Analysis of Squeeze Films Between Rough Circular Plates Lubricated with Rabinowitsch Fluids

The present theoretical work investigates the combined impacts of non-Newtonian (pseudoplastic and dilatant) lubricants and surface roughness on the performance of squeeze films lubrication between two rough circular plates. The modified Reynolds equation has been derived on the basis of Christensen’s stochastic theory of hydrodynamic lubrication for rough surfaces. The lubricant model adopted for the analysis is Rabinowitsch fluid model – an experimentally verified fluid model for lubricated bearing systems. Two types of one-dimensional roughness patterns (radial and azimuthal) have been considered in the analysis. An asymptotic solution for squeeze film pressure, load carrying capacity and squeeze film time are obtained. The numerical results for dimensionless film pressure, load carrying capacity and film squeezing time have been calculated for various values of fluid and operating parameters. The results for dimensionless film pressure, load capacity and squeezing time of the lubricant film have been discussed with clear graphical presentation for different values of parameters of pseudoplasticity and roughness. It was observed that the radial roughness decreases the film pressure, load capacity and squeezing time of lubricant, while increased values of these properties were observed for azimuthal roughness. It was also observed that the pseudoplastic lubricants decrease the film pressure and load capacity, while the dilatant lubricants increase these properties. Also, the variations in these results are highly significant.


Introduction
Study of squeeze film behaviour between two approaching surfaces is essential in many engineering applications and other allied areas. The effects of non-Newtonian lubricants on various performance properties of squeeze film, annular disks and externally pressurized bearings have been studied by many researchers [1,2]. Bakker and Ostayen presented recess shape optimization of annular disks [3]. Effects of surface roughness on pressure distribution of spherical bearing lubricated with Ellis fluids was studied by Jurczak and Falicki [4]. Effects of non-Newtonian lubricants on the performance of porus thrust bearings was investigated by Walicka et al. [5]. Many investigators also used power-law and couple stress fluid models to analyse steady and dynamic properties of film lubricated bearing systems [6,7]. Singh at el. adopted Rabinowitsch fluid model for theoretical investigation of performance properties of many film lubricated bearings in different lubrication regimes [8][9][10][11]. To study the characteristics of bearings lubricated with non-Newtonian lubricants, Rabinowitsch fluid model is one of the best and experimentally verified fluid models [12]. The properties of slider bearings lubricated with non-Newtonian Rabinowitsch type lubricants were studied by Hsu and Saibel and it was shown that the pressure distribution of pseudoplastic fluids decreases below that of Newtonian fluids [13].
In Rabinowitsch fluid model, the relationship between shearing stress and shearing strain rate is given as: where, µ is the initial viscosity of Newtonian fluid and k is the non-linear factor responsible for non-Newtonian nature of the lubricant [12]. By using this cubic equation model, one can analyse the dilatant fluids for k < 0, Newtonian fluids for k = 0 and pseudoplastic fluids for k > 0, respectively. In an experimental investigation, Wada and Hayashi obtained a range of values of µ and k. Later on, this fluid model was used by several investigators to study the properties of hydrodynamic and hydrostatic bearings lubricated with non-Newtonian properties [12]. Problems concerning hydrostatic journal bearings were investigated by Sharma et al. [14], Sinhasan and Sah [15] and, Bourgin and Gay [16]. Slider bearings were analysed by Hashimoto [17] and Lin [18]. In recent years, Singh et al. [10,[19][20][21] used Rabinowitsch fluid model to investigate the static and dynamic properties of various hydrodynamic, hydrostatic and squeeze film bearings. Singh et al. [22][23][24] expanded the span of this fluid model to study the peristaltic motion of physiological fluids and got excellent results. Recently, Walicka et al. [25] analysed curvilinear squeeze film bearings using stochastic model for hydrodynamic lubrication [26]. But none of the researchers used Rabinowitsch fluid model to study the performance of squeeze films between rough circular plates. The objective of the present paper is to extend the results of Rabinowitsch model to squeeze films between rough circular plates.

Mathematical Formulation and Solution
Consider a squeeze film configuration ( Figure 1) between rough circular plates lubricated with a non-Newtonian fluid. The plates are approaching each other with a normal velocity v H t = − ∂ ∂ . In present study, it is assumed that the thin film theory of hydrodynamic lubrication is applicable.
Under these assumptions, the equations governing the one dimensional flow of incompressible non-Newtonian fluids are given by: The boundary conditions for velocity components are: to y, we get the velocity component as follows: From equations (5-7), radial velocity component u is obtained as: where, ( ) (2) and integrating with respect to y between the limits 0 to h, the modified Reynolds equation can be obtained as: Taking the stochastic average of Eq. (10) with respect to ( ) s f h , the averaged modified Reynolds type equation can be obtained as: where the expectancy operator ( ) and the probability density function of the stochastic film thickness is given as: where, c is the parameter of roughness [26]. Following Christensen's theory, radial and azimuthal roughness patterns are adopted in the present theoretical analysis.

