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We consider the long-term evolution of an axisymmetric bubble and explore the ways in which it may develop. Linearised inviscid analysis is used to predict the stability of the bubble with a small disturbance while a nonlinear inviscid extension shows that the growth of unstable modes is ultimately limited by the formation of axisymmetric curvature singularities. The addition of surface tension is shown to delay, but not entirely prevent, these singularities. Our results are found to agree well with a viscous Boussinesq theory at least to early times. The inclusion of viscosity means that the development of the bubble structure is not limited by the creation of singularities, and the bubble may ultimately adopt one of a wide range of possible large-scale deformations. Among these, perhaps the most exotic are jet-like structures which can pinch off and break into several distinct parts. Spectral methods are employed to solve the inviscid and Boussinesq models while the linearised inviscid model admits a closed-form series solution. [ABSTRACT FROM AUTHOR]

A viscous fluid is confined between two smooth horizontal walls, in a vertical channel. The upper wall may move with constant speed, but the lower wall is stationary and a portion of it is heated. A plume of heated fluid develops, and may also be swept downstream by the motion of the upper wall. When the heating effect is small and the upper plate does not move, a closed-form solution for the temperature profile is presented. A numerical spectral method is then presented, and allows highly accurate nonlinear solutions to be obtained, for the temperature and the fluid motion. These are compared against the closed-form solution in the linearized case, and the effects of nonlinearity on temperature and velocity are revealed. The results also show that periodic plume shedding from the heated region can occur in the nonlinear case. [ABSTRACT FROM AUTHOR]