Stephen Childress

Nonlinear aspects of chemotaxis

Abstract A simplified Keller-Segel model for the chemotactic movements of cellular slime mold is reconsidered. In particular, we ask for the circumstances under which the cell distribution can autonomously develop a δ-function singularity. By the use of suitable differential inequalities, we show that this cannot happen in the case of one-dimensional aggregation. For three or more dimensions, we produce time developments which do become singular, while in the important special case of two-dimensional motion, we advance arguments that the possibility of chemotactic collapse requires a threshold number of cells in the system.

Stretch, Twist, Fold: The Fast Dynamo

Introduction: the fast dynamo problem fast dynamo action in flows fast dynamo in maps. Methods and their applications: dynamos and non-dynamos magnetic structure in steady integrable flows upper bounds magnetic structure in chaotic flows nearly integrable flows spectra and Eigenfunctions strongly chaotic systems random fast dynamos dynamics.

Pattern formation in a suspension of swimming microorganisms: equations and stability theory

A model for collective movement and pattern formation in layered suspensions of negatively geotactic micro-organisms is presented. The motility of the organism is described by an average upward swimming speed U and a diffusivity tensor D. It is shown that the equilibrium suspension is unstable to infinitesimal perturbations when either the layer depth or the mean concentration of the organisms exceeds a critical value. For deep layers the maximum growth rate determines a preferred pattern size explicitly in terms of U and D. The results are compared with observations of patterns formed by the ciliated protozoan Tetrahymena pyriformis.

Symmetry breaking leads to forward flapping flight

Flapping flight is ubiquitous in Nature, yet cilia and flagella, not wings, prevail in the world of micro-organisms. This paper addresses this dichotomy. We investigate experimentally the dynamics of a wing, flapped up and down and free to move horizontally. The wing begins to move forward spontaneously as a critical frequency is exceeded, indicating that ‘flapping flight’ occurs as a symmetry-breaking bifurcation from a pure flapping state with no horizontal motion. A dimensionless parameter, the Reynolds number based on the flapping frequency, characterizes the point of bifurcation. Above this bifurcation, we observe that the forward speed increases linearly with the flapping frequency. Visualization of the flow field around the heaving and plunging foil shows a symmetric pattern below transition. Above threshold, an inverted von Karman vortex street is observed in the wake of the wing. The results of our model experiment, namely the critical Reynolds number and the behaviour above threshold, are consistent with observations of the flapping-based locomotion of swimming and flying animals.

New Solutions of the Kinematic Dynamo Problem

The steady‐state kinematic dynamo problem in a homogeneous 3‐dimensional core is studied. The existence of a class of smooth solenoidal dynamos, satisfying a no‐slip condition on the core boundary, is proved using perturbation theory. The dynamos are of the form q = q(1) + q(2) + q(3), where q(1) is spatially periodic on a sufficiently small scale of length, q(2) is zero except near the core boundary, and q(3) is an arbitrary sufficiently small motion. The term q(1) is also a spatially periodic dynamo in an appropriate sense for an infinite core. The last property allows a simple characterization of the bounded dynamos in terms of the admissible q(1).

Viscous Flow Past a Random Array of Spheres

The steady, slow motion of a viscous fluid past a random, uniform array of identical nonoverlapping finite spheres is studied. The expected force F acting on any given sphere is obtained by averaging over the ensemble of possible positions of all other spheres. It is shown that, if the Stokes equations hold in the fluid, then for small particle volume concentration c, F has an expansion of the form F=[ 1 + (3 / 2) c1/2 + (135/64)c logc ] F0 + cD · F0 + 0 (c3/2logc), where F0 is the Stokes force on a single isolated sphere in the same flow and D is a tensor which is given in terms of the two‐sphere distribution function for the array and the Stokes solution for two spheres oriented arbitrarily in an unbounded flow. A physical model for the solution is presented and compared with the model of Brinkman.

Alpha-effect in flux ropes and sheets

Abstract The alpha-effect is studied at large magnetic Reynolds numbers (R) for certain simple steady flows. The flux sheets created by a two-dimensional cellular flow sustain an α of order R − 1 2 at large values of R. For an analogous helical axisymmetric motion the mean electromotive force gives a finite nonzero limit of α as R → ∞, this being a consequence of the existence of an intense axial flux rope. The latter estimate may be typical of steady three-dimensional motion.

The slow motion of a sphere in a rotating, viscous fluid

The uniform, slow motion of a sphere in a viscous fluid is examined in the case where the undisturbed fluid rotates with constant angular velocity Ω and the axis of rotation is taken to coincide with the line of motion. The various modifications of the classical problem for small Reynolds numbers are discussed. The main analytical result is a correction to Stokess drag formula, valid for small values of the Reynolds number and Taylor number and tending to the classical Oseen correction as the last parameter tends to zero. The rotation of a free sphere relative to the fluid at infinity is also deduced.

Stirring by squirmers

We analyse a simple ‘Stokesian squirmer’ model for the enhanced mixing due to swimming micro-organisms. The model is based on a calculation of Thiffeault & Childress ( Phys. Lett. A, vol. 374, 2010, p. 3487), where fluid particle displacements due to inviscid swimmers are added to produce an effective diffusivity. Here we show that, for the viscous case, the swimmers cannot be assumed to swim an infinite distance, even though their total mass displacement is finite. Instead, the largest contributions to particle displacement, and hence to mixing, arise from random changes of direction of swimming and are dominated by the far-field stresslet term in our simple model. We validate the results by numerical simulation. We also calculate non-zero Reynolds number corrections to the effective diffusivity. Finally, we show that displacements due to randomly swimming squirmers exhibit probability distribution functions with exponential tails and a short-time superdiffusive regime, as found previously by several authors. In our case, the exponential tails are due to ‘sticking’ near the stagnation points on the squirmers surface.

Scalar transport and alpha-effect for a family of cat's-eye flows

In this paper we study advection–diffusion of scalar and vector fields for the steady velocity field \[ (u, v, w) = \left(\frac{\partial \psi}{\partial y},-\frac{\partial\psi}{\partial x},K\psi\right),\quad \psi = \sin x \sin y + \delta \cos x \cos y. \] If δ > 0 the streamlines ψ = constant form a periodic array of oblique cats-eyes separated by continuous channels carrying finite fluid flux. In the problems treated, advection dominates diffusion, and fields are transported both in thin boundary layers and within the channels. Effective transport of a passive scalar and the alphaeffect generated by interaction of the flow with a uniform magnetic field are examined. For the latter problem, we determine the alpha-matrix as a function of δ. Our results consist of (i) numerical solution of steady problems in the limit of large Reynolds number R with β = δ R ½ held fixed and O (1), and (ii) analytic asymptotic solutions for large R , obtained using the Wiener–Hopf technique, which are valid for large β. The asymptotic method gives reliable values of the effective diffusion and of alpha-matrices down to β ≈ 1.5. When β > 0 the transport and alpha-effect are greatly enhanced by flux down the channels. Consequently, the alpha-effect found here may have application to the construction of efficient fast dynamos, but this requires spatial dependence of the mean field and the inclusion of three-dimensional effects, as in the established fast-dynamo analysis with δ = 0.