractional order along the y-direction (vertical) is the same as in the x-direction, see Fig. 12. Thevariable-fractional operator is defined as

D2s(x)φ(x) =d

dx

1Γ (2 − 2s(x))

x

a(x − τ)1−2s(x)φ′(τ )dτ +

b

x(τ − x)1−2s(x)φ′(τ )dτ

, (4.2)

with the boundary condition is given

−D2s(x)−1

x φ(x)+ x D2s(x)−1φ(x)

x=a = 0,D2s(x)−1

x φ(x)+ x D2s(x)−1φ(x)

x=b = 0, (4.3)

and all other simulation parameters are the same as in Section 4.2.

Fig. 13 shows the evolutions of the phase-field with variable-fractional order s(x, y). The diffusion rate in themiddle of the domain is slower than the diffusion rate near the boundary in the case of s(0) = 0.75. The bubble iselongated along the (vertical) y-direction because of the different diffusion rates in the domain. Fig. 14 shows thephase-field profiles along the lines y = 1 and x = 0 at t = 1.8, respectively. It is evident that the boundary layers areeliminated at all fractional orders, i.e., s(0) = 0.75, 0.85, 0.90.


(a) s = 0.85.

(b) s = 0.90.

(c) s = 1.00.

Fig. 9. Phase-field evolution from left to right at t = 0.1, 0.7, 1.1, 1.125, 1.175, 1.5 for fractional orders s = 0.85, 0.90, 1.00.

Fig. 10. Phase-field evolution from left to right at times t = 0.1, 0.7, 1.1, 1.4, 1.5, 1.8 for fractional order s = 0.75.

(a) y = 1. (b) x = 0.

Fig. 11. Phase-field profiles along the lines (left) y = 1 (upper wall) and (right) x = 0 (centerline) at t = 1.8, respectively.

5. Summary

This work is of exploratory nature and it represents the first attempt to develop a fractional phase-field modelcoupled to the fractional incompressible Navier–Stokes equation along with corresponding numerical methods. Atfirst, we considered the Cahn–Hilliard model since the standard Allen–Cahn model does not conserve mass. However,we found through simulations with the integer-order formulation that a constrained Allen–Cahn model not onlyconserves mass but it also leads to better accuracy for the same resolution compared to the Cahn–Hilliard model


Fig. 12. Distribution of the variable fractional orders along the horizontal axis; a similar distribution is employed along the vertical axis.

(a) s(0) = 0.75.

(b) s(0) = 0.85.

(c) s(0) = 0.90.

Fig. 13. Phase-field evolution from left to right at times t = 0.02, 0.52, 1.1, 1.12, 1.28, 1.8, for fractional order s(0) = 0.75, 0.85, 0.90, for thetop, middle and bottom rows, respectively.

as it requires numerical resolution of lower order operators at reduced computational cost. The primary objectiveof introducing fractional calculus in phase-field formulations is to investigate the possibility of controlling theinterface thickness by varying the fractional order. For integer-order formulations, this can be accomplished viathe nominal interface thickness η but this requires additional computational cost and possible adaptive refinementthat may weaken the advantages of the Eulerian-based phase-field methods. Surprisingly, we found that we canachieve sharper interfaces by simply tuning the fractional order even at relatively low resolution! However, theintroduction of fractional models led also to undesirable effects, e.g. the formation of erroneous boundary layersat the Dirichlet boundaries of the domain, presumably due to the non-local nature of the fractional operators. To thisend, we introduced the variable-fractional order approach that allows for a graceful transition of the rate of diffusionfrom the interior of the domain to the boundaries, and this resolved the problem. Another erroneous effect may bethe anisotropic diffusion due to the fractional operators introduced in the current work. We are currently workingon other implementations of the fractional Laplacian in bounded domains and we will report these results in future


(a) y = 1. (b) x = 0.

Fig. 14. Phase-field profiles along the lines y = 1 and x = 0 at t = 1.8.

work. A secondary objective of introducing fractional models to conservation laws is to investigate their general utilityand better understand the new possibilities enabled within the fractional formalism. For phase-field formulations, inparticular, an alternative approach would be to derive formally models “between” the Allen–Cahn and Cahn–Hilliardmodels by minimizing the free energy in the proper fractional Sobolev norm, instead of the L2 or H−1 minimizationfor the Allen–Cahn and Cahn–Hilliard models [9,58–60], respectively.

Acknowledgments

This work was supported by the OSD/ARO MURI. The second author is partially supported by NSF of China(Grant number 11471274). We thank the anonymous referees who prompted the analysis of Appendix D and for otheruseful suggestions.

Appendix A. Energy law

We first introduce some basic properties of the spaces related to fractional derivatives, and then we prove that thesystem (2.5)–(2.17)–(2.25) with the definition (2.13) satisfies the energy law.

