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. 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. 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 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. Let x {displaystyle x} , y {displaystyle y} be the directions parallel to the flat plates, and z {displaystyle z} the perpendicular direction, with 2 H {displaystyle 2H} being the gap between the plates (at z = ± H {displaystyle z=pm H} ).When the gap between plates is asymptotically small the velocity profile in the z {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 velocity is, where u {displaystyle {mathbf {u} }} is the velocity, p ( x , y , t ) {displaystyle p(x,y,t)} is the local pressure, μ {displaystyle mu } is the fluid viscosity. This relation and the uniformity of the pressure in the narrow direction z {displaystyle z} permits us to integrate the velocity with regard to z {displaystyle z} and thus to consider an effective velocity field in only the two dimensions x {displaystyle x} and y {displaystyle y} . When substituting this equation into the continuity equation and integrating over z {displaystyle z} we obtain the governing equation of Hele-Shaw flows,