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The combined effects of couple stress and surface roughness on the MHD squeeze-film lubrication between a sphere and a porous plane surface are analyzed, based upon the thin-film magnetohydrodynamic (MHD) theory. Using Stoke’s theory to account for the couple stresses due to the microstructure additives and the Christensen’s stochastic method developed for hydrodynamic lubrication of rough surfaces derives the stochastic MHD Reynolds-type equation. The expressions for the mean MHD squeeze-film pressure, mean load-carrying capacity, and mean squeeze-film time are obtained. The results indicate that the couple stress fluid in the film region enhances the mean MHD squeeze-film pressure, load-carrying capacity, and squeeze-film time. The effect of roughness parameter is to increase (decrease) the load-carrying capacity and lengthen the response time for azimuthal (radial) roughness patterns as compared to the smooth case. Also, the effect of porous parameter is to decrease the load-carrying capacity and increase the squeeze-film time as compared to the solid case.

With the development of modern machine equipments, the increasing use of fluids containing microstructures such as additives, suspensions, and long-chained polymers has received great attention. Newtonian fluid approximation (which neglects the size of fluid particles) is not a satisfactory engineering approach for the study fluids with microstructure additives. Since, the flow behaviours of non-Newtonian fluids cannot be described accurately by the classical Continuum theory, many microcontinuum theories have been proposed [

The use of liquid metals has recently become of interest due to thin highly conducting properties. There is a possibility of increasing the load-carrying capacity by using lubricants in the presence of an applied magnetic field. The effect of surface roughness plays an important role in engineering science and industrial applications. In bearings, surface roughness is a measure of the texture of a surface. It is quantified by the vertical deviations of a real surface from its ideal form. If these deviations are large, the surface is rough; if they are small, the surface is smooth. Rough surfaces usually wear more quickly and have higher friction coefficients than smooth surfaces. Roughness is often a good predictor of the performance of a mechanical component, since irregularities in the surface may form nucleation sites for cracks or corrosion. In bearings, the height of the roughness asperities and the mean separation of the sliding surfaces are of the same order. Thus, it appeared normal to view the film thickness in a bearing as a stochastic process characterized by a number of statistical parameters.

Traditionally, the analysis of porous squeeze-film bearings was based on the Darcy’s model, where the fluid flow in the porous matrix obeys Darcy’s law and at the bearing/film interface the no-slip condition was assumed. The term squeeze film applies to the case of two approaching surfaces which attempt to displace a viscous fluid between them. For very thin film, viscous forces offer a high resistance to such fluid motion, which, in turn, tends to inhibit the approach of the bounding surfaces. If one or both of the approaching surfaces are porous, the lubricant not only gets squeezed out from the sides but also bleeds into the pores of the porous matrix, thus reducing the time of approach of the surfaces considerably. Despite this advantage, porous bearings have proved to be useful because of their self-lubricating characteristics, low initial cost, and design simplicity.

In recent years, the magnetohydrodynamic lubrication phenomenon has many industrial applications, because of increased use of liquid metal lubricants in high temperature. A number of theoretical and experimental investigations have been made on the effects of magnetohydrodynamic lubrication [

Many workers have made investigations on the hydrodynamic lubrication of rough surfaces using stochastic approaches. Christensen [

The physical configuration of a rough porous squeeze-film bearing is shown in Figure

Squeeze-film geometry between a sphere and a porous plane surface with roughness in the presence of a transverse magnetic field.

