scholarly journals Bioconvection under uniform shear: linear stability analysis

2013 ◽  
Vol 738 ◽  
pp. 522-562 ◽  
Author(s):  
Yongyun Hwang ◽  
T. J. Pedley
Keyword(s):  
Stability Analysis ◽  
Shear Rate ◽  
Linear Stability ◽  
Couette Flow ◽  
Linear Stability Analysis ◽  
Plane Couette Flow ◽  
Shear Rates ◽  
Uniform Shear ◽  
Rayleigh Benard Convection ◽  
Rayleigh Benard

AbstractThe 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 destabilized. 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 behaviour 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 destabilization 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 stabilizing role as in Rayleigh–Bénard convection. However, at small shear rates, it destabilizes these perturbations through the mechanism of overstability discussed by Hill, Pedley and Kessler (J. Fluid Mech., vol. 208, 1989, pp. 509–543). Finally, the present findings are compared with a recent experiment by Croze, Ashraf and Bees (Phys. Biol., vol. 7, 2010, 046001) and they are in qualitative agreement.

Rayleigh–Bénard instability generated by a diffusion flame

2013 ◽  
Vol 736 ◽  
pp. 464-494 ◽  
Author(s):  
P. Pearce ◽  
J. Daou
Keyword(s):  
Stability Analysis ◽  
Rayleigh Number ◽  
Linear Stability ◽  
Linear Stability Analysis ◽  
Diffusion Flame ◽  
Damkohler Number ◽  
Benard Convection ◽  
The Stability ◽  
Rayleigh Benard Convection ◽  
Rayleigh Benard

AbstractWe investigate the Rayleigh–Bénard convection problem within the context of a diffusion flame formed in a horizontal channel where the fuel and oxidizer concentrations are prescribed at the porous walls. This problem seems to have received no attention in the literature. When formulated in the low-Mach-number approximation the model depends on two main non-dimensional parameters, the Rayleigh number and the Damköhler number, which govern gravitational strength and reaction speed respectively. In the steady state the system admits a planar diffusion flame solution; the aim is to find the critical Rayleigh number at which this solution becomes unstable to infinitesimal perturbations. In the Boussinesq approximation, a linear stability analysis reduces the system to a matrix equation with a solution comparable to that of the well-studied non-reactive case of Rayleigh–Bénard convection with a hot lower boundary. The planar Burke–Schumann diffusion flame, which has been previously considered unconditionally stable in studies disregarding gravity, is shown to become unstable when the Rayleigh number exceeds a critical value. A numerical treatment is performed to test the effects of compressibility and finite chemistry on the stability of the system. For weak values of the thermal expansion coefficient $\alpha $, the numerical results show strong agreement with those of the linear stability analysis. It is found that as $\alpha $ increases to a more realistic value the system becomes considerably more stable, and also exhibits hysteresis at the onset of instability. Finally, a reduction in the Damköhler number is found to decrease the stability of the system.

Linear stability analysis of cylindrical Rayleigh–Bénard convection

2012 ◽  
Vol 711 ◽  
pp. 27-39 ◽  
Author(s):  
Bo-Fu Wang ◽  
Dong-Jun Ma ◽  
Cheng Chen ◽  
De-Jun Sun
Keyword(s):  
Stability Analysis ◽  
Prandtl Number ◽  
Linear Stability ◽  
Linear Stability Analysis ◽  
Axisymmetric Flow ◽  
Base Flow ◽  
Prandtl Numbers ◽  
The Stability ◽  
Rayleigh Benard Convection ◽  
Rayleigh Benard

AbstractThe instabilities and transitions of flow in a vertical cylindrical cavity with heated bottom, cooled top and insulated sidewall are investigated by linear stability analysis. The stability boundaries for the axisymmetric flow are derived for Prandtl numbers from 0.02 to 1, for aspect ratio $A$ ($A= H/ R= \mathrm{height} / \mathrm{radius} $) equal to 1, 0.9, 0.8, 0.7, respectively. We found that there still exists stable non-trivial axisymmetric flow beyond the second bifurcation in certain ranges of Prandtl number for $A= 1$, $0. 9$ and 0.8, excluding the $A= 0. 7$ case. The finding for $A= 0. 7$ is that very frequent changes of critical mode (azimuthal Fourier mode) of the second bifurcation occur when the Prandtl number is changed, where five kinds of steady modes $m= 1, 2, 8, 9, 10$ and three kinds of oscillatory modes $m= 3, 4, 6$ are presented. These multiple modes indicate different flow structures triggered at the transitions. The instability mechanism of the flow is explained by kinetic energy transfer analysis, which shows that the radial or axial shear of base flow combined with buoyancy mechanism leads to the instability results.

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