The problem of locating stagnation points in the flow produced by a system of N interacting point vortices in two dimensions is considered. The general solution follows from an 1864 theorem by Siebeck, that the stagnation points are the foci of a certain plane curve of class N−1 that has all lines connecting vortices pairwise as tangents. The case N=3, for which Siebeck's curve is a conic, is considered in some detail. It is shown that the classification of the type of conic coincides with the known classification of regimes of motion for the three vortices. A similarity result for the triangular coordinates of the stagnation point in a flow produced by three vortices with sum of strengths zero is found. Using topological arguments the distinct streamline patterns for flow about three vortices are also determined. Partial results are given for two special sets of vortex strengths on the changes between these patterns as the motion evolves. The analysis requires a number of unfamiliar mathematical tools which are explained.

We consider a sheet flow in which heavy grains near a packed bed interact with a unidirectional turbulent shear flow of a fluid. We focus on sheet flows in which the particles are supported by their collisional interactions rather than by the velocity fluctuations of the turbulent fluid and introduce what we believe to be the simplest theory for the collisional regime that captures its essential features.

We employ a relatively simple model of the turbulent shearing of the fluid and use kinetic theory for the collisional grain flow to predict profiles of the mean fluid velocity, the mean particle velocity, the particle concentration, and the strength of the particle velocity fluctuations within the sheet. These profiles are obtained as solutions to the equations of balance of fluid and particle momentum and particle fluctuation energy over a range of Shields parameters between 0.5 and 2.5. We compare the predicted thickness of the concentrated region and the predicted features of the profile of the mean fluid velocity with those measured by Sumer et al. (1996). In addition, we calculate the volume flux of particles in the sheet as a function of Shields parameter.

Finally, we apply the theory to sand grains in air for the conditions of a sandstorm and calculate profiles of particle concentration, velocity, and local volume flux.

The general solution of the particle momentum equation for unsteady Stokes flows is obtained analytically. The method used to obtain the solution consists of applying a fractional-differential operator to the first-order integro-differential equation of motion in order to transform the original equation into a second-order non-homogeneous equation, and then solving this last equation by the method of variation of parameters. The fractional differential operator consists of a three-time-scale linear operator that stretches the order of the Riemann–Liouville fractional derivative associated with the history term in the equation of motion. In order to illustrate the application of the general solution to particular background flow fields, the particle velocity is calculated for three specific flow configurations. These flow configurations correspond to the gravitationally induced motion of a particle through an otherwise quiescent fluid, the motion of a particle caused by a background velocity field that accelerates linearly in time, and the motion of a particle in a fluid that undergoes an impulsive acceleration. The analytical solutions for these three specific cases are analysed and compared to other solutions found in the literature.

The steady-state flow generated by a rotating bottom in a closed cylindrical container and the resulting vortex breakdown bubbles have been studied experimentally. By comparing the flow inside two different container geometries, one with a rigid cover and the other with a free surface, we examined the way in which the formation and structure of the breakdown bubbles depend on the surrounding flow. Details of the flow were visualized by means of the electrolytic precipitation technique, whereas a particle tracking technique was used to characterize the whole flow field. We found that the breakdown bubbles inside the container flow are in many ways similar to those in vortex tubes. First, the bubbles are open with in- and outflow and second, their structure is, like in the case of vortex breakdown in pipe flows, highly axisymmetric on the upstream side of the bubble and asymmetric on their rear side. However, and surprisingly, we observed bubbles which are open and stationary at the same time. This shows that open breakdown bubbles are not necessarily the result of periodic oscillations of the recirculation zone. The asymmetry of the flow structure is found to be related to the existence of asymmetric flow separations on the container wall. If the angular velocity of the rotating bottom is increased the evolution of the breakdown bubbles is different in both configurations: in the rigid cover case the breakdown bubbles disappear but persist in the free surface case.

