Viscous dissipation and thermoconvective instabilities in a horizontal porous channel heated from below

Abstract

A linear stability analysis of the basic uniform flow in a horizontal porous channel with a rectangular cross section is carried out. The thermal boundary conditions at the impermeable channel walls are: uniform incoming heat flux at the bottom wall, uniform temperature at the top wall, adiabatic lateral walls. Thermoconvective instabilities are caused by the incoming heat flux at the bottom wall and by the internal viscous heating.Linear stability against transverse or longitudinal roll disturbances is investigated either analytically by a power series formulation and numerically by a fourth order Runge-Kutta method. The special cases of a negligible effect of viscous dissipation and of a vanishing incoming heat flux at the bottom wall are discussed. The analysis of these special cases reveals that each possible cause of the convective rolls, bottom heating and viscous heating, can be the unique cause of the instability under appropriate conditions. In all the cases examined, transverse rolls form the preferred mode of instability.

Nonlinear stability analysis of Darcy’s flow with viscous heating

“…This occurs, because growth in the base flow direction has been precluded owing to the removal of this coordinate in equations (4.5). Figure 4 displays the growth rate r of supercritical configurations with Λ = 65 as a function of the aspect ratio s. Once again, linear stability results from Barletta & Storesletten [34] (solid line) are used to validate the present nonlinear simulation results for Ge → 0 (plus symbol) and Ge = 1 (times symbol). This slightly supercritical configuration, i.e.…”

“…Here follows a list of the approximations employed to perform the nonlinear stability analysis -the tolerance is set to tol = 10 −5 ; -the truncation threshold is set to N = 40; and -the threshold velocity used to identify the onset of instability is set to v T = 10 −2 . Figure 3 displays the threshold values of the governing parameter Λ for the onset of convective instability as a function of the aspect ratio s. Different neutral stability curves and the benchmark linear stability analysis from Barletta & Storesletten [34] (solid lines) are drawn in figure 3 for comparison purposes. The vertical dashed lines are taken from this linear analysis as well to define the separation between two different convection cell patterns at the onset of instability.…”

“…Figure 3 displays the threshold values of the governing parameter Λ for the onset of convective instability as a function of the aspect ratio s. Different neutral stability curves and the benchmark linear stability analysis from Barletta & Storesletten [34] (solid lines) are drawn in figure 3 for comparison purposes. The vertical dashed lines are taken from this linear analysis as well to define the separation between two different convection cell patterns at the onset of instability.…”

Linear disturbance growth induced by viscous dissipation in Darcy–Bénard convection with throughflow

“…In their studies, no additional thermal forcing was imposed either internally or externally, namely through the walls or otherwise. The combined effect of internal heating through viscous dissipation and external heating through different thermal boundary conditions was studied by Barletta & Storesletten (2010) as well as Barletta, Celli & Nield (2010) and Nield, Barletta & Celli (2011). Nield & Barletta (2010) also explored two different models for the viscous dissipation effect.…”