ALBERT

Description: A swimming micro-organism is modelled as a squirming sphere with prescribed tangential surface velocity and referred to as a squirmer. The centre of mass of the sphere may be displaced from the geometric centre, and the effects of inertia and Brownian motion are neglected. The well-known Stokesian dynamics method is modified in order to simulate squirmer motions in a concentrated suspension. The movement of 216 identical squirmers in a concentrated suspension without any imposed flow is simulated in a cubic domain with periodic boundary conditions, and the coherent structures within the suspension are investigated. The results show that (a) a weak aggregation of cells appears as a result of the hydrodynamic interaction between cells; (b) the cells generate collective motions by the hydrodynamic interaction between themselves; and (c) the range and duration of the collective motions depend on the volume fraction and the squirmers' stresslet strengths. These tendencies show good qualitative agreement with previous experiments. © 2008 Cambridge University Press.

Description: An experimental study has been performed on the dynamics of a large turbulent buoyanthelium plume. Two-dimensional velocity fields were measured using particle image velocimetry (PIV) while helium mass fraction was determined by planar laser-induced fluorescence (PLIF). PIV and PLIF were performed simultaneously in order to obtain velocity and mass fraction data over a plane that encompassed the plume core, the near-field mixing zones and the surrounding air. The Rayleigh-Taylor instability at the base of the plume leads to the vortex that grows to dominate the flow. This process repeats in a cyclical manner. The temporally and spatially resolved data show a strong negative correlation between density and vertical velocity, as well as a strong 90° phase lag between peaks in the vertical and horizontal velocities throughout the flow field owing to large coherent structures associated with puffing of the turbulent plume. The joint velocity an mass fraction data are used to calculate Favre-averaged statistics in addition to Reynolds-(time) averaged statistics. Unexpectedly, the difference between both the Favre-averaged and Reynolds-averaged velocities and second-order turbulent statistics is less than the uncertainty in the data throughout the flow field. A simple analysis was performed to determine the expected differences between Favre and Reynolds statistics for flows with periodic fluctuations in which the density and velocity fields are perfectly correlated, but have the phase relations as suggested by the data. The analytical results agreewith the data, showing that the Favre and Reynolds statistics will be the same to lead order. The combination of observation and simple analysis suggests that for buoyancy-dominated flows in which it can be expected that density and velocity are strongly correlated,phase relations will result in only second-order differences between Favre- and Reynolds-averaged data in spite of strong fluctuations in both density and velocity. © 2005 Cambridge University Press.

Description: We have studied steady flow in a two-dimensional channel in which a section of one wall has been replaced by an elastic membrane under dimensionless longitudinal tension T but possessing no bending stiffness. The dimensionless upstream transmural pressure takes a value Pext, the membrane section is assumed to be long compared with the channel width and its deformation is assumed to remain within the viscous boundary layers. Standard high-Reynolds-number asymptotic methods are applied to arrive at a coupled boundary-layer-membrane problem. A non-zero cross-stream pressure gradient, leading to flow perturbations upstream of the membrane, is included in the analysis. Linearization of the boundary-layer problem yields firstly an analytic solution at non-zero Pext and asymptotically high T. This takes the form of an expansion in T-1 for which the membrane shape and the flow decouple at each order. Extension of this solution branch to smaller values of the tension suggests a singularity at finite tension, where the deformation of the membrane becomes very large. Secondly, when the upstream transmural pressure is zero the trivial solution is valid for all values of the tension. However, we also obtain eigensolutions where the membrane tension plays the role of eigenvalue. There are thus non-trivial solutions of the problem at these particular values of the tension. The nonlinear coupled boundary-layer-membrane problem is then solved numerically. A finite-difference, Keller-box, marching scheme is used, together with a shooting algorithm to satisfy the boundary condition at the downstream end of the membrane. This reveals a variety of different solutions, showing the relation between the two cases captured by the linearized analysis and demonstrating the existence of parameter ranges for which no solutions exist under the specified constraints. Such parameter ranges appear not to exist if the downstream, rather than the upstream, transmural pressure is held constant. The relation to our results of solutions obtained by solving the two-dimensional Navier-Stokes equations directly is discussed. Reasonable agreement between parameters is obtained, once allowance is made for the finite Reynolds number and membrane length in those computations. © 2006 Cambridge University Press.

