RAYLEIGH-BÉNARD INSTABILITY OF THE POWER-LAW FLUID FLOW IN A POROUS MEDIUM: NUMERICAL AND EXPERIMENTAL ANALYSIS

The thermal instability of fluid saturated porous media is a topic extensively investigated in the last decades. The scientific community has focused its efforts especially on the study of the stability of Newtonian fluids [1]. The investigation of the stability of non-Newtonian fluids is a topic poorly explored. Only a few papers extended the results reported by Horton and Rogers [2], Lapwood [3], and Prats [4] to non-Newtonian fluids. The aim of this paper is to further develop the work done by Barletta and Nield [5] with reference to a fluid characterised by a temperature-dependent consistency index. A two dimensional fluid saturated horizontal porous layer is thus investigated with respect to the onset of thermal instability. The porous layer is subject to a horizontal throughflow and it is heated from below as in the classical Prats problem. A power-law fluid is considered. An extended Darcy's law and Oberbeck-Boussinesq approximation are assumed while the solid phase and the fluid phase are in local thermal equilibrium. A temperature-dependent viscosity model is employed. The porous layer is impermeable. Two different temperatures are imposed, with heating from below. The problem here described is investigated both numerically and experimentally. From the theoretical viewpoint, the stationary basic throughflow is perturbed by plane wave disturbances. A linear stability analysis is thus carried out using the normal mode method. The eigenvalue problem obtained is solved numerically by means of the Runge-Kutta method coupled with the shooting method. The neutral stability curves are obtained along with the critical values of the stability parameters that identify the threshold for the onset of thermal instability. The experimental configuration employed for the analysis of this problem consists of a Hele-Shaw cell. The cell is composed by an aluminium frame with a central window obtained by using two polycarbonate plates that ensure the optical access to the cell for velocity measurements. These measurements are performed with a Particle Image Velocimetry (PIV). The isothermal boundary conditions are obtained by circulating water for the hot lower boundary, and by circulating coolant for the cold upper boundary. Two free surface wells are connected to the cell, allowing injection and extraction of fluid in order to generate a constant horizontal flow. A syringe pump is used to inject the requested discharge inside the cell. The pump discharge is modulated by a Proportional Integral Derivative (PID) control system. In order to obtain a purely vertical temperature gradient, the cell is thermally insulated in all its components with foam rubber and thermal insulating tape. The experimental investigation poses several challenges, such as the choice of tracers for PIV measurements, the discrimination between horizontal velocity components induced by convection and the imposed throughflow, the presence of hysteretic effects in cell appearance and disappearance, and the correct reproduction of shear-thickening behaviour