We consider near-inertial waves continuously excited by a localized source and their subsequent radiation and evolution on a two-dimensional -plane. Numerical simulations are used to quantify the wave propagation and the energy flux in a realistically stratified ocean basin. We focus on the dynamics near and poleward of the inertial latitude where the local value of the Coriolis parameter matches the forcing frequency , contrasting the behaviour of waves under the traditional approximation (TA), where only the component of the Earth’s rotation aligned with gravity is retained in the dynamics, with that obtained under the non-traditional approach (non-TA) in which the horizontal component of rotation is retained. Under the TA, assuming inviscid linear wave propagation in the WKB limit, all energy radiated from the source eventually propagates toward the equator, with the initially poleward propagation being internally reflected at the inertial latitude. Under the non-TA however, these waves propagate sub-inertially beyond their inertial latitude, exhibiting multiple reflections between internal turning points that lie poleward of the inertial latitude and the bottom. The numerical experiments complement and extend existing theory by relaxing the linearity and WKB approximations, and by illustrating the time development of the steadily forced flow and the spatial patterns of energy flux and flux divergence. The flux divergence of the flow at both the forcing frequency and its first harmonic reveal the spatial patterns of nonlinear energy transfer and highlight the importance of nonlinearity in the vicinity of near-critical bottom reflection at the inertial latitude of the forced waves.

In the laboratory experiments by Fultz & Riehl (1957) and by Hide (1958) on heated rotating liquids contained in the annulus between two cylinders, it has been observed that a strongly marked jet stream appears on the free surface of the liquid under certain conditions of rotating and heating. This jet stream meanders around the annulus in a regular wave-like pattern alternately approaching the outer and inner cylindrical boundaries. The present paper puts forward an analytical theory for this jet, or Rossby regime, in the course of which exact solutions are presented of certain fundamental non-linear partial differential equations. The assumption is made that the flow is geostrophic at the first approximation and that the heat transfer across the stream lines of geostrophic flow (that is, the isobars) is due to molecular conduction. From a calculation of the heat flow it appears that this leads to values of the heat transfer which are too small, so that the ageostrophic terms must be of importance in the actual heat transfer; never-theless, the exact solution obtained here probably reveals the mechanism of the change from one wave pattern to another and certainly provides an explanation for the observed upper limit to the number of waves in a given geometrical configuration, as discussed by Hide. It has been established that the mean zonal flow and mean zonal temperature field are dependent upon the amplitude function of a finite amplitude wave solution. In this exact solution it is found that the amplitude and phase functions of the wave patterns are themselves interdependent and that the shape of the wave depends on the quantity of heat and angular momentum being transferred. It is shown that the wave pattern consisting of an integral number m lobes or petals can exist only in a restricted range of the Rossby number S—this is well-known from the experimental work of Fultz and Hide.

Using the method of matched asymptotic expansions, the interaction between axisymmetric laminar boundary layers and supersonic external flows is investigated in the limit of large Reynolds numbers. Numerical solutions to the interaction equations are presented for flare angles α that are moderately large. If α > 0 the boundary layer separates upstream of the corner and the formation of a plateau structure similar to the two-dimensional case is observed. In contrast to the case of planar flow, however, separation can occur also if α < 0, owing to the axisymmetric effect of overexpansion and recompression. The separation point then is located downstream of the corner and, most remarkable, a hysteresis phenomenon is observed.

In this paper we consider the flow field generated by a uniform electrostatic field in and about a pair of identical liquid drops immersed in a conducting fluid. It is assumed that the undisturbed electric field is parallel to the line joining the centres of the two drops. The flow field is due to the tangential electric stress over the surfaces of the drops and here this stress and the flow field are expressed in terms of bispherical harmonics. When the distance between the centres of the drops is many drop diameters the tangential electric stress and the flow field in and about one drop are unaffected by the presence of the other drop, as expected. When the distance between the centres of the drops is of the order of two drop diameters there is a substantial modification in the tangential electric stress at the surfaces of the drops and in the induced flow field, especially in the region between the planes through the drop centres perpendicular to the undisturbed electric field.

