B Heat Transfer by Conduction Pe

When the dimensionless form of the thermal energy equation is compared with the dimensionless Navier-Stokes equation, it is clear that the Peclet number plays a role for heat transfer that is analogous to the Reynolds number for fluid motion. Thus it is natural to seek approximate solutions for asymptotically small values of the Peclet number, analogous to the low-Reynolds-number approximation of Chaps. 7 and 8. 

Let us suppose, then, that an asymptotic solution exists for Pe C I in the form of a regular perturbation expansion  

If we substitute (9-15) into (9-7) and (9-8) and remember that u is independent of Pe, we see that the leading term satisfies 

Equation (9 16) is known as the steady-state heat conduction equation and is completely analogous to the creeping-motion equation of Chaps. 7 and 8. It can be seen that convection plays no role in the heat transfer process described by (9 16) and (9 17). Thus the form of the velocity field is not relevant, and in spite of the initial assumption (9 15), there is no dependence of 0o on the Reynolds number of the flow. The solution of (9 16) and (9-17) depends on only the geometry of the body surface, represented in (9 17) by S. 

A simple example for which problems (9-16) and (9 17) can be solved easily is the case of a heated sphere. In this case, we may choose the sphere radius as the characteristic length scale for nondimensionalization, and the problem is to solve (9 16) subject to the boundary condition 0 = 1 at r = 1. This can be done easily. We may first note that a general solution of Laplace s equation in spherical coordinates is 

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