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<p>We are concerned with infinite Prandtl number Rayleigh–Bénard convection with Navier-slip boundary conditions. The goal of this work is to estimate the average upward heat flux measured by the non-dimensional Nusselt number <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\textit{Nu}\)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="italic">Nu</mi> </math></EquationSource> </InlineEquation> in terms of the Rayleigh number <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\textit{Ra}\)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="italic">Ra</mi> </math></EquationSource> </InlineEquation>, which is a non-dimensional quantity measuring the imposed temperature gradient. We derive bounds on the Nusselt number that coincide for relatively small slip lengths with the optimal Nusselt number scaling for no-slip boundaries, <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\textit{Nu}\lesssim \textit{Ra}^{1/3}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="italic">Nu</mi> <mo>≲</mo> <msup> <mi mathvariant="italic">Ra</mi> <mrow> <mn>1</mn> <mo stretchy="false">/</mo> <mn>3</mn> </mrow> </msup> </mrow> </math></EquationSource> </InlineEquation>; for relatively large slip lengths, we recover scaling estimates for free-slip boundaries, <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(\textit{Nu}\lesssim \textit{Ra}^{5/12}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="italic">Nu</mi> <mo>≲</mo> <msup> <mi mathvariant="italic">Ra</mi> <mrow> <mn>5</mn> <mo stretchy="false">/</mo> <mn>12</mn> </mrow> </msup> </mrow> </math></EquationSource> </InlineEquation>.</p>

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Infinite Prandtl Number Convection with Navier-Slip Boundary Conditions

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  • 发表日期：

    2026年5月2日

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  • Christian Seis

摘要

We are concerned with infinite Prandtl number Rayleigh–Bénard convection with Navier-slip boundary conditions. The goal of this work is to estimate the average upward heat flux measured by the non-dimensional Nusselt number \(\textit{Nu}\) Nu in terms of the Rayleigh number \(\textit{Ra}\) Ra , which is a non-dimensional quantity measuring the imposed temperature gradient. We derive bounds on the Nusselt number that coincide for relatively small slip lengths with the optimal Nusselt number scaling for no-slip boundaries, \(\textit{Nu}\lesssim \textit{Ra}^{1/3}\) Nu ≲ Ra 1 / 3 ; for relatively large slip lengths, we recover scaling estimates for free-slip boundaries, \(\textit{Nu}\lesssim \textit{Ra}^{5/12}\) Nu ≲ Ra 5 / 12 .