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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 $f=f\left(\eta ;\lambda \right)$ of the Cheng-Minkowycz free convection flow over a vertical plate with power law temperature distribution ${T}_{w}\left(x\right)={T}_{\infty }+A{x}^{\lambda }$ 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 ${u}_{w}\left(x\right)\sim {x}^{\lambda }$.
##### MSC:
 76R10 Free convection (fluid mechanics) 76M55 Dimensional analysis and similarity (fluid mechanics) 76S05 Flows in porous media; filtration; seepage 76D10 Boundary-layer theory, separation and reattachment, etc. (incompressible viscous fluids) 80A20 Heat and mass transfer, heat flow