THE INFLUENCE OF WALL'S POROSITYON THE PRESSURE DISTRIBUTION IN THE CURVILINEAR BEARING LUBRICATED BY POWER - LAW FLUID

The influence of porosity of the wall on the pressure distribution of a power-law fluid in a clearance of the curvilinear thrust bearing is considered. As a result one obtains the formulae expressing the pressure distribution. The example of a squeeze film between parallel disks is discussed in detail. INTRODUCTION Advances in technology and severe operational requirements of machines necessitated the development of improved lubricants for smooth and safe operation. Generally the viscosity of lubricating oils decreases with temperature. For operations under high speeds and heavy loads, oils containing high molecular-weight polymers as viscosity index improves are used to prevent viscosity variation with temperature. The increase in viscosity increases the load caring capacity of the modified lubricants. Most lubricants are polymer solutions thus the characteristics of the bearings change when such rheological substances, known as non-Newtonian fluids, are used as lubricants. Many authors have studied the characteristics of various bearings by considering power-law [2], Bingham plastic [7], viscoplastic [4,7] and micropolar [3J models of lubricants. The flows of Newtonian fluids in the clearance of a thrust bearing with impermeable surfaces have been examined theoretically and experimentally [5]. The bearing walls have been modelled as two disks, two conical or spherical surfaces. The more general case is established by the bearing formed by two surfaces of revolution [5-7]. Porous bearings have been widely used in industry for a long time. Basing on the Darcy model Morgan and Cameron [ 1 ] first presented theoretical research on these bearings. The purpose of this study is to investigate the pressure distribution in the clearance of the thrust bearing formed by two surfaces of revolution, having parallel axes, shown in Fig.l; the lower one is connected with a porous layer. The analysis is based on the assumption that the porous matrix consists of a system of capillaries of very small radii restricting the floty of the lubricant through the matrix in only one direction. ANALYSIS OF A FLUID FLOW IN THE BEARING CLEARANCE The bearing floty configuration is shown in Fig. 1. The upper bound of a porous layer is described by function R(x) which denotes the radius of this bound. The fluid film thickness is given by function h(x, t) . An intrinsic curvilinear orthogonal co-ordinate system (x, Θ, y) is also depicted in Fig. l . Fig. 1. Co-ordinate system in the bearing clearance Rys. 1. Układ współrzędnych w szczelinie łożyska By using the assumptions of hydrodynamic lubrication the equations of motion of a powerlaw fluid for axial symmetry one can present in the form [1,5,7]: The problem statement is complete after specification of boundary conditions which are: Here xi denotes the inlet co-ordinate and xo the outlet co-ordinate. Integrating Eq. (2) with respect to y in the interval 0 ≤ y ≤ h and determining the arbitrary constants from the boundary conditions (4) we obtain: Next, integrating the continuity equation (1) across the film thickness and taking finto account the boundary conditions (5) we have: where V fis the velocity of lubricant on the upper bound of the porous matrix. This velocity may be determined as follows. Consider that the porous matrix consists of a system of capillaries the axes of which are directed towards the film. The motion of the lubricant in a typical capillary fis governed by (see Fig.2): Fig.2. Flow of power-law lubricant in a thin capillary of porous layer Rys.2. Przepływ potęgowego płynu w cienkiej kapilarze warstwy porowatej The flux may be defined as where Q' is a constant as seen from the equation of continuity. The average velocity υ0 of the lubricant can then be given as: This equation may be interpreted as a modified from of Darcy's law for power-law fluids. Assume now that the porous layer is homogeneous and isotropic and the floty in this layer satisfies the modified Darcy's law. Thus we have: The equation of continuity in the porous region has the same form as Eq. (1): Since the cross velocity component must be continuous at the porous wall-film interface, one obtains from Eqs (8) and (15) the modified Reynolds equation: By substituting Eqs (14) and (15) finto Eq. (16) one obtains the following equation for pressure distribution in the porous region: Integrating this equation with respect to y over the porous layer and using the MorganCameron approximation [ 1 ] one obtains When Eq. (19) is substituted finto Eq. (17) the modified Reynolds equation takes the form: SOLUTION. EXAMPLE OF APPLICATION The form of solution of the Reynolds equation (20) depends on the bearing type. For the externally pressurised hydrostatic bearing       = ∂ ∂ 0 t h and for the bearing with a squeeze film       = ∂ ∂ = 0 0 x x p the solutions are, respectively: Introducing the following parameters: we may present Eqs (22) and (23) in the non dimensional forms: