Viscoelastic Fluid Flow with Heat and Mass Transfer in the Presence of Induced Magnetic Field

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INTRODUCTION
Heat transfer is of considerable interest because it can occur in many geothermal, geophysical, technological and engineering processes such as nuclear reactors. Geothermal gases are electrically conducting and are affected by the presence of magnetic field. Viscoelastic flows arise in numerous areas of chemical industrial process, biosystems, food processing and biomedical engineering (Lima and Rey, 2006). Viscoelastic fluid flows are important in petroleum industry and in the purification of crude oils, dust particles in boundary layers. The study of viscoelastic fluid has become important in the last few years. Qualitative analyses of these studies have significant bearing on several industrial applications such as polymer sheet extrusion from a dye, drawing of plastic firms etc. When manufacturing processes at high temperature need cooling, the flows may need viscoelastic fluid to produce a good effect or reduce the temperature (Kai-Long, 2010). Seddeek (2007) studied the heat and mass transfer problems about the viscoelastic boundary layer flow over a stretching sheet with magnetic effect. Kai-long (2010) investigated heat and mass transfer for viscous flow with radiation effect past a non-linear stretching sheet. Muthucumaraswamy and Senthil(2004) presented heat and mass transfer effects on moving vertical plate in the presence of thermal radiation. We find several industrial applications of magnetic field such as polymer technology and metallurgy. Sarpakaya (1961) is the first researcher to investigate MHD flows of non-Newtonian fluids. Chaudharyand Jha (2008) considered heat and mass transfer in elastic viscous fluid past an impulsively ISSN: 2488-9229 FEDERAL UNIVERSITY GUSAU-NIGERIA started infinite vertical plate with Hall Effect. Mohiddinet al. (2010) investigated the numerical study of unsteady free convective heat and mass transfer in a walters B viscoelastic flow along a vertical cone. Computational treatment of free convection effects on perfectly conducting viscoelastic fluid was studied by El-Bary (2005). The effect of magnetic field on an oscillatory flow of a viscoclastic fluid with thermal radiation was investigated by Ambethkar and Singh (2011). Hayat et al. (2010) examined two-dimensional mixed convection boundary layer magnetohydrodynamics (MHD) stagnation-point flow through a porous medium bounded by a stretching vertical plate with thermal radiation and magnetic field. They obtained exact solution using the method of homotopy analysis. Kai-long (2007) studied the heat transfer on a stretching sheet cooled or heated by a high or low prandtl number, the buoyancy parameter, the magnetic parameter, radiation parameter, and conduction convection coefficient for second grade viscoclastic fluid. Kishore et al. (2010) studied viscoelastic buoyancy driven MHD free convective heat and mass transfer past a vertical cone with thermal radiation and viscous dissipation effects. Afify (2004) studied the effects of radiation and chemical reaction on steady free convective flow and mass transfer of an optically dense viscous, incompressible and electrically conducting fluid past a vertical isothermal cone in the presence of magnetic field. Raptis and Perdikis (2006)  Motivated by all these studies and applications, we investigated the flow of a viscoelastic fluid with heat and mass transfer in the presence of induced magnetic field analytically by perturbation method.

MATHEMATICAL FORMULATION
Consider the flow of incompressible memory fluid in a fixed plane with heat and mass transfer, under the influence of an induced magnetic field and constant suction. The x-axis is taken along the plane in the upward direction and a straight line perpendicular to that of the y-axis. All fluid properties are assumed constant. Since the fluid is conducting, the magnetic Reynolds number is much less than unity and hence the induced magnetic field is not neglected.
The governing equations for the momentum, induced magnetic, energy and concentration is as follows; where u and v are the components of velocity in the x and y direction respectively, g is the acceleration due to gravity,  and 2  are the coefficient of volume expansion, 0 k is the kinematic viscoelasticity,  is the density,  is the viscosity, is the kinematic viscosity, KT is the thermal conductivity, Cp is the specific heat in the fluid at constant pressure, is the electrical conductivity of the fluid, e  is the magnetic permeability, D is the molecular diffusivity, T  is the temperature of the plane and T  is the temperature of the fluid far away from plane. C  is the concentration of the plane and C  is the concentration of the fluid far away from the plane.
And 0 0 v  , the negative sign indicate that the suction is towards the plane.
The boundary conditions of the problem are: Introducing the following non-dimensional quantities ; ; Substituting the dimensionless variables in (6) into (1) to (5),we get (dropping the bars) with boundary conditions 0 : 0, 1 , 1 , 1 where Gr is the thermal Grashof number, Gc is the mass Grashof number, Sc is the Schmidt number, Pr is the Prandtl number, Rm is theReynold number, K is the viscoelastic Parameter and S is the pearmeability.