The fractional operators have following relationships:

RLDsxφ(x) =

1Γ (1 − s)

(x − a)−sφ(a)+ Dsxφ(x), (A.1)

RLxDsφ(x) =

1Γ (1 − s)

(b − x)−sφ(b)+ x Dsφ(x), (A.2)

and

RLDsx (φ(x)− φ(a)) = Ds

xφ(x), (A.3)RL

xDs(φ(x)− φ(b))=x Dsφ(x), (A.4)

where the derivatives Dsx is the Caputo derivative defined in the main text.

Moreover, we list the following useful lemmas.

Lemma A.1. For all functions φ, ϕ ∈ H s(Λ) with φ satisfying the boundary condition (2.19), it holds(RLDs

x Dsx +

RLxDs

x Ds)φ, ϕ

= (Dsxφ ,x Dsϕ)+ (x Dsφ, Ds

xϕ). (A.5)


Proof. Using integration by parts, we obtain

(RLDsx Ds

xφ, ϕ) =

b

a

1Γ (1 − s)

d

dx

x

a(x − ξ)−s 1

Γ (1 − s)

ξ

a(ξ − η)−sφ′(η)dηdξϕ(x)dx

= ϕ(x)I 1−sx Ds

xφ(x)b

a−

b

a

1Γ (1 − s)

x

a(x − ξ)−s 1

Γ (1 − s)

ξ

a(ξ − η)−sφ′(η)dηdξϕ′(x)dx

= ϕ(x)I 1−sx Ds

xφ(x)b

a−

b

a

1Γ (1 − s)

ξ

a

φ′(η)

(ξ − η)sdη

1Γ (1 − s)

b

ξ

ϕ′(x)

(x − ξ)sdxdξ

= (Dsξφ(ξ), ξ Dsϕ(ξ))+ ϕ(x)I 1−s

x Dsxφ(x)

b

a. (A.6)

Similarly, the right fractional operator RLxDs

x Ds can be obtained by the same technique yielding

(RLxDs

x Dsφ, ϕ) = (ξ Dsφ(ξ), Dsξϕ(ξ))+ ϕ(x)xI

1−sxDsφ(x)

b

a. (A.7)

Combining the above two identities and with the homogeneous fractional Neumann boundary conditions which aredefined in (2.19), we can get the identity (A.5).

In the next theorem, we establish the energy law of the Allen–Cahn phase-field system (2.5)–(2.17)–(2.25) for thecase of low density ratio.

Theorem A.1. If u, φ, and ξ are solutions of (2.5)–(2.17)–(2.25) subject to the boundary conditions (2.18)–(2.20),then

d

dt

Ω

12ρ0|u|

2+ λF(φ)

dx +

λ

2|∇

sφ|2∗

=

Ω

g · udx −

Ω

µ2

|D(u)|2 + λγ(∆sφ − f (φ))+ ξ(t)

2

dx, (A.8)

where |∇sφ|

2∗ = −(Ds

xφ, xDsφ)− (Dsyφ, yDsφ).

Proof. Taking the inner product of (2.5) with λ∆sφ − f (φ)+ ξ(t)

, we obtain

φt + (u · ∇)φ, λ(∆sφ − f (φ)+ ξ(t))

= γ λ∆sφ − f (φ)+ ξ(t)

20.

(A.9)

For the terms on the left-hand-side, we have the following basic facts:φt , f (φ)

=

d

dt

Ω

F(φ)dx,(u · ∇)φ, f (φ)

= −(φ f (φ),∇ · u)+

∂Ωφ f (φ)u · ndσ = 0,

(φt , ξ(t)) = ξ(t)d

dt

Ωφdx = 0,

(u · ∇)φ, ξ(t)

= −ξ(t)(φ,∇ · u)+ ξ(t)∂Ωφu · ndσ = 0.

Bringing all these into (A.9), we get

λφt + (u · ∇)φ,∆sφ

−

d

dt

ΩλF(φ)dx = γ λ

∆sφ − f (φ)+ ξ(t)2

0. (A.10)

Next, taking the inner product of (2.25) with u, and using the boundary conditions for u, we get

ρ0(ut + (u · ∇)u,u) = (∇ · µD(u),u)− (∇ p,u)− λ∆sφ∇φ +

12∇|∇

sφ|2,u

+ (g,u).


Taking into account the fact that

ρ0(ut + (u · ∇)u,u) =d

dt

Ω

12ρ0|u|

2dx,

(∇ · µD(u),u) = −

Ω

µ

2|D(u)|2dx,

(A.11)

and

−(∇ p,u)−∇|∇

sφ|2,u

= 0,

we obtain

d

dt

Ω

12ρ0|u|

2dx = −

Ω

µ

2|D(u)|2dx − λ

∆sφ∇φ,u

+ (g,u). (A.12)

Subtracting (A.12) from (A.10) gives

d

dt

Ω

12ρ0|u|

2dx − λφt ,∆sφ

+

d

dt

ΩλF(φ)dx

= −

Ω

µ

2|D(u)|2dx + (g,u)− γ λ

∆sφ − f (φ)+ ξ(t)2

0. (A.13)

Finally, in virtue of a result established in Lemma A.1, we have

− (∆sφ, φ) = −12

(Ds

xφ, x Dsφ)+ (x Dsφ, Dsxφ)+ (Ds

yφ, y Dsφ)+ (y Dsφ, Dsyφ)

. (A.14)

(According to a result given in [61] the right-hand-side of (A.14) defines a coercive bilinear form in the H s(Ω)-seminorm.)

