The Generalized Reynolds Equation

Equations 3-21 can also be obtained from a direct engineering analysis of the forces on an element of fluid. Consider an element of fluid dxdydz with its center at the point x, y, z. By summing up the forces in the x- and z-directions (Fig. 3-3) we get 

The symbology of Eqns 3-22, 3-23 and 3-24 has been simplified from strict tensor notation, since the meaning is clear from Fig. 3-3. 

On making use of assumption 7 which was employed in simplifying the Navier-Stokes equations (Section 3.2), the second term of each left-hand side of Eqns 3 25 drops out, leaving them identical with Eqns 3-21. 

In the integration of Eqn 3-21b we use the boundary conditions w = 0 at y = 0 and at y = h, which implies that the bearing surfaces move only in the x-direction. We then get 

For a fluid whose density does not change with time the continuity equation can be written as 

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R. M. Crone, P. R. Peck, M. S. Jhon, and T. E. Karis, Scaling criteria for slider miniaturization using the generalized Reynolds equation, J. Tribol. 115, 566-572 (1993). 

Since ti/R is a quantity very much smaller than 1, usually of the order of 0.001, it follows that 6p(i[3/3x(U + U2)] must also be very small and therefore can be neglected. The final form of the generalized Reynolds equation then becomes... 

Treatments of the elastohydrodynamic problem for such cases have been published by Archard and Cowking [20], by Cheng [21] and by Hamrock and Dowson [22]. Of these, the work of Hamrock and Dowson is the most comprehensive. The general Reynolds equation is written as... 

In Chapter 3 it was demonstrated that one form of the generalized Reynolds equation is given by the expression below (c . Eqn 3-34),... 

Sternlicht and Elwell [2] started with the following form of the generalized Reynolds equation. 

Equations 5-12 and 5-13 both indicate departure from ideal gas behavior, the one by intermolecular attraction and the other by an occupied volume effect. Such departure from ideal behavior has a direct influence on density and hence on the transformation of the generalized Reynolds equation. An equation of state for the gas such as the van der Waals equation,... 

Under these conditions, the generalized Reynolds equation in non dimensional form can be written [32]... 

Writing the equation of continuity, it is then quite easy to obtain the generalized Reynolds equation ... 

For one-dimensional analysis, the general Reynolds equation for rough svirfaces is... 

Most screw channels and flight lands have shell-like geom cal configmations in which its thickness is much smaller than the other physical dimensions. Under the lubrication approximation, the generalized Reynolds equation and energy equation are described as follows ... 

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