Hele-Shaw flow

Hele-Shaw flow (named after Henry Selby Hele-Shaw) is defined as Stokes flow between two parallel flat plates separated by an infinitesimally small gap. Various problems in fluid mechanics can be approximated to Hele-Shaw flows and thus the research of these flows is of importance. Approximation to Hele-Shaw flow is specifically important to micro-flows. This is due to manufacturing techniques, which creates shallow planar configurations, and the typically low Reynolds numbers of micro-flows.

The governing equation of Hele-Shaw flows is identical to that of the inviscid potential flow and to the flow of fluid through a porous medium (Darcy's law). It thus permits visualization of this kind of flow in two dimensions.

Let ${\displaystyle x}$, ${\displaystyle y}$ be the directions parallel to the flat plates, and ${\displaystyle z}$ the perpendicular direction, with ${\displaystyle 2H}$ being the gap between the plates (at ${\displaystyle z=\pm H}$). When the gap between plates is asymptotically small

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