# 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 ${\displaystyle H}$  being the gap between the plates (at ${\displaystyle z=0,H}$ ). 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 horizontal velocity ${\displaystyle {\mathbf {u} }=(u,v)}$  is,

${\displaystyle u=-{\frac {1}{2\mu }}{\frac {\partial p}{\partial x}}z(H-z)\,}$
${\displaystyle v=-{\frac {1}{2\mu }}{\frac {\partial p}{\partial y}}z(H-z)\,}$

${\displaystyle p(x,y,t)}$  is the local pressure, ${\displaystyle \mu }$  is the fluid viscosity. While the velocity magnitude ${\displaystyle {\sqrt {u^{2}+v^{2}}}}$  varies in the ${\displaystyle z}$  direction, the velocity-vector direction ${\displaystyle \tan ^{-1}(v/u)}$  is independent of ${\displaystyle z}$  direction, that is to say, streamline patterns at each level are similar. Eliminating pressure in the above equation, one obtains[6]

${\displaystyle \omega _{z}={\frac {\partial v}{\partial x}}-{\frac {\partial u}{\partial y}}=0}$

where ${\displaystyle \omega _{z}}$  is the vorticity in the ${\displaystyle z}$  direction. The streamline patterns thus correspond to potential flow (irrotational flow). Unlike potential flow, here the circulation ${\displaystyle \Gamma }$  around any closed contour ${\displaystyle C}$ , whether it encloses a solid object or not, is zero,

${\displaystyle \Gamma =\oint _{C}udx+vdy=-{\frac {1}{2\mu }}z(H-z)\oint _{C}{\frac {\partial p}{\partial x}}dx+{\frac {\partial p}{\partial y}}dy=0}$

where the last integral is set to zero because ${\displaystyle p}$  is a single-valued function and the integration is done over a closed contour.

The vertical velocity is ${\displaystyle w=0}$  as can shown from the continuity equation. Integrating over ${\displaystyle z}$  the continuity 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.[7] 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

## 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.[page needed]
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.[page needed]
4. ^ L. M. Milne-Thomson (1996). Theoretical Hydrodynamics. Dover Publications, Inc.
5. ^ Horace Lamb, Hydrodynamics (1934).[page needed]
6. ^ Acheson, D. J. (1991). Elementary fluid dynamics.
7. ^ 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.