Stokes equations under nonlinear slip boundary conditions coupled with the heat equation: A priori error analysis

Abstract

In this work, we consider the heat equation coupled with Stokes equations under threshold type boundary condition. The conditions for existence and uniqueness of the weak solution are made clear. Next we formulate the finite element problem, recall the conditions of its solvability, and study its convergence by making use of Babuska–Brezzi's conditions for mixed problems. Third we formulate an Uzawa's type iterative algorithm that separates the fluid from heat conduction, study its feasibility, and convergence. Finally the theoretical findings are validated by numerical simulations.

Iterative methods for Stokes flow under nonlinear slip boundary condition coupled with the heat equation

“…We conclude this section by recalling the following convergence result obtained recently [13] Theorem 2.1. Let (u, p, θ) be the solution of (2.3) such that u ∈ W 1,p2 (Ω) and (2.14) is valid.…”

“…It should be noted that similar additional terms have been incorporated to numerical schemes in different contexts in [16,17]. This work is the follow up of the other contribution [13], and we believe that these contributions are to the best of our knowledge the first ones towards the numerically analysis from the mathematical viewpoint of fluid flows under nonlinear slip boundary condition coupled with the heat equation. The rest of the paper is organized as follows:…”

“…Thus our work combined the difficulties of the "Navier Stokes equations" and the ones generated by the non classical boundary condition (1.4) together with ν(θ)Du. It should be noted that the unique weak solution of (1.1)...(1.5) can be constructed following closely the arguments in [8](see theorem 2.2 and proposition 2.3), while the convergence of the finite element approximation associated to it has been studied recently and convergence of the following Uzawa's type algorithm established in [13] Initialization:…”

Non-isothermal non-Newtonian fluids: the stationary case

“…We also point out that the approximation of steady isothermal non-Newtonian incompressible Stokes fluid problems on polytopal meshes have been addressed in [19]. Concerning non-isothermal Newtonian fluid problems, we refer to [6], where spectral elements have been employed, and to [30] which considers the finite element approximation of the heat equation coupled with Stokes equations under threshold type boundary condition. It is worth mentioning that a very recent contribution is devoted to the numerical approximation of the steady state problem for an isothermal incompressible heat-conducting fluid with implicit non-Newtonian rheology (see [33]).…”

Virtual Element Method for the Navier--Stokes Equation coupled with the Heat Equation

“…Finite Element approximation of the time dependent Boussinesq model with nonlinear viscosity depending on the temperature has been studied in [6]. Recently, the finite element approximation of the heat equation coupled with Stokes equations with nonlinear slip boundary conditions has been analyzed in [35]. Finite element methods for Darcy's problem coupled with the heat equation has been studied in [23].…”