The orthogonal rheometer consisting of two parallel disks rotating with the same angular velocity about distinct axes was developed by Maxwell and Chartoff  . An exact solution corresponding to the flow of a Newtonian fluid in this instrument was obtained by Abbott and Walters  . Rajagopal and Gupta  were the first to obtain an exact solution for a second-grade fluid and also studied the stability of the flow. Rajagopal  also studied the problem for a second order fluid whose normal stress moduli do not obey the relations and . Rajagopal  showed that this motion is one with constant principal relative stretch history.
The steady flow between eccentric rotating disks under the effect of a magnetic field has been studied by many researchers. Mohanty  was the first to obtain an exact solution to the MHD flow for a Newtonian fluid. Rao and Rao  investigated the MHD flow for a second-grade fluid. Ersoy  studied the MHD flow in the case of an Oldroyd-B fluid. Guria et al.  examined the effect of the heat transfer for a Newtonian fluid in the presence of a magnetic field for porous disks. Siddiqui et al.  investigated the MHD flow of a Burger’s fluid. Ersoy  studied the flow of a Maxwell fluid under the effect of a magnetic field when the disks are porous.
The aim of this paper is to study the steady flow of an incompressible second-grade fluid between two insulated and porous disks rotating with the same angular velocity about two parallel axes under the effect of a magnetic field. It is a well-known fact that the governing equations of non-Newtonian fluids are in general of higher order than the Navier-Stokes equations. Even so, it is possible to find an exact solution for a second-grade fluid in this geometry when the disks are non-porous  . However, the order of the governing equation for porous disks exceeds the number of boundary conditions. The perturbation solution is obtained by taking the second-grade fluid parameter β as the perturbation parameter. The velocity field and the horizontal components of the dimensionless force per unit area exerted by the fluid on the top and bottom disks are constructed for small values of the second-grade fluid parameter.
2. Description of the Problem
Let us consider an incompressible second-grade fluid between two porous and insulated disks located at . The top and bottom disks rotate about the - and -axes with the same angular velocity , respectively. The distance between the axes of rotation is defined by 2ℓ in the y-direction. A uniform magnetic induction is applied in the positive z-direction. It is assumed that the magnetic Reynolds number for the flow is very small so that the induced magnetic field is negligible in comparison with the applied magnetic field. A uniform suction and injection are applied perpendicular to the top and bottom disks, respectively. The schematic configuration of the problem is illustrated in Figure 1. Therefore, the appropriate boundary conditions for the velocity field are
, , at , (1)
, , at , (2)
, , at , (3)
where u, v, w denote the velocity components along the x, y, z-directions, respectively. We look for a solution of the velocity field of the form
, , . (4)
Figure 1. Flow geometry.
3. Solution of the Problem
The equations governing the flow are
where ρ is the density, the velocity vector, the current density, the magnetic induction and the overdot denotes material time differentiation. The velocity field in Equation (4) is compatible with the continuity Equation (6).
The stress in a homogeneous second-grade fluid is given by 
where p is the pressure, the identity tensor, μ the dynamic viscosity coefficient, and the normal stress moduli. and stand for the first two Rivlin-Ericksen tensors defined through
The coefficients μ, , and must satisfy
, , . (10)
We refer the reader to    for more information about second-grade fluids.
Inserting Equation (4) into Equations (5), (7)-(9), we obtain
where a prime denotes differentiation with respect to z. Using , we have
, , , (14)
where σ is the electrical conductivity of the fluid, the electric
field and the magnitude of the applied magnetic induction . Since the disks are insulated, we obtain and . Using , we have
where and are constants. Combining Equations (15)-(16) and using , we get
where C is a constant. The boundary conditions for are as follows:
, , . (18)
Introducing the dimensionless quantities,
, , , ,
, , (19)
the governing Equation (17) and the conditions (18) reduce to the following dimensionless forms:
, , , (21)
where a prime denotes differentiation with respect to ζ, and is a constant.
We look for a perturbation solution for small values of the second-grade fluid parameter β as follows:
where , and are constants. Substituting Equations (22)-(23) into Equations (20)-(21) and equating the coefficients of different powers of β, we obtain
, , , (27)
, , , (28)
, , . (29)
The solution of Equation (24) with Equation (27) is
The solution of Equation (25) with Equation (28) is
, . (33)
Finally, the solution of Equation (26) subject to Equation (29) is
, . (35)
The shear stress components and in the fluid are presented in the dimensionless complex form as follows:
, . (37)
For the small values of the second-grade fluid parameter, we obtain
4. Results and Discussions
The present flow is defined by the superposition of a helical motion and a rigid body translation that is different from plane to plane in each z = constant plane. The axial component of velocity chosen in the positive 𝑧-direction is constant as a result of the continuity equation. Thus, the top and bottom disks are subjected to uniform suction and injection, respectively. The velocity field is presented by obtaining the functions and that represent the dimensionless x- and y-components of the translational velocity. The variations of and with the Hartmann number M, the second-grade fluid parameter β, the suction/injection velocity parameter P, the Reynolds number R are drawn in Figures 2-5.
Figure 2. Variations of and for ( ).
Figure 3. Variations of and for ( ).
Figure 4. Variations of and for ( ).
Figure 5. Variations of and for ( ).
The determination of the horizontal force acting on the disks is also important.
and represent the x- and y-components of the dimensionless
force per unit area exerted by the top and bottom disks on the fluid, respectively. When the disks are non-porous, it is clear that the x- and y-components of the
force on the top disk are equal to those on the bottom disk (i.e., and ). In view of the fact that the axial velocity of the fluid is chosen in the pozitive 𝑧-direction, the components of the horizontal force on the top disk are larger than those on the bottom disk (i.e., and ). The effects of all the parameters on and are illustrated in Figures 6-9.
Figure 6. Variations of and with M ( ).
Figure 7. Variations of and with β ( ).
Figure 8. Variations of and with P ( ).
Figure 9. Variations of and with R ( ).
The conclusions that can be drawn from the performed analysis are pointed out as follows:
・ When the Hartmann number M increases, the curves about which the fluid layers rotate start to get closer to the z-axis. These curves are closer to the plane than the plane . The dimensionless force in the x-direction increases but that in the y-direction decreases.
・ When the second-grade fluid parameter β increases, the curves mentioned tend to move away from the z-axis. The effect of β on the translational velocity in the y-direction is greater than that in the x-direction. The x-component of the dimensionless force increases but the y-component decreases. It may be noticed that the change is almost linear.
・ The functions and representing the dimensionless x- and y-components of the translational velocity vector are anti-symmetric for non-porous disks; however, they are not anti-symmetric for porous disks. When the suction/injection velocity parameter P increases, the space curves get closer to the plane in the region between the top disk and the plane , but move away from the plane . An adverse effect is observed in the region between the bottom disk and the plane . Both the x- and y-components of the force on the top disk increase but those on the bottom disk decrease. It is observed that the change is almost linear.
・ When the Reynolds number R increases, the space curves in the core region become closer to the z-axis, and thus the boundary layers developing on the disks lead to an increase in the horizontal force on the top and bottom disks.