SQUEEZE FILM LUBRICATION OF ASYMMETRIC ROLLERS BY BINGHAM PLASTIC FLUID

An attempt has been made to investigate hydrodynamic lubrication characteristics of asymmetric roller bearings lubricated by thin fluid film under the operating behavior of line contact for a heavily loaded rigid system for normal squeezing motion with cavitation points. The lubricant follows non-Newtonian incompressible Bingham plastic fluid model where the fluid viscosity is supposed to vary with hydrodynamic pressure . The equations which govern the fluid flow such as continuity and momentum equation are solved first analytically and later numerically using MATLAB. The numerical results are achieved for the velocity, pressure, load, and traction forces by varying different physical parameters like rolling ratio, squeezing parameter, and yield stress parameter. The obtained results are investigated numerically and graphically. Finally, it is concluded that a notable change is observed in velocity, pressure, load and traction with Newtonian and non-Newtonian fluids. Results follow good agreement with available literature.


INTRODUCTION
T he squeeze film lubrication process is of great importance in many engineering fields and is normally found in the course of motors, plane motors, machine devices, super gadget, and skeletal joints. Expectedly, the forecast of squeeze film movement recognizes that the salve goes probably as a viscous Newtonian fluid. Precisely, the forecast of squeeze film motion assumes that the lubricant acts as a Newtonian viscous fluid. However, experimental results show that the inclusion of small quantity of long-chained additives to a Newtonian fluid reduces the sensitivity of the lubricant to change in shear rate and dispenses beneficial effects on the load-carrying and frictional characteristics .
Hydrodynamic lubrication is a strategy which is utilized to diminish friction and wear of scouring surfaces with fluid. T he aim of hydrodynamic lubrication is typically to add a proper liquid, with the goal that it enters into the contact zone between scouring surface and makes a fluid thin film. T his film separates the surfaces from contacting one another and for the most part of it diminishes friction and wear (Revathi et al., 2020).
Bearing are consistently reliant upon unbelievably profound loads, top speeds and heavy slip conditions. T he effect of the high load on the treatment in the concentrated contact causes high pressure age in the fluid film. In the high pressure territory, material property of the oil, particularly viscosity, isn't, now consistent yet changes continually with pressure and temperature . Prasad et al. (1987) introduced a new approach which manages the issue of a couple of inelastic intensely highly fitted rollers with rolling and squeezing motions, greased up by power-law liquid accepting the fluid consistency to change dramatically with temperature and pressure.
At the same time, it was processed mathematically by settling this issue with adjusted Reynolds and energy conditions. Patrice Estellé & Christophe Lanos (2007) derived an analytical solution of a Bingham plastic squeeze film flow with friction at the plates under slip in a particular zone. T he result was determined accepting the sample gap to be little when contrasted with the plate radius and from the rate of flow energy scattering.  studied a journal bearing problem with non-Newtonian lubrication considering thermal and squeezing effects. An appreciable change in pressure and temperature was observed. Also, a non-Newtonian fluid was considered by highlighting the importance of the work due to non-dimensional scheme for pressure, temperature and consistency. Prasad et al. (2014) investigated another hydrodynamic problem with rigid system of roller bearings, greased up by a flimsy compressible liquid for normal squeezing movement that focused on expecting the consistency and density of the fluid to differ corresponding to the temperature and the pressure.
In modern engineering, many fluids show non -Newtonian behavior, therefore many researchers are more int erested in those industrial non-Newtonian fluids and their dynamics (Banerjee et al., 2018). T he exhibited qualities of non-Newtonian Bingham plastic fluid flow are used to depict the way of fluids, explicitly, fluids for quite a while; and besides they are used to portray the progression of melts and slurries in molds (Christopher and T ichy, 1992). Speculative examinations of grease with Bingham model return to Milne (1954). He investigated the basic 1-D journal and slider bearings, and infers inflexible 'cores' potentially added to either surfaces. Besides, Christopher Dorier and John T ichy presented a model of the way of Bingham-like fluids which show a yield stress (Christopher and T ichy, 1992). Recently,  studied non-Newtonian

Frontiers in Heat and Mass Transfer
Available at www.ThermalFluidsCentral.org Frontiers in Heat andMass Transfer (FHMT), 16, 7 (2021) DOI: 10.5098/hmt.16.7 Global Digital Central ISSN: 2151-8629 lubrication of asymmetric rollers for highly stacked rigid system with incompressible Bingham plastic fluid in rolling/sliding line contact , considering the fluid viscosity to vary with the hydrodynamic pressure. T he fluid velocity trends were obtained and the results, particularly, pressure, load and traction forces were found to agree with available results. Further, Revathi et al. (2020) investigated a similar problem on lubrication characteristics of anti-symmetric rollers using incompressible Bingham plastic fluid. Results particularly temperatures, pressure, including traction and load were in good agreement with available results.
In view of above discussion, the purpose of this paper is to study the squeezing effects in hydrodynamic lubrication of asymmetric rollers by an incompressible Bingham plastic fluid for heavily lo aded rigid rolling/sliding line contact problem assuming the viscosity of the lubricant to be varied with hydrodynamic pressure; the rolling ratios are used to study the rolling/sliding effects of surfaces on pressure, load and traction.

