A similarity solution for the flow and heat transfer over a moving permeable flat plate in an external free stream: case
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A similarity solution for the flow and heat transfer over a moving permeable flat plate in an external free stream: case of strong injection J.H. Merkin · N. Bachok · M.A. Jaradat · I. Pop
Received: 2 July 2012 / Accepted: 6 September 2012 / Published online: 10 October 2012 © Springer Science+Business Media Dordrecht 2012
Abstract The previous work of Bachok et al. (Heat Mass Transf. 47:1643–1649, 2011) on the forced convection heat transfer on an isothermal moving surface in an external free stream is extended to the case when fluid injection through the surface, characterized by the parameter γ , is large. The asymptotic solution derived in this limit shows that the boundary layer has a double region structure, with an inviscid inner region of thickness O(γ ) and an outer shear layer. Some further aspects of the original problem not treated in Bachok et al. (Heat Mass Transf. 47:1643–1649, 2011) are discussed as well as the analogous problem for a constant surface heat flux, where relatively small injection rates are seen to give rise to large increases in the surface temperature.
J.H. Merkin () Department of Applied Mathematics, University of Leeds, Leeds, LS2 9JT, UK e-mail: [email protected] N. Bachok Department of Mathematics, Faculty of Science, Universiti Putra Malaysia, 43400 Serdang, Malaysia M.A. Jaradat Bogdan Voda University, Cluj-Napoca, Romania I. Pop Faculty of Mathematics, University of Cluj, 3400 Cluj, CP 253, Romania
Keywords Heat and mass transfer · Forced convection · Moving surface · Strong injection · Asymptotic analysis
1 Introduction In a recent paper Bachok et al. [1] have studied the problem of the steady boundary-layer flow over a moving permeable flat surface in an external free stream with the nonlinear varying velocities uw (x) ∼ x 1/3 and u∞ (x) ∼ x 1/3 , where x is the coordinate measured along the plate and, uw and u∞ are the velocities of the moving flat surface and of the free stream respectively. These choices for uw and u∞ enabled the problem to be reduced to similarity form. In [1] the forced convection heat transfer was also considered where it was assumed that the moving surface was at a constant temperature different to ambient, again allowing the problem to be reduced to similarity form. A mass flux velocity through the surface at a rate proportional to γ x −1/3 , where γ is the dimensionless transpiration parameter, was also included in the discussion in [1]. This form for the transpiration velocity was also taken to keep the similarity form and where γ < 0 for suction and γ > 0 for injection, respectively. Although many of these boundary conditions may be difficult to realize fully in practice, they can constitute an important part of more general engineering problems as they can describe the initial development or asymptotic behavior of a more general flow. Also the
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Meccanica (2013) 48:297–306
detailed understanding of these relatively simple similarity systems can give insights into more complex processes, see for example Rosenhead [2], Schlichting
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