A silicone oil layer deep has a free upper surface open to air and is heated from below

PHYSICS · Multi-part · Fermi University Physics · Fluids & Continuum Physics · Convection, rotation and geophysical flows · Convection, rotation and geophysical flows

Fluid Dynamics, Ch Thermal convection and the Rayleigh-Benard problem

Problem

100\,\mathrm{cSt}

silicone oil layer

3.0\,\mathrm{mm}

deep has a free upper surface open to air and is heated from below. For a

1.0\,\mathrm{mm}

layer of this oil,

Ma_{\gamma} = 5.7\,\mathrm{K^{-1}}

and

Ra = 0.86\,\mathrm{K^{-1}}

. The Marangoni number scales as

Ma_{\gamma}\propto d

and the Rayleigh number as

Ra\propto d^3

at fixed

\Delta T

. Take the onset thresholds

Ma_{\gamma,c} \approx 80

and, for a layer with one free surface,

Ra_c \approx 1100

. ((a)) Find

Ma_{\gamma}

and

Ra

per kelvin for the

3.0\,\mathrm{mm}

layer. ((b)) Find the temperature difference at which each mechanism, taken alone, would reach its threshold. ((c)) State which mechanism starts convection first, and by roughly what factor in

\Delta T

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