Analysis and numerical simulation of a generalized compressible Cahn-Hilliard-Navier-Stokes model with friction effects

TL;DR

This work proposes a new generalized compressible diphasic Navier-Stokes Cahn-Hilliard model that is derived rigorously and satisfies basic mechanics of fluids and thermodynamics of particles and proves the existence of global weak solutions.

Abstract

We propose a new generalized compressible diphasic Navier-Stokes Cahn-Hilliard model that we name G-NSCH. This new G-NSCH model takes into account important properties of diphasic compressible fluids such as possible non-matching densities and contrast in mechanical properties (viscosity, friction) between the two phases of the fluid. the model also comprises a term to account for possible exchange of mass between the two phases. Our G-NSCH system is derived rigorously and satisfies basic mechanics of fluids and thermodynamics of particles. Under some simplifying assumptions, we prove the existence of global weak solutions. We also propose a structure preserving numerical scheme based on the scalar auxiliary variable method to simulate our system and present some numerical simulations validating the properties of the numerical scheme and illustrating the solutions of the G-NSCH model.

Figures (9)

• Figure 1: Simulation of compressible Navier-Stokes-Cahn-Hilliard model (with $\kappa = F_c = 0$). Matching densities (Top row) and non-matching densities (bottom row) for the two phases of the fluid.
• Figure 2: Temporal evolution of the dissipation of the energy $\frac{\mathop{\mathrm{d}}\!{} E}{\mathop{\mathrm{d}}\!{} t}$, mass of the fluid $1$ given by $\int_\Omega \rho c \,\mathop{\mathrm{d}}\!{} x$, scalar variable $\xi$, and of the minimum and maximal values of the mass fraction $c$ for matching densities (solid lines) and non-matching densities (dash-dotted lines).
• Figure 3: Simulation of compressible Navier-Stokes-Cahn-Hilliard model nonmatching densities, exchange term ($F_c(\rho,c)\neq 0$), same friction effects for both fluids (top row) and contrast of friction forces (bottom row).
• Figure 4: Temporal evolution of the dissipation of the energy $\frac{\mathop{\mathrm{d}}\!{} E}{\mathop{\mathrm{d}}\!{} t}$, mass of the fluid $1$ given by $\int_\Omega \rho c \,\mathop{\mathrm{d}}\!{} x$, scalar variable $\xi$, and of the minimum and maximal values of the mass fraction $c$ for matching densities (solid lines) and non-matching densities (dash-dotted lines).
• Figure 5: Two dimensional simulations of compressible Navier-Stokes-Cahn-Hilliard model with nonmatching densities, same friction effects for both fluids (top row) and contrast of friction (bottom row).

Theorems & Definitions (25)

• Lemma 2.2
• Proposition 3.1
• proof
• Theorem 3.3: Existence of weak solutions
• Proposition 3.4
• Lemma 3.5
• Proposition 3.6
• Lemma 3.7: Lemma 3.1 in MR2384572
• Lemma 3.8: Theorem 10.17 in MR2499296
• proof : Proof of Proposition \ref{['prop:mass_fraction_reg']}