Thermo-viscous fingering instability in cooling and spreading flows

Dr Shailesh Naire (Keele University)

Thursday 25th January 14:00-15:00 Maths 311B / Zoom (Meeting ID: 894 0173 1730)

Abstract

Molten fluid flows that cool as they spread are important in a wide variety of contexts, e.g., lava domes in geophysical flows and coolant in nuclear reactors. The interplay between the flow and cooling can also 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 drop extruding from a source and spreading over an inclined plane that is covered initially with a thin liquid precursor film. Lubrication theory is employed to model the one-dimensional spreading flow using coupled nonlinear evolution equations for the film 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 drop-air and drop-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 solutions illustrating the dynamics of a spreading flow undergoing cooling. These solutions are compared to a simpler model that results due to a further approximation of the temperature equation in the limit of small Péclet number.

The stability of the one-dimensional solutions to small-amplitude variations in the thickness and temperature in the transverse direction is also investigated using linear stability and transient growth analysis, and numerical simulations. The existence of a thermo-viscous fingering instability is revealed. Two-dimensional numerical simulations confirm the stability analysis elucidating the underlying thermo-viscous mechanism.

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