The constitutive equation and flow dynamics of bubbly magmas

A generalized constitutive equation for bubbly liquids is presented which successfully reproduces the expected viscosity response for both steady flows with varying capillary number Ca (a measure of the bubble deformation) and unsteady flows with varying dynamic capillary number Cd (a measure of the steadiness of the flow) previously given in separate studies. The constitutive equation is given in terms of observable material and flow parameters and is valid at least up to Casimilar toO(1000) and Cdsimilar toO(10). Analytical solutions are presented for steady, simple-shearing flow (Ca variable) and oscillatory flow with small total-strains (Cd variable). The special case of steady flow in a circular pipe-analogous to magma flow in a volcanic conduit-is investigated. Velocity profiles across the conduit are found to be parabolic or plug-flow depending on a dimensionless number, the conduit capillary number Cc. Plug flow is predicted for Ccapproximate to4 which is in the mid-range for volcanic-eruption conditions.