The Thin Gap Approximation

The analysis of nearly parallel flows originated in the study of problems of lubrication, and the approach is often called the lubrication approximation. The terminology is unfortunate from our perspective, given that this approach is at the heart of all analytical treatments of polymer processing operations - we would prefer that it be called the polymer processing approximation - but the historical name is well established. The major figure in the analysis of lubrication flows was Osborne Reynolds, and one widely used form of the resulting equations is often called the Reynolds lubrication equation. 

We restrict ourselves to two-dimensional flows, where all changes occur in the xy plane and there is no flow in the neutral z direction. We found in Section 3.2 that the following equations apply for parallel flow of a Newtonian fluid with a moving surface at y = 0 and a stationary surface at y = //  

Because we are dealing with incompressible liquids, is a constant for all x. Unlike flow between parallel planes, however, where we showed that the pressure gradient 9 P/9x is a constant, we see from Equation 5.4 that 9(P/9x is itself a function of x indeed, we may consider Equation 5.4 to be an equation that defines the spatial distribution of pressure, which we know is not linear, and write 

Equation 5.3 can then be rewritten, after some algebraic manipulation, as 

Vx will take on negative values in a region adjacent to the upper surface whenever H(x) 3q/V. 

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Chapter 6 The Thin-Gap Approximation - Films with a Free Surface... 

To see how the thin-gap approximation e <governing equations and boundary conditions. The characteristic velocity for the polar velocity component is clearly... 

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