Bioconvection in suspensions of oxytactic bacteria: linear theory

1996 ◽  
Vol 324 ◽  
pp. 223-259 ◽  
Author(s):  
A. J. Hillesdon ◽  
T. J. Pedley
Keyword(s):  
Rayleigh Number ◽  
Cell Concentration ◽  
Critical Rayleigh Number ◽  
Linear Instability ◽  
Instability Problem ◽  
Vertical Density ◽  
Linear Instability Analysis ◽  
Definition Of ◽  
Independent Parameters ◽  
Critical Wavenumber

When a suspension of the bacteriumBacillus subtilisis placed in a chamber with its upper surface open to the atmosphere, complex bioconvection patterns form. These arise because the cells (a) are denser than water, and (b) swim upwards on average so that the density of an initially uniform suspension becomes greater at the top than at the bottom. When the vertical density gradient becomes large enough an overturning instability occurs which evolves ultimately into the observed patterns. The cells swim upwards because they are oxytactic, i.e. they swim up gradients of oxygen, and they consume oxygen. These properties are incorporated in conservation equations for the cell and oxygen concentrations, which, for the pre-instability stage of the pattern formation process, have been solved in a previous paper (Hillesdon, Pedley & Kessler 1995). In this paper we carry out a linear instability analysis of the steady-state cell and oxygen concentration distributions. There are intrinsic differences between the shallow-and deep-chamber cell concentration distributions, with the consequence that the instability is non-oscillatory in shallow chambers, but must be oscillatory in deep chambers whenever the critical wavenumber is non-zero. We investigate how the critical Rayleigh number for the suspension varies with the three independent parameters of the problem and discuss the most appropriate definition of the Rayleigh number. Several qualitative aspects of the solution of the linear instability problem agree with experimental observation.

scholarly journals Thermal convection for a Darcy-Brinkman rotating anisotropic porous layer in local thermal non-equilibrium

2021 ◽  
Author(s):  
Florinda Capone ◽  
Jacopo A. Gianfrani
Keyword(s):  
Natural Convection ◽  
Rayleigh Number ◽  
Porous Layer ◽  
Nonlinear Stability ◽  
Critical Rayleigh Number ◽  
Vertical Axis ◽  
Linear Instability ◽  
Brinkman Model ◽  
Linear Instability Analysis ◽  
Non Equilibrium

AbstractThe onset of natural convection in a fluid-saturated anisotropic porous layer, which rotates about the vertical axis, under the hypothesis of local thermal non-equilibrium, is analysed. Since the porosity of the medium is assumed to be high, the more suitable Darcy-Brinkman model is adopted. Linear instability analysis of the conduction solution is carried out. Nonlinear stability with respect to $$L^2$$ L 2 -norm is performed in order to prove the coincidence between the linear instability and the global nonlinear stability thresholds. The effect of both rotation and thermal and mechanical anisotropies on the critical Rayleigh number for the onset of instability is discussed.

The stability of statically unstable layers

1994 ◽  
Vol 260 ◽  
pp. 315-331 ◽  
Author(s):  
S. A. Thorpe
Keyword(s):  
Rayleigh Number ◽  
Critical Rayleigh Number ◽  
Internal Gravity Waves ◽  
Companion Paper ◽  
Linear Instability ◽  
Buoyancy Frequency ◽  
Vertical Coordinate ◽  
Uniform Shear ◽  
Vertical Scale ◽  
The Stability

