Approximation by an iterative method of a low‐Mach model with temperature dependent viscosity

Abstract

In this work, we prove the existence and the uniqueness of the strong solution of a low‐Mach model, for which the dynamic viscosity of the fluid is a given function of its temperature. The method is based on the convergence study of a sequence towards the solution, for which the rates are also given. The originality of the approach is to consider the system in terms of the temperature and the velocity, leading to a nonlinear temperature equation and the development of some specific tools and results.

Discrete Gagliardo–Nirenberg inequality and application to the finite volume approximation of a convection–diffusion equation with a Joule effect term

“…vpx, tq " 0 for all x P BΩ and t P r0, T s. The system (30) can be seen as a particular case of a global low-Mach model with temperature dependent viscosity, in the case where v is a given datum of the problem (see e.g. [19,8]). A local wellposedness result for strong solutions to (30) has been established in [8,Theorem 1].…”

“…Results on local and global well-posedness of this system have been recently obtained under some smallness assumptions on the initial data, see e.g. [19,8], still based on a formulation using the temperature as a primary variable.…”

A Finite Volume Method for a Convection-Diffusion Equation Involving a Joule Term

“…In the case of the ideal gas law P 0 = R ρ u, where u(x, t), P 0 > 0 and R > 0 stand respectively for the temperature, the constant thermodynamic pressure and the ideal gas constant, a solenoidal velocity field v(x, t) can be introduced. It is shown in [3] that the change of variable v = V − λ ∇u leads to equation 1, where λ > 0 is a fixed constant which depends on the constant heat conductivity k > 0 in the nonstandard constraint…”

“…introduced in the low-Mach model. In [3] a particular dynamic viscosity is also introduced, defined by µ(u) = −λ ln u, in order to remove the O(λ 2 ) terms in the momentum equation. With this choice, µ(u) is strictly positive if and only if u ∈ (0, 1).…”

“…From the theoretical point of view, several results have been obtained for the system (1)-(2)-(3). For instance, the local-in-time existence of strong solutions has been established in [3] in the framework of a coupling with the Navier-Stokes system. In particular, a maximum-principle has been derived (see [6], Theorem 5.1).…”