On the derivation of the symmetric and asymmetric Hele–Shaw flow equations for viscous and viscoplastic fluids using the viscometric fluidity function

In viscometric flows, the viscometric fluidity function is used to define the shear rate as a function of the shear stress. This relationship is unique even if the viscosity function is constant or has a plateau. As far as a viscoplastic fluid is concerned, the viscometric function is zero when the magnitude of the shear stress is below the yield stress and is positive when the fluid has yielded. Using this fluidity function and integration by parts, it is quite simple to integrate the equations of motion defining a Hele–Shaw flow in viscous and viscoplastic fluids for both symmetric and asymmetric flow conditions. From this, the average velocity field is derived in a manner which is superior to those based on the viscosity–shear rate relationship, especially since such a relation is not invertible in all cases.