Non-linear bioconvection in a deep suspension of gyrotactic swimming micro-organisms

Abstract

Abstract. The non-linear structure of deep, stochastic, gyrotactic bioconvection is explored. A linear analysis is reviewed and a weakly non-linear analysis justifies its application by revealing the supercritical nature of the bifurcation. An asymptotic expansion is used to derive systems of partial differential equations for long plume structures which vary slowly with depth. Steady state and travelling wave solutions are found for the first order system of partial differential equations and the second order system is manipulated to calculate the speed of vertically travelling pulses. Implications of the results and possibilities of experimental validation are discussed.

Stability of downflowing gyrotactic microorganism suspensions in a two-dimensional vertical channel

“…stability of the state bifurcated via a primary instability), which clearly differs from the present analysis. It should finally be mentioned that this interpretation is consistent with Bees & Hill (1999) where the emergence and stability of stationary gyrotactic plumes in an unbounded and uniform suspension are analysed. This paper is organised as follows: In §2, the equations of motion are introduced and formulated for a linear stability analysis.…”

Analytical approximations for the orientation distribution of small dipolar particles in steady shear flows

“…It could be easily extended to the time dependent problem and the resulting linear dynamical system studied. These expressions, together with Equations (18) and (28), are used in the non-linear analysis of Bees and Hill (1997b). Figures 2 and 3 show the graphs of A and A after truncating at orders 2, 3 and 4.…”

“…The implementation in Maple is straightforward (see Appendix C of Bees (1996) for the Maple code). There are a number of input parameters in the problem.…”

“…The integral is unusual in that only min(p, n) is relevant and not max(p, n) Bees 1996). This procedure would ultimately give an infinite set of infinite-length recursion relations for the AK L .…”

“…In many ways, spherical harmonics are the natural eigenfunctions to use in such an expansion, exemplified by the compact expressions for the above two quantities. Only the first five coefficients in the spherical harmonic expansion are required (Bees 1996). The manageable expressions obtained from this work have been applied to non-linear studies of bioconvection which will be presented in another paper (Bees and Hill 1997b).…”