Stability analysis of stratified Rayleigh–Bénard–Poiseuille convection. Part II: Influence of thermocapillary forces

2016 ◽  
Vol 155 ◽  
pp. 99-110 ◽  
Author(s):  
Éliton Fontana ◽  
Erasmo Mancusi ◽  
Antônio A.U. de Souza ◽  
Selene M.A.G.U. de Souza
Keyword(s):  
Stability Analysis ◽  
Thermocapillary Forces ◽  
Rayleigh Benard

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.

scholarly journals Rayleigh–Bénard stability and the validity of quasi-Boussinesq or quasi-anelastic liquid approximations

2017 ◽  
Vol 817 ◽  
pp. 264-305 ◽  
Author(s):  
Thierry Alboussière ◽  
Yanick Ricard
Keyword(s):  
Stability Analysis ◽  
Equation Of State ◽  
Rayleigh Number ◽  
Equations Of State ◽  
Critical Rayleigh Number ◽  
Ideal Gas ◽  
Dimensionless Temperature ◽  
Boussinesq Model ◽  
Onset Of Convection ◽  
Rayleigh Benard

The linear stability threshold of the Rayleigh–Bénard configuration is analysed with compressible effects taken into account. It is assumed that the fluid under investigation obeys a Newtonian rheology and Fourier’s law of thermal transport with constant, uniform (dynamic) viscosity and thermal conductivity in a uniform gravity field. Top and bottom boundaries are maintained at different constant temperatures and we consider here mechanical boundary conditions of zero tangential stress and impermeable walls. Under these conditions, and with the Boussinesq approximation, Rayleigh (Phil. Mag., vol. 32 (192), 1916, pp. 529–546) first obtained analytically the critical value $27\unicode[STIX]{x03C0}^{4}/4$ for a dimensionless parameter, now known as the Rayleigh number, at the onset of convection. This paper describes the changes of the critical Rayleigh number due to the compressibility of the fluid, measured by the dimensionless dissipation parameter ${\mathcal{D}}$ and due to a finite temperature difference between the hot and cold boundaries, measured by a dimensionless temperature gradient $a$. Different equations of state are examined: ideal gas equation, Murnaghan’s model (often used to describe the interiors of solid but convective planets) and a generic equation of state with adjustable parameters, which can represent any possible equation of state. In the perspective to assess approximations often made in convective models, we also consider two variations of this stability analysis. In a so-called quasi-Boussinesq model, we consider that density perturbations are solely due to temperature perturbations. In a so-called quasi-anelastic liquid approximation model, we consider that entropy perturbations are solely due to temperature perturbations. In addition to the numerical Chebyshev-based stability analysis, an analytical approximation is obtained when temperature fluctuations are written as a combination of only two modes, one being the original symmetrical (between top and bottom) mode introduced by Rayleigh, the other one being antisymmetrical. The analytical solution allows us to show that the antisymmetrical part of the critical eigenmode increases linearly with the parameters $a$ and ${\mathcal{D}}$, while the superadiabatic critical Rayleigh number departs quadratically in $a$ and ${\mathcal{D}}$ from $27\unicode[STIX]{x03C0}^{4}/4$. For any arbitrary equation of state, the coefficients of the quadratic departure are determined analytically from the coefficients of the expansion of density up to degree three in terms of pressure and temperature.

scholarly journals Effects of Suction–Injection–Combination (SIC) on the onset of Rayleigh–Bénard Electroconvection in a Micropolar Fluid

2015 ◽  
Vol 14 (3) ◽  
pp. 23-42 ◽  
Author(s):  
S Pranesh ◽  
Tarannum Sameena ◽  
Baby Riya
Keyword(s):  
Stability Analysis ◽  
Micropolar Fluid ◽  
Fluid Layer ◽  
Suspended Particles ◽  
Wave Number ◽  
Critical Wave Number ◽  
Onset Of Convection ◽  
Temperature Boundary ◽  
The Stability ◽  
Rayleigh Benard

The effect of Suction – injection combination on the onset of Rayleigh – Bénard electroconvection micropolar fluid is investigated by making a linear stability analysis. The Rayleigh-Ritz technique is used to obtain the eigenvalues for different velocity and temperature boundary combinations. The influence of various parameters on the onset of convection has been analysed. It is found that the effect of Prandtl number on the stability of the system is dependent on the SIC being pro-gravity or anti-gravity. A similar Pe-sensitivity is found in respect of the critical wave number. It is observed that the fluid layer with suspended particles heated from below is more stable compared to the classical fluid layer without suspended particles.

Sign in / Sign up

Export Citation Format

Share Document