How ideal is the Hele-Shaw flow?*

Potential flow is perhaps the most studied subject of early fluid dynamics, whose powerful results have allowed us to understand heavier-than-air flight and much more. However such an ideal flow is rare in nature, as a boundary layer develops near any obstacles and the potential flow must be matched to a no-slip velocity. In 1899 H. S. Hele-Shaw published his famous photographs that demonstrated how an ideal flow can exist in an extremely viscous environment: By bounding the fluid in a thin gap between two plates, the flow has a Poiseuille profile in the thin direction but becomes irrotational in the perpendicular plane. In his carefully crafted study, the potential flow around a cylinder was demonstrated experimentally. A paradox thus arises: The potential flow has a slip velocity at the cylinder, yet this should be impossible as the fluid has a high viscosity. How about the fluid drag on the cylinder? Should it be zero? We examine these questions through a joint theoretical, numerical, and experimental effort, and develop a boundary layer analysis of how Hele-Shaw flow can be matched to a no-slip solid boundary. Through such an investigation, we show how a separation of scales can result in a nearly potential flow, whose deviation from ideal Hele-Shaw flow is only visible at an extremely small scale.