We consider the effects of slowly varying depth and current on the evolution of a packet of Stokes waves. The lengthscale of one-dimensional depth variation is assumed to be much greater than that of the wave envelope, and the direction of the current is perpendicular to the depth contours. By the method of multiple scales, the wave envelope is found to satisfy a cubic Schrödinger equation with slowly varying coefficients The criterion of spatial instability to small sidebands is extended. Numerical integration shows that the nonlinear evolution of a wavepacket is directly related to the instability parameter, which depends strongly on the current and depth variation. The heuristic hypothesis of Djordjevic & Redekopp on the soliton evolution over a slope in the absence of current is assessed.

Numerical calculations by Drobyshevski & Yuferev (1974) of the redistribution of magnetic flux by a Bénard layer with cells of square planform have been extended to higher values of electrical conductivity and to other velocity patterns, using a computer code developed for another purpose. Reconnection does not proceed as they supposed, but leads to overall field enhancement, and although the energy is greater at the bottom, there is as much unsigned flux in the upper half as in the lower half of the layer. However, compressible velocity patterns can concentrate flux at their bases.

A study of thermal instability for the two-dimensional stagnation flow for Prandtl numbers ranging from 0.7 to infinity is presented. The analysis represents an exact solution since neither the boundary-layer approximation nor the parallel-flow assumption was invoked. Of particular interest is that the critical Rayleigh number and the critical wavenumber, when defined on the basis of the thermal boundary-layer lengthscale, are found to be relatively insensitive to the Prandtl number for the range of Prandtl numbers studied.

We reconsider the second approximation for the height of the meniscus on a slender vertical cylinder, which has been calculated by James using the method of matched asymptotic expansions. Fraenkel has warned that the asymptotic matching principle may fail in certain cases, and we confirm that failure here. This meniscus problem is used to discuss the kind of failures that may occur if Fraenkel's restricted form of the matching principle is not used. A less restricted matching principle is also suggested in this paper. With this matching principle, we show that the last known term of order Re3 In Re in the drag on a sphere in low-Reynolds-number flow is correct, even though it was found in a way that violates Fraenkel's warning.

Three modes of instabilities in the slit-jet flow field are recognized. Additional evidence for the universality of the Strouhal number for the second mode ST = fw/U0 ≈ 0.43 and additional information on the wavelength (λ) between, and convection speed uc of the symmetrically placed, large-scale motions that result from this instability are presented. Specifically λ/w ≈ 1.2 and uc/U0 ≈ 0.51. The third instability mode is initiated at a Reynolds number U0w/v of approximately 1600; this instability results in a loss of the regular pattern associated with the large-scale motions.

The bursting frequency in turbulent boundary layers has been measured over the Reynolds-number range 103 < U∞/ν < 104. When scaled with the variables appropriate for the wall region, the non-dimensional frequency was constant independent of Reynolds number. A strong effect of the sensor size was noted on the measured bursting frequency. Only sensors having a spatial scale less than twenty viscous lengthscales were free from spatial-averaging effects and yielded consistent results. The spatial-resolution problem was apparently the reason for erroneous results reported in the past.

The three-dimensional problem of wave trapping above a submerged round sill was first analysed by Longuet-Higgins on the basis of a linear shallow-water theory. The large responses predicted by the theory were, however, not well borne out by the experiments of Barnard, Pritchard & Provis, and this has motivated a more detailed study of the problem. A full linear theory for both inviscid and weakly viscous fluid, without any shallow-water assumptions, is presented here. It reveals important limitations on the use of shallow-water theory and the reasons for them. In particular, while the qualitative features of wave trapping are similar to those of shallow-water theory, the nearly resonant frequencies differ significantly, and, since the resonances are narrow, the observed amplitudes at a given frequency differ greatly. The geometry is strongly indicative of long waves, and the dispersion relation appears quite consistent with that, but the part of the motion at wavenumbers that are not small has, despite the small amplitude, a substantial effect on the response to excitation.

