Colonies of the green alga Volvox are spheres that swim through the beating of pairs of flagella on their surface somatic cells. The somatic cells themselves are mounted rigidly in a polymeric extracellular matrix, fixing the orientation of the flagella so that they beat approximately in a meridional plane, with axis of symmetry in the swimming direction, but with a roughly $20^{\circ }$ azimuthal offset which results in the eponymous rotation of the colonies about a body-fixed axis. Experiments on colonies of Volvox carteri held stationary on a micropipette show that the beating pattern takes the form of a symplectic metachronal wave (Brumley et al. Phys. Rev. Lett., vol. 109, 2012, 268102). Here we extend the Lighthill/Blake axisymmetric, Stokes-flow model of a free-swimming spherical squirmer (Lighthill Commun. Pure Appl. Maths, vol. 5, 1952, pp. 109–118; Blake J. Fluid Mech., vol. 46, 1971b, pp. 199–208) to include azimuthal swirl. The measured kinematics of the metachronal wave for 60 different colonies are used to calculate the coefficients in the eigenfunction expansions and hence predict the mean swimming speeds and rotation rates, proportional to the square of the beating amplitude, as functions of colony radius. As a test of the squirmer model, the results are compared with measurements (Drescher et al. Phys. Rev. Lett., vol. 102, 2009, 168101) of the mean swimming speeds and angular velocities of a different set of 220 colonies, also given as functions of colony radius. The predicted variation with radius is qualitatively correct, but the model underestimates both the mean swimming speed and the mean angular velocity unless the amplitude of the flagellar beat is taken to be larger than previously thought. The reasons for this discrepancy are discussed.

The diffusive behaviour of swimming micro-organisms should be clarified in order to obtain a better continuum model for cell suspensions. In this paper, a swimming micro-organism is modelled as a squirming sphere with prescribed tangential surface velocity, in which the centre of mass of the sphere may be displaced from the geometric centre (bottom-heaviness). Effects of inertia and Brownian motion are neglected, because real micro-organisms swim at very low Reynolds numbers but are too large for Brownian effects to be important. The three-dimensional movement of 64 or 27 identical squirmers in a fluid otherwise at rest, contained in a cube with periodic boundary conditions, is dynamically computed, for random initial positions and orientations. The computation utilizes a database of pairwise interactions that has been constructed by the boundary element method. In the case of (non-bottom-heavy) squirmers, both the translational and the orientational spreading of squirmers is correctly described as a diffusive process over a sufficiently long time scale, even though all the movements of the squirmers were deterministically calculated. Scaling of the results on the assumption that the squirmer trajectories are unbiased random walks is shown to capture some but not all of the main features of the results. In the case of (bottom-heavy) squirmers, the diffusive behaviour in squirmers' orientations can be described by a biased random walk model, but only when the effect of hydrodynamic interaction dominates that of the bottom-heaviness. The spreading of bottom-heavy squirmers in the horizontal directions show diffusive behaviour, and that in the vertical direction also does when the average upward velocity is subtracted. The rotational diffusivity in this case, at a volume fraction c=0.1, is shown to be at least as large as that previously measured in very dilute populations of swimming algal cells (Chlamydomonas nivalis).

In order to understand the rheological and transport properties of a suspension of swimming micro-organisms, it is necessary to analyse the fluid-dynamical interaction of pairs of such swimming cells. In this paper, a swimming micro-organism is modelled as a squirming sphere with prescribed tangential surface velocity, referred to as a squirmer. The centre of mass of the sphere may be displaced from the geometric centre (bottom-heaviness). The effects of inertia and Brownian motion are neglected, because real micro-organisms swim at very low Reynolds numbers but are too large for Brownian effects to be important. The interaction of two squirmers is calculated analytically for the limits of small and large separations and is also calculated numerically using a boundary-element method. The analytical and the numerical results for the translational–rotational velocities and for the stresslet of two squirmers correspond very well. We sought to generate a database for an interacting pair of squirmers from which one can easily predict the motion of a collection of squirmers. The behaviour of two interacting squirmers is discussed phenomenologically, too. The results for the trajectories of two squirmers show that first the squirmers attract each other, then they change their orientation dramatically when they are in near contact and finally they separate from each other. The effect of bottom-heaviness is considerable. Restricting the trajectories to two dimensions is shown to give misleading results. Some movies of interacting squirmers are available with the online version of the paper.

