This paper considers the effect of transpiration on hybrid nanofluid flow and heat transfer over a stretching/shrinking sheet for uniform shear flow. The similarity equations are attained from the governing equations by using similarity transformation technique, and their solutions are obtained using the bvp4c solver in Matlab software. The effect of the several governing parameters on the flow and heat transfer characteristics are presented and interpreted theoretically. Results elucidate that dual solutions exist up to a certain range of the stretching/shrinking parameter. Moreover, the dual solutions are observed to occur for both stretching or shrinking cases in the present of suction parameter. An increasing of copper nanoparticle volume fractions generates an enhancement in the heat transfer coefficient, however dual behaviour is observed in the skin friction coefficient for the upper branch solutions. The temporal stability analysis is conducted to determine the stability of the dual solutions, and it is discovered that only one of them is stable and physically reliable.

Transpiration effects on hybrid nanofluid flow and heat transfer over a stretching/shrinking sheet with uniform shear flow

The standard boundary-layer equations play a central role in many aspects of fluid mechanics as they describe the motion of a slightly viscous fluid close to a surface. A number of exact solutions of these equations have been found and here we extend the class of known solutions by implementing the Clarkson-Kruskal direct method (1) for finding similarity reductions. We demonstrate the existence of reductions which take the boundary-layer forms to lower-order partial differential systems and others which simplify them to the solution of ordinary differential equations. It is shown how not only do we recover many of the previously known exact solutions but also find some completely new forms.

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New similarity solutions of the unsteady incompressible boundary-layer equations

Flow and heat transfer characteristics of the boundary layers over a stretching surface with a uniform-shear free stream

The possible existence of a self-similar wall jet over a permeable plane surface in the presence of a (suitable) lateral suction of the fluid is proved. In contrast to its well known exponentially decaying counterpart (the Glauert-jet) formed over an impermeable wall, the suction-controlled wall jet decays algebraically with increasing distance from the wall. However, as the suction becomes vanishingly small, it approaches the profile of the Glauert-jet gradually.

The algebraically decaying wall jet

The boundary-layer similarity flow driven over a semi-infinite permeable flat plate by a power-law shear with asymptotic velocity profile U
∞(y)=βy
α(y→∞,β>0) is considered in the presence of lateral suction or injection of the fluid (y denotes the coordinate normal to the plate). The analytically tractable cases α=−2/3 and α=−1/2 are examined in detail. It is shown that while for α=−2/3 the adjustment of the flow over an impermeable plate to the power-law shear is not possible, in the permeable cases the presence of suction allows for a family of boundary-layer solutions with the proper algebraic decay. The value of the skin friction corresponding to this family of solutions is given by the parameter s=9β3/(4f

 w
 ), where f

 w
 denotes the suction parameter. In the limiting case of a vanishing suction and a properly vanishing value of the parameter β (such that s=finite), this family of algebraically decaying solutions goes over into the (exponentially decaying) Glauert-jet. In the case α=−1/2, solutions showing the proper algebraic decay were found both for suction ( f

 w
 > 0) and injection ( f

 w
 <0) in the whole range −∞<f

 w
 <+∞. In this case the skin friction parameters s=2β2/3 is independent of the suction/injection parameter f

 w
 .

Boundary-layer similarity flows driven by a power-law shear over a permeable plane surface

Similarity solution of the Prandtl boundary layer equations describing wallbounded flows and symmetric free-shear flows driven by rotational velocitiesU(y)=βyα are determined for a range of exponents α and amplitudes β Asymptotic analysis of the equations shows that for α −2/3 the shear stressf″(0) parameter is determined as a function of α and β Symmetric free-shear flow solutions become singular as α→α0 = −1/2 and no solutions are found in the range −1 −1/2 the centerline velocityf′(0) is determined as a function of α and β An asymptotic analysis of the singular behavior of these two problems as α→α0, given in a separate Appendix, shows excellent comparison with the numerical results Similarity solutions at the critical values α0 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

D. G. Kubitschek

Papers

Boundary layer similarity flow driven by power-law shear