This study addresses the electronic cooling problem from the perspective of scaling laws applied to a simple conjugate heat transfer geometry, steady shear Bow over a heated strip on a flat plate. The results are reported in a compact form in terms of a modified Nusselt number and a combined parameter (k(6)/k(s))Pe(1/3). Numerical results are reported that apply to a wide range of values for fluid flow, thermal conductivity, and thickness of the Bat plate, and the results include simple design correlations. A new dimensionless parameter is suggested for determining when the fluid axial heat conduction can be neglected.