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.
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