Asymptotic models of solidification in cooling thin-layer flows of a highly viscous fluid

Asymptotic models are constructed for the solidification process in a highly viscous film flow on the surface of a cone with a given mass supply at the cone apex. In the thin-layer approximation, the problem is reduced to two parabolic equations for the temperatures of the liquid and the solid coupled with an ordinary differential equation for the solidification front. For large Péclet numbers, an analytical steady-state solution for the solidification front is found. A nondimensional parameter which makes it possible to distinguish flows (i) without a solid crust, (ii) with a steady-state solid crust, and (iii) with complete solidification is determined. For finite Péclet numbers and large Stefan numbers, an analytical transient solution is found and the time of complete flow solidification is determined. In the general case, when all the governing parameters are of the order of unity, the original system of equations is studied numerically. The solutions obtained are qualitatively compared with the data of field observations for lava flows produced by extrusive volcanic eruptions.