Hele-Shaw flow

Hele-Shaw flow is defined as flow taking place between two parallel flat plates separated by a narrow gap satisfying certain conditions, 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 conditions that needs to be satisfied are

${\displaystyle {\frac {h}{l}}\ll 1,\qquad {\frac {Uh}{\nu }}{\frac {h}{l}}\ll 1}$

where ${\displaystyle h}$ is the gap width between the plates, ${\displaystyle U}$ is the characteristic velocity scale, ${\displaystyle l}$ is the characteristic length scale in directions parallel to the plate and ${\displaystyle \nu }$ is the kinematic viscosity. Specifically, the Reynolds number ${\displaystyle Re=Uh/\nu }$ need not always be small, but can be order unity or greater as long as it satisfies the condition ${\displaystyle Re(h/l)\ll 1.}$ In terms of the Reynolds number ${\displaystyle Re_{l}=Ul/\nu }$ based on ${\displaystyle l}$, the condition becomes ${\displaystyle Re_{l}(h/l)^{2}\ll 1.}$

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]