Hele-Shaw cell

A Hele-Shaw cell confines fluid between two closely spaced parallel plates. When the gap is small relative to the flow’s in-plane dimensions, depth-averaged motion can approximate two-dimensional potential flow.

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Hele-Shaw flow: A depth-averaged flow between closely spaced parallel plates, governed by a Darcy-like law. It describes the motion the cell is designed to produce.
Saffman–Taylor instability: An interfacial instability in which a less viscous fluid displaces a more viscous one in a porous medium or narrow gap. The cell provides a classic setting for observing viscous fingering at a moving interface.
Incompressible flow: Fluid motion in which density is treated as constant, so velocity has zero divergence. This condition helps make the depth-averaged velocity field divergence-free.
Henry Selby Hele-Shaw: An English engineer known for developing a method to visualize flow between closely spaced plates. His experiments gave the apparatus its name.
Darcy's law: A relation connecting fluid flux in a porous medium to pressure gradient and permeability. Depth-averaging viscous flow across the cell’s gap yields a Darcy-like pressure-velocity relation.
Viscous fingering: Branching patterns formed when one fluid displaces another under conditions that destabilize their interface. The plates make the evolving fingers visible from above.
Navier–Stokes equations: The equations governing the motion of viscous fluids through conservation of momentum and mass. Hele-Shaw flow follows from simplifying these equations for a thin gap and slow motion.
Hele-Shaw's flow of water: A late nineteenth-century experimental method for visualizing fluid motion in a thin space between glass plates. This is the foundational plate-based technique behind the modern cell.
Laplace's equation: A partial differential equation stating that a function’s Laplacian is zero. For incompressible Hele-Shaw flow, pressure is harmonic away from sources and boundaries.
Diffusion-limited aggregation: A growth model in which particles undergoing random motion aggregate upon reaching a growing cluster. Hele-Shaw displacement patterns can resemble branching clusters produced by this model.