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Zbl 1101.76056
Liao, Shijun; Magyari, Eugen
Exponentially decaying boundary layers as limiting cases of families of algebraically decaying ones. (English)
[J] Z. Angew. Math. Phys. 57, No. 5, 777-792 (2006). ISSN 0044-2275; ISSN 1420-9039/e

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 $u_w(x) \sim x^\lambda$.

MSC 2000:: *76R10 Free convection
76M55 Dimensional analysis and similarity
76S05 Flows in porous media
76D10 Boundary-layer theory (incompressible fluids)
80A20 Heat and mass transfer

Keywords: similarity; Cheng-Minkowycz free convection flow; porous medium

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