We investigate the development of instability in a fluid with density locally of the form ρ0[1 −(N2 / g)z + A sin Kz], composed of an overall stable uniform gradient of buoyancy frequency, N, but with a superimposed sinusoidal variation of vertical wavenumber, K, and amplitude, A [Lt ] 1; g is the acceleration due to gravity and z is the upward vertical coordinate. Layers exist in which the fluid is statically unstable when the parameter r = N2 / gKA, is less than unity.When r is zero, the density is sinusoidal in z and the problem reduces to one studied by Batchelor & Nitsche (1991). Their solution, which finds a gravest mode of linear instability with terms having vertical motions independent of z and with horizontal scales large in comparison with K−1, is extended to non-zero r. An effect of a small, but finite, r is to stabilize the fluid, increasing the critical Rayleigh number and the corresponding non-dimensional horizontal wavenumber. The vertical scale of the mode which first becomes unstable is reduced as r increases. A small sinusoidal shear destabilizes the fluid.When r approaches unity, the density field contains regions of static instability which are of thickness small compared to K−1. The problem then approximates to one studied by Matthews (1988). Consistent solutions for the growth of disturbances are obtained by truncated series and, following Matthews, by the solution of a Fourier-transformed equation. A small uniform shear, characterized by a flow Reynolds number, Re > O, is found to stabilize the fluid, in that it increases the critical Rayleigh number of the onset of instability. This suggests that convective Rayleigh–Taylor instability, with constant phase lines parallel to the flow, is then the favoured mode of onset of instability. At very large Rayleigh numbers and at a Prandtl number of 700, however, the growth rate of the most rapidly growing linear disturbances may increase as Re increases from zero, and the form of the evolving flow is then less certain.The theory is used to estimate the scale and growth rates of instability in overturning internal gravity waves in the laboratory experiment described in a companion paper (Thorpe 1994).

scholarly journals Anisotropic bidispersive convection

2019 ◽  
Vol 475 (2227) ◽  
pp. 20190206 ◽  
Author(s):  
B. Straughan
Keyword(s):  
Porous Medium ◽  
Stability Analysis ◽  
Rayleigh Number ◽  
Nonlinear Stability ◽  
Thermal Convection ◽  
Linear Instability ◽  
Solid Skeleton ◽  
Nonlinear Stability Analysis ◽  
Cell Structures ◽  
Linear Instability Analysis

This paper investigates thermal convection in an anisotropic bidisperse porous medium. A bidisperse porous medium is one which possesses the usual pores, but in addition, there are cracks or fissures in the solid skeleton and these give rise to a second porosity known as micro porosity. The novelty of this paper is that the macro permeability and the micro permeability are each diagonal tensors but the three components in the vertical and in the horizontal directions may be distinct in both the macro and micro phases. Thus, there are six independent permeability coefficients. A linear instability analysis is presented and a fully nonlinear stability analysis is inferred. Several Rayleigh number and wavenumber calculations are presented and it is found that novel cell structures are predicted which are not present in the single porosity case.

scholarly journals Rayleigh–Bénard magnetoconvection with temperature modulation

2020 ◽  
Vol 476 (2242) ◽  
pp. 20200289
Author(s):  
Suparna Hazra ◽  
Krishna Kumar ◽  
Saheli Mitra
Keyword(s):  
Rayleigh Number ◽  
Modulation Frequency ◽  
Critical Rayleigh Number ◽  
Temperature Modulation ◽  
Modulation Amplitude ◽  
Local Minima ◽  
External Modulation ◽  
New Type ◽  
Rayleigh Benard ◽  
Critical Wavenumber

Floquet analysis of modulated magnetoconvection in Rayleigh–Bénard geometry is performed. A sinusoidally varying temperature is imposed on the lower plate. As Rayleigh number Ra is increased above a critical value Ra o , the oscillatory magnetoconvection begins. The flow at the onset of magnetoconvection may oscillate either subhar- monically or harmonically with the external modulation. The critical Rayleigh number Ra o varies non-monotonically with the modulation frequency ω for appreciable value of the modulation amplitude a . The temperature modulation may either postpone or prepone the appearance of magnetoconvection. The magnetoconvective flow always oscillates harmonically at larger values of ω . The threshold Ra o and the corresponding wavenumber k o approach to their values for the stationary magnetoconvection in the absence of modulation ( a  = 0), as ω  → ∞. Two different zones of harmonic instability merge to form a single instability zone with two local minima for higher values of Chandrasekhar’s number Q , which is qualitatively new. We have also observed a new type of bicritical point, which involves two different sets of harmonic oscillations. The effects of variation of Q and Pr on the threshold Ra o and critical wavenumber k o are also investigated.

