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
In fluid dynamics, potential flow (or ideal 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 app ...
and to the flow of fluid through a porous medium (
Darcy's law
Darcy's law is an equation that describes the flow of a fluid through a porous medium. The law was formulated by Henry Darcy based on results of experiments on the flow of water through beds of sand, forming the basis of hydrogeology, a branch of ...
). 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 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
In fluid dynamics, potential flow (or ideal 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 app ...
(irrotational flow). Unlike
potential flow
In fluid dynamics, potential flow (or ideal 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 app ...
, 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
In mathematics and physics, Laplace's equation is a second-order partial differential equation named after Pierre-Simon Laplace, who first studied its properties. This is often written as
\nabla^2\! f = 0 or \Delta f = 0,
where \Delta = \n ...
:
:
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
*
Diffusion-limited aggregation
Diffusion-limited aggregation (DLA) is the process whereby particles undergoing a random walk due to Brownian motion cluster together to form aggregates of such particles. This theory, proposed by T.A. Witten Jr. and L.M. Sander in 1981, is app ...
*
Lubrication theory
In fluid dynamics, lubrication theory describes the flow of fluids (liquids or gases) in a geometry in which one dimension is significantly smaller than the others. An example is the flow above air hockey tables, where the thickness of the air ...
*
Thin-film equation
*
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 pla ...
: A mechanical transmission clutch invented by Prof. Hele-Shaw, using the principles of a Hele-Shaw flow
References
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Fluid dynamics