Viscous Displacement in a Hele-Shaw Cell

TLDR: In this article, the authors show that the Darcian flow through a porous medium was the original motivation for the study of the Hele-Shaw cell geometry, and the geometry consists of a long rectilinear channel where the width of the cell is 2a, with b << a.

Abstract: A Hele-Shaw cell is a pair of parallel plates that are separated by a small gap 6 . The motion of a less viscous fluid displacing a more viscous fluid in this gap under the action of some imposed pressure gradient or gravity or fluid injection has been the study of intensive research over the last decade. (See Saffman1, Bensimon et al2, Homsy3, Pelee4 and Kessler, Koplik & Levine5 for reviews from a range of perspectives). This has been spurred by the newly discovered mathematical analogies between this flow and dendritic crystal growth, directional solidification and diffusion limited aggregation (see references 4,5), though Darcian flow through a porous medium was the original motivation6. In most Hele-Shaw cell studies to date, the geometry consists of a long rectilinear channel where the width of the cell is 2a , with b << a (Figure 1). In this case, the interfacial motion is caused by an imposed pressure gradient which causes the more viscous fluid at infinity to be displaced with velocity V . Alternately, gravity can effect the interfacial displacement.