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.

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