The surfactant concentration distribution on a planar uniform flow with a surface-piercing barrier was measured via the nonlinear optical technique of second-harmonic generation. The measurements were performed for an insoluble surfactant monolayer on the air/water interface. A theoretical model balancing surface elasticity and bulk shear at the interface was developed to predict the concentration profile for any insoluble monolayer. Measured equations of state, relating the surface tension to the surfactant concentration, were used in the model along with velocity data obtained using boundary-fitted digital particle image velocimetry. Theoretical concentration profiles were in agreement with experimental results. Additionally, global predictions from the model for four different insoluble surfactant systems also showed agreement with experimental measurements.

Mean momentum equation based analysis of polymer drag-reduced channel flow is performed to evaluate the redistribution of mean momentum and the mechanisms underlying the redistribution processes. Similar to channel flow of Newtonian fluids, polymer drag-reduced channel flow is shown to exhibit a four layer structure in the mean balance of forces that also connects, via the mean momentum equation, to an underlying scaling layer hierarchy. The self-similar properties of the flow related to the layer hierarchy appear to persist, but in an altered form (different from the Newtonian fluid flow), and dependent on the level of drag reduction. With increasing drag reduction, polymer stress usurps the role of the inertial mechanism, and because of this the wall-normal position where inertially dominated mean dynamics occurs moves outward, and viscous effects become increasingly important farther from the wall. For the high drag reduction flows of the present study, viscous effects become non-negligible across the entire hierarchy and an inertially dominated logarithmic scaling region ceases to exist. It follows that the state of maximum drag reduction is attained only after the inertial sublayer is eradicated. According to the present mean equation theory, this coincides with the loss of a region of logarithmic dependence in the mean profile.

The subject of the present study is the question of how the sound power of a jet of constant exit velocity would vary if the jet exit density were varied. Changes in jet exit density would inevitably be accompanied in a real experiment by changes in the speed of sound (temperature) in the jet, so that both effects must be considered simultaneously. The point of view advanced at the end of the study is that experimentally observed results in this area seem to admit an explanation based on how the radiative efficiency of moving acoustic sources is affected by the shrouding effect of a jet flow whose velocity, temperature and density differ from those of the ambient fluid. This change in efficiency is calculated with the aid of a simple model as follows. We determine the acoustic power output of a convected monopole source, simple harmonic in its own frame of reference, moving along the axis of a plug-flow round jet whose velocity is the same as that of the source. The jet is doubly infinite and the source is assumed to have an infinite lifetime. The density and temperature of the jet are allowed to differ from those of the ambient fluid though the specific-heat ratio of the jet fluid is assumed to be the same as that of the ambient. The requirement of equality of the static pressure inside and outside the jet then calls for a certain restraint on how the jet density and temperature vary. For a specific value of the jet exit velocity, the variation of acoustic power with the ratio of jet to ambient density along with a simple assumption on how the source strength varies with jet density are employed to deduce theoretically the ‘jet density exponent for jets which are subsonic with respect to the ambient speed of sound. The jet density exponent is found to depend both on the jet Mach number and even more strongly on a source frequency parameter. The theoretical results are compared with some experimental studies of this problem. Encouraging agreement is obtained both for the detailed observed effects on the power spectrum and the exponent for the overall power.

The reorientation phenomenon of a single red blood cell during sedimentation is simulated using the boundary element method. The cell settles downwards due to a density difference between the internal and external fluids, and it changes orientation toward a vertical orientation regardless of Bond number or viscosity ratio. The reorientation phenomenon is explained by a shape asymmetry caused by the gravitational driving force, and the shape asymmetry increases almost linearly with the Bond number. When velocities are normalised by the driving force, settling/drifting velocities are weak functions of the Bond number and the viscosity ratio, while the angular velocity of the reorientation drastically changes with these parameters: the angular velocity is smaller for lower Bond number or higher viscosity ratio. As a consequence, trajectories of the sedimentation are also affected by the angular velocity, and blood cells with slower reorientation travel longer distances in the drifting direction. We also explain the mechanism of the reorientation using an asymmetric dumbbell. From the analysis, we show that the magnitude of the angular velocity is explained by two main factors: the shape asymmetry and the instantaneous orientation angle.

