Abstract
The performance of several engineering applications are strictly connected to the rheology of the working fluids and the Oldroyd-B model is widely employed to describe a linear viscoelastic behaviour. In the present paper, a buoyant Oldroyd-B flow in a vertical porous layer with permeable and isothermal boundaries is investigated. Seepage flow is modelled through an extended version of Darcy’s law which accounts for the Oldroyd-B rheology. The basic stationary flow is parallel to the vertical axis and describes a single-cell pattern where the cell has an infinite height. A linear stability analysis of such a basic flow is carried out to determine the onset conditions for a multicellular pattern. This analysis is performed numerically by employing the shooting method. The neutral stability curves and the values of the critical Rayleigh number are evaluated for different retardation time and relaxation time characteristics of the fluid. The study highlights the extent to which the viscoelasticity has a destabilising effect on the buoyant flow. For the limiting case of a Newtonian fluid, the known results available in the literature are recovered, namely a critical value of the Darcy–Rayleigh number equal to 197.081 and a corresponding critical wavenumber of 1.05950.
Highlights
The convective instability in a plane vertical porous layer saturated by a fluid is a topic of great interest for its applications spanning from geophysical systems to building insulation
Following the path devised by Barletta [8], the aim of this paper is to relax the assumption of impermeable boundaries and reconsider the linear stability analysis of the vertical buoyant flow in a vertical layer saturated by a viscoelastic Oldroyd-B fluid
It is worth noticing that the vertical axis has the range 0 6 S 6 197.081, since this value corresponds to the onset of instability for a Newtonian fluid, namely the maximum possible critical value of S
Summary
The convective instability in a plane vertical porous layer saturated by a fluid is a topic of great interest for its applications spanning from geophysical systems to building insulation. The classical paper by Gill [1] offered a rigorous mathematical proof that a vertical porous layer, saturated by a Newtonian fluid endowed with impermeable boundaries, kept at different uniform temperatures, displays a stationary conduction regime with a stable parallel vertical buoyant flow, regardless of the temperature gap between the boundaries. In other words, such basic buoyant flow is always stable for every value of the Rayleigh number.
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