A physical mechanism for the long-wave instability of thin liquid films is presented. We show that the many diverse systems that exhibit this instability can be classified into two large groups. Each group is studied using the model of a thin liquid film with a deformable top surface flowing down a rigid inclined plane. In the first group, the top surface has an imposed stress, while in the other, an imposed velocity. The proposed mechanism shows how the details of the energy transfer from the basic state to the disturbance are handled differently in each of these cases, and how a common growth mechanism produces the unstable motion of the disturbance.

A one-dimensional model is derived for natural convection in a closed loop. The physical model can be reduced to a set of nonlinear ordinary differential equations of the Lorenz type. The model is based on a realistic heat transfer law and also accounts for a non-symmetric arrangement of heat sources and sinks. A nonlinear analysis of these equations is performed as well as experiments to validate the model predictions.

Both the experimental and the analytical data show that natural convection in a loop is characterized by strong nonlinear effects. Distinct subcritical regions are observed in addition to a variety of stable steady flow regimes. Thus, in certain ranges of the forcing parameter the flow stability depends significantly on the presence of finite perturbation amplitudes. An absolutely unstable range also exists which is characterized by a chaotic time behaviour of the flow quantities. It is also shown that the steady solutions are subject to an imperfect forward bifurcation if heating of the loop is performed non-symmetrically. In such a case one flow direction is preferred at the onset of convection and, moreover, the corresponding steady solution is more stable than a second, isolated, steady solution. The second solution has the opposite flow direction and is stable only in a relatively small, isolated interval. The preferred steady solution becomes unstable against time-periodic perturbations at a higher value of the forcing parameter. A backward- or a forward-directed bifurcation of the periodic solutions is found depending on the non-symmetry parameter of the system.

Finite-amplitude solutions of plane Couette flow are discovered. They take a steady three-dimensional form. The solutions are obtained numerically by extending the bifurcation problem of a circular Couette system between co-rotating cylinders with a narrow gap to the case with zero average rotation rate.

When a body of fluid with a vertical salinity gradient is heated from a vertical sidewall instabilities are sometimes observed. The linear stability of this basic state has been investigated by Kerr (1989). This linear theory predicts the onset of instability well when compared with experimental results; however, the form of the observed nonlinear instabilities does not coincide with the linear predictions (cf. Chen, Briggs & Wirtz 1971; Tsinober & Tanny 1986; Tanny & Tsinober 1988). In this paper we investigate some of the nonlinear aspects of the problem. A weakly nonlinear analysis reveals that the bifurcation into instability is subcritical, and that the initial trend along this branch of solutions is towards the co-rotating cells observed in experiments. The heating levels for which instabilities are absent are investigated by the use of energy stability analysis. This yields a weak result for arbitrary disturbances, showing that disturbances will decay for sufficiently low wall heating. This bound is greatly strengthened by imposing a vertical periodicity on the lengthscale proposed by Chen et al.

The deformation of a viscous drop, driven by buoyancy towards a solid surface or a deformable interface, is analysed in the asymptotic limit of small Bond number, for which the deformation becomes important only when the drop is close to the solid surface or interface. Lubrication theory is used to describe the flow in the thin gap between the drop and the solid surface or interface, and boundary-integral theory is used in the fluid phases on either side of the gap. The evolution of the drop shape is traced from a relatively undeformed state until a dimple is formed and a long-time quasi-steady-state pattern is established. A wide range of drop to suspending phase viscosity ratios is examined. It is shown that a dimple is always formed, independently of the viscosity ratio, and that the long-time thinning rates take simple forms as inverse fractional powers of time.

When a coiled tube is rotated about the coil axis, the effects of rotation interact with centrifugal and viscous effects to complicate the flow characteristics beyond those seen in stationary curved ducts. The phenomena encountered are examined for steady, fully developed Newtonian flow in circular tubes of small curvature. The governing equations are solved using orthogonal collocation, and the results presented cover both the nature of the flow and the bifurcation structure. When rotation is in the same direction as the axial flow imposed by a pressure gradient, the flow structure remains similar to that seen in stationary ducts, i.e. with two- or four-vortex secondary flows in addition to the axial flow. There are, however, quantitative changes, which are due to the Coriolis forces resulting from rotation. The bifurcation structure also shows only quantitative changes from that for stationary ducts at all values of Taylor number examined. More complex behaviour is possible when rotation opposes the flow due to the pressure gradient. In particular, the direction of the secondary flow may be reversed at higher rotational strengths, and the mechanism of the flow reversal is explored. The flow reversal occurs smoothly at low Taylor numbers, but at higher rotational strengths a cusp appears in the primary solution branch in the vicinity of the flow reversal.

By using the multiple-scales perturbation method, analytical solutions are obtained for the second-order low-frequency oscillations inside a rectangular harbour excited by incident wave groups. The water depth is a constant. The width of the harbour entrance is of the same order of magnitude as the wavelength of incident carrier (short) waves, but small in comparison with the wavelength of the wave envelope. Because of the modulations in the wave envelope, a second-order long wave is locked in with the wave envelope and propagates with the speed of the group velocity. Outside the harbour, locked long waves also exist in the reflected wave groups, but not in the radiated wave groups. Inside the harbour, the analytical expressions for the locked long waves are obtained. Owing to the discontinuity of the locked long waves across the harbour mouth, second-order free long waves are generated. The free long waves propagate with a speed of (gh)½ inside and outside the harbour. The free long waves inside the harbour may be resonated in a low-frequency range which is relevant to the harbour resonance.

The nonlinear Rayleigh–Taylor instability of a liquid layer resting on a plane wall below a second liquid of higher density is considered. Under the assumption of creeping flow, the motion is studied as a function of surface tension and the ratio of the viscosities of the two fluids. The flow induced by the deformation of the layer is represented by an interfacial distribution of Green's functions. A Fredholm integral equation of the second kind is derived for the density of the distribution, and is solved by successive iteration. The results show that for small and moderate surface tension, the instability of the layer leads to the formation of a periodic array of viscous plumes which penetrate into the overlying fluid. The morphology of these plumes strongly depends upon the viscosity ratio and surface tension. When the viscosity of the overlying fluid is comparable with or larger than that of the layer, the plumes are composed of a well-defined leading drop on top of a narrow stem. When the viscosity of the overlying fluid is smaller than that of the layer, the plumes take the form of a compact column of rising fluid. The size of the drop leading a plume is roughly proportional to the initial thickness of the layer. When surface tension is sufficiently small, ambient fluid is entrained into the leading drop and circulates in a spiral pattern. Convection currents generated by the rising plumes are visualized with streamline patterns, and the rate of thinning of the remnant layer, as well as the speed of the rising drop or plumes, are discussed.

Energy stability theory has been applied to a basic state of thermocapillary convection occurring in a cylindrical half-zone of finite length to determine conditions under which the flow will be stable. Because of the finite length of the zone, the basic state must be determined numerically. Instead of obtaining stability criteria by solving the related Euler–Lagrange equations, the variational problem is attacked directly by discretization of the integrals in the energy identity using finite differences. Results of the analysis are values of the Marangoni number, MaE, below which axisymmetric disturbances to the basic state will decay, for various values of the other parameters governing the problem.