OSCILLATORY MOTION OF AN ELECTRICALLY CONDUCTING VISCOELASTIC FLUID OVER A STRETCHING SHEET IN A SATURATED POROUS MEDIUM WITH SUCTION / BLOWING

The effect of an oscillatory motion of a viscoelastic fluid over an infinite stretching sheet through porous media in the presence of magnetic field with applied suction has been studied. The surface absorbs the fluid in a porous medium in the presence of magnetic field and the velocity oscillates depending on the stretching rate (b). Analytical expressions for the velocity and the coefficient of skin friction have been studied, first by the perturbation method and then by power series method. The effect of viscoelastic parameter k1, porous parameter k2, magnetic parameter Mn, and the vertical distance x in the presence of suction/blowing on the velocity and the flow characteristics are discussed. The velocity of the viscoelastic fluid is found to decrease in the presence of magnetic field and porous media, as compared to the study of viscous fluid. It is also found that the effect of unsteadiness in the wall velocity and skin friction are found to be appreciable in the presence of suction/blowing parameter.


Introduction
Boundary layer flow on continuous moving accelerating surfaces is an important type of flow occurring in a number of technical processes which have applications in fuel industries, in an aerodynamic extrusion of plastic sheets and boundary layer along liquid film in condensation processes.Drag, heat, and mass transfer are governed by the structure of the layer.Flows due to a continuously moving surface involve continuous pulling of a sheet through a reaction zone as in metallurgy, in textile and in paper industries, and in the manufacture of polymer sheets, sheet glass, and crystalline materials.
In a series of three articles entitled "Boundary layer behaviour on continuous solid surfaces" Sakiadis [20] pointed out the differences in boundary conditions between a moving flat plate of finite length and a continuous surface.The governing equations for both two-dimensional and axisymmetric flows were determined.In a subsequent paper he gave exact and approximate solutions for the flat sheet and the cylinder.This was followed by the work of Crane [2], P. S. Gupta and A. S. Gupta [7] and Rajagopal et al. [18].
A theoretical and experimental treatment for the moving flat plate was made by Tsou et al. [27].They determined heat transfer rates for certain values of the Prandtl numbers.
Siddappa and Abel [22] have extended Crane's flow problem to the viscoelastic fluid of Walters' liquid B model, and obtained the solutions of equations of motion for boundary layer flow past a stretching sheet.
One of the most popular models for non-Newtonian fluids is the model that is called the second-order fluid or fluid of second grade.It is reasonable to use the second-order fluid model to do numerical calculations.The constitutive assumption for the fluids of second grade is in the following form: where T is the Cauchy stress, −pI the spherical stress due to the constraint of incompressibility, μ the coefficient of viscosity, α 1 and α 2 material moduli and A 1 and A 2 are the first two Rivlin-Ericksen tensors which are discussed in detail by Fosdick and Rajagopal [6].
Further, a comprehensive discussion on the restrictions for μ, α 1 , and α 2 can be found in the work by Dunn and Rajagopal [4].The sign of the material moduli α 1 and α 2 is the subject of much controversy which was discussed by Rajagopal [14].In the experiments on several non-Newtonian fluids, the experimentalists have not confirmed these restrictions on α 1 and α 2 .Thus, the conclusion is that the fluids that have been tested are not fluids of second grade and they are characterised by a different constitutive structure.The equation of motion of incompressible second-grade fluid, in general, is of higher order than of the Navier-Stokes equations.The Navier-Stokes equation is a second-order partial differential equation, but the equation of motion of a second-order fluid is a third-order partial differential equation.But in the present article we considered momentum equation for a Viscoelastic incompressible fluid of Walters' liquid B model which is of fourth-order highly nonlinear differential equation having four boundary conditions which is solved analytically first by perturbation method and then by power series solution.A marked difference between the case of the Navier-Stoke's theory and that for fluids of second grade is that ignoring the nonlinearity in the Navier-Stoke's equation does not lower the order of the equation, however ignoring the higher-order nonlinearities in the case of the second-grade fluid reduces the order of the equation.The no-slip boundary condition is sufficient for a Newtonian fluid but for a second-order fluid it may not be sufficient and therefore one needs an additional condition at boundary.Therefore a critical review on the boundary conditions and the existence and uniqueness of the solution has been given by Rajagopal [15].In order to clarify these points, Rajagopal and Gupta [16] have presented a paper on the flow of a second-order fluid past an infinite porous plate with velocity component along the x-axis tending to U as y approaches infinity.The augumentation of the boundary conditions has also been discussed by Rajagopal and Gupta [16].
K. Rajagopal et al. 3 Subhas and Veena [25] have studied about the viscoelastic fluid flow and heat transfer characteristics in a saturated porous medium over a stretching surface with frictional heating and internal heat generation or absorption for both PST and PHF cases considering steady-state boundary layer equation.
A similar situation is obtained when the porous plate is bounded by another porous plate.A similar work has been provided by Rajagopal and Kaloni [17].
Unsteady flows of a second-order fluid in a bounded region have been studied by Ting [26].Some unsteady uni-directional flows of second order fluids have been considered by Rajagopal [13].These works showed that the no-slip condition at the boundary for this type of flow suffices.A recent work of Rajagopal et al. [1] showed that the solutions for unsteady flows of a second-order fluid occupying the space above a plate are bounded if the coefficient of higher-order derivative is positive.However, this is not necessary for steady flows of a second-order fluid.
Erdogan [5] has studied the unsteady motions of a second-order fluid over a plane wall and he has shown that a Newtonian fluid induced by a flat plate that applies a constant stress to the fluid flows faster than a second-order fluid.
Besides these excellent reviews of the literature dealing with nonsteady flows presented by Lighthill [10] and Stuart [24], the behavior of viscoelastic fluids in laminar flow through porous media has been the subject by numerous investigators, including Pilitisis and Beirs [12], H. Pascal and F. Pascal [11], Jones and Walters' [8], Rudraiah et al. [19], and Vafai and Kim [28].
Siddappa et al. [23] have investigated the oscillatory motion of a viscoelastic fluid past a stretching sheet and have shown that the unsteadiness in the wall velocity and skin friction are found to be appreciable.Devi and Nath [3] have discussed the similar solutions of the unsteady boundary layer equations for a moving wall and they predicted their results as the Prandtl number strongly affects the heat transfer, but the skin friction is unaffected by it.
Nonlinear streaming due to the oscillatory stretching of a sheet in a viscous fluid was discussed by Wang [29].He considered an elastic sheet which was stretched back and forth in a viscous fluid.His problem was governed by a nondimensional parameter S which represents the relative magnitude of frequency to stretching rate.
As unsteady flows of viscoelastic fluids through porous media in the presence of magnetic field are of great interest and have several applications such as in electromagnetic propulsion and in the flow of nuclear fuel slurries, flow of fluid metals, and alloys, Sarpakaya [21] was the first who has studied the MHD flows in non-Newtonian fluids.He studied about the MHD flow in Bingham plastic and Ostwald fluids and presented his results with the conclusion that as the intensity of the magnetic field increases, the distribution of velocity is increasingly more uniform.
The nonuniqueness of MHD flow of a second-order fluid past a stretching sheet was presented by Lawrence and Rao [9].
Motivated by all the above analyses, in the present paper we therefore investigate the oscillatory motion of a viscoelastic Walters' liquid B model over a stretching sheet through porous media in the presence of magnetic field considering the nonsteady boundary layer equations with suction/injection.The effects of velocity profiles u on different parameters such as viscoelastic parameter k 1 , permeability permeter k 2 , and magnetic parameter Mn are discussed.