Linear stability of rotating convection in an imposed shear flow

1997 ◽  
Vol 350 ◽  
pp. 271-293 ◽  
Author(s):  
PAUL MATTHEWS ◽  
STEPHEN COX
Keyword(s):  
Rayleigh Number ◽  
Shear Flow ◽  
Eigenvalue Problem ◽  
Linear Stability ◽  
Thermal Convection ◽  
Critical Rayleigh Number ◽  
Rotation Vector ◽  
Linear Stability Theory ◽  
Fixed Temperature ◽  
Differential Motion

In many geophysical and astrophysical contexts, thermal convection is influenced by both rotation and an underlying shear flow. The linear theory for thermal convection is presented, with attention restricted to a layer of fluid rotating about a horizontal axis, and plane Couette flow driven by differential motion of the horizontal boundaries.The eigenvalue problem to determine the critical Rayleigh number is solved numerically assuming rigid, fixed-temperature boundaries. The preferred orientation of the convection rolls is found, for different orientations of the rotation vector with respect to the shear flow. For moderate rates of shear and rotation, the preferred roll orientation depends only on their ratio, the Rossby number.It is well known that rotation alone acts to favour rolls aligned with the rotation vector, and to suppress rolls of other orientations. Similarly, in a shear flow, rolls parallel to the shear flow are preferred. However, it is found that when the rotation vector and shear flow are parallel, the two effects lead counter-intuitively (as in other, analogous convection problems) to a preference for oblique rolls, and a critical Rayleigh number below that for Rayleigh–Bénard convection.When the boundaries are poorly conducting, the eigenvalue problem is solved analytically by means of an asymptotic expansion in the aspect ratio of the rolls. The behaviour of the stability problem is found to be qualitatively similar to that for fixed-temperature boundaries.Fully nonlinear numerical simulations of the convection are also carried out. These are generally consistent with the linear stability theory, showing convection in the form of rolls near the onset of motion, with the appropriate orientation. More complicated states are found further from critical.

Heat transport by parallel-roll convection in a rectangular container

1987 ◽  
Vol 185 ◽  
pp. 205-234 ◽  
Author(s):  
R. W. Walden ◽  
Paul Kolodner ◽  
A. Passner ◽  
C. M. Surko
Keyword(s):  
Rayleigh Number ◽  
Heat Transport ◽  
Critical Rayleigh Number ◽  
Equation Model ◽  
Onset Of Convection ◽  
Infinite Layer ◽  
Lateral Extent ◽  
Prandtl Numbers ◽  
Rayleigh Numbers ◽  
Good Agreement

Heat-transport measurements are reported for thermal convection in a rectangular box of aspect’ ratio 10 x 5. Results are presented for Rayleigh numbers up to 35Rc, Prandtl numbers between 2 and 20, and wavenumbers between 0.6 and 1.0kc, where Rc and kc are the critical Rayleigh number and wavenumber for the onset of convection in a layer of infinite lateral extent. The measurements are in good agreement with a phenomenological model which combines the calculations of Nusselt number, as a function of Rayleigh number and roll wavenumber for two-dimensional convection in an infinite layer, with a nonlinear amplitude-equation model developed to account for sidewell attenuation. The appearance of bimodal convection increases the heat transport above that expected for simple parallel-roll convection.

Linear Instability Analysis of an Annular Swirling Viscous Liquid Jet

2011 ◽  
Vol 66-68 ◽  
pp. 1556-1561 ◽  
Author(s):  
Kai Yan ◽  
Ming Lv ◽  
Zhi Ning ◽  
Yun Chao Song
Keyword(s):  
Viscous Liquid ◽  
Liquid Velocity ◽  
Aerodynamic Force ◽  
Numerical Procedure ◽  
Liquid Sheet ◽  
Linear Instability ◽  
Viscous Force ◽  
Liquid Jet ◽  
Instability Analysis ◽  
Linear Instability Analysis

A three-dimensional linear instability analysis was carried out for an annular swirling viscous liquid jet with solid vortex swirl velocity profile. An analytical form of dispersion relation was derived and then solved by a direct numerical procedure. A parametric study was performed to explore the instability mechanisms that affect the maximum spatial growth rate. It is observed that the liquid swirl enhances the breakup of liquid sheet. The surface tension stabilizes the jet in the low velocity regime. The aerodynamic force intensifies the developing of disturbance and makes the jet unstable. Liquid viscous force holds back the growing of disturbance and the makes the jet stable, especially in high liquid velocity regime.

