Nonlinear phenomena in fluids with temperature-dependent viscosity: an hysteresis model for magma flow in conduits

Magma viscosity is strongly temperature-dependent. When hot magma flows in a conduit, heat is lost through the walls and the temperature decreases along the flow causing a viscosity increase. For particular values of the controlling parameters the steady-flow regime in a conduit shows two stable solutions belonging either to the slow or to the fast branch. As a consequence, this system may show an hysteresis effect, and the transition between the two branches can occur quickly when certain critical points are reached. In this paper we describe a model to study the relation between the pressure at the inlet and the volumetric magma flow rate in a conduit. We apply this model to explain an hysteric jump observed during the dome growth at Soufri\`ere Hills volcano (Montserrat), and described by Melnik and Sparks [1999] using a different model.