Consider the free convection past a vertical flat plate or wall maintained at a constant temperature with > , where is the fluid temperature far from the wall. W. M. Deen derived the governing equations for such a problem (see the reference):
and with and ,
where is the dimensionless temperature, is the Prandtl number (a dimensionless number that gives the rate of viscous momentum transfer relative to heat conduction), and is proportional to the fluid velocity ( is a modified stream function). One has to use the shooting technique to solve this split boundary value problem. This Demonstration displays the temperature and velocity profiles for various values of the Prandtl number versus a similarity variable , where is the Grashof number. When is reduced, the dimensionless temperature variations extend farther from the wall (an indication of higher rates of heat conduction).