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Mixed convection boundary layer similarity solutions: Prescribed wall heat flux. (English) Zbl 0667.76126
The similarity solutions for mixed convection boundary-layer flow when the wall heat flux is prescribed are analysed in detail in terms of a buoyancy parameter $\alpha$ and m the exponent of the free stream flow. It is shown that for $\alpha >0$ the solution approaches the free convection limit, and for $\alpha <0$, there is a range of $\alpha$, ${\alpha }_{s}<\alpha <0$, over which dual solutions exist. The nature of the bifurcation at $\alpha ={\alpha }_{s}$ and how the lower branch of solutions behaves as $\alpha \to {0}^{-}$ are also considered. It is established that the solution becomes singular as $m\to 1/5$ and the nature of this singularity is also discussed, where it is shown that two separate cases have to be treated, namely when $\alpha$ is of O(1) and when $\alpha$ is small. Finally it is shown that for m large the solution approaches that corresponding to exponential forms for the free stream and prescribed wall heat flux. Taken all together this information enables a complete description of how the solution behaves over all possible ranges of the parameters $\alpha$ and m to be deduced.

##### MSC:
 76R05 Forced convection (fluid mechanics) 76R10 Free convection (fluid mechanics) 35Q99 PDE of mathematical physics and other areas 76M99 Basic methods in fluid mechanics
##### References:
 [1] E. M. Sparrow, R. Eichhorn and J. L. Gregg,Combined forced and free convection in boundary layer flow. Phys. Fluids2, 319-328 (1959). · Zbl 0086.40601 · doi:10.1063/1.1705928 [2] C. B. Cohen and E. Reshotko,Similar solutions for the compressible boundary layer with heat transfer and pressure gradient. NACA Report 1293 (1956). [3] G. Wilks and J. S. Bramley,Dual solutions in mixed convection. Proc. Roy. Soc. Edinburgh87A, 349-358 (1981). [4] W. H. H. Banks and P. G. Drazin,Perturbation methods in boundary-layer theory. J. Fluid Mech.58, 763-775 (1973). · Zbl 0263.76026 · doi:10.1017/S002211207300248X [5] T. Mahmood and J. H. Merkin,Similarity solutions in axisymmetric mixed convection boundary-layer flow. J. Eng. Math.22, 73-92 (1988). · Zbl 0656.76073 · doi:10.1007/BF00044366 [6] R. Eichhorn and M. M. Hasan,Mixed convection about a vertical surface in cross-flow: a similarity solution. J. Heat Transfer102, 775-777 (1980). · doi:10.1115/1.3244391 [7] E. M. Sparrow and J. L. Gregg,Laminar free convection from a vertical plate with uniform surface heat flux. Trans. ASME78, 435-440 (1956). [8] J. H. Merkin,A note on the similarity solutions for free convection on a vertical plate. J. Eng. Math.19, 189-201 (1985). · Zbl 0577.76083 · doi:10.1007/BF00042533 [9] C. W. Jones and E. J. Watson,Two-dimensional boundary layers, InLaminar boundary layers, (Ed L. Rosenhead). Clarendon Press, Oxford (1963).