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

Yahya Alnashri

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.

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

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.

Paper Structure

This paper contains 4 sections, 3 theorems, 40 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Continuous and Discrete Problems
  3. Discrete Energy Estimates And Main Results
  4. Numerical Tests

Key Result

Lemma 3.1

Assume that the conditions assump-nsp hold and $({\boldsymbol u},p,S)$ is a solution to the discrete problem nsp-weak. Then there exist constants $C_1,C_2>0$ not depending on the discrete solution, such that, for all $n=0,...,N$, and

Figures (5)

  • Figure 4.1: Sample of the polygonal meshes.
  • Figure 4.2: The relative errors for the case $V(S)=1$.
  • Figure 4.3: The relative errors for the case $V(S)=1$.
  • Figure 4.4: The relative errors for the case $V(S)=\sqrt{S^2+1}+2$.
  • Figure 4.5: The relative errors for the case $V(S)=\sqrt{S^2+1}+2$.

Theorems & Definitions (10)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Lemma 3.1
  • proof
  • Definition 3.2
  • Lemma 3.3
  • proof
  • Theorem 3.4
  • proof