A new and systematic approach is proposed to determine the migration of torque-free spherical bubbles immersed in a steady and non-uniform unbounded Stokes flow and subject to arbitrary capillary effects. The advocated procedure appeals to only a very few quantities on the surface of each bubble and is therefore suitable for a future numerical treatment of arbitrary clusters of bubbles. For a single bubble, the theory allows a straightforward analytical implementation and the predicted results agree well with Hetsroni & Haber (1970), Hetsroni et al. (1971) and Subramanian (1985). The thermocapillary motion of non-conducting spherical bubbles freely suspended in a quiescent liquid in the presence of an arbitrary ambient temperature $T_{\infty}$ is considered and it is shown that it is futile to determine the disturbed temperature field, whatever $T_{\infty},$ once bubbles are equivalent (i.e. experience the same velocity in a given uniform temperature gradient ${\bm \nabla}T_{\infty},$ as obtained by Acrivos et al. 1990 and Wang et al. 1994).

A global analysis of steady states of low Mach number inviscid premixed reacting swirling flows in a straight circular finite-length open pipe is developed. We focus on modelling the basic interaction between the swirl and heat release of the reaction. For analytic simplicity, a one-step first-order Arrhenious reaction kinetics is considered in the limit of high activation energy and infinite Peclet number. Assuming a complete reaction with chemical equilibrium upstream and downstream of the reaction zone, a nonlinear partial differential equation is derived for the solution of the flow stream function downstream of the reaction zone in terms of the specific total enthalpy, specific entropy and circulation functions prescribed at the inlet. Several types of solutions of the nonlinear ordinary differential equation for the columnar flow case describe the outlet states of the flow in a long pipe. These solutions are used to form the bifurcation diagram of steady reacting flows with swirl as the inlet swirl level is increased at a fixed heat release from the reaction. The approach is applied to two profiles of inlet flows, the solid-body rotation and the Lamb–Oseen vortex, both with constant profiles of the axial velocity, temperature and mixture reactant mass fraction. The computed results provide theoretical predictions of the critical inlet swirl levels for the appearance of vortex breakdown states and for the size of the breakdown zone as a function of the inlet flow swirl level, Mach number and heat release of the reaction. For the inlet solid-body rotation, flow is decelerated to breakdown as the inlet swirl is increased above the critical swirl level, and there is a delay in the appearance of breakdown with the increase of the heat release of the reaction. For the inlet Lamb–Oseen vortex at low values of heat release, the critical swirl for breakdown is decreased with the increase of heat release while, at high values of heat release, the appearance of breakdown is delayed to higher incoming flow swirl levels with the increase of heat release. The analysis sheds light on the global dynamics of low Mach number reacting flows with swirl and vortex breakdown and on the interaction between vortex breakdown and heat release that affects the shape of the reaction zone in the domain.

Defining a link between wind-tunnel settling chamber screens, flow quality and test section boundary-layer spanwise variation is necessary for accurate transition prediction. The aim of this work is to begin establishing this link. The computed, steady, laminar wake of a zither (screen model) with imperfect wire spacing is tracked through a contraction and into a model test section. The contraction converts the zither wake into streamwise vorticity which then creates spanwise variation (streaks) in the test-section boundary layer. The magnitude of the spanwise variation is sensitive to the zither open-area ratio and imperfections, but the observed wavelength is relatively insensitive to the zither wire spacing. Increased spanwise variation is attributed to large wavelength variation of drag across the zither, and not the coalescence of jets phenomena. The linear stability of the streaks is predicted using the parabolized stability equations with the $\text{e}^{N}$ method. A standard deviation of zither wire position error of 38.1 ${\rm\mu}$m (15 % of wire diameter) for a zither of 50 % open-area ratio is found to suppress Tollmien–Schlichting wave growth significantly.

The fluid velocity far from a translating body in two-dimensional irrotational flow is generally dipolar. This is a classical result. Here we ask when the dipolar component vanishes. Lamb (1945, § 126) provides symmetry conditions on the virtual mass tensor for this to be the case. We show that these conditions are not necessary and obtain the sufficient and necessary condition in terms of the shape of the body using conformal maps. Some explicit examples are constructed based on this condition.

