A novel analytic solution of MHD flow for two classes of visco-elastic fluid over a sheet stretched with non-linearly (q
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A novel analytic solution of MHD flow for two classes of visco-elastic fluid over a sheet stretched with non-linearly (quadratic) velocity Rafael Cortell
Received: 3 June 2011 / Accepted: 20 April 2013 © Springer Science+Business Media Dordrecht 2013
Abstract The laminar boundary layer flow induced in a quiescent visco-elastic fluid by a permeable stretched flat surface with non-linearly (quadratic) velocity and appropriate wall transpiration under the influence of a magnetic field is investigated. It is shown that the problem permits a complete analytic exponentially decaying solution for the set of continuity and momentum equations with both magnetic field and visco-elasticity influences for two classes of viscoelastic fluid, namely, the second grade and Walters’ liquid B fluids. The effects on both the skin friction parameter α and velocity profiles of various physical parameters such as visco-elasticity, suction/blowing parameter and magnetic parameter are studied. The results for the velocity field are presented through graphs and discussed in detail. Keywords Laminar boundary layer · Magnetic field · Non-linear (quadratic) stretching surfaces
1 Introduction The magneto-hydrodynamic (MHD) boundary layer behaviour of a visco-elastic fluid over a continuously R. Cortell () Departamento de Física Aplicada, Escuela Técnica Superior de Ingenieros de Caminos, Canales y Puertos, Universidad Politécnica de Valencia, 46022 Valencia, Spain e-mail: [email protected]
stretching plate is a type of investigation which finds increasing importance in a great deal of processing industries such as petroleum engineering, metallurgy industry, chemical engineering, etc. These types of flows have a definite bearing in metallurgical and polymer extrusion processing, manufacturing of rubber sheets, drawing of plastic films and wires, glass-fiber production, the cooling and drying of paper and textiles, cooling of metallic sheets or electronic chips, among which we also cite many applications in petroleum in drilling, manufacturing of foods and slurry transporting. Blasius solution, which is the solution for a laminar viscous flow past a semi-infinite stationary flat plate, was investigated by Cortell [1] numerically. These numerical results have very recently been used for comparison [2]. On the other hand, Sakiadis [3] initiated the study of the boundary layer flow over a continuous solid surface moving with constant speed. It seems that Crane [4] was the first one who studied the stretching problem taking into account the fluid flow over a linearly stretched surface. The elastic sheet moves in its own plane with a velocity varying linearly with the distance from a fixed point (slit). Later, Gupta and Gupta [5] extended the Crane’s work [4] to include suction or blowing, and Nataraja et al. [6] analyzed the stretching problem for a second-order (Walters’ liquid B) fluid with variable surface temperature. The present author [7, 8] studied heat transfer in an incompressible second grade fluid caused by a stretching sheet with a
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