Hele-Shaw flow is defined as Stokes flow between two parallel flat plates separated by an infinitesimally small gap, named after Henry Selby Hele-Shaw, who studied the problem in 1898. 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.
Mathematical formulation of Hele-Shaw flowsEdit
Let , be the directions parallel to the flat plates, and the perpendicular direction, with being the gap between the plates (at ). When the gap between plates is asymptotically small
the velocity profile in the direction is parabolic (i.e. is a quadratic function of the coordinate in this direction). The equation relating the pressure gradient to the horizontal velocity is,
is the local pressure, is the fluid viscosity. While the velocity magnitude varies in the direction, the velocity-vector direction is independent of direction, that is to say, streamline patterns at each level are similar. Eliminating pressure in the above equation, one obtains
where is the vorticity in the direction. The streamline patterns thus correspond to potential flow (irrotational flow). Unlike potential flow, here the circulation around any closed contour , whether it encloses a solid object or not, is zero,
where the last integral is set to zero because is a single-valued function and the integration is done over a closed contour.
The vertical velocity is as can shown from the continuity equation. Integrating over the continuity we obtain the governing equation of Hele-Shaw flows, the Laplace Equation:
This equation is supplemented by the no-penetration boundary conditions on the side walls of the geometry,
where is a unit vector perpendicular to the side wall.
The term Hele-Shaw cell is commonly used for cases in which a fluid is injected into the shallow geometry from above or below the geometry, and when the fluid is bounded by another liquid or gas. For such flows the boundary conditions are defined by pressures and surface tensions.
- A mechanical transmission clutch invented by Prof. Hele-Shaw, using the principles of a Hele-Shaw flow
- ^ Shaw, Henry S. H. (1898). Investigation of the nature of surface resistance of water and of stream-line motion under certain experimental conditions. Inst. N.A. OCLC 17929897.[page needed]
- ^ Hele-Shaw, H. S. (1 May 1898). "The Flow of Water". Nature. 58 (1489): 34–36. Bibcode:1898Natur..58...34H. doi:10.1038/058034a0.
- ^ Hermann Schlichting,Boundary Layer Theory, 7th ed. New York: McGraw-Hill, 1979.[page needed]
- ^ L. M. Milne-Thomson (1996). Theoretical Hydrodynamics. Dover Publications, Inc.
- ^ Horace Lamb, Hydrodynamics (1934).[page needed]
- ^ Acheson, D. J. (1991). Elementary fluid dynamics.
- ^ Saffman, P. G. (21 April 2006). "Viscous fingering in Hele-Shaw cells" (PDF). Journal of Fluid Mechanics. 173: 73–94. doi:10.1017/s0022112086001088. S2CID 17003612.