Construction of upper and lower solutions for flow past a non-uniformly heated plate

Abstract

The Oseen linearization and the modified Oseen linearization are often used in studying fluid mechanical problems, but whether the linearized solution is accurate is usually difficult to assess. For the sample problem of uniform flow past a plate, we use a comparison theorem to show that the Oseen linearization, used in two ways, gives both an upper and a lower solution. Further, we make use of the comparison theorem and the modified Oseen linearization to construct a sharper upper solution valid in the boundary layer. We then go on to consider the case when the plate temperature increases along the plate according to a power law. Upper and lower solutions for the temperature equation are constructed, and bounds on the temperature gradient at the plate are obtained. With a minor modification, similar results for the case when the logarithmic derivative of the plate temperature lies between two power law curves are obtained.