Hele-Shaw flow

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.[1][2] 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.[3][4][5]

Mathematical formulation of Hele-Shaw flows

A schematic description of a Hele-Shaw configuration.

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

${\displaystyle H\rightarrow 0,\,}$

the velocity profile in the ${\displaystyle z}$ direction is parabolic (i.e. is a quadratic function of the coordinate in this direction). The equation relating the pressure gradient to the velocity is,

${\displaystyle {\mathbf {u} }={\mathbf {\nabla } }p{\frac {z^{2}-H^{2}}{2\mu }}\,}$

where ${\displaystyle {\mathbf {u} }}$ is the velocity, ${\displaystyle p(x,y,t)}$ is the local pressure, ${\displaystyle \mu }$ is the fluid viscosity.

This relation and the uniformity of the pressure in the narrow direction ${\displaystyle z}$ permits us to integrate the velocity with regard to ${\displaystyle z}$ and thus to consider an effective velocity field in only the two dimensions ${\displaystyle x}$ and ${\displaystyle y}$. When substituting this equation into the continuity equation and integrating over ${\displaystyle z}$ we obtain the governing equation of Hele-Shaw flows, the Laplace Equation:

${\displaystyle {\frac {\partial ^{2}p}{\partial x^{2}}}+{\frac {\partial ^{2}p}{\partial y^{2}}}=0.}$

This equation is supplemented by the no-penetration boundary conditions on the side walls of the geometry,

${\displaystyle {\mathbf {\nabla } }p\cdot {\hat {n}}=0,\,}$

where ${\displaystyle {\hat {n}}}$ is a unit vector perpendicular to the side wall.

Hele-Shaw cell

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.[6] For such flows the boundary conditions are defined by pressures and surface tensions.

See also

• Diffusion-limited aggregation
• Lubrication theory
• Thin-film equation
• Hele-Shaw clutch

A mechanical transmission clutch invented by Prof. Hele-Shaw, using the principles of a Hele-Shaw flow

References

1. 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.
2. Hele-Shaw, H. S. (1 May 1898). "The Flow of Water". Nature. 58 (1489): 34–36. Bibcode:1898Natur..58...34H. doi:10.1038/058034a0.
3. Hermann Schlichting,Boundary Layer Theory, 7th ed. New York: McGraw-Hill, 1979.
4. L. M. Milne-Thomson (1996). Theoretical Hydrodynamics. Dover Publications, Inc.
5. Horace Lamb, Hydrodynamics (1934).
6. Saffman, P. G. (21 April 2006). "Viscous fingering in Hele-Shaw cells" (PDF). Journal of Fluid Mechanics. 173: 73–94. doi:10.1017/s0022112086001088.