The stochastic film thickness

Under the usual assumption of hydrodynamic lubrication theory applicable to thin-film, the continuity equation and the magnetohydrodynamic (MHD) momentum equations in polar coordinates

The flow of conducting lubricant in the porous region is governed by the modified Darcy’s law [

The relevant boundary conditions for the velocity components are

The boundary conditions given in (

The radial velocity component

The non-Newtonian MHD couple stress Reynolds-type equation for the squeeze-film pressure can be obtained by substituting for

Let

Taking the stochastic average of (

Introducing the nondimensional quantities

The nondimensional non-Newtonian MHD couple stress Reynolds-type equation is obtained in the form:

In accordance with the Christensen [

For the one-dimensional radial roughness pattern, the roughness striations are in the form of ridges and valleys in the

For the one-dimensional radial roughness pattern, the roughness striations are in the form of ridges and valleys in the

The nondimensional MHD mean load-carrying capacity is derived by integrating the nondimensional MHD mean squeeze-film pressure acting on the sphere as

Following are the limiting cases of the present study.

When

When

As

The combined effects of couple stress and surface roughness on the MHD squeeze-film characteristics of a sphere and a porous plane surface lubricated with an electrically conducting fluid in the presence of a transverse magnetic field are analyzed in this study. The Hartmann number,

Figure

Variation of nondimensional mean film pressure

The variation of

Variation of nondimensional mean film pressure

Figure

Variation of nondimensional mean film pressure

The variation of

Variation of nondimensional mean film pressure

Figure

Variation of nondimensional mean load-carrying capacity

The variation of nondimensional mean load-carrying capacity

Variation of nondimensional mean load-carrying capacity

Figure

Variation of nondimensional mean load-carrying capacity

The variation of

Variation of nondimensional mean load-carrying capacity

The response time of the squeeze film is one of the significant factors in the design of bearings. The response time is the time that will elapse for a squeeze film to be reduced to some minimum permeability height. Figure

Variation of nondimensional response time

The initial condition

The variation of

Variation of nondimensional response time

Figure

Variation of nondimensional response time

Variation of nondimensional response time

The following mathematical demonstration of MHD couple stress squeeze-film characteristics between a sphere and a rough porous plane surface is considered, for the illustration of engineering design application. By using the values of various dimensional parameters, one can obtain a design example for the engineering design application as follows:

The effect of surface roughness on the MHD squeeze-film characteristics between a sphere and a porous plane surface is presented on the basis of Christensen stochastic theory for rough surfaces. On the basis of results discussed above, the following conclusions can be drawn.

The presence of an externally applied transverse magnetic field provides an enhancement in the load-carrying capacity and response time as compared to the non-conducting case for both types of roughness patterns.

The couple stress effects are more pronounced for azimuthal roughness pattern as compared to the radial roughness pattern. Due to the presence of additives in the fluid, the significant increase in load-carrying capacity and squeeze-film time is observed than the Newtonian case for both types of roughness patterns.

The roughness of the surface causes a reasonable effect on the characteristics of the bearing. As the surface asperity increases, the large amount of load is delivered in the bearing and lengthens the response time of squeeze-film motion as compared to the smooth case. The effect of surface roughness is more accentuated in the case of azimuthal roughness pattern than that of radial roughness pattern.

The permeability of the porous layer diminishes the squeeze-film characteristics as compared to the nonporous case. The raise in the height of the porous layer motivates the fall in the squeeze-film pressure and the load-carrying capacity and thus shortens the response time and also the possibility of sphere-to-plane contact.

Applied magnetic field

Roughness parameter

Nondimensional roughness parameter (

Film thickness

Minimum film thickness

Initial minimum film thickness

Nondimensional film thickness

Nondimensional minimum film thickness

Stochastic film thickness

Thickness of porous facing

Permeability of porous facing

Porosity

Magnetic Hartmann number

Nondimensional film pressure

Radius of the sphere

Radial and axial coordinates

Velocity components of the fluid in the film region

Velocity components of the fluid in the porous region

Load-carrying capacity

Non-dimensional load-carrying capacity

Time

Nondimensional response time.

Nondimensional central film thickness

Fluid conductivity

Standard deviation

Fluid viscosity

Porous parameter

Random variable.

The authors are thankful to Dr. N. B. Naduvinamani, Professor, Department of Mathematics, Gulbarga University, Gulbarga for his valuable discussions.