The separated flow past a zero-thickness flat plate held normal to a free stream at Re=250 has been investigated through numerical experiments. The long-time signatures of the drag and lift coefficients clearly capture a low-frequency unsteadiness with a period of approximately 10 times the primary shedding period. The amplitude and frequency of drag and lift variations during the shedding process are strongly modulated by the low frequency. A physical interpretation of the low-frequency behaviour is that the flow gradually varies between two different regimes: a regime H of high mean drag and a regime L of low mean drag. It is observed that in regime H the shear layer rolls up closer to the plate to form coherent spanwise vortices, while in regime L the shear layer extends farther downstream and the rolled-up Kármán vortices are less coherent. In the high-drag regime three-dimensionality is characterized by coherent Kármán vortices and reasonably well-organized streamwise vortices connecting the Kármán vortices. With a non-dimensional spanwise wavelength of about 1.2, the three-dimensionality in this regime is reminiscent of mode-B three-dimensionality. It is observed that the high degree of spanwise coherence that exists in regime H breaks down in regime L. Based on detailed numerical flow visualization we conjecture that the formation of streamwise and spanwise vortices is not in perfect synchronization and that the low-frequency unsteadiness is the result of this imbalance (or phase mismatch).

It is well-known that sound generated by localized sources embedded in a jet undergoes refraction as the acoustic waves propagate through the jet mean flow. For isothermal or hot jets, the effect of refraction causes the deflection of the radiated sound waves away from the jet flow direction. This gives rise to a cone of silence around the jet axis where there is a significant reduction in the radiated sound intensity. In this work, the mean flow refraction problem is investigated through the use of the reciprocity principle. Instead of the direct source Green's function, the adjoint Green's function with the source and observation points interchanged is used to quantify the effect of mean flow on sound radiation. One advantage of the adjoint Green's function is that the Green's functions for all the source locations in the jet radiating to a given direction in the far field can be obtained in a single calculation. This provides great savings in computational effort. Another advantage of the adjoint Green's function is that there is no singularity in the jet flow so that the problem can be solved numerically with axial as well as radial mean flow gradients included in a fairly straightforward manner. Extensive numerical computations have been carried out for realistic jet flow profiles with and without exercising the locally parallel flow approximation. It is concluded that the locally parallel flow approximation is valid as long as the direction of radiation is outside the cone of silence.

The steady flow is considered of a Newtonian fluid, of viscosity μ, between contra-rotating cylinders with peripheral speeds U1 and U2. The two-dimensional velocity field is determined correct to O(H0/2R)1/2, where 2H0 is the minimum separation of the cylinders and R an ‘averaged’ cylinder radius. For flooded/moderately starved inlets there are two stagnation–saddle points, located symmetrically about the nip, and separated by quasi-unidirectional flow. These stagnation–saddle points are shown to divide the gap in the ratio U1[ratio ]U2 and arise at [mid ]X[mid ]=A where the semi-gap thickness is H(A) and the streamwise pressure gradient is given by dP/dX =μ(U1+U2)/H2(A). Several additional results then follow.

(i) The effect of non-dimensional flow rate, λ: A2=2RH0(3λ−1) and so the stagnation–saddle points are absent for λ<1/3, coincident for λ=1/3 and separated for λ>1/3.

(ii) The effect of speed ratio, S=U1/U2: stagnation–saddle points are located on the boundary of recirculating flow and are coincident with its leading edge only for symmetric flows (S=1). The effect of unequal cylinder speeds is to introduce a displacement that increases to a maximum of O(RH0)1/2 as S→0.

Five distinct flow patterns are identified between the nip and the downstream meniscus. Three are asymmetric flows with a transfer jet conveying fluid across the recirculation region and arising due to unequal cylinder speeds, unequal cylinder radii, gravity or a combination of these. Two others exhibit no transfer jet and correspond to symmetric (S=1) or asymmetric (S≠1) flow with two asymmetric effects in balance. Film splitting at the downstream stagnation–saddle point produces uniform films, attached to the cylinders, of thickness H1 and H2, where

formula here

provided the flux in the transfer jet is assumed to be negligible.

(iii) The effect of capillary number, Ca: as Ca is increased the downstream meniscus advances towards the nip and the stagnation–saddle point either attaches itself to the meniscus or disappears via a saddle–node annihilation according to the flow topology.

Theoretical predictions are supported by experimental data and finite element computations.

Complex bioconvection patterns are observed when a suspension of the oxytactic bacterium Bacillus subtilis is placed in a chamber with its upper surface open to the atmosphere. The patterns form because the bacteria are denser than water and swim upwards (up an oxygen gradient) on average. This results in an unstable density distribution and an overturning instability. The pattern formation is dependent on depth and experiments in a tilted chamber have shown that as the depth increases the first patterns formed are hexagons in which the fluid flows down in the centre.

