Ecoulement laminaire et transfert de chaleur entre deux parois rectangulaires poreuses

Abstract

In this work, we study fluid flow and heat transfer between two parallel rectangular planes, which are uniformly porous and have a temperature difference that varies the thermal conductivity according to a linear law. The formulation of the problem is carried out using the continuity equation, the Navier-Stokes equations and the energy equation, which respectively translate the conservation of mass, momentum and energy. These equations are nondimensionalized to highlight the parameters that control flow and heat transfer, including Reynolds number, Péclet number, and the parameter that measures the sensitivity of thermal conductivity to temperature variations. Given the incompressibility of the fluid and the fact that the flow is described by two velocity components, the stream function is prescribed in the problem and is the solution to the vorticity equation. The geometrical configuration of the flow and heat transfer involves a change of variable inspired by a similar solution method which is increasingly used and which transforms the partial differential equations into ordinary differential equations describing the same problem. The dynamics of the fluids and heat transfer in the channel are studied along the branches of a hydrodynamic bifurcation diagram, highlighted in the analysis of the behaviour of the wall friction coefficient as a function of Reynolds number values. The bifurcation diagram shows five types of solutions, namely types I, I1, I1', II and III. A flow boundary layer and a thermal boundary layer were detected through branches I and II. Types I1 and I1' solutions are images of each other. Type III solutions are characterised by the absence of the boundary layer.