This leads to

− λφt ,∆sφ

= −

λ

2

d

dt(Ds

xφ, x Dsφ)+d

dt(Ds

yφ, y Dsφ). (A.15)

Bringing (A.15) into (A.13) gives (A.8).

For a large density ratio, there is no energy law available for the original system (2.5)–(2.14)–(2.17). This isalso true in case of integer order Allen–Cahn phase-field model [8]. Following the idea in [62], we can modify theNavier–Stokes equations (2.14)–(2.17) in a physically consistent way to obtain a system which admits an energy law.A critical property of the nonlinear term for the Naiver–Stokes equations (with constant density) is the skew-symmetricproperty

Ωρ0(u · ∇)v · vdx = 0 if ∇ · u = 0, u · n|∂Ω = 0, and v is sufficiently smooth, (A.16)

which does not hold if the constant ρ0 in the above is replaced by a nonconstant function ρ. To overcome this difficulty,Guermond and Quartapelle [62] introduced a new variable τ =

√ρ in place of ρ. Using the mass conservation

ρt + ∇ · (ρu) = 0, (A.17)

one derives

τ(τu)t = ρut +12ρt u = ρut −

12∇ · (ρu)u. (A.18)

Therefore, it is physically consistent to replace ρut in (2.14) by τ(τu)t +12∇ · (ρu)u, leading to the modified

momentum equation

τ(τu)t + (ρu · ∇)u +12∇ · (ρu)u = ∇ · µD(u)− ∇ p − λ

∆sφ∇φ +

12∇|∇

sφ|2. (A.19)


The main advantage of the new formulation is that the following desired property holds:Ω(ρu · ∇)v · vdx +

12

Ω

∇ · (ρu)v · vdx = 0, if u · n|∂Ω = 0. (A.20)

Given the above property, we can then derive that the modified Allen–Cahn phase-field system (2.5)–(2.17)–(A.19)admits the following energy law

d

dt

Ω

12|τu|

2+λ

2|∇

sφ|2∗ + λF(φ)

dx = −

Ω

µ2

|D(u)|2 + λγ∆sφ − f (φ)+ ξ(t)

2

dx. (A.21)

Appendix B. A spectral ADI scheme for the incompressible Navier–Stokes equations

In this appendix, we will demonstrate that the spectral ADI scheme is an effective method for the Navier–Stokesequation of second-order accuracy in velocity. Specifically, we consider the two-dimensional Navier–Stokes equationwith constant density and viscosity, i.e.,

∂u∂t

− µ∇2u + (u · ∇)u + ∇ p = f (x, t), in Ω × (0, T ], (B.1)

∇ · u = 0, in Ω × (0, T ], (B.2)

u = 0, on ∂Ω × (0, T ], (B.3)

with initial condition u(x, 0) = u0(x), and Ω = [−1, 1]2.

We use a direction splitting scheme [40] as follows: Setting u0= u(x, 0), u−1

= u0, p−12 = p0, and ψ−

12 = 0,

we find the numerical solutions of (un+1, pn+12 ) for time level n.

• Velocity equation:

ξn+1− un

∆t− µ∆un

+ (u∗,n+1· ∇)u∗,n+1

+ ∇ p⋆,n−12 = f n+

12 ,

ηn+1− ξn+1

12∆t

− µ∂xx (ηn+1

− un) = 0, ηn+1|x=±1 = 0,

un+1− ηn+1

12∆t

− µ∂yy(un+1− un) = 0, un+1

|y=±1 = 0.

(B.4)

• Pressure equation:

ϱn+12 − ∂xxϱ

n+12 = −

∇ · un+1

∆t, ∂xϱ

n+12 |x=±1 = 0,

ψn+12 − ∂yyψ

n+12 = ϱn+

12 , ∂yψ

n+12 |y=±1 = 0,

(B.5)

and

pn+12 = pn−

12 + ψn+

12 − χµ∇ · un+

12 . (B.6)

Next, we check the numerical convergence of the above algorithm both in time and space. The spectral methodis used for space discretization and it is based on (PN , PN−2), i.e., the polynomial degree of velocity is N and thepolynomial degree of pressure is N − 2. We assume the following exact solutions of velocity u and pressure p:

u(x, t) = π sin(t)sin(2πy) sin2(πx),− sin(2πx) sin2(πy)

,

p(x, t) = sin(t) cos(πx) sin(πy),

and the viscosity is µ = 0.025, and the fixed parameter is χ = 0.5. The forcing term f (x, t) is derived from the exactsolutions substituting into Eq. (B.1).