A planar liquid layer is bounded below by a rigid plate and above by an interface with a passive gas. A steady shear flow is set up by imposing a temperature gradient along the layer and driving the motion by thermocapillarity. This dynamic state is susceptible to two types of thermal-convective instabilities: (i) stationary longitudinal rolls, which involve the classical Marangoni instability studied by Pearson; and (ii) unsteady hydrothermal waves, which involve a new mechanism of instability deriving its energy from the horizontal temperature gradients. Thermal stability characteristics for liquid layers with and without return-flow profiles are presented as functions of the Prandtl number of the liquid and the Biot number of the interface. Comparisons are made with available experimental observations.

A planar liquid layer is bounded below by a rigid plate and above by an interface with a passive gas. A steady shear flow is set up by imposing a temperature gradient along the layer and driving the motion by thermocapillarity. This dynamic state is susceptible to surface-wave instabilities that couple the interfacial deflection to the underlying shear flow. These instabilities are found to be directly related to the two-dimensional waves on an isothermal layer subject to wind shear as described by Miles and by Smith & Davis. Hence the surface-tension gradients are important only in that they drive the basic shear flow. The surface-wave stability characteristics for liquid layers with and without return-flow profiles are presented, and special attention is paid to long-wave instabilities. Comparisons are made with available experimental observations.

This paper proposes a high-Reynolds-number theory for the approximate analysis of timewise steady viscous flows. Its distinguishing feature is linearity. But it differs fundamentally from Oseen's (1910) well-known linear theory. Oseen flow is a variation on Stokes flow at the low-Reynolds-number limit.

The theory is developed for a %dimensional body moving through an infinite incompressible fluid. The velocity-vorticity formulation is employed. A boundary integral expressing the body contour velocity is written in terms of Green functions of the approximate governing differential equations. The boundary integral contains three unknown boundary distributions. These are a velocity source density, the boundary vorticity, and the normal gradient of the boundary vorticity. The unknown distributions are determined as the solutions to a boundary-integral equation formed from the velocity integral by the prescription of zero relative fluid velocity on the body boundary.

The linear integral-equation formulation is applied specifically to the case of thin bodies, such that the boundary condition is satisfied approximately on the streamwise coordinate axis. The integral equation is then reduced to its leading-order contribution in the limit of infinite Reynolds number. The unknown distributions uncouple in the first-order formulation, and analytic solutions are obtained. A most interesting result appears at this point: the theory recovers linearized airfoil theory in the first-order infinite-Reynolds-number limit; the airfoil integral equation determines one of the three contour distributions sought.

The first-order theory is then demonstrated by application to two classical cases: the zero-thickness flat plate at zero incidence, and the circular cylinder.

For the flat plate, the streamwise velocity near the plate predicted by the proposed linear theory is compared with that of Blasius's solution to the laminar boundary-layer equations (Schlichting 1968). The linear theory predicts a fuller profile, tending more toward the character expected of the timewise steady turbulent profile. This character is also exhibited in the predicted velocity distribution across the plate wake, which is compared with Goldstein's asymptotic boundary-layer solution (Schlichting 1968). The wake defect is more severe according to the proposed theory.

For the case of the circular cylinder, application of the formulation is not truly valid, since the circular cylinder is not a thin body. The theory does, however, predict that the flow separates. The separation points are predicted to lie at position angles of approximately ± 135°, with angle measured from the forward stagnation point. This compares with the prediction of 109O from the Blasius series solution to the laminar boundary-layer equations (Schlichting 1968).

The theory is next applied to the case of a non-zero-thickness flat plate with incidence. From the fully attached flow at zero incidence, the theory predicts that both Ieading-edge separation and reattachment and trailing-edge separation appear on the suction side at small angle. On increasing incidence, the forward reattachment point moves aft, and the aft separation point moves forward. Coalescence occurs near midchord, and the foil is thereafter fully separated.

Finally, the first-order contribution to the far-field velocity at high Reynolds number is shown to be identically that corresponding to the ideal flow. This result, coupled with the recovery of linearized thin-foil theory in the near-field limit, is argued to support strongly the physical idea that the ideal flow is, in fact, the limiting state of the complete field flow at infinite Reynolds number. Flow separation can be viewed as present in the ideal flow limit; i t is simply embedded in the infinitesimally thin body-surface vortex sheets into which the entire viscous field collapses as vorticity convection overwhelms vorticity diffusion at the infinite-Reynolds-number limit.