A theoretical study is carried out for bubble oscillation in a compressible liquid with significant acoustic radiation based on the Keller–Miksis equation using a multi-scaled perturbation method. The leading-order analytical solution of the bubble radius history is obtained to the Keller–Miksis equation in a closed form including both compressible and surface tension effects. Some important formulae are derived including: the average energy loss rate of the bubble system for each cycle of oscillation, an explicit formula for the dependence of the oscillation frequency on the energy, and an implicit formula for the amplitude envelope of the bubble radius as a function of the energy. Our theory shows that the frequency of oscillation does not change on the inertial time scale at leading order, the energy loss rate on the long compressible time scale being proportional to the Mach number. These asymptotic predictions have excellent agreement with experimental results and the numerical solutions of the Keller–Miksis equation over very long times. A parametric analysis is undertaken using the above formula for the energy of the bubble system, frequency of oscillation and minimum/maximum bubble radii in terms of the dimensionless initial pressure of the bubble gases (or, equivalently, the dimensionless equilibrium radius), Weber number and polytropic index of the bubble gas.

The inviscid and viscous stability problems are addressed for a boundary layer which can support both Görtler and crossflow vortices. The change in structure of Görtler vortices is found when the parameter representing the degree of three-dimensionality of the basic boundary-layer flow under consideration is increased. It is shown that crossflow vortices emerge naturally as this parameter is increased and ultimately become the only possible vortex instability of the flow. It is shown conclusively that at sufficiently large values of the crossflow there are no unstable Görtler vortices present in a boundary layer which, in the zero-crossflow case, is centrifugally unstable. The results suggest that in many practical applications Görtler vortices cannot be a cause of transition because they are destroyed by the three-dimensional nature of the basic state. In swept-wing flows the Görtler mechanism is probably not present for typical angles of sweep of about 20°.

Some discussion of the receptivity problem for vortex instabilities in weakly three-dimensional boundary layers is given; it is shown that inviscid modes have a coupling coefficient marginally smaller than those of the fastest growing viscous modes discussed recently by Denier, Hall & Seddougui (1991). However, the fact that the growth rates of the inviscid modes are the larger in most situations means that they are probably the more likely source of transition.

The problem of an unsteady axisymmetric expansion of a monatomic gas into a vacuum is considered in the limit of small source Knudsen number. It is shown that a solution of the Boltzmann equation for Maxwell molecules valid for large time can be constructed, which matches with the known equilibrium solution for an inviscid expansion of a fixed mass of gas into a vacuum provided that the region near the zero density front is excluded. This solution is formally the same as that obtained for the similar problem of steady spherical expansion into a vacuum—the variations along each particle path of the unsteady flow being the same as that in the steady flow.

Near the front, the expansion procedure breaks down and the equations require a different scaling. A modified form of the Boltzmann equation is obtained which leads to a corresponding set of moment equations. Unfortunately, the set of moment equations is no longer closed and no essential simplification has been made.

The aim of this paper is to develop a stability theory and a numerical model for three density-stratified electrically conductive liquid layers. Using regular perturbation methods to reduce the full three-dimensional problem to the shallow layer model, the coupled wave and electric current equations are derived. The problem set-up allows for weakly nonlinear velocity field action and an arbitrary vertical magnetic field. Further linearisation of the coupled equations is used for the linear stability analysis in the case of a uniform vertical magnetic field. New analytical stability criteria accounting for the viscous damping are derived for particular cases of practical interest and compared to the numerical solutions for a variety of materials used in batteries. These new criteria are equally applicable to the aluminium electrolysis cell magnetohydrodynamic (MHD) stability estimates.

