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: For very small samples, it is difficult to prepare graphitic targets that will yield a useful and steady sputtered ion beam. Working with materials separated by preparative capillary gas chromatography, we have succeeded with amounts as small as 20 μg C. This seems to be a practical limit, as it involves 1) multiple chromatographic runs with trapping of effluent fractions, 2) recovery and combustion of the fractions, 3) graphitization and 4) compression of the resultant graphite/cobalt matrix into a good sputter target. Through such slow and intricate work, radiocarbon ages of lignin derivatives and hydrocarbons from coastal sediments have been determined. If this could be accomplished as an “online” measurement by flowing the analytes directly into a microwave gas ion source, with a carrier gas, then the number of processing steps could be minimized. Such a system would be useful not just for chromatographic effluents, but for any gaseous material, such as CO2 produced from carbonates. We describe tests using such an ion source.

Description: Motivated by the study of blood flow in the major coronary arteries, which are situated on the outer surface of the pumping heart, we analyse flow of an incompressible Newtonian fluid in a tube whose curvature varies both along the tube and with time. Attention is restricted to the case in which the tube radius is fixed and its centreline moves in a plane. Nevertheless, the governing equations are very complicated, because the natural coordinate system involves acceleration, rotation and deformation of the frame of reference, and their derivation forms a major part of the paper. Then they are applied to two particular, relatively simple examples: a tube of uniform but time-dependent curvature; and a sinuous tube, representing a small-amplitude oscillation about a straight pipe. In the former case the curvature is taken to be small and to vary by a small amount, and the solution is developed as a triple power series in mean curvature ratio δ0, curvature variation ε and Dean number D. In the latter case the Reynolds number is taken to be large and a linearized solution for the perturbation to the flow in the boundary layer at the tube wall is obtained, following Smith (1976a). In each case the solution is taken far enough that the first non-trivial effects of the variable curvature can be determined. Results are presented in terms of the oscillatory wall shear stress distribution and, in the uniform curvature case, the contribution of steady streaming to the mean wall shear stress is calculated. Estimation of the parameters for the human heart indicates that the present results are not directly applicable, but point the way for future work.

Description: The effect of wall inertia on the self-excited oscillations in a collapsible channel flow is investigated by solving the full coupled two-dimensional membrane-flow equations. This is the continuation of a previous study in which self-excited oscillations were predicted in an asymmetric channel with a tensioned massless elastic membrane (Luo & Pedley 1996). It is found that a different type of self-excited oscillation, a form of flutter, is superposed on the original large-amplitude, low-frequency oscillations. Unlike the tension-induced oscillations, the flutter has high frequency, and grows with time from a small amplitude until it dominates the original slower mode. The critical value of tension below which oscillations arise (at fixed Reynolds number) is found to increase as the wall inertia is increased. The rate at which energy is (a) dissipated in the flow field and (b) transferred to the wall during the flutter is discussed, and results at different parameter values are compared with those of a massless membrane. There is also a discussion of whether the onset of flutter, or that of the slower oscillations, is correlated with the appearance of flow limitation, as is thought to be the case in the context of wheezing during forced expiration of air from the lungs.

Description: Motivated by the study of blood flow in the coronary arteries, this paper examines the flow of an incompressible Newtonian fluid in a tube of time-dependent curvature. The flow is driven by an oscillatory pressure gradient with the same dimensionless frequency, α, as the curvature variation. The dimensionless governing parameters of the flow are α, the curvature ratio δ0 a secondary streaming Reynolds number Rs and a parameter Rt representing the time-dependence of curvature. We consider the parameter regime δ0 ≪ Rt ≪ 1 (Rs and α remain O(1) initially) in which the effect of introducing time-dependent curvature is to perturb the flow driven by an oscillatory pressure gradient in a fixed curved tube. Flows driven by low- and high-frequency pressure gradients are then considered. At low frequency (δ0 ≪ Rt ≪ α ≪ 1) the flow is determined by using a sequence of power series expansions (Rs = O(1)). At high frequency (δ0 ≪ Rt ≪ 1/α2 ≪ 1) the solution is obtained using matched asymptotic expansions for the region near the wall (Stokes layer) and the region away from the wall in the interior of the pipe. The behaviour of the flow in the interior is then determined at both small and intermediate values of Rs. For both the low and high frequency cases, we find the principal corrections introduced by the time-varying curvature to the primary and secondary flows, and hence to the wall shear stress. The physiological application to flow in the coronary arteries is discussed.

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.