A similarity solution of the boundary layer equations for a wall jet on a heated horizontal surface at constant temperature taking into account the coupling of the temperature and velocity fields by buoyancy is described. This similarity solution exists for any value of ${\it\Lambda}=Gr/Re^{2}$, characterizing this coupling between natural and forced convection over the horizontal plate, where $Gr$ is a Grashof number and $Re$ is a Reynolds number, provided that the plate temperature is higher than the ambient temperature (${\it\Lambda}>0$, say). Two main qualitative differences are found in the flow structure in relation to the well-known Glauert’s similarity solution for a wall jet without natural convection effects (i.e. when ${\it\Lambda}=0$): the first is that the similarity variable and structure of the horizontal velocity and temperature have the same functional form for both a radially spreading jet and a two-dimensional jet; the second is that the maximum of the horizontal velocity increases as the jet spreads over the surface, instead of decreasing like in Glauert’s solution, as the radial or horizontal distance to the power $1/5$. To check this similarity solution we solve numerically the boundary layer equations for the particular case of a jet with constant velocity and temperature emerging from a slot of height ${\it\delta}$ and radius $r_{0}$ (in the radially spreading case). An approximate, analytical similarity solution near the jet exit is also found that helps to start the numerical integration. Far from the jet exit the numerical solution tends to the similarity solution for any set of values of the non-dimensional parameters governing the problem, provided that the plate is heated (${\it\Lambda}>0$). No similarity solution is found numerically for the case of a cooled plate (${\it\Lambda}<0$). For ${\it\Lambda}=0$ Glauert’s similarity solution is recovered.

Stably stratified sheared flows are ubiquitous in geophysical flows from the ocean to the stars, and the route to turbulence in these flows remains an open question. The article by Lefauve et al. (J. Fluid Mech., vol. 848, 2018, pp. 508–544) is an invitation to this journey. With impressive experimental precision mastered by few teams in the world, the nature of the coherent structure that dominates the flow on the verge of turbulent breakdown is revealed and analysed through one- or two-dimensional modern stability analysis of an experimentally obtained base flow. The effect of confinement is surprisingly strong, advocating for leaving the textbook flows, inhomogeneous in only one direction, for the more complex shores of real flows, now accessible to analysis of multidimensional stability problems. The route explored by Lefauve et al. (2018) renews with the long tradition of the supercritical bifurcation scenario, it revisits the linear stability theory with possibility of resonances, critical layers and more to be imagined, since complex base flows are now available to explore both experimentally and analytically.

The empirical description of turbulent wall layers across which the shear stress varies is considered. The description given by Townsend for zero-stress layers is found to be inapplicable to uniform pressure flows in pipes and two-dimensional channels, and to a boundary layer developing in the absence of a pressure gradient.

The hydrodynamic stability of an ideal mixture of two viscous, dissimilar liquids contained between two concentric rotating cylinders is analyzed. The basic flow of the mixture is determined by coupling the mass and momentum equations with an equation for the equilibrium concentration distribution. Infinitesimal, axisymmetric disturbances are assumed, and the disturbance equations are written for the limiting case of large Schmidt numbers (no diffusion).

The presence of density and viscosity variations leads to a twelfth-order eigenvalue problem with two-point boundary conditions that has the appearance of a combined Taylor and density-stratified shear flow problem. A numerical technique is devised to determine the stability boundary and to calculate Taylor numbers and oscillation frequencies for different growth rates.

It is found that very small mean density gradients alter the critical Taylor number and that oscillations occurring in both the growing and neutral solutions are the dominant mode.

The boundary-layer flow over a semi-infinite horizontal circular cylinder heated to a constant temperature and immersed in a uniform axial free stream is discussed in five situations corresponding to successively greater displacements from the leading edge. In the first three cases the drift velocity due to buoyancy is assumed small compared to the axial velocity component. Close to the leading edge of the cylinder the techniques of Seban & Bond are extended to include the drift velocity; far from the leading edge the asymptotic series methods of Stewartson, of Glauert & Lighthill, and of Eshghy & Hornbeck are employed to obtain a solution for the drift velocity. In the intermediate zone where the series solutions do not apply the appropriate partial differential equations are solved numerically. Still further downstream than the region where the ‘asymptotic’ solutions hold it is assumed that the boundary-layer flow is primarily convective and that the boundary layer is thin compared with the radius of the cylinder. A series solution is obtained which is valid near the lowest generator of the cylinder. Numerical methods are used to advance this solution upwards around the cylinder by solving the full boundary-layer equations step-by-step.

The energetics of a linear array of hollow or stagnant-cored vortices of finite cross-section in an ideal fluid is studied in this paper. The results are useful in a discussion of the amalgamation of vortex structures in a turbulent mixing layer.

