Diffuse Interface Models for Incompressible Two-Phase Flows with Different Densities

Abstract

Diffuse interface models have become an important analytical and numerical method to model two-phase flows. In this contribution we review the subject and discuss in detail a thermodynamically consistent model with a divergence free velocity field for two-phase flows with different densities. The model is derived using basic thermodynamical principles, its sharp interface limits are stated, existence results are given, different numerical approaches are discussed and computations showing features of the model are presented.

Appendix

1.1 Discretization in Space and Time

We assume $\mathscr{T}_{h}$ to be a quasiuniform triangulation of Ω with simplicial elements in the sense of [19].

Concerning discretization with respect to time, we assume that

(T):

the time interval I: = [0, T) is subdivided in intervals I _k = [t _k , t _k+1) with t _k+1 = t _k + τ _k for time increments τ _k > 0 and k = 0, ⋯ , N − 1. For simplicity, we take τ _k ≡ τ for k = 0, ⋯ , N − 1.

We write v ^k for v(⋅ , kτ), $k \in \mathbb{N}$, and we denote step functions in time mapping I = [0, T] onto one of the discrete function spaces X _h , …by an index τh.

For the approximation of both the phase-field φ and the chemical potential μ, we introduce the space U _h of continuous, piecewise linear finite element functions on $\mathscr{T}_{h}$. The expression $\mathscr{I}_{h}$ stands for the nodal interpolation operator from C ⁰(Ω) to U _h defined by $\mathscr{I}_{h}u:=\sum _{ j=1}^{\dim U_{h}}u(x_{ j})\theta _{j}$, where the functions θ _j form a dual basis to the nodes x _j , i.e. θ _i (x _j ) = δ _ij , i, j = 1, …, dimU _h .

Let us furthermore introduce the well-known lumped masses scalar product corresponding to the integration formula

$$\displaystyle{ \left (\varTheta,\varPsi \right )_{h}:=\int _{\varOmega }\mathscr{I}_{h}(\varTheta \varPsi ). }$$

For the discretization of the velocity field v and the pressure p, we use function spaces W _h ⊂ X _h ⊂ W ₀ ^1,2(Ω) and S _h ⊂ L ₀ ²(Ω): = {v ∈ L ²(Ω) | ∫ _Ω v = 0} which form an admissible pair of discretization spaces in hydrodynamics. This means that besides the conditions

(S1):

$\mathbf{W}_{h}:=\{ \mathbf{v}_{h} \in \mathbf{X}_{h}\vert \int _{\varOmega }q_{h}\mathop{ \mathrm{div}}\nolimits \mathbf{v}_{h} = 0\quad \forall q_{h} \in S_{h}\},$

(S2):

The Babuška-Brezzi condition is satisfied, i.e. a positive constant β exists such that

$$\displaystyle{\sup _{\mathbf{v}_{h}\in \mathbf{X}_{h}}\frac{(q_{h},\mathop{\mathrm{div}}\nolimits \mathbf{v}_{h})} {\left \|\mathbf{v}_{h}\right \|_{\mathbf{W}_{0}^{1,2}(\varOmega )}} \geq \beta \left \|q_{h}\right \|_{L^{2}(\varOmega )}}$$

for all q _h ∈ S _h

a number of additional conditions hold true which are specified in [26] and [29].

Taylor-Hood elements (i.e. P ₂ P ₁-elements) and P ₂ P ₀-elements are examples in agreement with these conditions. In both cases, X _h is given as

$$\displaystyle{ \mathbf{X}_{h}:= \left \{\mathbf{w} \in (\mathbf{C}_{0}^{0}(\bar{\varOmega })): \left (\mathbf{w}\right )_{ j}\vert _{K} \in P_{2}(K),K \in \mathscr{T}_{h},j = 1,\ldots,d\right \},\quad d = 2,3. }$$

For Taylor-Hood elements, S _h is defined to be the subset of functions in U _h with vanishing mean value. In the case of P ₂ P ₀-elements, S _h is given by the set of elementwise constant functions with mean value zero. Let us specify the assumptions on initial data and the double-well potential.

Definition 1

Let $\psi \in C^{1}(\mathbb{R}; \mathbb{R}_{0}^{+})$ be given such that ψ′ is piecewise C ¹ with at most quadratic growth of the derivatives for $\left \vert x\right \vert \rightarrow \infty$. We call $\psi _{\tau h}^{{\prime}} \in C^{0}(\mathbb{R}^{2}; \mathbb{R})$ an admissible discretization of ψ′ if the following is satisfied.