When a suspension of the bacterium Bacillus subtilis is placed in a chamber with its upper surface open to the atmosphere, complex bioconvection patterns form. These arise because the cells (a) are denser than water, and (b) swim upwards on average so that the density of an initially uniform suspension becomes greater at the top than at the bottom. When the vertical density gradient becomes large enough an overturning instability occurs which evolves ultimately into the observed patterns. The cells swim upwards because they are oxytactic, i.e. they swim up gradients of oxygen, and they consume oxygen. These properties are incorporated in conservation equations for the cell and oxygen concentrations, which, for the pre-instability stage of the pattern formation process, have been solved in a previous paper (Hillesdon, Pedley & Kessler 1995). In this paper we carry out a linear instability analysis of the steady-state cell and oxygen concentration distributions. There are intrinsic differences between the shallow-and deep-chamber cell concentration distributions, with the consequence that the instability is non-oscillatory in shallow chambers, but must be oscillatory in deep chambers whenever the critical wavenumber is non-zero. We investigate how the critical Rayleigh number for the suspension varies with the three independent parameters of the problem and discuss the most appropriate definition of the Rayleigh number. Several qualitative aspects of the solution of the linear instability problem agree with experimental observation.

A new continuum model is formulated for dilute suspensions of swimming microorganisms with asymmetric mass distributions. Account is taken of randomness in a cell's swimming direction, p, by postulating that the probability density function for p satisfies a Fokker–Planck equation analogous to that obtained for colloid suspensions in the presence of rotational Brownian motion. The deterministic torques on a cell, viscous and gravitational, are balanced by diffusion, represented by an isotropic rotary diffusivity Dr, which is unknown a priori, but presumably reflects stochastic influences on the cell's internal workings. When the Fokker-Planck equation is solved, macroscopic quantities such as the average cell velocity Vc, the particle diffusivity tensor D and the effective stress tensor Σ can be computed; Vc and D are required in the cell conservation equation, and Σ in the momentum equation. The Fokker-Planck equation contains two dimensionless parameters, λ and ε; λ is the ratio of the rotary diffusion time D-1r to the torque relaxation time B (balancing gravitational and viscous torques), while ε is a scale for the local vorticity or strain rate made dimensionless with B. In this paper we solve the Fokker–Planck equation exactly for ε = 0 (λ arbitrary) and also obtain the first-order solution for small ε. Using experimental data on Vc and D obtained with the swimming alga, Chamydomonas nivalis, in the absence of bulk flow, the ε = 0 results can be used to estimate the value of λ for that species (λ ≈ 2.2; Dr ≈ 0.13 s−1). The continuum model for small ε is then used to reanalyse the instability of a uniform suspension, previously investigated by Pedley, Hill & Kessler (1988). The only qualitatively different result is that there no longer seem to be circumstances in which disturbances with a non-zero vertical wavenumber are more unstable than purely horizontal disturbances. On the way, it is demonstrated that the only significant contribution to Σ, other than the basic Newtonian stress, is that derived from the stresslets associated with the cells’ intrinsic swimming motions.

The collapse of a compressed elastic tube conveying a flow occurs in several physiological applications and has become a problem of considerable interest. Laboratory experiments on a finite length of collapsible tube reveal a rich variety of self-excited oscillations, indicating that the system is a complex, nonlinear dynamical system. Following our previous study on steady flow in a two-dimensional model of the collapsible tube problem (Luo & Pedley 1995), we here investigate the instability of the steady solution, and details of the resulting oscillations when it is unstable, by studying the time-dependent problem. For this purpose, we have developed a time-dependent simulation of the coupled flow – membrane problem, using the Spine method to treat the moving boundary and a second-order time integration scheme with variable time increments.

It is found that the steady solutions become unstable as tension falls below a certain value, say Tu, which decreases as the Reynolds number increases. As a consequence, steady flow gives way to self-excited oscillations, which become increasingly complicated as tension is decreased from Tu. A sequence of bifurcations going through regular oscillations to irregular oscillations is found, showing some interesting dynamic features similar to those observed in experiments. In addition, vorticity waves are found downstream of the elastic section, with associated recirculating eddies which sometimes split into two. These are similar to the vorticity waves found previously for flow past prescribed, time-dependent indentations. It is speculated that the mechanism of the oscillation is crucially dependent on the details of energy dissipation and flow separation at the indentation.

As tension is reduced even further, the membrane is sucked underneath the downstream rigid wall and, although this causes the numerical scheme to break down, it in fact agrees with another experimental observation for flow in thin tubes.

The unsteady flow of a viscous, incompressible fluid in a channel with a moving indentation in one wall has been studied by numerical solution of the Navier-Stokes equations. The solution was obtained in stream-function-vorticity form using finite differences. Leapfrog time-differencing and the Dufort-Frankel substitution were used in the vorticity transport equation, and the Poisson equation for the stream function was solved by multigrid methods. In order to resolve the boundary-condition difficulties arising from the presence of the moving wall, a time-dependent transformation was applied, complicating the equations but ensuring that the computational domain remained a fixed rectangle.

