A generic Scheme For the time-dependent Navier-Stokes Equation Coupled With The Heat Equation

TL;DR

This paper develops a convergence-proof gradient discretisation scheme for the coupled time-dependent Navier-Stokes and heat equations with temperature-dependent viscosity, validated through numerical experiments.

Contribution

It introduces a novel discrete method for the coupled equations and proves its convergence without non-physical assumptions.

Findings

Convergence of the proposed scheme is theoretically established.

Numerical experiments confirm the accuracy and stability of the method.

The scheme effectively handles temperature-dependent viscosity in fluid flow simulations.

Abstract

In this work, we study the gradient discretisation method (GDM) of the time-dependent Navier-Stokes equations coupled with the heat equation, where the viscosity depends on the temperature. We design the discrete method and prove its convergence without non-physical conditions. The paper is closed with numerical experiments that confirm the theoretical results.