The linear stability of this system was analysed by Hillesdon & Pedley (1996) who found that the system is unstable if the Rayleigh number Γ exceeds a critical value, which depends on the wavenumber k of the disturbance as well as on the values of other parameters. Hillesdon & Pedley found that the critical wavenumber kc could be either zero or non-zero, depending on the parameter values.

In this paper we carry out a weakly nonlinear analysis to determine the relative stability of hexagon and roll patterns formed at the onset of bioconvection. The analysis is different in the two cases kc≠0 and kc=0. For the kc≠0 case (which appears to be more relevant experimentally) the model does predict down hexagons, but only for a certain range of parameter values. Hence the analysis allows us to refine previous parameter estimates. For the kc=0 case we carry out a two-dimensional analysis and derive an equation describing the evolution of the horizontal planform function.

The generalized Lagrangian mean (GLM) formulation is used to describe the interaction of waves and currents. In contrast to the more conventional Eulerian formulation the GLM description enables splitting of the mean and oscillating motion over the whole depth in an unambiguous and unique way, also in the region between wave crest and trough. The present paper deals with non-breaking long-crested regular waves on a current using the GLM formulation coupled with a WKBJ-type perturbation-series approach. The waves propagate under an arbitrary angle with the current direction. The primary interest concerns nonlinear changes in the vertical distribution of the mean velocity due to the presence of the waves, but modifications of the orbital velocity profiles, due to the presence of a current, are considered as well. The special case of no initial current, where waves induce a so-called drift velocity or mass-transport velocity, is also studied.

Fully developed flow in an infinite helically coiled pipe is studied, motivated by physiological applications. Most of the bends in the mammalian arterial system curve in a genuinely three-dimensional way, so that the arterial centreline has not only curvature but torsion and can be modelled by a helix. Flow in a helically symmetric pipe generalizes related problems in axisymmetry (Dean flow) and two-dimensionality, but the geometry ensures that even irrotational flow has a cross-pipe component. Fully developed helical flows driven by a steady pressure gradient are studied analytically and numerically. Varying the radius and pitch of the helical pipe, the effects of curvature and torsion on the flow are investigated.

Fully developed flow in a helical pipe is investigated with a view to modelling blood flow around the commonly non-planar bends in the arterial system. Medical research suggests that the formation of atherosclerotic lesions is strongly correlated with regions of low wall shear and it has been suggested that the observed non-planar geometry may result in a more uniform shear distribution. Helical flows driven by an oscillating pressure gradient are studied analytically and numerically. In the high-frequency limit an expression is derived for the second-order steady flow driven by streaming from the Stokes layers. Finite difference methods are used to calculate flows driven by sinusoidal or physiological pressure gradients in various geometries. Possible advantages of the observed helical rather than planar arterial bends are discussed in terms of wall shear distribution and the inhibition of boundary layer separation.

A continuously unstable precessing flow within a short cylindrical chamber following a large sudden expansion is described. The investigation relates to a nozzle designed to produce a jet which achieves large-scale mixing in the downstream field. The inlet flow in the plane of the sudden expansion is well defined and free from asymmetry. Qualitative flow visualization in water and semi-quantitative surface flow visualization in air are reported which identify this precession within the chamber. Quantitative simultaneous measurements from fast-response pressure transducers at four tapping points on the internal walls of the nozzle chamber confirm the presence of the precessing field. The investigation focuses on the flow within the nozzle chamber rather than that in the emerging jet, although the emerging flow is also visualized.

Two flow modes are identified: a ‘precessing jet’ mode which is instantaneously highly asymmetric, and a quasi-symmetric ‘axial jet’ mode. The precessing jet mode, on which the investigation concentrates, predominates in the geometric configuration investigated here. A topologically consistent flow field, derived from the visualization and from the fluctuating pressure data, which describes a three-dimensional and time-dependent precessing motion of the jet within the chamber is proposed. The surface flow visualization quantifies the axial distances to lines of positive and negative bifurcation allowing comparison with related flows involving large-scale precession or flapping reported by others. The Strouhal numbers (dimensionless frequencies) of these flows are shown to be two orders of magnitude lower than that measured in the shear layer of the jet entering the chamber. The phenomenon is demonstrated to be unrelated to acoustic coupling.