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<p>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 <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\({\widetilde{\textrm{Ge}}}\)</EquationSource> <EquationSource Format="MATHML"><math> <mover accent="true"> <mtext>Ge</mtext> <mo stretchy="true">~</mo> </mover> </math></EquationSource> </InlineEquation>, 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 <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\({\widetilde{\textrm{Ge}}}\)</EquationSource> <EquationSource Format="MATHML"><math> <mover accent="true"> <mtext>Ge</mtext> <mo stretchy="true">~</mo> </mover> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\textrm{Ra}\)</EquationSource> <EquationSource Format="MATHML"><math> <mtext>Ra</mtext> </math></EquationSource> </InlineEquation> (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.</p>

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

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  • 发表日期：

    2025年11月9日

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  • D. Andrew S. Rees,
  • P. G. Siddheshwar

摘要

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 \({\widetilde{\textrm{Ge}}}\) Ge ~ , 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 \({\widetilde{\textrm{Ge}}}\) Ge ~ and \(\textrm{Ra}\) Ra (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.