On Existence and Uniqueness of the Weak Solution of a Generalized Boussinesq Equation with Press and

TL;DR

This paper investigates the existence and uniqueness of weak solutions for a generalized Boussinesq equation modeling coupled mass and heat flow in a viscous incompressible fluid with temperature-dependent properties and complex boundary conditions.

Contribution

It establishes the existence and uniqueness of weak solutions for a generalized Boussinesq equation with non-standard boundary conditions using Galerkin approximation.

Findings

Existence of weak solutions proved using Galerkin scheme

Uniqueness established under specific conditions related to Reynolds and Rayleigh numbers

Handles temperature-dependent viscosity and heat conductivity in complex boundary settings

Abstract

In this paper, a generalized Boussinesq equation that couples the mass and heat flows in a viscous incompressible uid is considered. The kinematic viscosity and the heat conductivity are assumed to be dependent on the temperature. The boundary condition on the velocity of fluid is non-standard where the dynamical pressure is given on some part of the boundary, and the temperature of fluid is represented in a mixed boundary condition. The existence of the weak solution is proved by the Galerkin approximation scheme, and the uniqueness is also obtained under the condition on the weak solution that is somehow like the restriction on the Reynold number and the Raleigh number in hydrodynamics.