The waves generated by the steady motion of an obstacle along the axis of a uniformly rotating, electrically conducting homogeneous fluid have been studied by Lighthill's technique. The wave-number surface consists of a sphere and four coincident planes. The waves corresponding to points on the sphere travel ahead or trail behind the obstacle according as a1 the Alfvén velocity, is greater or less than U, the velocity of the obstacle. By drawing the appropriate normals to the four planes, it is seen that the formation of a Taylor column ahead of the obstacle is possible even at large Rossby numbers when U < a1 in contrast with the non-magnetic case and the case with U > a1.

A method for obtaining analytic solutions to the problem of blunt bodies in the supersonic stream of an ideal gas is presented. The solutions are written in terms of power series whose coefficients are elementary functions. These solutions are approximate, but the approximation is rational, i.e. any higher approximation can, in principle, be obtained. Some of these higher approximations have been calculated. Examples are presented for various free-stream conditions and prescribed body shapes. These are compared with results from standard numerical procedure and with available experimental measurements.

The slow uniform motion, after an impulsive start from relative rest, of a paraboloid of revolution along the axis of a rotating fluid is investigated by using a perturbation method. The principal purpose of the note is to illustrate the mechanism by which the fluid is not subjected to any substantial radial displacement, which is a direct consequence of the requirement that the circulation round material circuits should be constant when the perturbation velocities due to the motion of the paraboloid remain small. It appears that the mechanism is an oscillatory one in which the distance between any fluid particle and the axis of rotation oscillates sinusoidally in time with small amplitude. As time progresses, the amplitude of the oscillation decays to zero everywhere except on the paraboloid. The ultimate motion is then a rigid body rotation everywhere except on the paraboloid and the axis of rotation, where the perturbation velocities continue to oscillate indefinitely with small amplitude.

The Hunt–Stewartson technique of estimating fluid velocity and magnetic flux profiles in rectangular ducts is generalized for the entire secondary boundary layer.

The results of calculations of the compression of air by repeated shock waves are compared with the perfect-gas values given by Evans & Evans (1956). The comparison emphasizes the increasing divergence of real from perfect-gas results as shock strengths are raised. The equations relating conditions across a shock wave are obtained in a convenient form for solution using a Mollier diagram.

The results of Ackerberg (1970, 1971a), on the two-dimensional boundary-layer separation at a sharp trailing edge where a free streamline is attached, are extended to the axisymmetric case, with and without swirl. When the flow has swirl, the boundary-layer swirl velocity close to the wall may be opposite to that in the external flow; this may help explain the ‘bathtub vortex paradox’ observed by Sibulkin (1962). The solutions for the detached shear layers downstream of separation for two-dimensional and axisymmetric flows, with and without swirl, have been obtained. Some misprints in parts 1 and 2 are corrected in the appendix.

A lifting-surface theory is presented for a cascade in subsonic shear flow by applying Fourier integral methods to the expressions of the perturbed flow field. The pressure distribution on the blade surface is determined by means of the socalled singularity method. Some numerical examples are presented and discussed in comparison with the results according to the lifting-line theory.

A significant difference is found in the effect of compressibility between a shear flow and a uniform flow. In shear flows with the maximum Mach number close to one, no such great local lift force is found near the sonic station as would be predicted by the linearized subsonic uniform flow theory. The correlation between the local lift and the local effective angle of attack at high Mach number span-stations shows a great deviation from that according to the uniform flow theory.

The magnetohydrodynamic lubrication flow in a step-type thrust bearing has been investigated both theoretically and experimentally. The experimental configuration consisted of nine identical segments arranged on a circular plate. Flow was induced by rotating the upper plate. Pressures, voltages and torques were recorded. Provision was made for passing current radially through the segments, and it was shown that the torque could be reduced to zero and even reversed by this means. Good agreement has been obtained between the analysis and the experimental results.

The general equations of motion are developed for a compressible, inviscid flow in which a non-uniform distribution of heat transfer is applied to the fluid or a non-uniform generation of heat per unit volume occurs. In general, vorticity can be created if the cross-products of the temperature and entropy gradients are finite. If the temperature gradients in the flow are small (first-order), then a non-uniform heat addition across the stream will produce a second-order change in vorticity. For this type of flow, solutions are obtained for the variations of velocity and density that occur in a two-dimensional plane flow and an axially symmetric three-dimensional flow. A simple expression is also obtained for the streamline displacement caused by the non-uniform addition of heat.