Fig. B.1. Velocity (a) and pressure (b) errors in L2-norm as a function of ∆t in log–log scale; here the space polynomial degree is N = 30.

Fig. B.2. Velocity (a) and pressure (b) errors in L2-norm as a function of ∆t in log–log scale; here the space polynomial degree is N = 64.

First, we investigate the direction splitting error. We plot in Fig. B.1 the results for the space polynomial degree ofvelocity N = 30 and Fig. B.2 for space polynomial degree of velocity N = 64. Here we consider the Navier–Stokesequation without the advection term (i.e., Stokes flow). The L2-velocity errors (sub-figures (a)) is second-orderconvergence for any time step ∆t when the space polynomial degree N is sufficiently large. We also observe thatthe pressure needs smaller time steps to achieve second-order convergence with the larger polynomial degree N ;however, it exhibits 3

2 -order convergence if the time step is not small enough. In Fig. B.3 we add the advectionterm into the Navier–Stokes equation. In this case, the errors of velocity and pressure are both larger than theerrors of the first test. However, the L2-velocity and even the pressure errors are both second-order for any timestep ∆t .

Next, we investigate the spectral (exponential) accuracy in space. We plot the results in Fig. B.4 for time step∆t = 10−4. We see that the errors of the velocity and pressure decay exponentially fast with the polynomial degreeN . Then, we use a spectral approximation with (PN , PN ) (i.e. the polynomial degrees of velocity and pressure areequal). Although, Fig. B.5 shows that the velocity and pressure errors decay exponentially fast with the polynomialdegree N , the solutions are less accurate than the approximation corresponding to (PN , PN−2).

Finally, we compute the extra error of the above splitting method. As we know, this method will add an extra term∆tµ2

4 ∂xx∂yyun+1

− un

compared with the standard Crank–Nicolson scheme. Table B.1 shows the extra terms andthe numerical errors of our whole scheme. The extra term is second-order and comparable to the temporal integrationerrors, so the overall accuracy of the scheme is second-order in time and exponential in space.


Fig. B.3. Velocity (a) and pressure (b) error in L2-norm as a function of ∆t in log–log scale; here the space polynomial degree is N = 64.

Fig. B.4. (PN , PN−2) formulation: Velocity (a) and pressure (b) error in L2-norm as a function of polynomial degree N ; here we set the time step∆t = 10−4.

Fig. B.5. (PN , PN ) formulation: Velocity (a) and pressure (b) error in L2-norm as a function of polynomial degree N ; here we set the time step∆t = 10−4.


Table B.1

The errors of the splitting method and extra terms in L2-norm with different time steps ∆t .

∆t ∥u∥L2(Ω) Extra-u ∥p∥L2(Ω) Extra-p

5.00e−02 5.0984e−02 5.0873e−02 2.2215e−01 5.0828e−031.00e−02 2.7092e−03 1.9367e−03 2.0389e−02 2.9425e−045.00e−03 6.7141e−04 4.8209e−04 6.2794e−03 7.8816e−051.00e−03 2.6796e−05 1.9173e−05 4.6092e−04 4.0411e−065.00e−04 6.7094e−06 4.7899e−06 1.5638e−04 1.1000e−061.00e−04 2.8771e−07 1.9152e−07 1.2956e−05 5.3404e−08

Appendix C. Spectral method for the spatial discretization of fractional phase-field equation

In this appendix we present a spectral method for the spatial discretization, and details of the fully discrete schemeof the fractional phase-field.

Let N be a non-negative integer, and we denote by PN (Λ) the set of all polynomials of degree less thanor equal to N defined in Λ, and set P0

N (Λ) := φ ∈ PN (Λ) : φ(±1) = 0, PN (Ω) = PN (Λ) ⊗ PN (Λ),P0

N (Ω) := φ ∈ PN (Ω) : φ|∂Ω = 0. Let SN = P0N (Ω).

Let xi Ni=0 (also denoted, when the variable is y, by y j

Nj=0 with yi = xi for 0 ≤ i ≤ N ) and ωi

Ni=0 be the nodes

and associated weights of the Legendre–Gauss–Lobatto quadrature in Λ, i.e., xi Ni=0 are zeros of (1 − x2)L ′

N (x),where L N is the Legendre polynomial of degree N , and ωi

Ni=0 are such that

1

−1ϕ(x)dx =

Ni=0

ϕ(xi )ωi , ∀ϕ ∈ P2N−1(Λ). (C.1)

