Onset of Primary and Secondary Instabilities of Viscoelastic Fluids Saturating a Porous Layer Heated from below by a Constant Flux

Abstract

We analyze the thermal convection thresholds and linear characteristics of the primary and secondary instabilities for viscoelastic fluids saturating a porous horizontal layer heated from below by a constant flux. Galerkin method is used to solve the eigenvalue problem by taking into account the elasticity of the fluid, the ratio between the viscosity of the solvent and the total viscosity of the fluid and the lateral confinement of the medium. For the primary instability, we found out that depending on the rheological parameters, two types of convective structures may appear when the basic conductive solution loses its stability: stationary long wavelength instability as for Newtonian fluids and oscillatory convection. The effect of the lateral confinement of the porous medium by adiabatic walls is to stabilize the oblique and longitudinal rolls and therefore selects transverse rolls at the onset of convection. In the range of the rheological parameters where stationary long wave instability develops first, we use a parallel flow approximation to determine analytically the velocity and temperature fields associated to the monocellular convective flow. The linear stability analysis of the monocellular flow is performed, and the critical conditions above which the flow becomes unstable are determined. The combined influence of the viscoelastic parameters and the lateral confinement on the characteristics of the secondary instability is quantified. The major new findings concerning the secondary instabilities may be summarized as follows: (i) For concentrated viscoelastic fluids, computations showed that the most amplified mode of convection corresponds to oscillatory transverse rolls which appears via a Hopf bifurcation. This pattern selection is independent of both the fluid elasticity and the lateral confinement of the porous medium; (ii) For diluted viscoelastic fluids, the preferred mode of convection is found to be oscillatory transverse rolls for a very laterally confined medium. Otherwise stationary or oscillatory longitudinal rolls may develop depending on the fluid elasticity. Results also showed the destabilizing effect of the relaxation fluid elasticity and the stabilizing effect of the viscosity ratio for the onset of both primary and secondary instabilities.

Heat and mass transfer of triple diffusive convection in viscoelastic liquids under internal heat source modulations

“…Research on viscoelastic liquids is done by various researchers. [13][14][15][16][17][18][19][20][21][22][23][24] Kayodé and Khayat 25 compared the solutions of Newtonian and non-Newtonian liquids and it was noted that the effect of elasticity in the liquid is found to be greater for higher Rayleigh numbers. Siddheshwar et al 26 concluded that the Maxwell liquids are comparatively more stable than other liquids in the study.…”

Effect of heat source on Rayleigh–Bénard convection in rotating viscoelastic liquids

“…Literature on thermal convection/heat transfer in viscoelastic/viscoelastic ferromagnetic liquid with rotation/gravity modulation/temperature modulation/spacial modulation is available in large numbers (see Laroze et al, 19 Siddheshwar et al, 20 Pérez et al, 21 Melson et al, 22 Kayodé and Khayat, 23 Jayalatha et al, 24 Jayalatha and Suma, 25 and Jayalatha and Muchikel 26 ). Convection in viscoelastic liquids in porous medium has been studied by Siddheshwar and Srikrishna, 27,28 Gueye et al, 29 and Mahmud et al 30 Convection in a porous layer of viscoelastic/couple stress liquid is studied by Hemanthkumar et al, 31 Shivakumara and Raghunath, 32 and Raghunath et al 33 …”