Ghanim Mohammed Salih Algwauish
The non-Newtonian and non-isothermal spreading of liquid domes using mathematical and numerical methods
Algwauish, Ghanim Mohammed Salih
This thesis is concerned with the spreading of a large mass of fluid under the influence of gravity and viscous forces, referred to as a viscous-gravity current. The focus is on a particular class of viscous-gravity currents which involve the spreading of a hot fluid undergoing cooling as it flows. The flow of lava is a primary example and is what motivates this work. The ow and cooling are strongly coupled. The fluid properties, such as the viscosity, are temperature-dependent and flows also exhibit non-Newtonian behaviour due to compositional changes as a result of cooling, such as an apparent viscosity and a yield stress. Conversely, the flow convects the heat causing cooling.
A consequence of this is the development of dynamic flow patterns, such as fingering-type instabilities, e.g., toe-shaped protrusions at an advancing lava flow front. These behaviours have motivated theoreticians to understand the interplay between flow and cooling and the mechanisms behind these instabilities.
This work develops a theoretical model of a planar liquid dome spreading down an inclined substrate due to gravity. This model incorporates non-Newtonian and viscoplastic behaviour, a temperature-dependent viscosity and heat transfer boundary conditions at the dome's free surface and the underlying substrate. A power-law and Carreau constitutive law is used to describe the non-Newtonian behaviour and a Herschel-Bulkley constitutive law to model the viscoplastic behaviour. Two viscosity-temperature relationships, an exponential and a bi viscosity model, are considered. We combine numerical simulations and similarity solutions to perform a parameter study on the influence of key parameters on the free surface shapes and spreading behaviour, such as the apparent viscosity, yield stress, Peclet number (compares conductive and convective heat transport), temperature viscosity coupling constant and the surface and substrate heat transfer coefficients. Our one-dimensional results reveal a variety of free surface shape profiles, such as symmetric domes, slumped domes, pancake domes and overriding fluid humps. A two-dimensional numerical linear stability analysis reveals the stability characteristics of the above one-dimensional shapes to a small-amplitude transverse perturbation. We have identified a fingering instability based on a thermo-viscous mechanism. The viscosity-temperature coupling is identified as the key parameter that controls the growth rate of the instability and the band of unstable wavenumbers. We provide the necessary conditions on the base state for the onset of the instability. The preliminary work undertaken here provides the basis for doing a thorough theoretical analysis of the instability and for exploring the nonlinear stability of the flow.
|Jun 1, 2019