THEORETICAL MODEL
T he system in this paper is considered in such a way that both the surfaces are having same radius but moving with different velocities. Further it can be noted that the upper surface is moving faster than lower surface. T he complete flow configuration is shown in Fig. (1).

Mathematical formulation
T he equations such as continuity and momentum which govern the incompressible fluid flow are considered under usual assumptions (Prasad and Singh, 1987) as follows: where "p" and "  " are hydrodynamic pressure and shear stress of the fluid respectively. T he constitutive equation for Bingham plastic fluid is given by Sasaki et al. (1962) where  is the viscosity of the fluid taken by R, being the radius of the equivalent cylinder.

Boundary conditions
T he boundary conditions for this problem at both upper and lowers surfaces are taken as where 2 1 U and U are velocities of the rolling cylinders as shown in Fig.(1). Integration of the equation (2) using the above boundary conditions gives the fluid velocity expression as below Now the volume flux "Q" for the fluid flow can be obtained by integrating the velocity within the gap between the surfaces is presented below And the volume flux at the point of maximum pressure is Where the film thickness

Reynolds equation
Solving the equation (2) using the boundary conditions gives the pressure Reynolds equation which is presented below

Dimensionless scheme
T he following dimensionless scheme for roller bearings is applied throughout this paper T he velocity expression and pressure Reynolds equation are written in dimensionless form using the above mentioned dimensionless scheme. (Subrahmanyam and Prasad, 2015) Since load capacity produces complete estimation of the bearings efficiency, so load capacity is the one of the significant qualities. Hence, Integration of the pressure across the film thickness gives the xcomponent of the load W x per unit length of the cylinder as

Load and Traction
The dimensionless normal load y W is given by

Velocity profile
Velocity u of the fluid is calculated and presented in Figs.(2-4) respectively for the regions before, after, and at the point of maximum pressure. T he profiles in first two graphs look like parabolas having vertices upward and downward directions in the regions before and after the point of maximum pressure.    Fig.(2). Reverse flow was also shown by . As the fluid progresses, the back flow is over ; Revathi et al, (2020);Prasada et al. (1991). But the velocity profile, at the point of maximum pressure, seems to be linearly increasing  and can be observed in Fig. (4).

Viscosity (  ) Profile
T he lubricant viscosity  is computed numerically for different values of Ū and V and depicted in Fig. (9) and Fig. (10). T he lubricant viscosity for different values of squeezing parameter for rolling/sliding case is presented in Fig. (9) and can be observed that the viscosity has impact in the minimum gap between the rollers. Further, Fig. (10) shows that viscosity profile for different values of rolling ratio when . From this figure one can observe that viscosity increases as rolling ratio increases. T he same trend can be observed in Subrahmanyam and Prasad (2015).

Load and Traction
T he dimensionless load values in both x and y-direction are numerically computed for different values of rolling ratio and squeezing parameters and presented in T able-1. T he table shows that load increases in both directions with rolling ratio Ū for fixed value of V . T hese results are in good agreement with earlier published results. Prasad et al., (1988); Frontiers in Heat and Mass Transfer (FHMT), 16, 7 (2021) DOI: 10.5098/hmt.16.7 Global Digital Central ISSN: 2151-8629 Prasad et al., (1991; . Further, the load decreases in both directions as squeezing parameter increases for fixed value of Ū.  Prasad et al., (1988); Prasad et al., (1991); . Both the surfaces have the same traction force when Ū=1.

CONCLUSION
An attempt is made to study the squeeze film lubrication characteristics of line contact problem by an incompressible non -Newtonian Bingham plastic fluid. T he continuity and momentum equations both are solved for pressure and the fluid velocity for different values of 0  and the sliding parameter Ū. T he following points may be obtained from the results of this work:  Both the velocity and pressure of the lubricant are independent of 0   Velocity of the lubricant at point of max imum pressure increases linearly.  T he increase in rolling ratio Ū causes the rise in pressure with and without squeezing.  Pressure increases with decreasing value of squeezing parameter for pure rolling case.  T he increase in rolling ratio causes increase of load in both x and y-directions.  T he traction at the upper surface is higher than that of the lower surface due to more speed of upper surface.  T he increase in 0  causes the increase in traction forces at both the surfaces.