Unsteady convective heat transfer from a flat plate with heat flux that varies in space and time

Convective heat transfer due to laminar fluid flow past a flat plate is a standard problem in heat transfer. While constant heat flux or temperature along the plate is often assumed for solving such problems, there may be several practical scenarios where the heat flux along the plate varies as a function of both space and time. Developing an analytical solution for the resulting plate temperature distribution is important for understanding and optimizing the thermal performance of such systems. While some work exists on analyzing problems with time-dependent or space-dependent heat flux, there is a lack of work on the general problem where the heat flux is a function of both space and time. This paper presents a solution for this problem by solving the integral form of the energy equation, along with the use of fourth-order Karman-Pohlhausen polynomials for velocity and temperature distributions in the momentum and thermal boundary layers. A non-linear, first order, hyperbolic partial differential equation for the plate temperature is derived in response to the time- and space-varying plate heat flux. This equation is shown to agree well with results from past work for several special cases. Numerical solutions for the generalized equation are presented. Based on this approach, the plate temperature distribution is predicted for several heat flux profiles that may be of interest in practical applications. Results from this work improve our understanding of unsteady convective heat transfer, and contribute towards modeling and optimization of practical engineering systems where such phenomena occur.