(H1):

There is a positive constant C, such that

$$\displaystyle{ \left \vert \psi _{\tau h}^{{\prime}}\left (a,b\right )\right \vert \leq C\left (1 + \left \vert a\right \vert ^{3} + \left \vert b\right \vert ^{3}\right ). }$$

(H2):

$\psi _{\tau h}^{{\prime}}\left (a,b\right )\left (a - b\right ) \geq F\left (a\right ) - F\left (b\right )$ for all $a,b \in \mathbb{R}$.

(H3):

$\psi _{\tau h}^{{\prime}}\left (a,b\right ) = F'\left (a\right )$ for all $a \in \mathbb{R}$.

(H4):

There is a positive constant C such that

$$\displaystyle{\left \vert \psi _{\tau h}^{{\prime}}(a,b) -\psi _{\tau h}^{{\prime}}(b,c)\right \vert \leq C\left (a^{2} + b^{2} + c^{2}\right )\left (\vert a - b\vert + \vert b - c\vert \right )}$$

for all $a,b,c \in \mathbb{R}.$

Moreover, we make the following assumptions on initial data and the regularized mass density $\rho \left (\varphi \right )$.

(H5):

Let initial data Φ ₀ ∈ H ²(Ω; [−1, 1]) and $\mathbf{V}_{0} \in \mathbf{W}_{0,\mathop{\mathrm{div}}\nolimits }^{1,2}(\varOmega )$ be given such that we have for discrete initial data $\varphi _{h}^{0}:=\mathscr{ I}_{h}\varPhi _{0}$ and v _h ⁰—given by the orthogonal projection of V ₀ onto $\mathscr{W}_{h}$—uniformly in h > 0 that

$$\displaystyle{\int _{\varOmega }\left \vert \varDelta _{h}\varphi _{h}^{0}\right \vert ^{2} \leq C\left \|\varPhi _{ 0}\right \|_{H^{2}(\varOmega )}^{2}}$$

and that

$$\displaystyle{\int _{\varOmega }\rho (\varphi _{h}^{0})\left \vert \mathbf{v}_{ h}^{0}\right \vert ^{2} + \frac{1} {2}\int _{\varOmega }\left \vert \nabla \varphi _{h}^{0}\right \vert ^{2} +\int _{\varOmega }\mathscr{I}_{ h}F(\varphi _{h}^{0}) \leq C\mathscr{E}(\mathbf{V}_{ 0},\varPhi _{0}).}$$

Here, the discrete Laplacian Δ _h w ∈ U _h ∩ H _∗ ¹(Ω) is defined by

$$\displaystyle{ \left (\varDelta _{h}w,\varTheta \right )_{h} = -\int _{\varOmega }\langle \nabla w,\nabla \varTheta \rangle \quad \forall \varTheta \in U_{h}. }$$

(8.45)

(H6):

Given mass densities $0 <\tilde{\rho } _{-}\leq \tilde{\rho }_{+} \in \mathbb{R}$ of the fluids involved and an arbitrary, but fixed regularization parameter $\bar{\varphi }\in \left ( \frac{\tilde{\rho }_{-}} {\tilde{\rho }_{+}-\tilde{\rho }_{-}}, \frac{2\tilde{\rho }_{-}} {\tilde{\rho }_{+}-\tilde{\rho }_{-}}\right )$, we define the regularized mass density of the two-phase fluid by a smooth, increasing, strictly positive function ρ of the phase-field φ which satisfies

$$\displaystyle\begin{array}{rcl} \rho (\varphi )\vert _{(-1-\bar{\varphi },1+\bar{\varphi })} = \frac{\tilde{\rho }_{+} -\tilde{\rho }_{+}} {2} \varphi + \frac{\tilde{\rho }_{-} +\tilde{\rho } _{+}} {2},& & {}\end{array}$$

(8.46)

$$\displaystyle\begin{array}{rcl} \rho (\varphi )\vert _{(-\infty,-1- \frac{2\tilde{\rho }_{-}} {\tilde{\rho }_{+}-\tilde{\rho }_{-}})} \equiv const.,& & {}\end{array}$$

(8.47)

$$\displaystyle\begin{array}{rcl} \rho (\varphi )\vert _{(1+ \frac{2\tilde{\rho }_{-}} {\tilde{\rho }_{+}-\tilde{\rho }_{-}},\infty )} \equiv const..& & {}\end{array}$$

(8.48)