Mhd Flow of a Second Order/grade Fluid Due to Non- Coaxial Rotation of a Porous Disk and the Fluid at Infinity

The magnetohydrodynamic (MHD) flow of an electrically conducting second order/grade fluid past a porous disk is studied when the disk and the fluid at infinity rotate with the same angular velocity about non-coincident axes. It is found that the existence of solutions is in connection with the sign of the material modulus 1 α for both suction and blowing cases. The effects of all the parameters on the flow are carefully examined.


INTRODUCTION
The flow induced by non-coaxial rotation of a disk and a fluid at infinity has attracted the interest of many investigators.Following Coirier [1], Erdoğan [2] examined the flow produced by the rotation non-coaxially of a porous disk and a Newtonian fluid at infinity with the same angular velocity.Murthy and Ram [3] extended the flow in [2] to the magnetohydrodynamic flow and studied the effect of heat transfer.Ersoy [4] analysed the flow of an Oldroyd-B fluid for a porous disk in the presence of a uniform magnetic field.The case of the flow of a second order/grade fluid past a porous disk was studied by Ersoy and Barış [5].Hayat et al. [6] examined the flow of a second grade fluid past a porous disk under the influence of an applied magnetic field, depending on the restrictions 0 1 ≥ α and 0 2 1
In this paper, we are concerned with the flow of an incompressible and electrically conducting second order/grade fluid caused by the non-coaxial rotation of a porous disk and the fluid at infinity with the common angular velocity under the application of a uniform magnetic field.It should be pointed out clearly that our main purpose is to examine the problem depending on the sign of the material modulus 1 α .The flow is characterized by non-dimensional parameters β (the elastic parameter), e (the suction-blowing parameter) and N (the magnetic parameter).All results we have found are drawn in the figures.

BASIC EQUATIONS AND SOLUTION
In a Cartesian coordinate system, let us consider a porous disk in the xy-plane rotating counterclockwise at a constant rate of Ω about the z-axis perpendicular to the disk.A second order/grade fluid is present in the upper half-space z ≥ 0. The axis of rotation of the fluid at infinity which rotates at equal angular velocity with the disk is parallel to Oz axis and passes through the point O′ (x=0, y= l ).A uniform magnetic induction 0 B acts normal to the insulated disk, i. e. along z-direction.We assume that the induced magnetic field is negligible in comparison with the applied magnetic field.
The Cauchy stress T in an incompressible and homogeneous second order/grade fluid is given by Rivlin and Ericksen [33] where p is the pressure, µ the dynamic viscosity of the fluid, 1 α and 2 α the material moduli which are usually referred to as the normal stress coefficients.In the above representation, I is the identity tensor, and the kinematical tensors 1 A and 2 A are defined through where v is the velocity vector and Dt D / the material time derivative.We notice that if 0

≠ + α α
), which is in good agreement with experimental results, if it is not required to be compatible with thermodynamics [35].Therefore, it is clear that the results established for the case 0 1 > α have more value than the solution 0 1 < α .In this study, we consider both positive and negative values of 1 α .Moreover, another result that emerges from this analysis is that there is no effect of the material modulus 2 α on the velocity field.
The governing equations are where ρ is the density, J the current density, B the magnetic induction, m µ the magnetic permeability, E the electric field, and σ is the electrical conductivity of the fluid.
The boundary conditions for the velocity field are taken to be where u, v, w denote the x, y, z components of the velocity, respectively.We seek a solution, compatible with the continuity equation (4b), such as to have the following form: The appropriate boundary conditions for ) (z f and ) (z g from Eqs.( 5) and ( 6) are From Eqs.(1), ( 2), (4a) and ( 6), one has where a prime denotes differentiation with respect to z.Using Eq.(4f), we obtain ( ) Bearing in mind that the disk is non-conducting, when we use the current conservation equation which is a consequence of Eq.(4d) with Eqs.(4e), (8a-c) and ( 9), we have with the conditions Furthermore, all derivatives of ) (z F go to zero as ∞ → z because the fluid at infinity is free of shear stress.Thus, we find that the constant in ( 11) is equal to . Let us make the variables non-dimensional by the following substitutions: Here ν denotes the kinematic viscosity of the fluid, β the elastic parameter, e the suction-blowing parameter, and N is the magnetic parameter.As seen from ) 2 /( ν Ω = w e , the case of suction corresponds to e<0 and the case of blowing to e>0.The non-dimensional equation becomes with the conditions as follows It is noticed that Eq.( 14) is one order higher than the Navier-Stokes equations due to the viscoelasticity of the fluid.It would thus appear that the additional boundary condition must be imposed to determine the solution completely.The issue of difficulties with regard to prescribing boundary conditions is discussed in detail by Rajagopal [36].Since the flow under consideration takes place in unbounded domain, we are able to overcome this difficulty by using asymptotic conditions and boundedness of solutions, as in the study of Rajagopal and Gupta [37].They examined the existence of solutions that is tied in with the sign of material modulus 1 α for the flow of a second order/grade fluid past an infinite porous plate subjected to either suction or blowing at the plate.They found that if the material modulus 0 1 > α it is possible to exhibit an exact solution which is asymptotic in nature for both suction and blowing at the plate.However, in the case of 0 1 < α , they found that such solutions cannot exist for the blowing case.The characteristic equation of Eq.( 14) is in the form of a cubic equation and has three roots.In order to obtain physically acceptable solutions to Eq.( 14) under the conditions (15) this characteristic equation must have only one complex root with negative real part.Otherwise, the conditions Eq.( 15) will not suffice to get physically acceptable solutions.It is for this reason that there exist above mentioned solutions for

DISCUSSION
When a porous disk and a fluid at infinity rotate eccentrically with the same angular velocity Ω , there exists a single point in each plane z=constant where the velocity vector has only the axial component and about which the fluid layer rotates as a rigid body with the angular velocity Ω .The coordinates of this point are given by . Figure 2 shows these space curves for various values of the parameters β , e and N. The graphs 1 are the projections of the above mentioned space curves on the yzplane and the xz-plane, respectively.
The following conclusions can be extracted from our analysis: 1.The positive sign of the material modulus 1 α brings out physically acceptable solutions for the suction case, whereas its negative sign is meaningful for the blowing case.2. It is a well-known fact that suction and blowing have opposite characteristics on the boundary layer flows.It is clearly shown that the suction causes a thinning of the boundary layer, whereas the blowing leads to a reverse effect.3. The presence of externally applied magnetic field brings about a thin boundary layer near the disk.4. The effect of magnetic field on any flow is an important problem related to many practical applications as in the case of boundary layer flow control.Since the blowing causes an increment in the boundary layer thickness, it is shown that the boundary layer can be controlled by applying a magnetic field.5. Increasing the fluid elasticity causes thickening of the boundary layer for the suction case, whereas the reverse is true for the blowing case.6.There is no effect of the material modulus 2 α on the velocity field since both the disk and the fluid at infinity rotate with the same speed.
the model Eq.(1) reduces to the classical linearly viscous fluid model.The thermodynamical principles impose some restrictions on 1 α and 2 α[34].In particular, the Clasius-Duhem inequality implies that by above restrictions are called the second grade fluids in the literature.On the other hand, the model Eq.(1) is called a second order fluid model

Figure 1 -Figure 2 -
Figure 1-Profiles of l Ω / f and l Ω / g for various values of β , e and N.
of parameters are plotted against ζ in Figure 1.