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. 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
h l ≪ 1 , U h ν h l ≪ 1 {\displaystyle {\frac {h}{l}}\ll 1,\qquad {\frac {Uh}{\nu }}{\frac {h}{l}}\ll 1}
where h {\displaystyle h} is the gap width between the plates, U {\displaystyle U} is the characteristic velocity scale, l {\displaystyle l} is the characteristic length scale in directions parallel to the plate and ν {\displaystyle \nu } is the kinematic viscosity. Specifically, the Reynolds number R e = U h / ν {\displaystyle \mathrm {Re} =Uh/\nu } need not always be small, but can be order unity or greater as long as it satisfies the condition R e ( h / l ) ≪ 1. {\displaystyle \mathrm {Re} (h/l)\ll 1.} In terms of the Reynolds number R e l = U l / ν {\displaystyle \mathrm {Re} _{l}=Ul/\nu } based on l {\displaystyle l} , the condition becomes R e l ( h / l ) 2 ≪ 1. {\displaystyle \mathrm {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.
Mathematical formulation
Let x {\displaystyle x} , y {\displaystyle y} be the directions parallel to the flat plates, and z {\displaystyle z} the perpendicular direction, with h {\displaystyle h} being the gap between the plates (at z = 0 , h {\displaystyle z=0,h} ) and l {\displaystyle l} be the relevant characteristic length scale in the x y {\displaystyle xy} -directions. Under the limits mentioned above, the incompressible Navier–Stokes equations, in the first approximation becomes
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