On the modified Reynolds equation for journal bearings in a case of non-Newtonian Rabinowitsch fluid model

In this paper, a theoretical analysis of hydrodynamic plain journal bearings with finite length at taking into account the effect of nonNewtonian lubricants is presented. Based upon the Rabinowitsch fluid model (cubic stress constitutive equation) and by integrating the continuity equation across the film, the nonlinear modified 2D Reynolds type equation is derived in details so that to study the dilatant and pseudoplastic nature of the lubricant in comparison with Newtonian fluid. A dimensionless equation of hydrodynamic pressure distribution in a form appropriate for numerical modeling is also presented. Some particular cases of 1D applications can be recovered from the present derivation.


Introduction
Hydrodynamic journal bearings are considered to be a vital component of all rotating machines. Journal bearings are used widely for thousands of years in many areas of mechanical engineering. Currently large numbers of them are applied successfully in machine tools, automotive and aircraft piston engines, turbo-machinery, micro-electromechanical systems, etc. Furthermore, it is well known that the performance of bearings is strongly influenced by the composition and rheological characteristics of the lubricant.
In order to meet requirements of the modern machinery systems, it is necessary to look for the enhancement in lubricating performance of all kind of bearings and other types of lubricated contacts. In this relation, currently, the use of Newtonian fluids blended with various additives increases, which is based on the effective improvement in the bearing characteristics as compared to the lubrication with Newtonian lubricants. These additives often are viscosity index improvers which represent high molecular weight polymers such as polyisobutylene, polymethacrylate, ethylene propylene, etc. These kinds of lubricants exhibit pseudoplastic and dilatant behaviour of non-Newtonian lubricants, in which the ratio between the shear stress and shear rate is no longer a constant. According to the experimental work of Wada and Hayashi [1] the non-Newtonian rheological behaviour of such kinds of lubricants with additives can be represented by an empiric cubic stress model, also called Rabinowitsch fluid model.

Modeling of the hydrodynamic lubrication in journal bearings
A radial journal bearing with finite length is considered under steady state conditions. The lubricant fluid in bearing clearance has non-Newtonian properties. The rheological fluid law described by the Rabinowitsch model [1, 13, 18, etc.] is presented with the following cubic equation where:  -shear stress in the fluid film ; k -coefficient of pseudoplasticity (parameter responsible for the lubricants non-Newtonian behaviour);  -initial viscosity of the lubricant; , , u v w -velocity components in the directions of Cartesian coordinates , , x y z , respectively.
In dependence of the values of the coefficient of pseudoplasticity k there are three different groups of lubricants: for 0 k  -pseudoplastic fluids, if 0 k  -Newtonian fluid, as for 0 k  -dilatant fluid. The flow characteristics of these kinds of lubricants are shown in Fig. 1. Here  is a tangent at the original point of flow curves because of which it is called "initial viscosity" by analogy with [1,13,17]. If values of  do not vary, the nonlinearity of the flow curve increases with the value of the coefficient of pseudoplasticity k (see Fig. 1 -on the right).

Bearing geometry
For the geometry of the considered plane journal bearing with 360° range are introduced the following assumptions (see Fig.2): The journal and the bearing have a round shape and parallel axes; their surfaces are perfectly smooth. The bearing gap is filled with a lubricating fluid with a constant pressure equal to the external. The journal rotates with a constant angular velocity  around its axis. The bearing radius R is approximately equal to the journal radius r   R r  . The radial bearing clearance c R r   is of the order Moreover, for all types of bearings the fluid film thickness h is very small compared to the other dimensions of the contact surfaces because of which the ratio where l is the fluid film length in a circumferential direction. All of the above mentioned about the bearing geometry justifies the commonly accepted hypothesis in the theory of hydrodynamic lubrication to neglect the curvature of the lubricated surfaces and, respectively, the curvature of the lubricant film. Fig. 2. Hydrodynamic journal bearing.

Boundary value problem
The constitutive equations between the shear stress and rate of share strain for two dimensional flow of the Rabinowitsch fluid obeys the following nonlinear relationships where xy  and zy  are shear stress components in x and z directions, respectively. According to the thin film theory of hydrodynamic lubrication [25,1,18], the momentum and continuity equations in Cartesian coordinates are represented by the following differential equations: where p is the hydrodynamic pressure. Related to the bearing configuration, the boundary conditions for the fluid velocity components are at 0 y  : at y h  : where:  is the journal angular velocity,

 
, h h x z  is the fluid film thickness.

Mathematical procedure
The integration of (3.a) and (3.c) with respect to y yields: where 1  and 2  are integration constants. By substitution of (6.a) and (6.b) in (2.a) and (2.b), respectively, it is obtained: where Here the couples   ..,.. are ordered and mapping between them one-to-one.
Integration of (7) with respect to where Applying boundary conditions (4) and (5) and performing some elementary transformations the following cubic equation for  is obtained: On the other hand, reordering (9) in ascending order by power of 2 qh     leads to a semi-cubic equation of  : Furthermore, by integrating (7) with respect to it is possible to find an antiderivative of f as where s -variable of integration. Afterwards, based on (11) the velocity component u is obtained in a form: Neglecting in (12) terms multiplying by second and third powers of  yields where:     The expression for w is similar to (13) without the term multiply by  . Therefore Integrating the continuity equation (3.d) with respect to Analogously, the integral of w in the forth term of (18) is obtained by (16) By using (14) the integrals of 1  , 2  , 3  in (19) and (20) are respectively equal to   Then after substitution of the velocity components (15), (16) in (18) and using (21) the following Reynolds type equation is obtained