Numerical Approximations of a Cahn-Hilliard-Navier-Stokes Model with Variable Densities and Degenerate Mobility

Core Concepts

The authors present coupled and decoupled numerical approximations for a Cahn-Hilliard-Navier-Stokes model with variable densities and degenerate mobility, preserving key properties like mass conservation, point-wise bounds, and energy stability.

Abstract

The content discusses the numerical approximation of a Cahn-Hilliard-Navier-Stokes (CHNS) model with variable densities and degenerate mobility. The authors propose two approaches:

A coupled structure-preserving scheme that conserves the mass of the mixture, preserves the point-wise bounds of the density and phase-field variables, and decreases an underlying energy functional. This scheme is based on a finite element approximation for the Navier-Stokes fluid flow with discontinuous pressure and an upwind discontinuous Galerkin scheme for the Cahn-Hilliard part.

A decoupled scheme that is more computationally efficient but cannot guarantee the discrete energy-decreasing property.

The key highlights and insights are:

The coupled scheme is shown to satisfy mass conservation, point-wise bounds, and energy stability at the discrete level.
The decoupled scheme lacks the energy-stability property but is more computationally efficient.
The authors use an upwind discontinuous Galerkin approach and a convex splitting technique to ensure the desired properties.
Numerical experiments are conducted to compare the two approaches, including convergence tests, qualitative comparisons, and benchmark problems involving gravitational forces.

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Property-preserving numerical approximations of a Cahn-Hilliard-Navier-Stokes model with variable densities and degenerate mobility

by Dani... at arxiv.org 04-29-2024

https://arxiv.org/pdf/2310.01522.pdf

Deeper Inquiries

How can the decoupled scheme be further improved to achieve energy stability while maintaining computational efficiency

To improve the decoupled scheme for energy stability while maintaining computational efficiency, one approach could be to incorporate a stabilization technique that ensures the discrete energy law is satisfied. One possible method is to introduce a consistent stabilization term in the decoupled scheme that helps control the numerical instabilities and maintains the energy stability property. This stabilization term can be designed to counteract the effects of the convective terms and ensure that the discrete energy decreases over time steps. By carefully selecting and implementing such a stabilization technique, the decoupled scheme can achieve energy stability without compromising its computational efficiency.

What are the potential applications of this CHNS model with variable densities and degenerate mobility, and how can the numerical schemes be adapted to those applications

The CHNS model with variable densities and degenerate mobility has various potential applications in the field of fluid dynamics and materials science. One application could be in simulating phase separation processes in materials with different densities, such as in the formation of microstructures in alloys or polymers. The numerical schemes developed for this model can be adapted to simulate and analyze the evolution of these phase transitions, capturing the intricate interplay between fluid flow, phase-field dynamics, and density variations. Additionally, the model can be extended to study complex fluid systems with multiple components and varying densities, providing insights into phenomena like droplet coalescence, phase segregation, and pattern formation in mixtures.

Can the ideas presented in this work be extended to other types of diffuse interface models or fluid-structure interaction problems

The ideas presented in this work can be extended to other types of diffuse interface models and fluid-structure interaction problems. By modifying the numerical schemes and adapting the formulations to different models, similar property-preserving approximations can be developed for a wide range of systems. For instance, the concepts of mass conservation, point-wise bounds, and energy stability can be applied to diffuse interface models in biological systems, environmental flows, and multiphase fluid dynamics. Additionally, the techniques used in this work can be extended to fluid-structure interaction problems, where the interaction between fluid flow and solid structures is modeled using coupled or decoupled approaches. By incorporating similar numerical approximations and preserving key properties, these models can provide accurate simulations of complex fluid-structure interactions in various engineering and scientific applications.