Formulation of the problem and mathematical solution
We consider the unsteady flow of an incompressible electrically conducting Viscoelastic fluid (Walters' liquid B ) through a porous medium over a stretching sheet with suction/injection that issues from a thin slit in which the flow approaches the sheet with zero angle coincidence.Two equal and opposite forces are introduced along the sheet so that the wall is stretched, keeping the origin fixed.The x-axis is taken along the sheet, the y-axis perpendicular to it, and the origin is the slit.The speed of a point on the sheet is assumed to be proportional to its distance from the slit.
The constitutive equations for the incompressible Viscoelastic (Walters' liquid B ) fluid are where where N(τ) is the distribution function of relaxation times, P ik the stress tensor, P is an isotropic pressure, g ik (x) is the metric tensor of fixed co-ordinate system x , and P ik is the rate of strain tensor.In the case of fluids with short memories, that is, short relaxation times, the above equation of state can be written in the following simplified form: The governing boundary layer equation for Walters' liquid B through porous media in the presence of magnetic field with suction or injection is of the form and the equation of continuity is where ν is the coefficient of viscosity, k the coefficient of porosity, σ electrical conductivity, B 0 the magnetic field strength, and ρ the density of the fluid.
The corresponding boundary conditions are Here b denotes the maximum rate of stretch with dimension (time) −1 .
We assume S = ω/b ≡ 1/ε 1, where ε is very small.S implies the small amplitude of oscillations.The boundary conditions suggest the following transformation: ( Substitution of (2.7) in (2.4) leads to where (2.9) The corresponding boundary conditions are (2.10) 6 Oscillatory motion of an electrically conducting viscoelastic fluid Here v w is the suction velocity across the stretching sheet when v w < 0 and it is blowing velocity when v w > 0.
(2.11) Substituting (2.11) in (2.8) and (2.10) and equating harmonic and nonharmonic terms to zero, we obtain fourth-order nonlinear ordinary differential equation for g 1 (η) and fourth-order nonlinear differential equation for g 2 (η) with variable coefficients (2.12) where (2.14) The corresponding boundary conditions are suffix denotes differentiation with respect to η. Making use of (2.15), we derive the exact analytical solution of (2.12) in the form (2.16) α is the positive root of the cubic equation (2.17) Further, we apply series method to obtain the solution for (2.13).Hence the solution of (2.13) subjected to the boundary conditions (2.15) is obtained as where (2.19)