One-Dimensional Radial Roughness
For one-dimensional radial roughness pattern, the roughness is in the form of long narrow ridges and valleys running through x-direction. In this case, the non-dimensional stochastic film thickness is given as ; and the stochastic modified Reynolds equation (11) takes the form:

One-Dimensional Azimuthal Roughness
For one-dimensional azimuthal roughness pattern, the roughness is in the form of long narrow ridges and valleys running through θ-direction. In this case, the non-dimensional stochastic film thickness is ; and the stochastic modified Reynolds equation (11) takes the form: For axisymmetric flow, Eqs. (14)(15) can be represented as: and, Introducing the dimensional quantities: (16) The squeeze film pressure can be perturbed as: Using Eq. (24) in the Reynolds equation (23) and neglecting higher order terms of β, following equations for squeeze film pressure 0 p * and 1 p * can be obtained: 20 Integrating Eqs. (24) and (25) with respect to r * with the boundary conditions: 0 dp dr * * = at 0 r * = The squeezing time can be calculated as follows: 3  27  5  2 , , where,

Results and Discussion
Based on the present theoretical analysis, the effects of non-Newtonian fluid and surface roughness on different properties of squeeze film between rough circular plates are presented. Results are analysed for different values of roughness parameter C and pseudoplasticity parameter β. Figure 2 shows the variation of non-dimensional pressure p * with non-dimensional radial coordinate r * for * 0.4 h = , C = 0.3 and different values of β. It is observed that the pressure p * decreases with increasing value of r * for each value of β in both the cases of radial and azimuthal patterns. However, pressure for azimuthal roughness is always higher in comparison to that for radial roughness. The variation of dimensionless pressure p * with respect to r * for different values of C with β = 0.001 and * 0.6 h = is shown in Figure 3. It is seen that an increase in С decreases the pressure for radial roughness, while increases the pressure for azimuthal roughness. The nature of variation of pressure sustains for all types of fluids.  . It is observed that the load carrying capacity W * decreases with an increase of h * , which satisfy the physical nature of the problem. In comparison to Newtonian fluids, the load capacity is higher for dilatant fluids, while lower for pseudoplastic fluids. The load capacity for azimuthal roughness is always higher than that for radial roughness. Figure 5 shows the variation of non-dimensional load carrying capacity W * with h * for different values of C and β = 0.001. It is observed that with an increase in value of C, the dimensionless load carrying capacity decreases in the case of radial roughness, while the load capacity increases in the case of azimuthal roughness.    The variation of squeeze film time t * with h * for different values of C is shown in Figure 7. It is easy to observe that with the increase in the value of C, the squeezing time decreases for radial roughness and decreases for azimuthal roughness. The variation of squeezing time with amount and nature of surface roughness is in agreement with physical nature of the problem.

Conclusions
In the present analysis, the combined effect of surface roughness and non-Newtonian lubricants on squeeze film characteristics between rough circular plates is presented on the basis of Rabinowitsch fluid model and Christensen's roughness model. The modified Reynolds equations for radial and azimuthal roughness are derived. An analytic solution for film pressure and squeezing time are obtained by means of small perturbation technique.
It is analysed that in comparison to the Newtonian lubricants, dimensionless film pressure, load carrying capacity and squeeze film time are smaller for pseudoplastic lubricants, while higher values of these properties are obtained for dilatant lubricants.
Film pressure, load carrying capacity and squeeze film time decrease with the increase in pseudoplasticity, while the results are reversed for an increase in dilatancy. In comparison to the smooth surfaces (c = 0), the film pressure, load capacity and squeezing time is smaller for radial roughness and greater for azimuthal roughness. In the case of radial roughness, the film pressure, load capacity and squeezing time decreases with an increase in surface roughness, while the results are reversed in case of azimuthal roughness.
Based on the present analysis, it is concluded that the stability and load sustaining capacity of a squeezing film bearing can be increased by reducing the radial roughness, introducing optimal azimuthal roughness and blending the proper additives to the lubricant to make it dilatant up to an optimal level.