Downstream of the moving indentation, the flow in the centre of the channel becomes wavy, and eddies are formed between the wave crests/troughs and the walls. Subsequently, certain of these eddies ‘double’, that is a second vortex centre appears upstream of the first. These observations are qualitatively similar to previous experimental findings (Stephanoff et al. 1983, and Pedley & Stephanoff 1985), and quantitative comparisons are also shown to be favourable. Plots of vorticity contours confirm that the wave generation process is essentially inviscid and reveal the vorticity dynamics of eddy doubling, in which viscous diffusion and advection are important at different stages. The maximum magnitude of wall vorticity is found to be much larger than in quasi-steady flow, with possibly important biomedical implications.

We describe flow-visualization experiments and theory on the two-dimensional unsteady flow of an incompressible fluid in a channel with a time-dependent indentation in one wall. There is steady Poiseuille flow far upstream, and the indentation moves in and out sinusoidally, its retracted position being flush with the wall. The governing parameters are Reynolds number Re, Strouhal number (frequency parameter) St and amplitude parameter ε (the maximum fraction of the channel width occupied by the indentation); most of the experiments were performed with ε ≈ 0.4. For St ≤ 0.005 the flow is quasi-steady throughout the observed range of Re (360 < Re < 1260). For St > 0.005 a propagating train of waves appears, during every cycle, in the core flow downstream of the indentation, and closed eddies form in the separated flow regions on the walls beneath their crests and above their troughs. Later in the cycle, a second, corotating eddy develops upstream of the first in the same separated-flow region (‘eddy doubling’), and, later still, three-dimensional disturbances appear, before being swept away downstream to leave undisturbed parallel flow at the end of the cycle. The longitudinal positions of the wave crests and troughs and of the vortex cores are measured as functions of time for many values of the parameters; they vary with St but not with Re. Our inviscid, long-wavelength, small-amplitude theory predicts the formation of a wavetrain during each cycle, in which the displacement of a core-flow streamline satisfies the linearized Kortewegde Vries equation downstream of the indentation. The waves owe their existence to the non-zero vorticity gradient in the oncoming flow. Eddy formation and doubling are not described by the theory. The predicted positions of the wave crests and troughs agree well with experiment for the larger values of St used (up to 0.077), but less well for small values. Analysis of the viscous boundary layers indicates that the inviscid theory is self-consistent for sufficiently small time, the time of validity increasing as St increases (for fixed ε).

Experiments are performed on steady and impulsively started flow in an approximately two-dimensional closed channel, with one wall locally indented. In plan the indentation is a long trapezium which halves the channel width: the inclination of the sloping walls is approximately 5.7°, and these tapered sections merge smoothly into the narrowest section via rounded corners. The Reynolds number $ Re = a_0\overline{u}_0/\nu $ (a0 = unindented channel width, $\overline{u}_0$ = steady mean velocity in the unindented channel) lies in the range 300 [les ] Re [les ] 1800. In steady flow, flow visualization reveals that separation occurs on the lee slope of the indentation, at a distance downstream of the convex corner which decreases (tending to a non-zero value) as Re increases. There is no upstream separation, and there is some evidence of three-dimensionality of the flow in the downstream separated eddy. Pressure measurements agree qualitatively but not quantitatively with theoretical predictions. Unsteady flow visualization reveals that, as in external flow, wall-shear reversal occurs over much of the lee slope (at dimensionless time $\tau = \overline{u}_0t/a_0 \approx 4$) before there is any evidence of severe boundary-layer thickening and breakaway. Then, at τ ≈ 5.5, a separated eddy develops, and its nose moves gradually upstream from the downstream end of the indentation to its eventual (τ ≈ 75) steady-state position on the lee slope. At about the same time as the wall-shear reversal, wavy vortices appear at the edge of the boundary layer on both walls of the channel, and (for Re < 750) subsequently disappear again; these are interpreted as manifestations of inflection-point instability and not as intrinsic aspects of boundary-layer separation. Pressure measurements are made to investigate the discrepancy between the actual pressure drop across the lee slope and that predicted on the assumption that energy dissipation is quasi-steady. This discrepancy has a maximum value of approximately $1.5\rho \overline{u}^2_0$ (ρ = fluid density), and decays to zero by the time τ ≈ 7.

