Dynamics of cooling viscoplastic domes

A variety of problems in engineering and geology involve spreading cooling non-Newtonian fluids. If the fluid is relatively shallow and spreads slowly, lubrication-style asymptotic approximations can be used to build reduced models for the spreading dynamics. The centrepiece of such models is a nonlinear diffusion equation for the local fluid thickness, and ideally this should become coupled to a correspondingly simple equation determining the local temperature field. However, when heat diffuses relatively slowly as the fluid flows, we cannot usefully reduce the temperature equation, and the asymptotic reduction couples the local thickness equation to an advection diffusion equation that crucially involves diffusion in the vertical. We present an efficient computational algorithm for numerically solving this more complicated type of lubrication model, and describe a suite of solutions that illustrate the dynamics captured by the model in the case of an expanding Bingham fluid with a temperature-dependent viscosity. Based on these solutions, we evaluate two simpler models that further approximate the temperature equation: a vertically isothermal theory, and a ‘skin theory’. The latter is based on the integral-balance method of heat-transfer theory, and demands that the vertical structure of the temperature field has the form of an advancing boundary layer, or skin. The vertically isothermal model performs well when the thermal conductivity is relatively large. The skin theory reproduces the full dynamics qualitatively, if not quantitatively, for all thermal conductivities. The main errors in both models arise near the fluid edge, where the numerical solutions show that chilled fluid is overridden as the fluid expands, creating an underlying collar of cold material. Encouraged by the success of the skin model, we extend the theory by incorporating extensional stresses in the skin, which emerge when cooling induces an extreme rheological change in the material, such as an exponential rise in the viscosity. The model predicts that when skin stresses are sufficiently strong, the skin is brought to rest, whilst hotter fluid expands underneath.