Corrections to Stokes’ law are determined to first order in a/b and a/h for a sphere of radius a in a one-dimensional array of identical spheres having centre-to-centre-spacing b and translating a distance h from a no-slip wall. When h/b is small the drag is greater than that given by Stokes’ law; as h/b increases, the drag generally decreases and becomes less than that given by Stokes’ law. Stability of the array is examined. Motion along the line of centres is found to be stable, but the other two motions are unstable. The wall is a stabilizing influence when motion is toward the wall and a destabilizing influence when motion is away from the wall. For motion parallel to the wall, the presence of the wall shifts the region of maximum instability to smaller wavelengths. Crowley's results, which neglect any influence of the wall, are approached for h/b greater than about 5.

The temporal and spatial evolution of large-scale modulations of weakly nonlinear edge waves on a uniformly sloping beach is studied using the full water-wave formulation for beach angles α = π/2N. Equations governing the evolution of envelopes of edge waves, excited by resonant interactions with incident wavetrains, are derived. It is deduced that a uniform train of free periodic edge waves is always unstable to large-scale variations, so that envelope solitons will develop; the resulting three-dimensional solitons are described in detail. In addition, it is shown that steady-state standing subharmonic edge waves, excited by incident wavetrains on a long, mildly sloping beach, can be unstable to large-scale modulations. The possible physical significance of these findings is discussed.

The critical Taylor number, phenomena accompanying the transition to turbulence, and the cellular structure of Taylor–Görtler vortex in the flow between two concentric spheres, of which the inner one is rotating and the outer is stationary, are investigated using three kinds of flow-visualization technique. The critical Taylor number generally increases with the ratio β of clearance to inner-sphere radius. For β [les ] 0.08, the critical Taylor number in spherical Couette flow is smaller than in circular Couette flow, but vice versa for β > 0.08. A pair of toroidal Taylor–Görtler vortices occurs first around the equator at the critical Reynolds number Rec (or critical Taylor number Tc). More Taylor–Görtler vortices are added with increasing Reynolds number Re. After reaching the maximum number of vortex cells, as Re is increased, the number of vortex cells decreases along with the various transition phenomena of Taylor–Görtler vortex flow, and the vortex finally disappears for very large Re, where the turbulent basic flow is developed. The instability mode of Taylor–Görtler vortex flow depends on both β and Re. The vortex flows encountered as Re is increased are toroidal, spiral, wavy, oscillating (quasiperiodic), chaotic and turbulent Taylor–Görtler vortex flows. Fourteen different flow regimes can be observed through the transition from the laminar basic flow to the turbulent basic flow. The number of toroidal and/or spiral cells and the location of toroidal and spiral cells are discussed as a means to clarify the spatial organization of the vortex. Toroidal cells are stationary. However, spiral cells move in relation to the rotating inner sphere, but in the reverse direction of its rotation and at about half its speed. The spiral vortices number about six, and the spiral angle is 2–10°.

Electrohydrodynamic instability is described in a planar region of gas, with unipolar injection through a linear temperature gradient, with corresponding mobility, viscosity, and density gradients. The role of charge-relaxation instability, which is incipient at a critical value of electric Rayleigh number in a uniform mobility layer, is distinguished from the mobility-gradient-driven mechanism, which can produce instability at much smaller voltage thresholds. Relative values of electroviscous time, viscous diffusion time and ion transit time determine the nature of the instability process. The instability is stationary or dynamic for wavenumbers respectively below or above a critical value. The fastest-growing wavenumber and corresponding growth rate are determined as functions of time constants and temperature gradient.

It is observed that an interface is established in any open-ended vertical cylinder when a plume flows in the direction of a mean flow through the cylinder. The location of the interface defines the volume flux of the plume because the total discharge in the environment has been entrained at this level. Experiments have been conducted to verify this relation and to define the limits within which the interface is established. It is found that, as the buoyancy in the plume is reduced or the area of the environment is reduced, limits do exist. An application of the interface in industrial-building ventilation is described.

