Free Convection of a Bingham Fluid in a Differentially-Heated Porous Cavity: The Effect of a Square Grid Microstructure

D Andrew S Rees

doi:10.3390/physics4010015

Abstract

We examine how a square-grid microstructure affects the manner in which a Bingham fluid is convected in a sidewall-heated rectangular porous cavity. When the porous microstructure is isotropic, flow arises only when the Darcy–Rayleigh number is higher than a critical value, and this corresponds to when buoyancy forces are sufficient to overcome the yield threshold of the Bingham fluid. In such cases, the flow domain consists of a flowing region and stagnant regions within which there is no flow. Here, we consider a special case where the constituent pores form a square grid pattern. First, we use a network model to write down the appropriate macroscopic momentum equations as a Darcy–Bingham law for this microstructure. Then detailed computations are used to determine strongly nonlinear states. It is found that the flow splits naturally into four different regions: (i) full flow, (ii) no-flow, (iii) flow solely in the horizontal direction and (iv) flow solely in the vertical direction. The variations in the rate of heat transfer and the strength of the flow with the three governing parameters, the Darcy–Rayleigh number, Ra, the Rees–Bingham number, Rb, and the aspect ratio, A, are obtained.

Highlights

There is considerable interest in determining how Bingham fluids flow through a porous medium
Showed that there is no general Darcy–Bingham law. Both the dependence of the rate of flow on the applied pressure gradient and the value of the pseudo-threshold compared with the yield threshold both vary in a way which is a function of the microstructure
Rees [8] showed that a Bingham fluid that occupies a square network that is composed of identical channels has an anisotropic response to an applied pressure gradient

Summary

Introduction

There is considerable interest in determining how Bingham fluids flow through a porous medium. The modelling of such flows is complicated greatly by the presence of a yield stress wherein the fluid remains stagnant whenever it is acted upon by an applied stress that is smaller than that yield stress. Showed that there is no general Darcy–Bingham law. Rather, both the dependence of the rate of flow on the applied pressure gradient and the value of the pseudo-threshold compared with the yield threshold both vary in a way which is a function of the microstructure. Rees [8] showed that a Bingham fluid that occupies a square network that is composed of identical channels has an anisotropic response to an applied pressure gradient

Methods

Results

Conclusion