A class of nonlinear boundary layers in a hydromagnetic flow under differential rotation is analysed. The function of these layers is to provide smooth transition from the conditions within the electrically conducting container to those in the region far away from the boundary through the flow regime. The structure of the double-decker boundary layer depends on the diffusivity of the fluid, the conductance of the rigid boundary and the relative strength of the applied magnetic field. The method of multiple scales is used to obtain uniformly valid solutions with the conductance of the container varying from zero to infinity. It is found that even a small differential rotation (ε → 0) can induce perturbations of order ε½ or of order unity in the field functions.

The flow field in the nose region of a blunt body in hypersonic flow is studied by considering the transport of vorticity and enthalpy. The entire region between body and shock is considered to be viscous, not necessarily thin in comparison with the nose radius of the body and to be of slowly varying density. The (given) post-shock vorticity need not be small and the density ratio ρ∞/ρs may either be small or near unity, the analysis being valid asymptotically at both limits.

It is found that the vorticity equation may be uncoupled from the total enthalpy equation if μ√ρ is constant. While the equations are not expected to be necessarily restricted to the immediate vicinity of the stagnation line, only there can the solution be written down explicitly; elsewhere, numerical integration is required.

A novel analytical equation for the assessment of the accuracy of filters used for the interpolation and differentiation of scattered experimental data is presented. The equation takes into account the statistical nature of the filter output resulting from both the arbitrary positions of the data points and the randomness and noise present in the experimental data. Numerical estimation of the accuracy of the filter, using a Monte Carlo procedure, shows good agreement with the deduced analytical equation. This numerical procedure was also used to determine the accuracy of variance filters aimed at calculating the mean-square fluctuation of experimental data. The combination of the numerical results and analytical equations reveals the exact sources of inaccuracy arising in scattered point filters, namely: (i) the spectral inaccuracy of the weighting function; (ii) the noise or stochastic signal amplification; and (iii) the error arising from the random collocation of points within the filter window. The results also demonstrate that the use of the local mean in the calculation of the quadratic fluctuation leads to smaller estimation errors than the central mean. Finally, all these filters are used and critically evaluated in the framework of the stochastic position, diameter, and velocity of bubbles in a gas-fluidized bed. It is shown that the empirical coefficient of bubble coalescence in the two-dimensional bed tested, $\bar {\lambda } $, is in the range 2.0–2.4 when incorporating only the visible flow of bubbles. Here, the vertical distance over which a bubble survives without coalescing is $\bar {\lambda } {L}_{c} $, where ${L}_{c} $ is the characteristic separation between neighbouring bubbles in the horizontal direction prior to coalescence. It was also seen that the relative mean-square-root fluctuation of both bubble diameter and velocity is more than 50 % at the centre of the bed and remains nearly constant along the height of the bed.

We study rotating homogeneous turbulent convection forced by a mean vertical temperature gradient by means of direct numerical simulations in the Boussinesq approximation in a rotating frame. In the absence of rotation, our results are in agreement with the ‘ultimate regime of thermal convection’ for the scaling of the Nusselt and Reynolds numbers versus Rayleigh and Prandtl numbers. Rotation is found to increase both $Nu$ and $Re$ at fixed $Ra$ with a maximum enhancement for intermediate values of the Rossby numbers, qualitatively similar, but with stronger intensity, to what is observed in Rayleigh–Bénard rotating convection. Our results are interpreted in terms of a quasi-bidimensionalization of the flow with the formation of columnar structures displaying strong correlation between the temperature and the vertical velocity fields.

The results of Ursell (1964) are confirmed by proving that there are no additional contributions to Ursell's integral from singularities of the integrand at infinity. The method consists of proving that the asymptotic expansion of a force coefficient Δ(N) is uniformly valid in a finite sector of the complex N-plane. This in turn requires that the kernel of an integral equation remains small in this sector.

