The thermo-viscous instability of a cooling spreading liquid dome on an inclined substrate

Molten liquid flows that cool as they spread are important in a wide variety of contexts, e.g., lava domes in geophysical flows, reactor core melt in nuclear engineering and molten metal coating flows in chemical and metallurgical engineering. The interplay between the flow and cooling can give rise to a variety of intriguing flow features and fingering instabilities.

Motivated by the above, we consider theoretically a model system of a molten viscous planar liquid dome extruding from a source and spreading over an inclined substrate. The liquid in the dome cools as it spreads, losing its heat to the surrounding colder air and substrate. Lubrication theory is employed to model the spreading flow using coupled nonlinear evolution equations for the dome’s thickness and temperature. The coupling between flow and cooling is via a constitutive relationship for the temperature-dependent viscosity. This model is parameterized by the heat transfer coefficients at both the dome-air and dome-substrate interfaces, the Péclet number, the viscosity-temperature coupling parameter and the substrate inclination angle. A systematic exploration of the parameter space reveals a variety of 1D free surface shapes illustrating the dynamics of a spreading flow undergoing cooling. In particular, we distinguish a new type of solution with the hotter and more mobile liquid piling up behind the dome's colder and less mobile leading edge, forming a distinct hump. A mapping in parameter space is constructed to characterize the existence of the humped solution.

The stability of the 1D hump solution to small-amplitude transverse variations is investigated using linear stability analysis and numerical simulations. The existence of a thermo-viscous fingering instability is revealed; for this instability to occur the presence of the hump is shown to be necessary. Two-dimensional simulations confirm the stability analysis elucidating the underlying thermo-viscous mechanism.