We present a theoretical stability analysis of the flow after the sudden release of a fixed mass of fluid on an inclined plane formally restricted to relatively long time scales, for which the kinematic regime is valid. Shallow-water equations for steep slopes with bed stress are employed to study the threshold for the onset of roll waves. An asymptotic solution for long-wave perturbations of small amplitude is found on background flows with a Froude number value of 2. Small disturbances are stable under this condition, with a linear decay rate independent of the wavelength and with a wavelength that increases linearly with time. For larger values of the Froude number it is shown that the basic flow moves at a different scale than the perturbations, and hence the wavelength of the unstable modes is characterized as a function of the plane-parallel Froude number Frp and a measure of the local slope of the free-surface height φ by means of a multiple-scale analysis in space and time. The linear stability results obtained in the presence of small non-uniformities in the flow, φ > 0, introduce substantial differences with respect to the plane-parallel flow with φ = 0. In particular, we find that instabilities do not occur at Froude numbers Frcr much larger than the critical value 2 of the parallel case for some wavelength ranges. These results differ from that previously reported by Lighthill & Whitham (Proc. R. Soc. A, vol. 229, 1955, pp. 281–345), because of the fundamental role that the non-parallel, time-dependent characteristics of the kinematic-wave play in the behaviour of small disturbances, which was neglected in their stability analyses. The present work concludes with supporting numerical simulations of the evolution of small disturbances, within the framework of the frictional shallow-water equations, that are superimposed on a base state which is essentially a kinematic wave, complementing the asymptotic theory relevant near the onset. The numerical simulations corroborate the cutoff in wavelength for the spectrum that stabilizes the tail of the dam-break flood.

Motivated by recent studies that have revealed the existence of trapped acoustic waves in subsonic jets (Towne et al., J. Fluid Mech., vol. 825, 2017, pp. 1113–1152), we undertake a more general exploration of the physics associated with acoustic modes in jets and wakes, using a double vortex-sheet model. These acoustic modes are associated with eigenvalues of the vortex-sheet dispersion relation; they are discrete modes, guided by the vortex sheet; they may be either propagative or evanescent; and under certain conditions they behave in the manner of acoustic-duct modes. By analysing these modes we show how jets and wakes may both behave as waveguides under certain conditions, emulating ducts with soft or hard walls, with the vortex-sheet impedance providing effective ‘wall’ conditions. We consider, in particular, the role that upstream-travelling acoustic modes play in the dispersion-relation saddle points that underpin the onset of absolute instability. The analysis illustrates how departure from duct-like behaviour is a necessary condition for absolute instability, and this provides a new perspective on the stabilising and destabilising effects of reverse flow, temperature ratio and compressibility; it also clarifies the differing symmetries of jet (symmetric) and wake (antisymmetric) instabilities. An energy balance, based on the vortex-sheet impedance, is used to determine stability conditions for the acoustic modes: these may become unstable in supersonic flow due to an energy influx through the shear layers. Finally, we construct the impulse response of flows with zero and finite shear-layer thickness. This allows us to show how the long-time wavepacket behaviour is indeed determined by interaction between Kelvin–Helmholtz and acoustic modes.

The effect of gravity modulation on long-wavelength disturbances at the free surface of a surfactant-covered thin liquid layer is analysed. The surfactants are assumed to be insoluble so that variations in their concentration along the free surface produce Marangoni flows in the underlying liquid. Lubrication theory is applied to obtain nonlinear partial differential equations that describe the behaviour of the free surface height and surfactant concentration, and the stability of these equations to small-amplitude disturbances is examined by applying Floquet theory. It is found that long-wavelength disturbances are destabilized by gravity modulation when surfactants are present, whereas such disturbances are stable when surfactants are absent. Results from additional calculations indicate that the instability becomes more difficult to excite as the Marangoni forces, body forces, capillary forces and surfactant diffusivity increase, and becomes easier to excite as the van der Waals forces increase.