An analysis of the structure of the wakes and waves in steady compressible magnetohydrodynamics is presented. No restriction is made on the equation of state of the gas or on the ratios of the various dissipative parameters. An asymptotic solution is obtained which furnishes directly the flow far from a body and which may be used in the construction of the entire flow field. The non-dissipative solutions are obtained as a non-uniform limit for vanishing dissipation; no matter how small the dissipation, one can go far enough from the origin that the flow is essentially dissipative. For non-aligned fields the wave pattern consists of a downstream wake and either two or four standing waves, depending on the flow regime. For aligned fields, two of these waves become wakes, so that the wake is a superposition of three structured layers, with either all downstream or two downstream and one up-stream. It is found that the non-dissipative limit of the wake is non-unique for the aligned fields case. Different limits are obtained depending on how the various dissipative parameters vanish.

This note provides numerical values for the long-wave limit of the virtual-mass coefficient relating to the heaving motion of a half-immersed circular cylinder on water of finite depth, found analytically by Ursell in the preceding paper; some preliminary analysis is needed, however.

The effect of a parallel magnetic field upon the stability of the plane interface between two conducting viscous fluids in uniform relative motion is considered. A parameter reduction, which has not previously been noted, is employed to facilitate the solution of the problem. Neutral stability curves for unrestricted ranges of the governing parameters are found, and the approximate solutions of other authors are examined in this light.

An investigation is made into the propagation of a one-dimensional combustion wave, which consists of a flame front and a precursor shock wave which pass down a tube closed at one end, in the presence of a transverse magnetic field in the undisturbed gas at rest. The shock wave is assumed to be of sufficient strength to ionize completely the initially non-electrically-conducting gas and the conditions at the flame front are taken to satisfy the Chapman–Jouguet condition. Details of the solution are compared with the corresponding results for ordinary gasdynamic deflagration.

A Lagrangian fluctuation–dissipation relation has been derived in a previous work to describe the dissipation rate of advected scalars, both passive and active, in wall-bounded flows. We apply this relation here to develop a Lagrangian description of thermal dissipation in turbulent Rayleigh–Bénard convection in a right-cylindrical cell of arbitrary cross-section, with either imposed temperature difference or imposed heat flux at the top and bottom walls. We obtain an exact relation between the steady-state thermal dissipation rate and the time $\unicode[STIX]{x1D70F}_{mix}$ for passive tracer particles released at the top or bottom wall to mix to their final uniform value near those walls. We show that an ‘ultimate regime’ with the Nusselt number scaling predicted by Spiegel (Annu. Rev. Astron., vol. 9, 1971, p. 323) or, with a log correction, by Kraichnan (Phys. Fluids, vol. 5 (11), 1962, pp. 1374–1389) will occur at high Rayleigh numbers, unless this near-wall mixing time is asymptotically much longer than the free-fall time $\unicode[STIX]{x1D70F}_{free}$. Precisely, we show that $\unicode[STIX]{x1D70F}_{mix}/\unicode[STIX]{x1D70F}_{free}=(RaPr)^{1/2}/Nu,$ with $Ra$ the Rayleigh number, $Pr$ the Prandtl number, and $Nu$ the Nusselt number. We suggest a new criterion for an ultimate regime in terms of transition to turbulence of a thermal ‘mixing zone’, which is much wider than the standard thermal boundary layer. Kraichnan–Spiegel scaling may, however, not hold if the intensity and volume of thermal plumes decrease sufficiently rapidly with increasing Rayleigh number. To help resolve this issue, we suggest a program to measure the near-wall mixing time $\unicode[STIX]{x1D70F}_{mix}$, which is precisely defined in the paper and which we argue is accessible both by laboratory experiment and by numerical simulation.

