Boundary layer similarity flow driven by power-law shear

Similarity solution of the Prandtl boundary layer equations describing wallbounded flows and symmetric free-shear flows driven by rotational velocities U(y) = beta y(alpha) are determined for a range of exponents alpha and amplitudes beta. Asymptotic analysis of the equations shows that for alpha < -1 no similarity solutions with proper algebraic decay exist. For wall-bounded flow, exact solutions found at alpha = -1/2 and alpha = 1 correspond to an Airy function wall jet and uniform planar Couette flow. Numerical integration of the governing similarity equation reveals singular behaviour for wall-bounded flows as alpha --> alpha(0) = -2/3, and no solutions are found in the range -1 < alpha less than or equal to -2/3. For alpha > -2/3 the shear stress f''(0) parameter is determined as a function of alpha and beta. Symmetric free-shear flow solutions become singular as alpha --> alpha(0) = -1/2 and no solutions are found in the range -1 < alpha less than or equal to -1/2. For alpha > -1/2 the centerline velocity f'(0) is determined as a function of alpha and beta. An asymptotic analysis of the singular behavior of these two problems as alpha --> alpha(0), given in a separate Appendix, shows excellent comparison with the numerical results. Similarity solutions at the critical values alpha(0), have exponential decay in the far field and correspond to the Glauert wall jet for wall-bounded flow and to the Schlichting/Bickley planar jet for symmetric free-shear flow.