Bioconvection under uniform shear: linear stability analysis

Abstract

The role of uniform shear in bioconvective instability in a shallow suspension of swimming gyrotactic cells is studied using linear stability analysis. The shear is introduced by applying a plane Couette flow, and it significantly disturbs gravitaxis of the cell. The unstably stratified basic state of the cell concentration is gradually relieved as the shear rate is increased, and it even becomes stably stratified at very large shear rates. Stability of the basic state is significantly changed. The instability at high wavenumbers is drastically damped out with the shear rate, while that at low wavenumbers is destabilised. However, at very large shear rates, the latter is also suppressed. The most unstable mode is found as a pair of streamwise uniform rolls aligned with the shear, analogous to Rayleigh-Bénard convection in plane Couette flow. To understand these findings, the physical mechanism of the bioconvective instability is reexamined with several sets of numerical experiments. It is shown that the bioconvective instability in a shallow suspension originates from three different physical processes: gravitational overturning, gyrotaxis of the cell, and negative cross diffusion flux. The first mechanism is found to rule the behavior of low-wavenumber instability whereas the last two mechanisms are mainly associated with high-wavenumber instability. With the increase of the shear rate, the former is enhanced, thereby leading to destabilisation at low wavenumbers, whereas the latter two mechanisms are significantly suppressed. For streamwise varying perturbations, shear with sufficiently large rates is also found to play a stabilising role as in Rayleigh-Bénard convection. However, at small shear rates, it destabilises these perturbations through the mechanism of overstability discussed by Hill et al. (1989). Finally, the present findings are compared with a recent experiment by Croze et al. (2010) and they are in qualitative agreement.

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

“…However, contrary to the classical Rayleigh-Bénard convection, the pattern in bioconvection is not a simple consequence of the gravitational instability. As recently shown by the authors (Hwang & Pedley 2014), the gyrotactic instability mechanism also plays an important cooperative role in generating the instability (particularly at high wavenumbers). Furthermore, under an unstably stratified background cell concentration, the gyrotaxis yields non-negligible amounts of cross-diffusion flux to locally denser regions of the cells, which also significantly contributes to the bioconvective instability (Hwang & Pedley 2014).…”

“…As recently shown by the authors (Hwang & Pedley 2014), the gyrotactic instability mechanism also plays an important cooperative role in generating the instability (particularly at high wavenumbers). Furthermore, under an unstably stratified background cell concentration, the gyrotaxis yields non-negligible amounts of cross-diffusion flux to locally denser regions of the cells, which also significantly contributes to the bioconvective instability (Hwang & Pedley 2014). The instabilities observed in the suspensions of gyrotactic cells are often a consequence of cooperation and/or competition of multiple physical processes, and this feature often hampers precise understanding of the origin of the instabilities.…”

“…Their swimming speed is V * c and they are assumed to perform random walks as a result of repeated reorientations. Following Hwang & Pedley (2014), where a more detailed discussion of the present model is given, the equations of motion are then written as…”

“…The blob therefore becomes denser and creates even stronger downflow, causing an instability in the suspension (Pedley et al 1988;Pedley & Kessler 1990;Pedley 2010a,b). Another example of the crucial role played by gyrotaxis in generating instability is bioconvection which appears in a shallow layer (d < 1cm) of the suspension (Childress et al 1975;Kessler 1985a;Hill et al 1989;Bees & Hill 1998;Hwang & Pedley 2014). Here, the gravitactic up-swimming cells quickly accumulate at the top end of the suspension and form a layer of cells heavier than the surroundings.…”

“…We start with a more complete form of mathematical model developed by Pedley (2010a) where the distribution of the randomly swimming cells in time, space, and the cell swimming orientation-space is described with a probability density function satisfying a Smoluchowski equation. We then follow the approximations given in Hwang & Pedley (2014) so that the computational cost of the present study is manageable. For simplicity, the flow configuration is set as a twodimensional vertical channel instead of the circular pipe (see figure 1).…”