The two-dimensional thermal boundary layer over a finite hot film embedded in a plane insulating wall, with a shear flow over it which reverses its direction, is analysed approximately using methods similar to those previously developed for viscous boundary layers (Pedley 1976). The heat transfer from the film is calculated both for uniformly decelerated and for oscillatory wall shear, and application is made to predict the response of hot-film anemometers actually used to measure oscillatory velocities in water and blood. The results predict that the velocity amplitude measured on the assumption of a quasi-steady response will depart from the actual amplitude at values of the frequency parameter St greater than about 0·3 (St = ΩX0/U0, where Ω = frequency, U0 = mean velocity, X0 = distance of hot film from the leading edge of the probe). This is in good agreement with experiment. So too is the shape of the predicted anemometer output as a function of time throughout a complete cycle, for cases when the response is not quasi-steady. However, there is a significant phase lead between the predicted and the experimental outputs. Various possible reasons for this are discussed; no firm conclusions are reached, but the most probable cause lies in the three-dimensionality of the velocity and temperature fields, since the experimental hot films are only about 2·5 times as broad as they are long, and are mounted on a cylinder not a flat plate.

The airways of the lung form a rapidly diverging system of branched tubes, and any discussion of their mechanics requires an understanding of the effects of the bifurcations on the flow downstream of them. Experiments have been carried out in models containing up to two generations of symmetrical junctions with fixed branching angle and diameter ratio, typical of the human lung. Flow visualization studies and velocity measurements in the daughter tubes of the first junction verified that secondary motions are set up, with peak axial velocities just outside the boundary layer on the inner wall of the junction, and that they decay slowly downstream. Axial velocity profiles were measured downstream of all junctions at a range of Reynolds numbers for which the flow was laminar.

In each case these velocity profiles were used to estimate the viscous dissipation in the daughter tubes, so that the mean pressure drop associated with each junction and its daughter tubes could be inferred. The dependence of the dissipation on the dimensional variables is expected to be the same as in the early part of a simple entrance region, because most of the dissipation will occur in the boundary layers. This is supported by the experimental results, and the ratio Z of the dissipation in a tube downstream of a bifurcation to the dissipation which would exist in the same tube if Poiseuille flow were present is given by \[ Z = (C/4\surd{2})(Re\,d/L)^{\frac{1}{2}}, \] where L and d are the length and diameter of the tube, Re is the Reynolds number in it, and the constant C (equal to one for simple entry flow) is equal to 1·85 (the average value from our experiments). In general, C is expected to depend on the branching angles and diameter ratios of the junctions used. No experiments were performed in which the flow was turbulent, but it is argued that turbulence will not greatly affect the above results at Reynolds numbers less than and of the order of 10000. Many more experiments are required to consolidate this approach, but predictions based upon it agree well with the limited number of physiological experiments available.

The viscous boundary layer on a finite flat plate in a stream which reverses its direction once (at t = 0) is analysed using an improved version of the approximate method described earlier (Pedley 1975). Long before reversal (t < −t1), the flow at a point on the plate will be quasi-steady; long after reversal (t > t2), the flow will again be quasi-steady, but with the leading edge at the other end of the plate. In between (−t1 < t < t2) the flow is governed approximately by the diffusion equation, and we choose a simple solution of that equation which ensures that the displacement thickness of the boundary layer remains constant at t = −t1. The results of the theory, in the form of the wall shear rate at a point as a function of time, are given both for a uniformly decelerating stream, and for a sinusoidally oscillating stream which reverses its direction twice every cycle. The theory is further modified to cover streams which do not reverse, but for which the quasi-steady solution breaks down because the velocity becomes very small. The analysis is also applied to predict the wall shear rate at the entrance to a straight pipe when the core velocity varies with time as in a dog's aorta. The results show positive and negative peak values of shear very much larger than the mean. They suggest that, if wall shear is implicated in the generation of atherosclerosis because it alters the permeability of the wall to large molecules, then an appropriate index of wall shear at a point is more likely to be the r.m.s. value than the mean.

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.

Unsteady flow in collapsible tubes has been widely studied for a number of different physiological applications; the principal motivation for the work of this paper is the study of blood flow in the jugular vein of an upright, long-necked subject (a giraffe). The one-dimensional equations governing gravity- or pressure-driven flow in collapsible tubes have been solved in the past using finite-difference (MacCormack) methods. Such schemes, however, produce numerical artifacts near discontinuities such as elastic jumps. This paper describes a numerical scheme developed to solve the one-dimensional equations using a more accurate upwind finite volume (Godunov) scheme that has been used successfully in gas dynamics and shallow water wave problems. The adapatation of the Godunov method to the present application is non-trivial due to the highly nonlinear nature of the pressure–area relation for collapsible tubes.

The code is tested by comparing both unsteady and converged solutions with analytical solutions where available. Further tests include comparison with solutions obtained from MacCormack methods which illustrate the accuracy of the present method.

Finally the possibility of roll waves occurring in collapsible tubes is also considered, both as a test case for the scheme and as an interesting phenomenon in its own right, arising out of the similarity of the collapsible tube equations to those governing shallow water flow.