The Effect of Internal and External Heating on the Free Convective Flow of a Bingham Fluid in a Vertical Porous Channel

D Andrew S Rees,Andrew P Bassom

doi:10.3390/fluids4020095

D Andrew S Rees, Andrew P Bassom

Open Access

https://doi.org/10.3390/fluids4020095

Copy DOI

Abstract

We study the steady free convective flow of a Bingham fluid in a porous channel where heat is supplied by both differential heating of the sidewalls and by means of a uniform internal heat generation. The detailed temperature profile is governing by an external and an internal Darcy-Rayleigh number. The presence of the Bingham fluid is characterised by means of a body force threshold as given by the Rees-Bingham number. The resulting flow field may then exhibit between two and four yield surfaces depending on the balance of magnitudes of the three nondimensional parameters. Some indication is given of how the locations of the yield surfaces evolve with the relative strength of the Darcy-Rayleigh numbers and the Rees-Bingham number. Finally, parameter space is delimited into those regions within which the different types of flow and stagnation patterns arise.

Highlights

This short paper considers the steady flow which is induced when a vertical porous channel is heated both externally and internally, and when that porous medium is saturated by a Bingham fluid
Within the context of a porous medium the natural yield threshold is expressed in terms of a pressure gradient and, in the context of convective flows, this includes the buoyancy force
Three cases are shown: one with purely external heating (Ra = 10, Rai = 0), one with purely internal heating (Ra = 0, Rai = 120), and an intermediate case which corresponds to the values of Ra and Rai used for the profile shown in Figure 1 (i.e., Ra = 10, Rai = 120)

Summary

Introduction

This short paper considers the steady flow which is induced when a vertical porous channel is heated both externally and internally, and when that porous medium is saturated by a Bingham fluid These fluids form, perhaps, the simplest model of a yield stress fluid where the fluid shears when the applied stresses are greater than a threshold value, the yield stress, but acts like a solid when the applied stresses are too small. This work is part of a study of different aspects of porous channel and boundary layer flows involving Bingham fluids; see Rees and Bassom [9,10,11] These earlier works consider the unsteady evolution of yield surfaces in such flows

Governing Equations

Numerical Solutions

Variation with Rb

Variation with Rai

Detailed

Conclusions