Exponentially decaying boundary layers as limiting cases of families of algebraically decaying ones

Abstract.

The boundary value problem for the similar stream function f  =  f(η;λ) of the Cheng–Minkowycz free convection flow over a vertical plate with a power law temperature distribution T _w (x)  =  T_∞ + Ax^λ in a porous medium is revisited. It is shown that in the λ-range  − 1/2  < λ  <  0 , the well known exponentially decaying “first branch” solutions for the velocity and temperature fields are not some isolated solutions as one has believed until now, but limiting cases of families of algebraically decaying multiple solutions. For these multiple solutions well converging analytical series expressions are given. This result yields a bridging to the historical quarreling concerning the feasibility of exponentially and algebraically decaying boundary layers. Owing to a mathematical analogy, our results also hold for the similar boundary layer flows induced by continuous surfaces stretched in viscous fluids with power-law velocities u _w (x)∼ x^λ.