On the Stability of Parallel Flow in a Vertical Porous Layer with Annular Cross Section

• Published in: Transport in porous media Vol. 134; no. 2; pp. 491 - 501
• Main Authors: Barletta, A., Celli, M., Rees, D. A. S.
• Format: Journal Article
• Language: English
• Published: Dordrecht Springer Netherlands 01.09.2020; Springer Nature B.V
• Subjects: Boundary conditions; Civil Engineering; Classical and Continuum Physics; Convection cells; Cross-sections; Earth and Environmental Science; Earth Sciences; Flow stability; Geotechnical Engineering & Applied Earth Sciences; Hydrogeology; Hydrology/Water Resources; Industrial Chemistry/Chemical Engineering; Parallel flow; Permeability; Theorems; Vertical layer; Flow instability; Convection; Porous medium; Gill’s theorem; Annular cross section
• ISSN: 0169-3913; 1573-1634
• DOI: 10.1007/s11242-020-01456-3

The linear stability of buoyant parallel flow in a vertical porous layer with an annular cross section is investigated. The vertical cylindrical boundaries are kept at different uniform temperatures, and they are assumed to be impermeable. The emergence of linear instability by convection cells is excluded on the basis of a numerical solution of the linearised governing equations. This result extends to the annular geometry the well-known Gill’s theorem regarding the impossibility of convective instability in a vertical porous plane slab whose boundaries are impermeable and isothermal with different temperatures. The extension of Gill’s theorem to the annular domain is approached numerically by evaluating the growth rate of normal mode perturbations and showing that its sign is negative, which means asymptotic stability of the basic flow. A concurring argument supporting the absence of linear instability arises from the investigation of cases where the impermeability condition at the vertical boundaries is relaxed and a partial permeability is modelled through Robin boundary conditions for the pressure. With partially permeable boundaries, an instability emerges which takes the form of axisymmetric normal modes. Then, as the boundary permeability is reduced towards zero, the critical Rayleigh number becomes infinite.