Development and stability of gyrotactic plumes in bioconvection

Abstract

Using the continuum model of Pedley, Hill & Kessler (1988) for bioconvection in a suspension of swimming, gyrotactic micro-organisms, we investigate the existence and stability of a two-dimensional plume in tall, narrow chambers with stress-free sidewalls. The system is governed by the Navier-Stokes equations for an incompressible fluid coupled with a micro-organism conservation equation. These equations are solved numerically using a conservative finite-difference scheme. In sufficiently deep chambers, the plume is always unstable to both varicose and meandering modes. A linear stability analysis for an infinitely long plume predicts the growth rates of these instabilities, explains the mechanisms, and is in good agreement with the numerical results.

Wavelengths of Gyrotactic Plumes in Bioconvection

“…To validate the code, written in terms of stretched coordinates, the code has been run for the heat convection problem and the agreement with the benchmark solution of De Vahl Davis (1983) (Ghorai, 1997) is good. The code has also been used to compute the critical Rayleigh numbers against wavelengths (Ghorai and Hill, 1999) and the agreement with the linear stability results of Hill et al (1989) is excellent.…”

“…If the vorticity is large ( |Bζ | > 1), the cell tumbles but swims on average in a fixed direction at an angle to the vertical (Kessler, 1985b). When the vorticity is large, the average swimming directionp is approximated by integrating the swimming direction over the tumbling period (Ghorai and Hill, 1999).…”

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

“…Although this configuration is not exactly the same as that of the experiment by Kessler (1985a) and Kessler (1986), we will see that many features of the instability found in the present study are remarkably similar to the blips observed in the pipe experiment. Finally, we should emphasise that the present study should be distinguished from previous work by Ghorai & Hill (1999, 2000 who performed a linear stability analysis of naturally arising 'standing' gyrotactic plumes. In the present study, the basic state, obtained in the form of the beam-like structure as in the experiment, is essentially caused by the imposed flow.…”

“…Although the role of this mechanism has not been discussed at all, the related term does appear e.g. in the equations of Ghorai & Hill (1999). We note that this mechanism is represented by the term D⟨e 2 ⟩ 0n in (2.21e), which describes production due to the gradient in the cell swimming vector field.…”

“…Therefore, the basic state given here is continuously connected to the uniform suspension of the cells in the absence of the imposed flow (see §3.1). In contrast, the basic state in Ghorai & Hill (1999, 2000 was given in the form of a vertically uniform stationary plume, generated as a consequence of the gyrotactic instability in a stationary suspension, the top and bottom ends of which are closed by the no-fluid-and no-cell-flux conditions. In this respect, their analysis is precisely a secondary stability analysis of a uniform suspension in the given flow configuration (i.e.…”