A second order in time, uniquely solvable, unconditionally stable numerical scheme for Cahn–Hilliard–Navier–Stokes equation

Abstract

We propose a novel second order in time numerical scheme for Cahn–Hilliard–Navier–Stokes phase field model with matched density. The scheme is based on second order convex-splitting for the Cahn–Hilliard equation and pressure-projection for the Navier–Stokes equation. We show that the scheme is mass-conservative, satisfies a modified energy law and is therefore unconditionally stable. Moreover, we prove that the scheme is unconditionally uniquely solvable at each time step by exploring the monotonicity associated with the scheme. Thanks to the simple coupling of the scheme, we design an efficient Picard iteration procedure to further decouple the computation of Cahn–Hilliard equation and Navier–Stokes equation. We implement the scheme by the mixed finite element method. Ample numerical experiments are performed to validate the accuracy and efficiency of the numerical scheme.

Introduction

In this work, we are interested in solving numerically the Cahn–Hilliard–Navier–Stokes (CHNS) phase field model that describes the interface dynamics of a binary incompressible and macroscopically immiscible Newtonian fluids with matched density and viscosity in a bounded domain $\mathrm{\Omega }\subseteq {\mathbb{R}}^d$, $d=2,3$. The non-dimensional system takes the explicit form as, cf. [1]

$${\varphi }_t+\mathrm{\nabla }\cdot (\varphi \mathbf{u})=\mathrm{\nabla }\cdot (M(\varphi )\mathrm{\nabla }\mu ),\text{in}{\mathrm{\Omega }}_T$$

$$\mu =f_0^{\prime }(\varphi )-{ϵ}^2\mathrm{\Delta }\varphi ,\text{in}{\mathrm{\Omega }}_T$$

$${\mathbf{u}}_t-\frac{1}{\mathit{Re}}\mathrm{\Delta }\mathbf{u}+\mathbf{u}\cdot \mathrm{\nabla }\mathbf{u}+\mathrm{\nabla }p=-\frac{{ϵ}^{-1}}{{\mathit{We}}^{⁎}}\varphi \mathrm{\nabla }\mu ,\text{in}{\mathrm{\Omega }}_T$$

$$\mathrm{\nabla }\cdot \mathbf{u}=0,\text{in}{\mathrm{\Omega }}_T$$

where u is the velocity field, p is a modified pressure, ϕ is the phase field variable (order parameter), μ is the chemical potential, $f_0(\varphi )$ is the quartic homogeneous free energy density function $f_0(\varphi )=\frac{1}{4}{(1-{\varphi }^2)}^2$, and ${\mathrm{\Omega }}_T:=\mathrm{\Omega }\times (0,T)$ with $T>0$ a fixed constant. Re is the Reynolds number; ${\mathit{We}}^{⁎}$ is the modified Weber number that measures the relative strengths of the kinetic and surface energies [1]; ϵ is a dimensionless parameter that measures capillary width of the diffuse interface; $M(\varphi )$ is the mobility function that incorporates the diffusional Peclet number Pe. We refer to [2], [3] for the detailed non-dimensionalization of the CHNS system.

We close the system with the following initial and boundary conditions

$$\mathbf{u}=0,\text{on}\partial \mathrm{\Omega }\times (0,T)$$

$$\mathrm{\nabla }\varphi \cdot \mathbf{n}=\mathrm{\nabla }\mu \cdot \mathbf{n}=0,\text{on}\partial \mathrm{\Omega }\times (0,T)$$

$$(\mathbf{u},\varphi ){|}_{t=0}=({\mathbf{u}}_0,{\varphi }_0),\text{in}\mathrm{\Omega }.$$

Here n denotes the unit outer normal vector of the boundary ∂Ω. It is clear that the CHNS system (1.1), (1.2), (1.3), (1.4) under the above boundary conditions is mass-conservative,

$$\frac{d}{dt}\underset{\mathrm{\Omega }}{\int }\varphi dx=0,$$

and energy-dissipative

$$\frac{d}{dt}{E}_{\mathit{tot}}(\mathbf{u},\varphi )=-\frac{1}{\mathit{Re}}\underset{\mathrm{\Omega }}{\int }{|\mathrm{\nabla }\mathbf{u}|}^{2}dx-\frac{{ϵ}^{-1}}{{\mathit{We}}^{⁎}}\underset{\mathrm{\Omega }}{\int }M(\varphi ){|\mathrm{\nabla }\mu |}^{2}dx,$$

where the total energy $E_{\mathit{tot}}$ is defined as

$$E_{\mathit{tot}}(\mathbf{u},\varphi )=\underset{\mathrm{\Omega }}{\int }\frac{1}{2}{|\mathbf{u}|}^{2}dx+\frac{1}{{\mathit{We}}^{⁎}}\underset{\mathrm{\Omega }}{\int }(\frac{1}{ϵ}{f}_0(\varphi )+\frac{ϵ}{2}{|\mathrm{\nabla }\varphi |}^2)dx.$$

The first term on the right hand side of Eq. (1.10) is the total kinetic energy, and the term, denoted throughout by $E_f$, is a measure of the surface energy of the fluid system.

The CHNS phase field model (1.1), (1.2), (1.3), (1.4) is proposed as an alternative of sharp interface model to describe the dynamics of two phase, incompressible, and macroscopically immiscible Newtonian fluids with matched density, cf. [3], [4], [5], [6], [7]. In contrast to the sharp interface model, the diffuse interface model recognizes the micro-scale mixing and hence treats the interface of two fluids as a transition layer with small but non-zero width ϵ. Although the region is thin, it may play an important role during topological transition like interface pinchoff or reconnection [3]. One then introduces an order parameter ϕ, for instance the concentration difference, which takes the value 1 in the bulk of one fluid and −1 in regions filled by the other fluid and varies continuously between 1 and −1 over the interfacial region. One can view the zero level set of the order parameter as the averaged interface of the mixture. Thus, the dynamics of the interface can be simulated on a fixed grid without explicit interface tracking, which renders the diffuse interface method an attractive numerical approach for deforming interface problems. The CHNS diffuse interface model has been successfully employed for the simulations of two-phase flow in various contexts. We refer the readers to [1], [6] and references therein for its diverse applications.

