Slider block, lubrication

Figure 5-4. Some common lubrication configurations (a) the slider block, (b) a sphere or cylinder moving in the vicinity of a plane wall, (c) a sphere translating axially in a circular tube.

In this section we consider the detailed analysis for two applications of lubrication theory the classic slider-block problem that was depicted in the previous section and the motion of a sphere toward an infinite plane wall when the sphere is very close to the wall. It is the usual practice in lubrication theory to focus directly on the motion in the thin gap using (5-69)-(5-72), or their solutions (5-74) and (5-79), without any mention of the asymptotic nature of the problem or of the fact that these equations (and their solutions) represent only a first approximation to the full solution in the lubrication layer. We adopt the same approach here but with the formal justification of the preceding section. 

The integration constants c and c2 are given by the boundary values for p( )) at x = 0 and x = 1 these come from the dynamic pressure values for a stationary fluid in the region outside the lubrication layer. We assume that the slider block is moving through a large body of fluid that completely surrounds it. In this case, it is obvious that... 

Let us begin by calculating the forces on the flat surface of the slider block that forms the upper boundary of the lubrication layer. To do this we require an expression for the outer normal n to this surface. In general, for a surface z = g (x), the unit normal is... 

A second example of the application of lubrication theory is its use in analyzing the motion of a sphere that is pushed by the action of an applied force / toward a solid plane boundary, when the gap between the sphere and the wall at the point of closest approach is small compared with the radius of the sphere.6 Although this problem is geometrically similar to the slider block in the sense that a body of finite dimensions is moving in the vicinity of an infinite plane boundary, the problem differs in that the motion is normal to the boundary rather than parallel to it and the gap width is time dependent. 

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