Description: A generalization of criticality - called secondary criticality - is introduced and applied to finite-amplitude Stokes waves. The theory shows that secondary criticality signals a bifurcation to a class of steady dark solitary waves which are biasymptotic to a Stokes wave with a phase jump in between, and synchronized with the Stokes wave. We find the that the bifurcation to these new solitary waves - from Stokes gravity waves in shallow water - is pervasive, even at low amplitude. The theory proceeds by generalizing concepts from hydraulics: three additional functionals are introduced which represent non-uniformity and extend the familiar mass flux, total head and flow force, the most important of which is the wave action flux. The theory works because the hydraulic quantities can be related to the governing equations in a precise way using the multi-symplectic Hamiltonian formulation of water waves. In this setting, uniform flows and Stokes waves coupled to a uniform flow are relative equilibria which have an attendant geometric theory using symmetry and conservation laws. A flow is then 'critical' if the relative equilibrium representation is degenerate. By characterizing successively non-uniform flows and unsteady flows as relative equilibria, a generalization of criticality is immediate. Recent results on the local nonlinear behaviour near a degenerate relative equilibrium are used to predict all the qualitative properties of the bifurcating dark solitary waves, including the phase shift. The theory of secondary criticality provides new insight into unsteady waves in shallow water as well. A new interpretation of the Benjamin-Feir instability from the viewpoint of hydraulics, and the connection with the creation of unsteady dark solitary waves, is given in Part 2. © 2006 Cambridge University Press.

Description: We present a numerical study of the structure and stability of laminar isothermal flows formed by two counterflowing jets of an incompressible Newtonian fluid. We demonstrate that symmetric counterflowing jets with identical mass flow rates exhibit multiple steady states and, in certain cases, time-dependent (periodic) steady states. Two geometric configurations were studied based on the inlet jet shapes: planar and axisymmetric. Stagnation flows formed by planar counterflowing jets exhibit both steady-state multiplicity and time-dependent behaviour, while axisymmetric jets exhibit only a steady-state multiplicity. A linearized bifurcation and stability analysis based on the continuity and Navier-Stokes equations revealed transitions between a single (symmetric) steady state and multiple steady states or periodic steady states. The dimensionless quantities forming the parameter space of this system are the inlet Reynolds number (Re) and a geometric aspect ratio α, equal to the jet inlet characteristic length (used for calculating Re) divided by the jet separation. The boundaries separating different flow regimes have been identified in the (Re, α) parameter space. The resulting flow maps are useful for the design and operation of counterflow jet reactors. © 2006 Cambridge University Press.

Description: 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). © 2007 Cambridge University Press.

Description: We present results for the average mass transfer to a spherical squirmer, a model micro-organism whose surface oscillates tangentially to itself. The surface motion drives a low-Reynolds-number flow which enables the squirmer either to swim relative to the fluid at infinity, at an average speed proportional to a streaming parameter, W, or to stir the fluid around it while remaining, on average, at rest (if W =0), as represented by a hovering parameter, b. We assume that the amplitude of the time-periodic surface distortions is scaled by a dimensionless small parameter ∈, and consider only high Péclet numbers P - a measure of convection versus diffusion - by setting P-1 = ∈2γ, where γ is a parameter of 0(1). It is shown that the average mass concentration distribution satisfies a steady convection-diffusion equation with an effective velocity field that is different from the actual mean velocity field. The model is used to calculate the mass transfer across the surface of the squirmer, measured by the mean Sherwood number Sh. We find asymptotic solutions for small and large γ and numerical results for the whole range of values. While the large-γ expansions are reproduced well by the numerical results, there is a discrepancy between the two at small γ. We believe this is due to very small recirculation regions, attached to the surface of the squirmer, which make boundary layer theory applicable only when 1/γ is immense. For the parameters chosen in this study, results indicate that both hovering and streaming contribute to the mass transfer, although streaming has a greater effect. Also, energy dissipation considerations show that an optimum swimming mode exists, at least at small and large γ, for any given uptake rate. However, other factors have still to be taken into account, and the model realism improved, if we want to make predictions for real aquatic micro-organisms. © 2005 Cambridge University Press.

Description: The rheological properties of a cell suspension may play an important role in the flow field generated by populations of swimming micro-organisms (e.g. in bioconvection). 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 identical squirmers in a simple shear flow field, 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. The restriction to pairwise additivity of forces is expected to be justified if the suspension is semi-dilute. The results for non-bottom-heavy squirmers show that the squirming does not have a direct influence on the apparent viscosity. However, it does change the probability density in configuration space, and thereby causes a slight decrease in the apparent viscosity at O(c2), where c is the volume fraction of spheres. In the case of bottom-heavy squirmers, on the other hand, the stresslet generated by the squirming motion directly contributes to the bulk stress at O(c), and the suspension shows strong non-Newtonian properties. When the background simple shear flow is directed vertically, the apparent viscosity of the semi-dilute suspension of bottom-heavy squirmers becomes smaller than that of inert spheres. When the shear flow is horizontal and varies with the vertical coordinate, on the other hand, the apparent viscosity becomes larger than that of inert spheres. In addition, significant normal stress differences appear for all relative orientations of gravity and the shear flow, in the case of bottom-heavy squirmers. © 2007 Cambridge University Press.

Description: 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. © 2006 Cambridge University Press.