In this work, we assume that $m_1\le M(\varphi )\le m_2$ for constants $0<m_1\le m_2$. We point out that the degenerate mobility function may be more physically relevant, as it guarantees the order parameter stays within the physical bound $\varphi \in [-1,1]$ [8], though uniqueness of weak solutions is still open even for the Cahn–Hilliard equation. Recent numerical experiments [9] also indicate that the Cahn–Hilliard equation with degenerate mobility may be more accurate for immiscible binary fluids. Numerical resolution of the degenerate case is a subtle matter and beyond the scope of our current work (cf. [10], [11] for the case of Cahn–Hilliard equation).

There are several challenges in solving the system (1.1), (1.2), (1.3), (1.4) numerically. First of all, the small interfacial width ϵ introduces tremendous amount of stiffness into the system (large spatial derivative within the interfacial region). It demands the numerical scheme to be unconditionally stable so that the stiffness can be handled with ease. The resulting numerical scheme tends to be nonlinear and therefore poses challenge in proving unconditionally unique solvability. A popular strategy in discretizing the Cahn–Hilliard equation (Eqs. (1.1), (1.2)) in time is based on the convex-splitting of the free energy functional $E_f$, i.e., treating the convex part of the functional implicitly and concave part explicitly, an idea dates back to Eyre [12]. The design of convex-splitting scheme yields not only unconditional stability, but also unconditionally unique solvability for systems with symmetric structures [13], [14]. However, the variational approach for proving unique solvability (see the references above) is not applicable to the CHNS system since the advection term in Navier–Stokes equation (Eq. (1.3)) breaks the symmetry. In addition, the stiffness issue naturally requires adaptive mesh refinement in order to reduce the computational cost. Secondly, when it comes to solving the Navier–Stokes equation, one always faces the difficulty of the coupling between velocity and pressure. The common practice is to use the well-known Chorin–Temam type pressure projection scheme, see [15] for a general review. Lastly, higher order scheme is always preferable from the accuracy point-of-view. Yet, it is a challenge to design higher order scheme for a nonlinear system while maintaining the unconditional stability.

There have been many works on the numerical resolution of the CHNS system, see a comprehensive summary by Shen [16]. Here we survey several papers that are especially relevant to ours. In [1], Kim, Kang and Lowengrub proposed a conservative, second-order accurate fully implicit discretization of the CHNS system. The update of the pressure in the Navier–Stokes equation is based on an approximate pressure projection method. To ensure the unconditional stability, they introduce a non-linear stabilization term to the Navier–Stokes solver. The scheme is strongly coupled and highly nonlinear, for which they design a multigrid iterative solver. The authors point out (without proof) that a restriction on the time-step size may be needed for the unique solvability of the scheme. In [17], Feng analyses a first-order in time, fully discrete finite element approximation of the CHNS system. He shows that his scheme is unconditionally energy-stable and convergent, but gives no analysis on unique solvability. Kay, Styles and Welford [18] also studied a first-order in time, finite element approximation of CHNS system. In contrast to Feng's scheme, the velocity in the Cahn–Hilliard equation (1.1) is discretized explicitly at the discrete time level. Thus the computation of the Cahn–Hilliard equation is fully decoupled from that of Navier–Stokes equation. Moreover, the unique solvability of the overall scheme can be established easily by exploring the gradient flow structure of the Cahn–Hilliard equation. However, a CFL condition has to be imposed for the scheme to be stable. See [19] for an operator-splitting strategy in decoupling the computation of Cahn–Hilliard equation and Navier–Stokes equation which still preserves the unconditional stability (without decoupling the pressure and velocity). Dong and Shen [20] recently derived a fully decoupled linear time stepping scheme for the CHNS system with variable density, which involves only constant matrices for all flow variables. However, there is no stability analysis on their numerical scheme.

In this paper, we propose a novel second order in time numerical scheme for Cahn–Hilliard–Navier–Stokes phase field model with matched density. The scheme is based on second order convex-splitting for the Cahn–Hilliard equation and pressure-projection for the Navier–Stokes equation. This scheme satisfies a modified energy law which mimics the continuous version of the energy law (1.9), and is therefore unconditionally stable. Moreover, we prove that the scheme is unconditionally uniquely solvable at each time step by exploring the monotonicity associated with the scheme. Thanks to the simple coupling of the scheme, we design an efficient Picard iteration procedure to further decouple the computation of Cahn–Hilliard equation and Navier–Stokes equation. We implement the scheme by the mixed finite element method. Ample numerical experiments are performed to validate the accuracy and efficiency of the numerical scheme. The possibility of such a scheme is alluded in Remark 5.5 [16]. A similar scheme without pressure-correction for Cahn–Hilliard–Brinkman equation is proposed in the concluding remarks of [14].

The rest of the paper is organized as follows. In Section 2, we give the discrete time, continuous space scheme. We prove the mass-conservation, unconditional stability and unconditionally unique solvability in Section 3. In Section 4, the scheme is further discretized in space by mixed finite element approximation. An efficient Picard iteration procedure is proposed to solve the fully discrete equations. Finally, we provide some numerical experiments in Section 5 to validate our numerical scheme.