Capillary rise of a viscoplastic fluid in a Hele-Shaw cell

The capillary-driven rise between two plates of a yield-stress fluid modelled by the Herschel-Bulkley constitutive law is considered. For the geometry of a relatively narrow (Hele-Shaw) cell, the mathematical problem simplifies considerably, the dynamics being captured by a viscoplastic generalization of Darcy's law. This formulation can be used to determine the height of rise within a cell with varying gap thickness. In the limit that the gap varies over a wider scale than the height to which the fluid can rise, the problem reduces to one-dimensional, viscoplastic capillary rise, the solution of which has been given previously and compared with experiments. More generally, the dynamics is richer, with the capillary pressures causing fluid to first rise and then plug up parts of the cell.