Let Σ be the set of all collocation points, i.e., Σ := (xi , y j ) : 0 ≤ i, j ≤ N , and ΣI the set of all interiorcollocation points, i.e., ΣI := (xi , y j ), 1 ≤ i, j ≤ N − 1. We denote by I N the polynomial interpolation operatorbased on the set Σ , i.e., I N : C(Ω) → PN (Ω), such that, for all f ∈ C(Ω),

I N f (xi , y j ) = f (xi , y j ), ∀(xi , y j ) ∈ Σ . (C.2)

We then define a number of discrete inner products:

(ϕ, ψ)N ,Λ :=

Ni=0

ϕ(xi )ψ(xi )ωi , ∀ϕ,ψ ∈ C0(Λ), (C.3)

(ϕ, ψ)N ,Ω :=

Ni, j=0

ϕ(xi , y j )ψ(xi , y j )ωiω j , ∀ϕ,ψ ∈ C0(Ω), (C.4)

(ϕ, ψ)y,N ,Ω :=

Nj=0

ϕ(x, y j ), ψ(x, y j )

Λω j , ∀ϕ,ψ ∈ C0(Ω), (C.5)

(ϕ, ψ)x,N ,Ω :=

Ni=0

ϕ(xi , y), ψ(xi , y)

Λωi , ∀ϕ,ψ ∈ C0(Ω), (C.6)

and the associated norm

∥ϕ∥N ,Ω := (ϕ, ϕ)1/2N ,Ω , ∥ϕ∥y,N ,Ω = (ϕ, ϕ)

1/2y,N ,Ω , ∥ϕ∥x,N ,Ω = (ϕ, ϕ)

1/2x,N ,Ω .

Now we present the following spectral method for (3.6) based on the weak formulation with Legendre–Gauss–Lobatto quadrature.


• Let φ0N = I Nφ0.

• Predictor for φn+1N : We compute the predictor ζ n+1

N by(

1∆t

+S∆t

η2 )(ζ n+1N − φn

N , vN )

N ,Ω=γ

2

(Ds

xφnN , x DsvN )y,N ,Ω + (x Dsφn

N , DsxvN )y,N ,Ω

+ (Dsyφ

nN , y DsvN )x,N ,Ω + (y Dsφn

N , DsyvN )x,N ,Ω

−

f (φ

∗,n+12

N ), vN

N ,Ω −(u

∗,n+12

N · ∇)φ∗,n+

12

N , vN

N ,Ω , ∀vN ∈ PN (Ω), (C.7)

here u∗,n+

12

N is the numerical solution of momentum equations at each time step.

• Direction splitting: First find ϕn+1N (x, y j ) ∈ PN (Λ), j = 0, . . . , N , such that, ∀vN ∈ PN (Λ), 1

12∆t

+S∆t

η2

((ϕn+1

N − ζ n+1N )(x, y j ), vN (x))

N ,Λ

=γ

4

Ds

x (ϕn+1N − φn

N )(x, y j ), x DsvN (x)Λ

+γ

4

x Ds(ϕn+1

N − φnN )(x, y j ), Ds

xvN (x)Λ. (C.8)

Then find φn+1N (xi , y) ∈ PN (Λ), i = 0, . . . , N , such that, ∀vN ∈ PN (Λ), 1

12∆t

+S∆t

η2

((φn+1

N − ϕn+1N )(xi , y), vN (y))

N ,Λ

=γ

4

Ds

y(φn+1N − φn

N )(xi , y), y DsvN (y)Λ

+γ

4

y Ds(φn+1

N − φnN )(xi , y), Ds

yvN (y)Λ. (C.9)

Remark C.1. On the right-hand-side numerical quadratures are only used in the direction in which non fractionalderivatives are applied. As it is known that the fractional derivative of a polynomial is no longer a polynomial, andnaive applications of Legendre–Gauss–Lobatto quadratures would result in a loss of accuracy. In our implementation,we use an efficient way to evaluate the integrals on the right-hand-sides of (C.7)–(C.9), which is described below. Adirect calculation shows

Dsx pN (x) =

1Γ (1 − s)

x

−1(x − τ)−s p′

N (τ )dτ = (1 + x)1−sϕ(pN (x)),

x Ds pN (x) = −1

Γ (1 − s)

1

x(τ − x)−s p′

N (τ )dτ = (1 − x)1−sψ(pN (x)),

where

ϕ(pN (x)) =1

Γ (1 − s)

1

−1(1 − θ)−s p′

N

x + 12

θ +x − 1

2

dθ,

ψ(pN (x)) = −1

Γ (1 − s)

1

−1(1 + θ)−s p′

N

1 − x

2θ +

x + 12

dθ.

