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An efficient numerical method is described for studying the combined conductive and convective transport from a surface film on a planar boundary to a fluid in simple shear flow. Such problems arise most commonly in the use of hot film anemometers, electrochemical shear probes, or in simple models of chemical reactions. The method is illustrated by calculating the total flux (Nusselt number) and variation of the flux along the surface for isothermal circular disks at arbitrary values of the Peclet number (dimensionless shear rate) and is compared with asymptotic results valid at high and low Peclet numbers. The theoretical low Peclet number results of Phillips Q. J. Mech. Appl. Math. (in press) lie within 2% of the numerical results for Peclet numbers as high as P=2. At high Peclet numbers, a theoretical estimate for the neglected flux from the edge regions is used, together with the numerical simulations, to propose a correction to the standard one-term asymptotic expression. This approximate relationship remains within 7% of the numerical calculations for Peclet numbers as small as P=5. In addition, results for the total heat transfer as a function of Peclet number are presented for isolated elliptical disks at two orthogonal orientations with respect to the flow and an asymptotic expression for high Peclet numbers is presented for arbitrary disk orientations.
Howard A. Stone (Sat,) studied this question.