Vorticity and Temperature Fields

If the surface over which the fluid is flowing is held at a different temperature from the bulk of the) fluid, there will also be temperature gradients in the fluid near the surface. The difference between the temperature of the fluid locally and that of the fluid far from the surface is, in general, highest at the surface and decreases with increasing distance from the surfac . The purpose of the present section is to show that there is a similarity between these vorticity and temperature fields. 

For the present purposes, attention will be restricted to steady, two-dimensional constant fluid property flow and, consistent with this assumption, the dissipation teim in the energy equation will be neglected. The result will be derived using the governing equations expressed in Cartesian coordinates although a similar result can, of course, be obtained in terms of other coordinate systems. 

Consider the velocity field first. It is governed by the continuity and Navier-Stokes equations which, subject to the assumptions introduced above, are, as previously presented  

Now when the dissipation term is neglected, the two-dimensional energy equation becomes 

the vorticity and temperature fields are governed by equations having the same basic form. When Pr is equal to 1, the equations have exactly the same form. Even in this case, however, the vorticity and temperature fields will not be identical because the boundary conditions on the two fields at the surface will not in general be identical. However, there will obviously be similarities between the two fields. Vorticity is generated in the flow by the action of viscosity due to the presence of the surface. The temperature differences arise in the flow because the surface is at a temperature which is different from the flowing fluid. Thus, Eq. (2.76) essentially describes the rate at which viscous effects spread into the fluid while Eq. (2.77) describes the rate at which the effects of the temperature changes at the surface spread into the fluid. It will be seen that the relative rates of spread depend on the value of the Prandtl number. 

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