Observations, numerical simulations and theoretical scaling arguments suggest that in mid-latitudes, away from the source regions and the equator, the meridional transport of abyssal water masses along a continental slope corresponds to planetary geostrophic flows that are gravity- or density-driven and topographically steered. We investigate these dynamics using a nonlinear reduced-gravity model that can describe grounded abyssal meridional flow over sloping topography that crosses the planetary vorticity gradient. Exact nonlinear steady and time-dependent solutions are obtained. The general steady theory is illustrated for a non-parallel equatorward flow that possesses a single along-slope grounding along the upslope flank of the current (complementing previous work). Four specific nonlinear time-dependent solutions are described. Two initial-value problems are solved exactly. The first initial configuration corresponds to an equatorward abyssal flow that has no cross-slope shear in the along-slope velocity and possesses a single grounding along the upslope flank of the current. The nonlinear time-dependent evolution of this initial current into a non-parallel shear flow is described. The second initial condition corresponds to an isolated radially symmetric grounded abyssal pool or dome. The nonlinear time-dependent evolution of this abyssal dome, which propagates equatorward with unsteady along- and cross-slope velocities while deforming into an elliptically shaped abyssal dome with $\unicode[STIX]{x1D6FD}$-induced diminishing height, is described. Finally, the nonlinear time-dependent boundary-value problem can be solved exactly in which the in-flow boundary condition on the poleward boundary of the mid-latitude domain corresponds to a time-dependent abyssal current with both an upslope and downslope grounding. Two specific time-dependent boundary conditions are examined. The first corresponds to a time-limited surge in the equatorward volume transport in the abyssal current along the poleward boundary. The second configuration corresponds to the nonlinear evolution of a finite-amplitude downslope plume or loop that forms in the abyssal current that is reminiscent of those seen in baroclinic instability simulations.

We study vibrational instabilities of the thermocapillary return flow driven by a constant temperature gradient along the free surface of an infinite layer that vibrates in its normal direction with acceleration of amplitude $g_1 $ and frequency $\omega _1 $. The layer is unstable to hydrothermal waves in the absence of vibrations beyond a critical Marangoni number $M$. Modulated gravitational instabilities with $M\,{=}\,0$ are also possible beyond a critical Rayleigh number $R$ based on $g_1 $. We employ two-time-scale high-frequency asymptotics to derive the equations governing the mean field. The influence of vibrations on the hydrothermal waves is found to be characterized by a dimensionless parameter $G$ that is proportional to $R^2.$ The return flow at $G\,{=}\,0$ is also a mean field basic flow and we study its linear instability at different Prandtl numbers $P$. The hydrothermal waves are stabilized with increasing $G$ and reverse their direction of propagation at particular values of $G$ that decrease with increasing $P$. At finite frequencies, a time-periodic base state exists and we study its linear instability by calculating the Floquet exponents. The stability boundaries in the $(R,M)$-plane are found to be composed of two intersecting branches emanating from the points of pure thermocapillary or buoyant instabilities. Three-dimensional modes are always preferred and the region of stability, while anchored at the point of hydrothermal waves corresponding to $R\,{=}\,0$, is found to grow without bound along the $R$-axis with increasing frequencies. Results from the two approaches are shown to be in asymptotic agreement at large frequencies.

When a semi-infinite flat plate is immersed parallel to an unbounded, plane, steady, asymmetric, constant shear flow of an incompressible viscous fluid, an interaction occurs between the surface-generated vorticity and the external vorticity. A physical assumption was made in a previous paper (Mark 1962) concerning this problem that the pressure field in a thin layer adjacent to the top-side of the plate may be accurately approximated by the undisturbed constant-pressure field—that which exists before the insertion of the plate into the flow. This means that the vorticity interaction is assumed to have no effect on the undisturbed pressure field. That this is a valid first approximation far downstream along the plate where the interaction is intense is given rigorous support in this paper.

The flow below the plate is examined on a heuristic basis. It is found that there is a strong possibility for the flow to separate from the lower surface near the leading edge of the plate. However, far downstream the flow settles down to a Stokes-type flow near the plate.