On radiation effects on hydromagnetic Newtonian liquid flow due to an exponential stretching sheet
© Kameswaran et al.; licensee Springer 2012
Received: 25 May 2012
Accepted: 23 August 2012
Published: 2 October 2012
The paper investigates the radiation effect on the magnetohydrodynamic Newtonian fluid flow over an exponentially stretching sheet. The effects of frictional heating and viscous dissipation on the heat transport are taken into account. The governing partial differential equations are transformed into ordinary differential equations using a suitable similarity transformation. Zero-order analytical solutions of the momentum equation and confluent hypergeometric solutions of heat and mass transport equations are obtained. The accuracy of analytical solutions is verified by numerical solutions obtained using a shooting technique that uses a Runge-Kutta-Felhberg integration scheme and a Newton-Raphson correction scheme. The effects of the radiation parameter, the magnetic parameter, Gebhart and Schmidt numbers on the momentum, heat and mass transports are discussed. The skin friction and heat and mass transfer coefficients for various physical parameters are discussed.
The study of laminar boundary layer flow over a stretching sheet has received considerable attention in the recent past due to its immense application in industry, for example, in extrusion processes such as the polymer extrusion from a dye and wire drawing. Other engineering applications of the stretching sheet problem include polymer sheet extrusion from a dye, drawing, tinning and annealing of copper wires, glass fiber and paper production, the cooling of a metallic plate in a cooling bath and so on. There has been tremendous amount of work on the stretching sheet problem in the past several decades (see Crane , Gupta and Gupta , Grubka and Bobba , Dutta and Gupta , Siddappa and Abel , Chen and Char , Laha et al. , Chakrabarti and Gupta , Anderson et al. , Siddheshwar and Mahabaleswar , Abel and Mahesha , Abel et al.  and the references therein).
The above studies concern the linear stretching sheet problem but most of the practical situations involve a non-linear stretching sheet such as an exponential one. With this in mind, several authors have considered the velocity of the sheet to vary exponentially with the distance from the slit. Elbashbeshy  was among the first to study the exponentially stretching sheet problem. He considered a perforated sheet and examined the effect of wall mass suction on the flow and heat transfer over an exponentially stretching surface. Using a suitable similarity transformation, he transformed the momentum equation into a non-linear Riccati type equation and solved it iteratively. Ishak  studied the MHD boundary layer flow due to an exponentially stretching sheet with radiation effect. He found that the local heat transfer rate at the surface decreased with increasing values of the magnetic and radiation parameters. The flow and heat transfer from an exponentially stretching surface was considered by Magyari and Keller . They examined the heat and mass transfer characteristics and compared with the well-known results of the power-law models. Sanjayanand and Khan  studied the heat and mass transfer in a viscoelastic boundary layer flow over an exponentially stretching sheet. They found that the viscoelastic parameter enhances the thermal boundary layer thickness. The effect of viscous dissipation on the mixed convection heat transfer from an exponentially stretching surface was studied by Partha et al. . They observed a rapid growth in the non-dimensional skin friction coefficient with the mixed convection parameter. The influence of thermal radiation on the boundary layer flow due to an exponentially stretching sheet is studied by Sajid and Hayat . Khan  presented an elegant solution of the viscoelastic boundary layer flow over an exponentially stretching sheet in terms of Whittaker’s function.
The characteristics desired of the final product in an extrusion process depend on the rate of stretching and cooling. Hence, it is very important to have a controlled cooling environment where the flow over the stretching sheet can be regulated by external agencies like a magnetic field. An exponential variation of a magnetic field is used, among other applications, to determine the diamagnetic susceptibility of plasma. Steenbeck  determined the diamagnetic susceptibility of a cylindrical plasma for axial magnetic fields with various gas pressure and magnetic field strengths. Tonks  studied the effects of a magnetic field in the plasma of an arc. Pavlov  considered the magnetohydrodynamic flow of an incompressible viscous fluid over a linearly stretching surface. Sarpakaya  extended Pavlov’s work to non-Newtonian fluids. Subsequent studies by Andersson , Lawrence and Rao , Abel et al. , Cortell  concerned the magnetohydrodynamic flow of viscoelastic liquids over a stretching sheet. Radiation effects on MHD flow past an exponentially accelerated isothermal vertical plate with uniform mass diffusion in the presence of a heat source was studied by Reddy et al. . They observed that the velocity decreases with an increase in the magnetic parameter due to a resistive drag force which tends to resist the fluid flow and thus reduces the velocity. The boundary layer thickness was also found to decrease with an increase in the magnetic parameter.
