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Hele-Shaw Flows

DOI: 10.1615/AtoZ.h.hele_shaw_flows

This term describes the flow of a real fluid between two closely spaced parallel plates. Figure 1 outlines a typical arrangement (usually referred to as a Hele Shaw cell) in which the fluid flow is generated by a pressure gradient applied across the ends of the plates, and the spacing d is sufficiently small to ensure that the viscous forces dominate (i.e., viscous forces >> inertial forces).

If some form of "obstruction" is placed between the plates (i.e., a vertical cylinder), then, provided the linear dimension of this object measured in the plane of the plates (L) is large in comparison to the spacing d, the flow behaves as if the local pressure gradient extended to infinity. As a result, the local velocity may be expressed in terms of the local pressure gradient:

where η is the dynamic viscosity, and the Cartesian coordinates are aligned so that z is perpendicular to the plane of the plates. The condition necessary for this to occur is:

The velocity components noted above define a two dimensional velocity field which is both irrotational and satisfies the zero normal flow condition at a solid boundary. As a result, the streamlines generated in a Hele Shaw cell are identical to the hypothetical streamlines associated with the two dimensional flow of an ideal fluid around a similar obstruction. The introduction of dye traces within a Hele Shaw cell thus provides a visual representation of an ideal fluid flow. Batchlor (1967) provides further investigation of this and other viscous dominated flows.

Hele Shaw cell.

Figure 1. Hele Shaw cell.

REFERENCES

Batchlor, G. K. (1967) An Introduction to Fluid Mechanics, Cambridge University Press, Cambridge.

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