Thin-gap averaging of variable-viscosity flows: application to thermoviscous fingering
A consistent averaging technique, using the weighted residual integral boundary layer (WRIBL) method, is presented for flow through a thin-gap geometry wherein the fluid's viscosity varies across the gap. In such situations, the flow has a non-parabolic cross-gap velocity profile -- an effect that is ignored by Darcy models conventionally used for such Hele-Shaw flows. The WRIBL technique systematically accounts for the cross-gap variation of viscosity and yields reduced-order equations for the gap-averaged fluid flow rate. As a test case, we consider a fluid with a temperature-dependent viscosity and analyse the previously-studied problem of thermoviscous fingering: a hot fluid flowing through a Hele-Shaw geometry with cold walls spontaneously forms channels of low-viscosity, hot fluid, separated by regions of high-viscosity, cold fluid. The temperature of the cold walls is assumed either to be constant, a scenario that mimics the upward flow of magma through fissures in the Earth's crust, or to vary linearly along the direction of flow. In both cases, the predictions of the WRIBL model, regarding the multiplicity of uniform steady flow states and their linear stability, are compared with that of the hitherto used, ad hoc, Darcy model (Helfrich, J. Fluid Mech., vol. 305, 1995, pp. 219-238), as well as with calculations of the full three-dimensional governing equations (Wylie and Lister, J. Fluid Mech., vol. 305, 1995, pp. 239-261). Though the results are qualitatively similar, the WRIBL model is found to be much more accurate than the Darcy model. The averaging method is presented in a general manner to facilitate its application to other physical situations where, for example, the viscosity depends on solute/particle concentration in a solution/suspension.