An Engineering Derivation of the Two-Dimensional Reynolds Equation

Consider a lubricant film between two plane converging surfaces, as shown in the cross-sectional diagram of Fig. 2-3, where x, ly and z are coordinate directions and u, v and w are velocities. Take an element of 

It has already been postulated that p is not a function of y (assumption 6, above) on applying the additional assumption that p does not depend on z, the partial derivative 3p/3x in Eqn 2-5 can be replaced by a simple derivative dp/dx. Then the integration is carried out twice with respect to y  

Equation 2-8 gives the velocity distribution across any section of the oil film, subject to the restrictions implicit in Eqns 2-6. 

Going back to the three-dimensional point of view, since the fluid has been assumed incompressible, the quantity of fluid leaving the element dKdydz must equal the quantity entering it  

On introducing u from Eqn 2-8, with the restrictions imposed there, 

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