The Lubrication Limit, Re

In this subsection, we begin by considering the limit of the thin-film equations in which Re 1. In this case, the problem reduces to solving a classical lubrication problem, and a slightly modified version of the Reynolds equation, (5-79), can be used to obtain the leading-order approximation to the pressure distribution in the thin gap. 

We seek an asymptotic solution of the leading-order thin-film equations, (5—134)— (5 136), in the form 

at the first approximation, we obtain the classic lubrication equations  

If we review the derivation of the Reynolds equation, (5 79), starting with the governing equations and boundary conditions, (5—69)—(5--72), we see that the present problem differs in that dh/dt = )d/ )t = 0, but there is still a normal velocity at z = 0 that is due to blowing of air through the porous tabletop. Hence we can see from (5—75)—(5—77) that the appropriate form of Reynolds equation for the present problem should be 

In writing (5-145), we have taken account of the fact that the dimensionless film thickness is h = 1. If we reexpress this equation in cylindrical coordinates, it can be written in the form 

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