| Summary: | The stability of wall driven and thermally driven cavity flow is investigated for a wide range of viscous and viscoelastic fluids. The effect of inertia, elasticity, temperature gradients, viscous heating and Biot boundary conditions are of particular interest. Both destabilisation and bifurcation phenomenon are found. For Newtonian constant viscosity flow the instabilities are characterised by a critical Reynolds number which represents the ratio of inertial forces to viscous forces, and instability occurs when the inertial forces become large. For non-Newtonian viscoelastic fluids the instability is characterised by a critical Weissenberg number, which represents the ratio of elastic forces to viscous forces, and instability also occurs when elastic forces dominate the viscous forces. For thermally driven flow the instability is characterised by a critical Rayleigh number, which represents the ratio of temperature gradient to viscosity, and instability occurs when the Rayleigh number become large. In this case the instability is also characterised by both Eckert and Biot number. The work has relevance to thermal convection and mixing processes which occur in the viscous and viscoelastic fluid within the Earth's mantle. Three-dimensional steady and transient flow in a cylindrical cavity and three dimensional steady flow in a spherical cavity, are also considered for both viscous and viscoelastic fluids. Instabilities in these three-dimensional flow depend on the same parameters as the flow in square cavity.
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