METHOD OF SOLUTIONS
To solve (7) to (10) where the primes denotes differentiation with respect to y. Solving (16) to (24) subject to the boundary conditions (20) and (25).And substituting the obtained solutions into (12) to (15) Figures 1-7 represent the velocity profiles, figure 8 depicts the temperature profiles and Figure9 shows the concentration profiles with varying parameters respectively. The effect of velocity for different values of (Sc = 0.2, 0.3, 0.6, 0.8, 1, 2.01) is presented in Figure 1,the graph shows that velocity decreases with increase in Sc. The effect of velocity for different values of (  =1,2,3, 4, 5) is given in Figure 2.From the graph it is seen that, thevelocity decreases with the increase in . Figure 3 denotes the effect of velocity for different values of (Gr =1, 2, 3, 4, 5), it is clear that velocity increases with the increase in Gr. The effect of velocity for different values of (Gc =1, 2, 3, 4, 5) is shown in Figure 4, it is found that velocity increases with increase in Gc. Figure 5 depicts the effect of velocity for (Pr = 0.71, 0.94, 3, 7, 10),the graph shows that velocity decreases with the increase in Pr. The effect of velocity for different values of(K = 0.01, 0.05, 0.08, 0.1, 0.3) is presented in Figure 6, it is seen that velocity increases with the increase in K.

Velocity Profiles
The effect of velocity for different values of (  = 0.2, 0.02, 0.002, 0.0002, 0.00002) is shown in Figure 7, it is observed that velocity decreases with the decrease in  .

Temperature Profiles
In figure 8, the effect of temperature for different values of (Pr = 0.71, 0.94, 1, 3, 7, 10) is given. The graph shows that the temperature increases with increasing Pr.  Figure 9 depicts the effect of concentration for (Sc = 0.22, 0.3, 0.6, 0.8, 1, 2.01), it is seen that the concentration increases with the increase in Sc.  Table 1 depicts that the Skin frictionincreases with increase in Gc, Gr, K and  anddecreases with increase in Sc, S,  and Pr. Table 2 represents that the rate of heat transfer decreases with increase in Pr, and increases with increase in  and  . Table 3 shows that the Sherwood number increases with increase in  and  and decreases with the increase in Sc.

SUMMARY AND CONCLUSION
We have examined viscoelastic fluid flow with heat and mass transfer in the presence of induced magnetic field analytically using perturbation technique; the governing equations in dimensionless form were solved analytically using perturbation method to obtain the velocity, temperature and concentration expressions. Computations was carried outin order to investigate the effect of physical parameters namely; thermal Grashof number, mass Grashof number, Schmidt number, Prandtl number, viscoelasticity parameter and the frequency of oscillation on the flow field. It was observed that, the velocity increases with the rise in Gc, Gr,  and K,and it decreases with increase in Sc, Pr and  . The temperature becomes lower asPr is increased and the concentration falls with higher values of Sc. TheSkin friction increases with increase in Gc, Gr, K and  and decreases with increase in Sc, S,  and Pr.The rate of heat transfer decreases with increase in Pr, and increases with increase in  and  .The Sherwood number increases with increase in  and  and decreases with the increase in Sc.

RECOMMENDATION
This work examined viscoelastic fluid flow with heat and mass transfer taking into consideration the presence of induced magnetic field analytically using perturbation method. It is however recommended that other methods could be used to examine and extend the study and also the solution provided in this study for various material effects would serve as an insight for understanding more problems; viscoelastic fluid flow in a vertical channel with constant heat flux, effects of radiation and variable thermal conductivity on viscoelastic fluid flow in a micro-channel.