scholarly journals Nonlinear feedback in a six-dimensional Lorenz Model: impact of an additional heating term

2015 ◽  
Vol 2 (2) ◽  
pp. 475-512
Author(s):  
B.-W. Shen
Keyword(s):  
Rayleigh Number ◽  
Numerical Solutions ◽  
Critical Rayleigh Number ◽  
Critical Value ◽  
Small Scale ◽  
Solution Stability ◽  
Nonlinear Feedback ◽  
Lorenz Model ◽  
Additional Heating ◽  
The Impact

Abstract. In this study, a six-dimensional Lorenz model (6DLM) is derived, based on a recent study using a five-dimensional (5-D) Lorenz model (LM), in order to examine the impact of an additional mode and its accompanying heating term on solution stability. The new mode added to improve the representation of the steamfunction is referred to as a secondary streamfunction mode, while the two additional modes, that appear in both the 6DLM and 5DLM but not in the original LM, are referred to as secondary temperature modes. Two energy conservation relationships of the 6DLM are first derived in the dissipationless limit. The impact of three additional modes on solution stability is examined by comparing numerical solutions and ensemble Lyapunov exponents of the 6DLM and 5DLM as well as the original LM. For the onset of chaos, the critical value of the normalized Rayleigh number (rc) is determined to be 41.1. The critical value is larger than that in the 3DLM (rc ~ 24.74), but slightly smaller than the one in the 5DLM (rc ~ 42.9). A stability analysis and numerical experiments obtained using generalized LMs, with or without simplifications, suggest the following: (1) negative nonlinear feedback in association with the secondary temperature modes, as first identified using the 5DLM, plays a dominant role in providing feedback for improving the solution's stability of the 6DLM, (2) the additional heating term in association with the secondary streamfunction mode may destabilize the solution, and (3) overall feedback due to the secondary streamfunction mode is much smaller than the feedback due to the secondary temperature modes; therefore, the critical Rayleigh number of the 6DLM is comparable to that of the 5DLM. The 5DLM and 6DLM collectively suggest different roles for small-scale processes (i.e., stabilization vs. destabilization), consistent with the following statement by Lorenz (1972): If the flap of a butterfly's wings can be instrumental in generating a tornado, it can equally well be instrumental in preventing a tornado. The implications of this and previous work, as well as future work, are also discussed.

Observed flow patterns at the initiation of convection in a horizontal liquid layer heated from below

1970 ◽  
Vol 42 (4) ◽  
pp. 755-768 ◽  
Author(s):  
E. F. C. Somerscales ◽  
T. S. Dougherty
Keyword(s):  
Rayleigh Number ◽  
Critical Rayleigh Number ◽  
Test Chamber ◽  
Lower Surface ◽  
Flow Patterns ◽  
Large Temperature ◽  
Small Temperature ◽  
Initial Flow ◽  
High Prandtl Number ◽  
Rigid Surfaces

An experimental investigation has been made of the flow patterns at the initiation of convection in a layer of a high Prandtl number liquid confined between rigid, horizontal surfaces and heated from below. The experiment was designed to overcome the limitations of earlier experiments and to correspond closely to the conditions of the theory. In particular, the upper and lower rigid surfaces which enclosed the layer were made of copper which has a high thermal conductivity. To aid in the analysis of the experimental results some supplementary observations of the flow patterns were made using a glass upper plate. For small fluid depths and large temperature differences between the upper and lower surface the initial flow was in the form of hexagonal cells as predicted theoretically. With increasing Rayleigh number the cellular flow appeared to transform into rolls as predicted. For large fluid depths and small temperature differences only circular plan-form rolls were observed. This is in agreement with the results of other experimenters. It is tentatively proposed that this non-appearance of an initial cellular flow is due to the shape of the test chamber having a dominating influence on the flow pattern when the temperature gradient in the fluid is small. Measurements were also made of the development time for the flow patterns and the critical Rayleigh number.

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