Obviously, if pN (x) ∈ PN (Λ), then ϕ(pN (x)), ψ(pN (x)) ∈ PN−1(Λ). Thus for pN , qN ∈ PN (Λ), we have

(Dsx pN (x), x DsqN (x))Λ =

Λ(1 − x)1−s(1 + x)1−sϕ(pN (x))ψ(qN (x))dx

=

Ni=0

ϕ(pN (x1−s,1−si ))ψ(qN (x

1−s,1−si ))ω

1−s,1−si , (C.10)

where xγ,γi Ni=0 and ω

γ,γ

i Ni=0 are the Jacobi–Gauss–Lobatto points and weights associated to the weight function

(1 − x)γ (1 + x)γ . The formula (C.10) will be used to evaluate the integrals in (C.7)–(C.9) in the direction wherefractional derivatives appear.


Remark C.2. We use Petrov–Galerkin method for variable-fractional order phase-field model. Then the weak formulaof the fractional operator which is defined as (4.2) in x-direction can be written as follows: d

dx(D2s(x)−1

x + x D2s(x)−1)φN (x), qN (x)Λ

= −(D2s(x)−1

x + x D2s(x)−1)φN (x), q ′

N (x)Λ

≈ −

Ni=0

(D2s(xi )−1xi

+ xi D2s(xi )−1)φN (xi )q′

N (xi )ωi . (C.11)

Appendix D. Numerical surface tension

In this appendix, we present a numerical method for computing the interface width and surface tension.Following the method in [9], we consider a one-dimensional model. Then the fractional mixing energy functional

takes form

E(φ,∇sφ) =

Ω

18(x Dsφ + Ds

xφ)2+ F(φ)

dx,

where

F(φ) =1

4η2 (φ2− 1)2.

We further assume that the diffuse interface is at equilibrium, which means zero chemical potential, i.e.,

δE

δφ= −

14(x Ds

+ Dsx )

2φ + f (φ) = 0, (D.1)

where (x Ds+ Ds

x )2

= (RLDsx +

RLxDs)(x Ds

+ Dsx ) and f (φ) =

1η2 (φ

2− 1)φ, and, in the calculation of the functional

derivative, we have used the integration by parts established in [61] for fractional derivatives. However, unlike theinteger case, we cannot obtain an analytical solution of the above nonlinear fractional equation due to very limittheoretical result concerning fractional PDEs. Nevertheless, it is possible to obtain a solution to Eq. (D.1) in somelimit cases The case s → 1 corresponds to the integer Allen–Cahn equation, the equilibrium profile for φ(x) wasgiven in [9]:

φ(x) = tanh x√

2η

. (D.2)

In case s → 0, solving (D.1) with s = 0 gives the unique non trivial solution:

φ2(x) = 1. (D.3)

This leads to φ(x) = ±1 if we require φ(±∞) = ±1, i.e., total separation of the phases into domains of the purecomponents with an infinitely thin interface. This may explain the decreasing effect of the interface thickness as thefractional order s decreases from 1 to 0.

In the following, we present some numerical results to show numerically how the interface thickness and surfacetension depend on the fractional order s.

We solve the 1D fractional Allen–Cahn equation:

∂φ

∂t= λ

14(x Ds

+ Dsx )

2φ − f (φ)

(D.4)

in the domain (−1, 1)× (0, T ]. The above equation is discretized in time by the following scheme:

φn+1− φn

δt= λ

14(x Ds

+ Dsx )

2φn+1− f (φn)

. (D.5)


The spatial discretization makes use of the method described in Appendix C. Inspired by the results in the two extremecases s = 0 and 1, it is reasonable to conjecture that the equilibrium solution, denoted by φ(x, 2s), would behave like

tanh

x

(2sη2)12s

, (D.6)

which is nothing than the solution in (D.2) in the case s = 1. On the other side, it is readily seen that the abovesolution converges to (D.3) as s tends to 0. The sub-figure Fig. D.1(a) plots the profiles of φ(x, 2s) given in Eq. (D.6)for several values of s between 0 and 1, showing clearly the thinning feature of the interface as s decreases. Ournumerical test in this appendix aims at verifying this conjecture. In the figures Fig. D.1(b), (c), and (d), we present thenumerical solutions of Eq. (D.1) at t = 1000 for s = 0.9, 0.85, 0.75 respectively. The plots with label s(x) correspondto the results from using the variable order 2s(x) =

r + 2s(0)

−

√r2 − x2. The parameters and the initial condition

used in the calculation are: λ = 1, η = 0.1, δ = 0.01, φ(x, 0)x∈[−1,0]

= −1 and φ(x, 0)x∈[0,1]

= 1. For ease ofcomparison the conjectured analytical solution is also plotted in these figures. It is observed that the numerical solutionmatches quite well with the conjectured analytical solution for all tested values of s. In order to have a quantitativecomparison, we evaluate the discrepancy between the conjectured solution and the numerical solution in the L2-normas follows:

eI (s) =

1

−1

φ

1000/δtN (x)− φ(x, 2s)

2dx, (D.7)

where φnN is the numerical solution at tn . It is found that eI (0.9) = 7 × 10−4, eI (0.85) = 2 × 10−4, eI (0.75) =

2×10−3. This serves as evidence that the conjectured function (D.6) is a good approximation to the exact equilibriumsolution of (D.4).

