The Effect of Viscous Dissipation on the Darcy–Bénard Problem: Weakly Nonlinear Analysis and Strongly Nonlinear Computations

Abstract We consider the effect of viscous dissipation on the onset and nonlinear development of two-dimensional convection in a unit enclosure heated from below. First, we show that the linear theory is unchanged from that which arises when viscous dissipation is absent. Second, a weakly nonlinear analysis shows that convection becomes weaker with increasing values of $${\widetilde{\textrm{Ge}}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mover> <mml:mtext>Ge</mml:mtext> <mml:mo>~</mml:mo> </mml:mover> </mml:math> , a modified Gebhart number. In addition, the rate of heat transfer at the lower and upper sufaces differ from one another. Third, nonlinear convection is found to lose both up/down and left/right symmetry as both $${\widetilde{\textrm{Ge}}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mover> <mml:mtext>Ge</mml:mtext> <mml:mo>~</mml:mo> </mml:mover> </mml:math> and $$\textrm{Ra}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mtext>Ra</mml:mtext> </mml:math> (the Darcy–Rayleigh number) increase. It is also found that once viscous dissipation increases in strength to unphysically large amounts, then the maximum temperature migrates from the lower boundary to the interior of the enclosure.