Flow in a Vertical Channel with Walls at Different Temperatures

Consider a vertical parallel-plate channel of width with walls at different temperatures and , . Two cases are studied: (1) pure free convection (i.e., the channel is closed at both ends and there is no net flow); and (2) mixed free and forced convection (a pressure gradient is present). The velocity (solution of the momentum equation) is given by
,
where is gravitational acceleration, is the thermal expansion coefficient, is the reference temperature taken equal to the mean temperature, is the kinematic viscosity, is the dynamic viscosity, is the pressure gradient, and is the dimensionless position.
The velocity can be expressed as , where is the mean velocity and is the maximum velocity when the pressure gradient is zero.
If and , one recovers pure free convection. When , the velocity profile is the expected parabolic profile corresponding to a Poiseuille flow. In general, mixed free and forced convection is observed.

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W. M. Deen, Analysis of Transport Phenomena, New York: Oxford University Press, 1998.
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