A priori bounds in the Bénard problem with boundaries of finite conductivity

1983 ◽  
Vol 32 (2) ◽  
pp. 208-216
Author(s):  
F. Franchi ◽  
B. Straughan
Keyword(s):  
A Priori ◽  
Finite Conductivity ◽  
A Priori Bounds ◽  
Bénard Problem ◽  
Benard Problem

On the solution of the Bénard problem with boundaries of finite conductivity

1967 ◽  
Vol 296 (1447) ◽  
pp. 469-475 ◽  
Keyword(s):  
Thermal Diffusivity ◽  
Rayleigh Number ◽  
Hydrodynamic Stability ◽  
Critical Rayleigh Number ◽  
Finite Conductivity ◽  
Stationary Convection ◽  
Bénard Problem ◽  
Benard Problem ◽  
The Media

The Bénard problem in hydrodynamic stability is formulated under conditions where the media bounding the fluid have finite thermal diffusivity. It is shown that the principle of the exchange of stabilities remains valid in this case so that instability in the fluid first sets in as stationary convection. Solutions are obtained for various values of the ratio of the thermal diffusivity of the fluid to that of the bounding media; the critical Rayleigh number at which the instability occurs is markedly reduced when this ratio is large.

scholarly journals A priori bounds of the solution of a one point IBVP for a singular fractional evolution equation

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Said Mesloub ◽  
Hassan Eltayeb Gadain
Keyword(s):  
Differential Equations ◽  
Evolution Equation ◽  
A Priori Estimates ◽  
Boundary Value ◽  
A Priori ◽  
Initial Boundary ◽  
A Priori Bounds ◽  
Fractional Evolution Equation ◽  
Priori Estimates ◽  
Initial Boundary Value

Abstract A priori bounds constitute a crucial and powerful tool in the investigation of initial boundary value problems for linear and nonlinear fractional and integer order differential equations in bounded domains. We present herein a collection of a priori estimates of the solution for an initial boundary value problem for a singular fractional evolution equation (generalized time-fractional wave equation) with mass absorption. The Riemann–Liouville derivative is employed. Results of uniqueness and dependence of the solution upon the data were obtained in two cases, the damped and the undamped case. The uniqueness and continuous dependence (stability of solution) of the solution follows from the obtained a priori estimates in fractional Sobolev spaces. These spaces give what are called weak solutions to our partial differential equations (they are based on the notion of the weak derivatives). The method of energy inequalities is used to obtain different a priori estimates.

On the two-component Benard problem a numerical solution

1975 ◽  
Vol 80 (1) ◽  
pp. 76-88 ◽  
Author(s):  
J.C. Legros ◽  
D. Longree ◽  
G. Chavepeyer ◽  
J.K. Platten
Keyword(s):  
Numerical Solution ◽  
Two Component ◽  
Bénard Problem ◽  
Benard Problem

Stabilization of the no-motion state in the Rayleigh–Bénard problem

1994 ◽  
Vol 447 (1931) ◽  
pp. 587-607 ◽  
Keyword(s):  
Fluid Layer ◽  
Critical Rayleigh Number ◽  
Control Flow ◽  
Feedback Controller ◽  
Linear Stability Theory ◽  
Motion State ◽  
Conduction State ◽  
Bénard Problem ◽  
Benard Problem ◽  
Rayleigh Benard

Using linear stability theory and numerical simulations, we demonstrate that the critical Rayleigh number for bifurcation from the no-motion (conduction) state to the motion state in the Rayleigh–Bénard problem of an infinite fluid layer heated from below and cooled from above can be significantly increased through the use of a feedback controller effectuating small perturbations in the boundary data. The controller consists of sensors which detect deviations in the fluid’s temperature from the motionless, conductive values and then direct actuators to respond to these deviations in such a way as to suppress the naturally occurring flow instabilities. Actuators which modify the boundary’s temperature or velocity are considered. The feedback controller can also be used to control flow patterns and generate complex dynamic behaviour at relatively low Rayleigh numbers.

Sign in / Sign up

Export Citation Format

Share Document