Abstract
We consider two unsteady free convection flows of a Bingham fluid when it saturates a porous medium contained within a vertical circular cylinder. The cylinder is initially at a uniform temperature, and such flows are then induced by suddenly applying either a new constant temperature or a nonzero heat flux to the exterior surface. As time progresses, heat conducts inwards and this may or may not overcome the yield threshold for flow. For the constant temperature case, flow begins immediately should the parameter, Rb, which is a nondimensional yield parameter, be sufficiently large. The ultimate fate, though, is full immobility as the cylinder eventually tends towards a new constant temperature. For the constant heat flux case, the fluid remains immobile but will begin to flow eventually should Rb be sufficiently large. The two cases have different critical values for Rb.
Highlights
The topic of the flow of a Bingham fluid when it is not saturating a porous matrix is a wellestablished field of study, and the literature is quite mature
We have considered where the outer impermeable surface of a porous circular cylinder is subject either to a sudden change in the surface temperature or in the applied heat flux
The porous medium within the cylinder is saturated with a Bingham fluid
Summary
The topic of the flow of a Bingham fluid when it is not saturating a porous matrix is a wellestablished field of study, and the literature is quite mature. The analysis can proceed analytically (such as for plane-Poiseuille and Hagen–Poiseuille flows), but in other one dimensional problems the analysis has to be completed by using a simple Newton–Raphson iteration equation on a transcendental equation which allows the positions of the yield surfaces to be found Examples of such works include those by Yang and Yeh (1965) and Bayazitoglu et al (2007) who studied steady free convection in a vertical channel which is heated from the side. Further realism is obtained by considering the porous medium as an assembly of identical channels or pores, which gives the well-known Buckingham–Reiner law (1921) In this case the initial rise in the flow is quadratic immediately post-threshold, as opposed to linear in Pascal’s model. Using Pascal’s piecewise-linear Darcy–Bingham law, the analysis proceeds analytically and again, a Newton–Raphson scheme has to be used in order to locate both the yield surfaces and, given that the flow domain is finite horizontally, the corresponding change to the initial hydrostatic pressure gradient
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