scholarly journals Transport coefficients for an inelastic gas around uniform shear flow: Linear stability analysis

2006 ◽  
Vol 73 (2) ◽  
Author(s):  
Vicente Garzó
Keyword(s):  
Stability Analysis ◽  
Shear Flow ◽  
Linear Stability ◽  
Linear Stability Analysis ◽  
Transport Coefficients ◽  
Uniform Shear ◽  
Uniform Shear Flow

Non-Newtonian stress, collisional dissipation and heat flux in the shear flow of inelastic disks: a reduction via Grad’s moment method

2014 ◽  
Vol 757 ◽  
pp. 251-296 ◽  
Author(s):  
Saikat Saha ◽  
Meheboob Alam
Keyword(s):  
Heat Flux ◽  
Shear Flow ◽  
Moment Tensor ◽  
Transport Coefficients ◽  
Navier Stokes ◽  
Restitution Coefficient ◽  
Moment Equations ◽  
Second Moment ◽  
Uniform Shear ◽  
Uniform Shear Flow

AbstractThe non-Newtonian stress tensor, collisional dissipation rate and heat flux in the plane shear flow of smooth inelastic disks are analysed from the Grad-level moment equations using the anisotropic Gaussian as a reference. For steady uniform shear flow, the balance equation for the second moment of velocity fluctuations is solved semi-analytically, yielding closed-form expressions for the shear viscosity $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}\mu $, pressure $p$, first normal stress difference ${\mathcal{N}}_1$ and dissipation rate ${\mathcal{D}}$ as functions of (i) density or area fraction $\nu $, (ii) restitution coefficient $e$, (iii) dimensionless shear rate $R$, (iv) temperature anisotropy $\eta $ (the difference between the principal eigenvalues of the second-moment tensor) and (v) angle $\phi $ between the principal directions of the shear tensor and the second-moment tensor. The last two parameters are zero at the Navier–Stokes order, recovering the known exact transport coefficients from the present analysis in the limit $\eta ,\phi \to 0$, and are therefore measures of the non-Newtonian rheology of the medium. An exact analytical solution for leading-order moment equations is given, which helped to determine the scaling relations of $R$, $\eta $ and $\phi $ with inelasticity. We show that the terms at super-Burnett order must be retained for a quantitative prediction of transport coefficients, especially at moderate to large densities for small values of the restitution coefficient ($e \ll 1$). Particle simulation data for a sheared inelastic hard-disk system are compared with theoretical results, with good agreement for $p$, $\mu $ and ${\mathcal{N}}_1$ over a range of densities spanning from the dilute to close to the freezing point. In contrast, the predictions from a constitutive model at Navier–Stokes order are found to deviate significantly from both the simulation and the moment theory even at moderate values of the restitution coefficient ($e\sim 0.9$). Lastly, a generalized Fourier law for the granular heat flux, which vanishes identically in the uniform shear state, is derived for a dilute granular gas by analysing the non-uniform shear flow via an expansion around the anisotropic Gaussian state. We show that the gradient of the deviatoric part of the kinetic stress drives a heat current and the thermal conductivity is characterized by an anisotropic second-rank tensor, for which explicit analytical expressions are given.

Linear stability analysis of the reacting shear flow

2006 ◽  
Vol 20 (8) ◽  
pp. 1309-1320 ◽  
Author(s):  
Yang Na ◽  
Seungbae Lee ◽  
Dongshin Shin
Keyword(s):  
Stability Analysis ◽  
Shear Flow ◽  
Linear Stability ◽  
Linear Stability Analysis

Shear-flow stability within the atmosphere of Venus

1974 ◽  
Vol 66 (2) ◽  
pp. 267-272 ◽  
Author(s):  
R. D. Cess ◽  
Harshvardhan
Keyword(s):  
Stability Analysis ◽  
Radiative Transfer ◽  
Shear Flow ◽  
Linear Stability ◽  
Linear Stability Analysis ◽  
Flow Instability ◽  
Flow Stability ◽  
Shear Flow Instability ◽  
Realistic Formulation ◽  
Atmosphere Of Venus

Employing a linear stability analysis, Dudis (1973) has recently suggested that shear-flow instability might exist within the upper stratosphere of Venus owing to destabilization by radiative transfer. We have incorporated a more realistic formulation for radiative transfer into his stability analysis and conclude that such an instability is unlikely.

Transient and asymptotic stability of granular shear flow

1994 ◽  
Vol 264 ◽  
pp. 255-275 ◽  
Author(s):  
P. J. Schmid ◽  
H. K. Kytömaa
Keyword(s):  
Shear Flow ◽  
Linear Stability ◽  
Equations Of Motion ◽  
Wave Structure ◽  
Finite Amplitude ◽  
Particulate Flows ◽  
Uniform Shear ◽  
Uniform Shear Flow ◽  
The Stability ◽  
Fast Mechanism

The linear stability of granular material in an unbounded uniform shear flow is considered. Linearized equations of motion derived from kinetic theories are used to arrive at a linear initial-value problem for the perturbation quantities. Two cases are investigated: (a) wavelike disturbances with time constant wavenumber vector, and (b) disturbances that will change their wave structure in time owing to a shear-induced tilting of the wavenumber vector. In both cases, the stability analysis is based on the solution operator whose norm represents the maximum possible amplification of initial perturbations. Significant transient growth is observed which has its origin in the non-normality of the involved linear operator. For case (a), regions of asymptotic instability are found in the two-dimensional wavenumber plane, whereas case (b) is found to be asymptotically stable for all physically meaningful parameter combinations. Transient linear stability phenomena may provide a viable and fast mechanism to trigger finite-amplitude effects, and therefore constitute an important part of pattern formation in rapid particulate flows.

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.

Convective instability of reaction fronts: Linear stability analysis

1998 ◽  
Vol 9 (5) ◽  
pp. 507-525 ◽  
Author(s):  
V. A. VOLPERT ◽  
A. I. VOLPERT
Keyword(s):  
Stability Analysis ◽  
Linear Stability ◽  
Linear Stability Analysis ◽  
Convective Instability ◽  
Eigenvalue Problems ◽  
Transport Coefficients ◽  
Stability Boundary ◽  
New Approaches ◽  
Reaction Fronts ◽  
The Stability

The paper is devoted to convective instability of reaction fronts. New approaches are developed to study some eigenvalue problems arising in chemical hydrodynamics. For gaseous combustion in the case of equality of transport coefficients, a linear stability analysis of an upward propagating front is carried out. A minimax representation of the stability boundary is obtained.

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