TABLE OF CONTENTS We turn our attention now to a class of flows in which inertial effects are negligible due to the fact that the curvature of the streamlines is sufficiently small. As a result, the flow may be assumed to be locally unidirectional, which means that the velocity profile may be assumed to depend only upon the local pressure gradient, direction of the body force, and geometry of the domain of flow. The union of these assumptions constitutes the premise of lubrication flow.
To study the structure of a nearly unidirectional flow in a mathematically rigorous manner, we must develop a formal asymptotic expansion for the flow variables in terms of the magnitude of the curvature of the streamlines, which is assumed to be comparable to that of the boundaries (Langlois, 1964). The results confirm that the assumption of locally unidirectional flow provides us with the correct leading-order approximation to the exact solution.
Consider flow in a channel with parallel-sided walls that are separated by a uniform distance h and are confined by side walls, known as the Hele Shaw cell. The clearance of the channel may be blocked by stationary or moving objects such as disks, or flattened air bubbles and liquid drops, as depicted in Figure 6.3.1. The flow may be driven by an imposed pressure gradient, gravity, or the motion of the objects.
When h is small compared to the global dimensions of the channel and size...
