Hele-Shaw flow is defined as Stokes Flow between two parallel flat plates separated by an infinitesimally small gap. 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

Potential flow

In fluid dynamics, potential flow describes the velocity field as the gradient of a scalar function: the velocity potential. As a result, a potential flow is characterized by an irrotational velocity field, which is a valid approximation for several applications...

 and to the flow of fluid through a porous medium (Darcy's law

Darcy's law

Darcy's law is a phenomenologically derived constitutive equation that describes the flow of a fluid through a porous medium. The law was formulated by Henry Darcy based on the results of experiments on the flow of water through beds of sand...

). It thus permits visualization of this kind of flow in two dimensions.

Mathematical formulation of Hele-Shaw flows

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 velocity is,

where is the velocity , is the local pressure, is the fluid viscosity.

This relation and the uniformity of the pressure in the narrow direction permits us to integrate the velocity with regard to and thus to consider an effective velocity field in only the two dimensions and . When substituting this equation into the continuity equation and integrating over we obtain the governing equation of Hele-Shaw flows,

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.

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

See also

• Lubrication theory
  Lubrication theory
  In fluid dynamics, lubrication theory describes the flow of fluids in a geometry in which one dimension is significantly smaller than the others....
  
  
• Stokes flow
• Hele-Shaw clutch
  Hele-Shaw clutch
  The Hele-Shaw clutch was an early form of multi-plate wet clutch, in use around 1900. It was named after its inventor, Professor Henry Selby Hele-Shaw, who was noted for his work in viscosity and flows through small gaps between parallel plates...
  
  

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