Flow and mass transfer on a stretching sheet with a magnetic field and chemically reactive species
Introduction
The heat, mass and momentum transport on a continuously moving or stretching sheet has several applications in, electro-chemistry and polymer processing [1], [2], [3], [4]. Most of the studies deal with flow induced by surfaces moving with a constant velocity. Crane [5] first studied the flow caused by an elastic sheet whose velocity varies linearly with the distance from a fixed point on the sheet. Since then, the flow, heat and mass transfer problem with or without suction (blowing) or magnetic field has been considered by several investigators [6], [7], [8], [9], [10], [11], [12], [13], who obtained self-similar solutions. It may be remarked that the assumption of linear variation of the surface velocity with the distance (uw=ax) gives the unrealistic surface velocity at x=0, to be zero. Jeng et al. [14] considered the non-similar flow where the velocity of the stretching sheet is uw=u0(1+x/L). Also, the self-similarity is destroyed by the presence of a chemical reaction in the mass diffusion equation except in the case of a stagnation-point flow [15]. Recently, Andersson et al. [15] have investigated the transport of mass and momentum of chemically reactive species in the laminar flow over a linearly stretching surface and solved the nonlinear ordinary differential equations governing the self-similar flow.
In this study, we have investigated the flow and mass diffusion of chemical species with first and higher order reactions over a continuously stretching sheet with an applied magnetic field. The velocity of the surface varies linearly with the distance x with non-zero velocity at . The reactive component resulting from the stretching surface undergoes an isothermal and homogeneous one-stage reaction as it diffuses into the surrounding fluid. The non-linear partial differential equations governing the non-similar flow and mass transfer are solved numerically using an implicit finite-difference scheme. The particular cases of the present results are compared with those of Jeng et al. [14] and Andersson et al. [15].
Section snippets
Problem formulation
Let us consider the steady incompressible flow of a viscous electrically conducting fluid over a stretching sheet (surface) with a magnetic field B which is applied normal to the surface. The physical system to be investigated is given in the inset of Fig. 1. We use the Cartesian x–y coordinate system which is fixed in space and at x=y=0 a thin solid surface is extruded which moves in the x-direction with surface velocity uw=u0(1+x/L). This introduces non-similarity in the equations. The motion
Method of solution
The partial differential equations , under conditions (8) are solved numerically by an implicit, iterative tridigonal finite-difference method similar to that of Blottner [17]. All the first-order derivatives with respect to ξ are replaced by two-point backward difference formulae of the formwhere H represents dependent variable f or g and i and j are the node locations along ξ and η directions, respectively. First the third-order differential equation (6) is converted
Results and discussion
In order to assess the accuracy of our method, we have compared our skin friction coefficient and the mass (heat) transfer coefficient in terms of the Sherwood (Nusselt) number for M=β=0, Sc=0.7 with those of Jeng et al. [14]. It may be noted that Eq. (7) is the same as that of Jeng et al. [14] if we replace Sc by Pr and Sh by Nu. Also, for M=ξ=0, λ=λ3=1, we have compared our mass transfer with that of Andersson et al. [15] (see Table 1). In both
Conclusions
An important result is that the surface skin friction is significantly increased by the magnetic field, but the surface mass transfer is slightly reduced. The effect of the magnetic field on the skin friction becomes more pronounced as the streamwise distance increases. Another important result is that the surface mass transfer for the first order reaction is more than that of the second or higher order reaction. The surface mass transfer strongly depends on the Schmidt number and the reaction
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