Combined effect of viscosity variation on rough porous rectangular plates with magneto hydrodynamic effect

The squeezing action between rough porous parallel rectangular plates under the effect of magneto hydrodynamic force and viscosity variation effect is considered in this paper. The modified Reynolds equation is derived . The load carrying capacity of the plate thus derived is found to increase with increase in the magnetic, roughness and viscosity variation parameter.


Introduction
Squeezing technology is applied in many engineering and medical fields. Squeezing produces a positive pressure which helps in the load carrying capacity of the system. The Stokes [1] is the simplest generalization of the classical theory of fluids which allows for polar effects like presence of couple stresses, body couples and non-symmetric tensors. Ramaniah , Gupta [2,3] and others analyzed the squeeze film action between finite plates of various geometries lubricated with couple stress fluid. Porous material allows the lubricant to seep in the pores and provides an cushioning effect to the lubricant. The effect of porosity and the boundary condition of a porous material was studied by Beavers , Joseph and others [4,5] who showed that the porosity helps in the load carrying capacity. Hartmann was the pioneer who studied the flow of an incompressible fluid between parallel plates with magnetic field acting normal to them. He investigated it both theoretically and experimentally. Hamza E.A [6] results show that the electromagnetic forces increases the load carrying capacity considerably. SundarammalKesavan et al [7] studied the effect of MHD on a porous rectangular plate and found that the MHD improves the performance. The lubricant which we choose must be suitable to function under various ranges of temperature. To enhance the performance of the fluid additives are added which makes the viscosity of the fluid to vary from layer to layer. Sinha et al., [8] studied the effects of viscosity variation due to lubricant additives in journal bearing. Raghavendra Rao and Prasad studied the effects of velocity slip and viscosity variation in rolling and normal motion, roller bearings and journal bearings [9,10,11] using multi layer technique. Continuous working of the machinery leads to the wear of the surface . Also the additives reaction and the contamination of the lubricants contributes to the degradation of the surface resulting in roughness. Christensen and Tonder [12,13,14] extensively studied the effect of surface roughness and modeled it mathematically. N.B Naduvinamani, S.T Fathima [15,16,17] studied the effect of the surface roughness on porous parallel plates . The effect of the velocity and friction of porous elliptic plates lubricated with couple stress fluid considering the effects of slip velocity and effect of viscosity variation in porous parallel rectangular plates lubricated with couple stress fluids was carried out by Sujatha and Sundarammal  [18,19]. In this paper we try to investigate the effect of surface roughness, magnetic effect and viscosity variation effect on a porous parallel rectangular plate . B is applied in the z direction. It is assumed that the inertial force and the induced magnetic force are too small compared to the applied magnetic field.

Geometry
Geometry of the plates

Mathematical formulation
The MHD momentum and continuity equation based upon the above assumptions are given by Here u and v are the velocity components in the x and y directions respectively,  is the viscosity of the lubricant , which in now replaced by the factor 1 between 0 to 1 depending upon the nature of the lubricant. It takes the value 0 for perfect Newtonian fluids and 1 for perfect gases. We also assume that a thermal equilibrium exists. p represents the fluid pressure and *  the electrical conductivity.
The boundary conditions for the velocity components are The flow in the porous medium is governed by the Darcy's law given by , k , the permeability of the porous region. Solving equations (1) and (2) using (3) and boundary conditions (5) and (6)   Substituting the velocity components obtained by equations (7) and (8) in the continuity equation (4) and integrating across the fluid film thickness using (5) , (6) , (7) Applying the surface roughness effect developed by Christensen and Tonder , who gave a stochastic approach to model the roughness governed by the probability density function 4 1234567890 ''"" The mean value  , the standard deviation  and the parameter  , used to measure the symmetry of Introducing the non dimensionless parameters With boundary conditions The mean pressure is thus obtained as         Figure 2 gives the changes in the load carrying capacity for variation in the viscosity variation parameter Q. It is observed that the value of the load carrying capacity increase with an increase in the viscosity variation parameter Q. Figure 3 gives the changes in the load carrying capacity with variation in the mean value  . We observe that the load carrying capacity increases with an increase in  . Figure 4 gives the changes in the load carrying capacity with variation in the standard deviation  . We notice that the load carrying capacity increases with an increase in  . Figure 5 gives the changes in the load carrying capacity with changes in the  . With increase in  we notice an increase in the load carrying capacity.

Conclusion
The investigation shows that the magnetic effect, the viscosity variation parameter and the roughness effect parameter increases that the load carrying capacity of the model under consideration. Thus it enhances the life of the bearing with such a configuration.