By using the conjectured equilibrium phase function (D.6), the impact of the fractional order s on the equilibriumsurface tension can be analyzed and numerically investigated in a similar way as above.

We first define the surface tension as follows:

σ = λ(σ1 + σ2), (D.8)

where σ1 =

∞

−∞

12

dφdx

2dx and σ2 =

∞

−∞F(φ)dx . Substituting Eq. (D.6) into σ1, σ2, we arrive at the following

matching conditions

σ1 =2(2s)−1/2s

3η−

1s ,

σ2 =(2s)1/2s

3η

1s −2.

(D.9)

If s = 1, we recover the traditional relationship σ = λ(σ1 + σ2) =2√

2λ3η , given in [9].

We may equally consider the following fractional surface tension:

σ s= λ(σ s

1 + σ2), (D.10)

where σ s1 =

∞

−∞

18 (x Dsφ+ Ds

xφ)2dx . By using the Fourier transform, we have σ s

1 = O(η1s −2) with φ(x, 2s) defined

as in Eq. (D.6). In our numerical simulation, the numerical surface tensions are defined as follows

σ1,N =

1

−1

12

dφN

dx

2dx,

σ s1,N =

1

−1

18(x Dsφ + Ds

xφN )2dx,

σ2,N =

1

−1F(φN )dx .

(D.11)


(a) Analytical φ(x, 2s) for several s. (b) Analytical and numerical φ for s = 0.9.

(c) Analytical and numerical φ for s = 0.85. (d) Analytical and numerical φ for s = 0.75.

Fig. D.1. Comparison of the analytical and numerical solutions φ(x, 2s) at t = 1000 for several values of s with λ = 1, η = 0.1.

Table D.1The surface tensions of different η with s = 1.00.

η σ1,N R11,N σ1 σ 1

1,N R13,N σ 1

1 σ2,N R12,N σ2

0.10 4.7140 (−1.00) 4.7140 4.7146 (−1.00) 4.7732 4.7140 (−1.00) 4.71400.09 5.2378 −0.99 5.2378 5.2384 −1.00 5.3063 5.2378 −1.00 5.23780.08 5.8925 −1.00 5.8925 5.8932 −1.00 5.9731 5.8925 −1.00 5.89250.07 6.7343 −1.00 6.7343 6.7351 −1.00 6.8310 6.7343 −1.00 6.73430.06 7.8567 −1.00 7.8567 7.8577 −1.00 7.9757 7.8567 −1.00 7.85670.05 9.4281 −1.00 9.4281 9.4292 −1.00 9.5797 9.4280 −1.00 9.42800.04 11.785 −1.00 11.785 11.786 −1.00 11.988 11.785 −1.00 11.785


η σ1,N R0.901,N σ1 σ 0.90

1,N R0.903,N σ 0.90

1 σ2,N R0.902,N σ2

0.10 6.1674 (−1.111) 6.2116 4.7407 (−0.888) 4.7936 3.6156 (−0.888) 3.57750.09 6.9326 −1.109 6.9830 5.2078 −0.891 5.2643 3.9700 −0.887 3.92870.08 7.9009 −1.110 7.9594 5.7842 −0.891 5.8453 4.4077 −0.887 4.36240.07 9.1636 −1.110 9.2324 6.5148 −0.890 6.5820 4.9626 −0.888 4.91220.06 10.874 −1.110 10.957 7.4731 −0.890 7.5486 5.6909 −0.888 5.63350.05 13.315 −1.110 13.417 8.7895 −0.889 8.8766 6.6916 −0.888 6.62470.04 17.060 −1.110 17.193 10.719 −0.889 10.824 8.1592 −0.888 8.0781



η σ1,N R0.851,N σ1 σ 0.85

1,N R0.853,N σ 0.85

1 σ2,N R0.852,N σ2

0.10 7.4071 (−1.176) 7.3252 4.8255 (−0.823) 4.8530 3.0303 (−0.823) 3.03360.09 8.3807 −1.172 8.2918 5.2672 −0.831 5.2929 3.3037 −0.819 3.30860.08 9.6222 −1.172 9.5242 5.8080 −0.829 5.8320 3.6390 −0.820 3.64560.07 11.254 −1.173 11.144 6.4874 −0.828 6.5100 4.0607 −0.821 4.06940.06 13.487 −1.174 13.360 7.3698 −0.827 7.3911 4.6092 −0.821 4.62020.05 16.709 −1.175 16.556 8.5684 −0.826 8.5886 5.3544 −0.822 5.36870.04 21.750 −1.181 21.528 10.310 −0.829 10.321 6.4243 −0.816 6.4509