Skin friction
Shearing stress at a point on the plate for the Viscoelastic fluid is Substituting similarity transforms in the above equation, the form of the equation will be (3.2) x/ √ ν and the calculated skin friction is

Results and discussion
In Figure 4.1(a) a graph of g 1 (η) versus η is drawn for various values of the elastic parameter k 1 = 0.4, 0.6, 0.8 in both cases of suction and blowing.It is observed from the figure that g 1 decreases with increasing values of k 1 .
In Figure 4.1(b) a graph of g 1 (η) versus η is drawn for various values of the permeability parameter k 2 = 1.0,100,1000 and it is seen from the figure that velocity increases as the permeability parameter k 2 increases owing to the inhibitive influence of the permeability parameter k 2 in both the cases of suction and blowing.
In of Mn in the presence of porous medium, that is, velocity increases with small values of elastic parameter k 1 , which might be regarded as a manifestation of the presence of normal stresses inside the boundary layer.Speaking in physical terms, the thickening of the boundary layer may be attributed to the tensile stress in the layer which causes an axial contraction.These results are well in agreement with the results of Rajagopal et al. [18].In Figure 4.2(a) velocity profiles for g 2 (η) versus η are drawn for various values of viscoelastic parameter k 1 and it is noticed from the figure that g 2 decreases with increasing values of k 1 in the presence of suction or blowing.
In Figure 4.2(b) velocity profiles for g 2 (η) versus η are drawn for different values of permeability parameter k 2 and it is noticed from the figure that g 2 increases as k 2 increases in the presence of both suction and blowing.
In Figure 4.2(c) a graph of g 2 (η) versus η is drawn for various values of magnetic parameter Mn in the presence of suction or blowing and it is seen from the figure that velocity increases as Mn increases.
In Figure 4.3(a), that is, from the graph of g 1 (η) versus η, it is found that the effects of impermeability of the boundary wall velocity increase as Mn increases in the case of suction and blowing on the horizontal velocity profiles in the boundary layer with permeability k 2 = 100 and the viscoelastic effect of k 1 = 0.2 being drawn in the absence of magnetic field and it is observed from the figure that velocity decreases with increase of distance.The effect of suction is to decrease the velocity and that of blowing is to increase the velocity.These results are consistent with the physical situation.
Figure 4.3(b) is plotted for the same set of parameters except in the absence of porous medium and with the magnetic effect of Mn = 100 and it is observed that g 1 decreases with increase in the values of magnetic parameter Mn.Comparison of these two graphs, 4.3(a) and 4.3(b), reveals the fact that the effect of porosity and magnetic parameter is to decrease the velocity for all cases of suction, blowing, and impermeability of the wall.This is because of porous medium's and magnetic field's obstruction to the flow over the oscillatory motion of a stretching sheet.
In Figure 4.4 velocity profiles for g 2 (η) versus η for a set of parameters like k 1 = 0.4, Mn = 10.0, b = 0.4, ν = 0.04 in the absence of porous medium are drawn and it is noticed from the figure that velocity decreases within the boundary layer of an oscillatory stretching sheet with the increase of distance from the boundary.The effect of suction is to decrease the velocity and injection increases the velocity.we also notice that velocity decreases as the distance increases from the boundary sheet and also observe that the oscillatory motion of the stretching sheet produces oscillating flow in the fluid.
We reveal the fact that for increasing values of stretching rate parameter (b) the oscillation also increases.In other words, that the mode of oscillation depends on stretching rate.
It is observed that the magnitude of skin friction decreases initially and later increases as the viscoelastic parameter k 1 increases.Similarly, for various values of magnetic parameter Mn and for fixed value of k 2 = 1.0, we observed that the effect of magnetic parameter decreases the magnitude of skin friction up to certain level and suddenly increases to the top level.Table 4.1 represents the values of shear stress for different values of viscoelastic parameter and it is observed that the magnitude of skin friction coefficient decreases with increasing values of viscoelastic parameter k 1 .For industrial applications, this result is of some importance, since the power expenditure in stretching the sheet decreases with increasing values of k 1 .The same idea has already been investigated by Rajagopal et al. [18] in the case of steady flows.