Capillary rise of yield-stress fluids

The rise of a viscous fluid between two plates driven by capillary effects is a classical problem in fluid mechanics, giving rise to Jurin's law for the elevation attained. In this talk I will reconsider this problem for a yield-stress fluid modelled by the Herschel-Bulkley constitutive law. Theoretically, the problem can be simplifed for the geometry of a relatively narrow (Hele-Shaw) cell, the non-Newtonian properties of the fluid being captured by a viscoplastic generalization of Darcy's law. This formulation leads to an interesting mathematical problem for the height to which a yield-stress fluid can rise within a cell with varying gap thickness, a problem that can be solved using the method of characteristics. In the limit that the gap varies over a wider scale than the height to which the fluid can rise, the problem reduces to a much simpler problem equivalent to one-dimensional, viscoplastic capillary rise, the solution of which has been given previously and compared with experiments. More generaliiy, the dynamics is richer, with the capillary pressures causing fluid to first rise and then plug up parts of the cell.