Exponentially decaying boundary layers as limiting cases of families of algebraically decaying ones. (English) Zbl 1101.76056

Summary: We revisit the boundary value problem for similar stream function \(f = f (\eta;\lambda)\) of the Cheng-Minkowycz free convection flow over a vertical plate with power law temperature distribution \(T_{w}(x) = T_{\infty} + Ax^{\lambda}\) in a porous medium. It is shown that in the \(\lambda\)-range \(-1/2 < \lambda < 0\), the well-known exponentially decaying “first branch” solutions for velocity and temperature fields are not isolated solutions as one has believed until now, but limiting cases of families of algebraically decaying multiple solutions. For these multiple solutions we give well-converging analytical series. 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 similar boundary layer flows induced by continuous surfaces stretched in viscous fluids with power-law velocities \(u_w(x) \sim x^\lambda\).


76R10 Free convection
76M55 Dimensional analysis and similarity applied to problems in fluid mechanics
76S05 Flows in porous media; filtration; seepage
76D10 Boundary-layer theory, separation and reattachment, higher-order effects
80A20 Heat and mass transfer, heat flow (MSC2010)
Full Text: DOI Link