Most of the earlier work neglected radiation effects. If the polymer extrusion process is placed in a thermally controlled environment, radiation could become important. As with magnetohydrodynamics, careful control of thermal radiative heat transfer has an effect on the characteristics of the final product. Many researchers have considered the effect of thermal radiation on flows over stretching sheets. Studies by Raptis , Raptis and Perdikis  address the effect of radiation in various situations. Siddheshwar and Mahabaleswar  studied the effects of radiation and heat source on MHD flow of a viscoelastic liquid and heat transfer over a stretching sheet. Bidin and Nazar  studied the effects of numerical solution of the boundary layer flow over an exponentially stretching sheet with thermal radiation. They observed that the temperature profiles and the thermal boundary layer thickness increase slightly with an increase in the Eckert number. They also showed that an increase in Pr causes a decrease in temperature profiles and the thermal boundary layer thickness. Physically, if Pr increases, the thermal diffusivity decreases, and these phenomena lead to the decreasing of energy ability that reduces the thermal boundary layer. Elbashbeshy and Dimian  analyzed boundary layer flow in the presence of radiation effect and heat transfer over the wedge with a viscous coefficient. Thermal radiation effects on hydro-magnetic flow due to an exponentially stretching sheet were studied by Reddy and Reddy . They found that as radiation increases, the temperature profiles and thermal boundary layer thickness also increase. They also observed that the temperature profiles and thermal boundary layer thickness increase slightly with an increase in the Eckert number. Raptis et al.  studied the effect of thermal radiation on the magnetohydrodynamic flow of a viscous fluid past semi-infinite stationary plate and Hayat et al.  extended the analysis for the second grade fluid.
In addition to a magnetic field and thermal radiation, one has to consider the viscous dissipation effects due to frictional heating between fluid layers. The effect of viscous dissipation in natural convection processes has been studied by Gebhart  and Gebhart and Mollendorf . They observed that the effect of viscous dissipation is predominant in vigorous natural convection and mixed convection processes. They also showed the existence of a similarity solution for the external flow over an infinite vertical surface with an exponential variation of surface temperature. Vajravelu and Hadjinicalaou  studied the heat transfer characteristics over a stretching surface with viscous dissipation in the presence of internal heat generation or absorption.
In this paper, we investigate the effects of various physical and fluid parameters such as the magnetic parameter, radiation parameter and viscous dissipation parameter on the flow and heat transfer characteristics of an exponentially stretching sheet. The momentum, energy and concentration equations are coupled and nonlinear. By using suitable similarity variables, these equations are converted into coupled ordinary differential equations and are solved analytically and numerically by using the Runge-Kutta-Fehlberg and Newton-Raphson schemes.
2 Mathematical formulation
Here the subscripts w, ∞ refer to the surface and ambient conditions respectively, , are positive constants, is the characteristic velocity, and L is the characteristic length.
where η is the similarity variable, is the dimensionless stream function, is the dimensionless temperature, and is the dimensionless concentration.
3 Skin friction, heat and mass transfer coefficients
The parameters of engineering interest in heat and mass transport problems are the skin friction coefficient , the local Nusselt number , and the local Sherwood number . These parameters respectively characterize the surface drag, wall heat and mass transfer rates.
In Equations (19), (22) and (25), represents the local Reynolds number and it is defined as .
4 Analytical solution
4.1 Solution of momentum equation
Further, we assume that the first-order iterate of f satisfies the boundary conditions on f as given in (14). The above non-linear Riccati type equation can be solved in terms of a confluent hypergeometric Whittaker function as discussed by Khan . However, we restrict ourselves to the zero-order solution, and similarly, to heat and mass transport equations.
4.2 Solution of heat transfer equation
4.3 Solution of mass transfer equation
5 Solution procedure
where , and are determined such that , and . Thus, to solve this system, we require six initial conditions. However, since we have only three initial conditions for f and two initial conditions for θ and ϕ, the conditions , , are to be determined by the shooting method using the initial guess values , and until the conditions , and are satisfied. In this paper, we employed the shooting technique with the Runge-Kutta-Fehlberg scheme to determine two more unknowns in order to convert the boundary value problem to an initial value problem. Once all the six initial conditions were determined, the resulting differential equations were integrated using an initial value solver. For this purpose, the fifth-order Runge-Kutta-Fehlberg integration scheme was used.
6 Results and discussion
A comparison of obtained by the analytical method with the shooting technique for different values of M
A comparison of obtained by the analytical method with the shooting technique for different values of M , Gb and K for fixed values of
A comparison of obtained by the analytical method with the shooting technique for different values of M , Sc
A comparison of for different values of M for fixed values of
Reddy and Reddy 
Figure 2 shows the variation of the velocity profile against the magnetic parameter. We notice that the effect of the magnetic parameter is to reduce the velocity of the fluid in the boundary layer region. This is due to an increase in the Lorentz force, similar to Darcy’s drag observed in the case of flow through a porous medium. This adverse force is responsible for slowing down the motion of the fluid in the boundary layer region. These results are similar to the results obtained by Reddy and Reddy .
The problem of hydromagnetic Newtonian liquid flow due to an exponentially stretching sheet in the presence of radiation and viscous dissipation effects has been analyzed. Exact solutions were found in terms of hypergeometric functions, and a comparison of analytical and numerical results was shown. We found that the effect of the magnetic parameter is to reduce the velocity of the fluid in the boundary layer region. It was also observed that the increase in values of M results in thickening of the species boundary layer. The combined and individual effects of the magnetic parameter M, the radiation parameter K, and the viscous dissipation parameter Gb are to increase the heat transfer rates. Under some limiting conditions when the parameters Pr, Sc, K, Gb are zero, the current results agree well with available results in the literature.
The authors are grateful to the National Research Foundation (NRF) and the University of KwaZulu-Natal for financial support.
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