η σ1,N R0.751,N σ1 σ 0.75

1,N R0.753,N σ 0.75

1 σ2,N R0.752,N σ2

0.10 12.5262 (−1.333) 10.9609 5.1325 (−0.667) 5.0576 1.8761 (−0.667) 2.02740.09 14.3659 −1.300 12.6141 5.5227 −0.695 5.4256 2.0094 −0.651 2.17490.08 16.7523 −1.304 14.7591 5.9910 −0.691 5.8688 2.1703 −0.654 2.35250.07 19.9517 −1.308 17.6353 6.5666 −0.687 6.4152 2.3691 −0.656 2.57160.06 24.4296 −1.313 21.6594 7.2962 −0.683 7.1096 2.6219 −0.657 2.84990.05 31.1064 −1.325 27.6197 8.2636 −0.682 8.0284 2.9534 −0.652 3.21830.04 42.4811 −1.396 37.1782 9.6759 −0.707 9.3152 3.3670 −0.587 3.7370

Table D.5The surface tensions of different η with s(0) = 0.90.

η σ1,N Rs(x)1,N σ1 σ

s(x)1,N Rs(x)

3,N σs(x)1 σ2,N Rs(x)

2,N σ2

0.10 6.1635 (−1.111) 6.2115 4.7474 (0.889) 4.7968 3.6149 (−0.889) 3.57750.09 6.9293 −1.111 6.9830 5.2145 −0.890 5.2675 3.9691 −0.887 3.92870.08 7.8984 −1.111 7.9593 5.7908 −0.890 5.8485 4.4065 −0.887 4.36240.07 9.1617 −1.111 9.2324 6.5212 −0.889 6.5850 4.9614 −0.888 4.91220.06 10.873 −1.111 10.957 7.4792 −0.889 7.5514 5.6896 −0.888 5.63350.05 13.314 −1.111 13.417 8.7951 −0.888 8.8792 6.6903 −0.888 6.62470.04 17.060 −1.111 17.193 10.724 −0.888 10.826 8.1579 −0.888 8.0781

For the variable order phase-field model, we define the surface tension σ = σ1 + σ2 as Eq. (D.8), and the variablefractional order tension is defined by

σ s(x)= λ(σ

s(x)1 + σ2), (D.12)

where σ s(x)1 =

1−1

18 (x Ds(x)φ + Ds(x)

x φ)2dx .The numerical surface tensions are shown in following tables with different fractional order. Here, the rates

Rs1,N , Rs

2,N , Rs3,N are defined as follows

Rs1,N =

logσ1,N (η1)/σ1,N (η2)

log(η1/η2)

,

Rs2,N =

logσ2,N (η1)/σ2,N (η2)

log(η1/η2)

,

Rs3,N =

logσ s

1,N (η1)/σs1,N (η2)

log(η1/η2)

.

(D.13)

Following Eqs. (D.9) and (D.10) we have the analytical value Rs1 = (− 1

s ), Rs2 = ( 1

s − 2), and Rs3 = ( 1

s − 2). Theexact values of the rates are presented as (∗) in all tables. It is observed from the tables Tables D.1–D.7 that in all




s(x)1,N Rs(x)


2,N σ2

0.10 7.3908 (−1.176) 7.3252 4.8454 (−0.824) 4.8585 3.0262 (−0.824) 3.03360.09 8.3656 −1.176 8.2918 5.2861 −0.826 5.2981 3.2996 −0.820 3.30860.08 9.6083 −1.175 9.5242 5.8257 −0.825 5.8369 3.6350 −0.821 3.64560.07 11.241 −1.175 11.144 6.5038 −0.824 6.5143 4.0569 −0.822 4.06940.06 13.475 −1.175 13.360 7.3846 −0.824 7.3950 4.6056 −0.822 4.62020.05 16.697 −1.175 16.556 8.5813 −0.823 8.5918 5.3514 −0.823 5.36870.04 21.713 −1.177 21.527 10.313 −0.824 10.323 6.4295 −0.822 6.4517



s(x)1,N Rs(x)


2,N σ2

0.10 12.1924 (−1.333) 10.9609 5.2110 (−0.667) 5.0576 1.8658 (−0.667) 2.02740.09 14.0299 −1.300 12.6141 5.5981 −0.695 5.4256 1.9991 −0.651 2.17490.08 16.4143 −1.304 14.7591 6.0628 −0.691 5.8688 2.1601 −0.654 2.35250.07 19.6118 −1.308 17.6353 6.6343 −0.687 6.4152 2.3591 −0.656 2.57160.06 24.0878 −1.313 21.6594 7.3590 −0.683 7.1096 2.6123 −0.657 2.84990.05 30.7618 −1.325 27.6197 8.3207 −0.682 8.0284 2.9445 −0.652 3.21830.04 42.1160 −1.396 37.1782 9.7249 −0.707 9.3152 3.3602 −0.587 3.7370

tested cases with a number of different η and s the numerical surface tensions match very well the predicted analyticalvalues.

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a fractional phase-field model for two-phase flows with ... · viscosity and density in simulations...

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