A study has been made of how initially planar shocks in air propagate around 90° bends in channels of nearly rectangular cross-section. In shallow bends for which the radius of curvature R is much greater than the radius r of the channel, the shock recovers from a highly curved profile at the start of the bend to regain planarity towards the end of the bend. This occurs on account of the acceleration of the triple point across the channel following its interaction with the expansion waves generated at the convex wall. In sharp bends the shock profiles retain their pronounced curvature for some distance downstream of the bend.

At the start of a shallow bend (R/r ≈ 6) the shock at the concave wall, initial Mach number M0, accelerates to Mw = 1.15M0 and remains at this value until towards the end of the bend it begins to attenuate. At the convex wall, shocks of M0 > 1.7 attenuate to Mw = 0.7M0 and propagate at this value for some distance around the bend. In the early stages of a sharper bend (R/r ≈ 3) the shock at the concave wall strengthens to Mw = 1.3M0, remaining at this value for some distance downstream of the bend. At the convex wall the shock decelerates to 0.6M0.

Whitham's (1974) ray theory is shown to predict with reasonable accuracy the Mach numbers of the wall shocks at both surfaces for both bends tested and the range of incident shock velocities used, 1.2 < M0 < 3. The agreement between the theory and experimental results is particularly close for stronger shocks propagating along the inner bend. Predictions from 3-shock theory (Courant & Friedrichs 1948) of the Mach number at the outer wall are consistently higher than those from Whitham's analysis. In turn, the latter tends to slightly overestimate the strength of the wall shock.

A model is developed, based on an extension of Whitham's analysis, and is shown to predict the length of the Mach stem produced by shocks of M0 > 2 over the initial stages of the bend.

This paper discusses the stability of the flow of a low-Prandtl-number liquid contained in a shallow slot with differentially heated vertical endwalls. The effect of thermally insulating boundaries at the top and bottom of the container on wavelength selection is emphasized. Stability calculations indicate that, for Prandtl number Pr in the range 0.015 < Pr < 0.27, the first perturbations to grow are overstable (oscillatory) longitudinal rolls with axes perpendicular to the endwalls, and with very large cross-stream wavelengths of about 9 to 15 layer depths. Previous studies using thermally conducting boundaries predict critical wavelengths of about three layer depths. The new results are in substantial agreement with an experiment using a differentially heated layer of mercury with aspect ratio (depth/length) 0.047 in both horizontal directions. The implications of the long-wavelength instability for the interpretation of thermal oscillations observed in other smaller-aspect-ratio configurations is discussed.

The linear stability of the stationary Ekman-layer flow near a plane boundary is considered. Analytical formulas for the eigenfunctions are derived by a spectral analysis. Standard optimization algorithms are used to calculate critical points, maximum growth rates and neutral-stability curves. The near approach provides a better basis for both a linear and a nonlinear stability analysis than the well-known methods have done. The method may also be applied to other boundary-layer problems.

In the present study we examine the steady axisymmetric creeping flow due to the motion of a liquid drop or a bubble which is partially covered by a thin immiscible fluid layer or film. The analysis is based on the assumption that surface-tension forces are large compared with viscous forces which deform the drop, and that the circulation in the film is weak. The latter assumption is satisfied provided that the film-fluid viscosity is not too small. A perturbation scheme based on the thinness of the fluid layer is used to construct the solution.

One of the principal results is an expression for the drag force on the complex drop. We also find that the extent to which the drop or bubble is covered the film has a maximum value depending on the magnitude of the driving force on the film. In addition, we find the rather interesting result that when the ratio of the primary drop viscosity and bulk fluid viscosity is greater than ½, the circulation within the film may have a double-cell structure.

We study experimentally the spanwise structure of the mean flow field of a plane shear layer. The field is dominated by a lateral undulation that persists downstream to form long longitudinal structures. Both the amplitude and spacing of these structures vary in such a way as to suggest that they are due to a secondary instability of the flow field. Some models for this instability are discussed and compared with the experimental results.