Linear Stability Analysis for the Wall Temperature Feedback Control of Planar Poiseuille Flows

2002 ◽  
Vol 124 (4) ◽  
pp. 617-624
Author(s):  
Herve´ Pabiou ◽  
Jun Liu ◽  
Christine Be´nard
Keyword(s):  
Stability Analysis ◽  
Feedback Control ◽  
Flow Control ◽  
Linear Stability ◽  
Wall Temperature ◽  
Linear Stability Analysis ◽  
Poiseuille Flow ◽  
Numerical Results ◽  
Coupled Equations ◽  
Linear Loss

Active control of a planar Poiseuille flow can be performed by increasing or decreasing the wall temperature in proportion to the observed wall shear stress perturbation. In continuation with the work of H. H. Hu and H. H. Bau (1994, Feedback Control to Delay or Advance Linear Loss of Stability in Planar Poiseuille Flow, Proc. R. Soc. London A, 447, pp. 299–312), a linear stability analysis of such a feedback control is developed in this paper. The Poiseuille flow control problem is reduced to a modified Orr-Sommerfeld equation coupled with a heat equation. By solving numerically the coupled equations with a finite element method, many numerical results about the stability of the flow control are obtained. We focus our attention on the interpretation of the numerical results. In particular, the role of two essential parameters—the Prandtl number Pr and the control gain K—is investigated in detail. When Pr>1.31, stabilizing K is negative; while, when Pr<1.31, stabilizing K is positive. And when Pr=1.31, the flow cannot be stabilized by a real K. A comparison between symmetric two-wall control and non-symmetric one-wall control is also made.

GENERATION OF BRIGHT MATTER-WAVE SOLITON PATTERNS IN MIXTURES OF BOSE–EINSTEIN CONDENSATES WITH CUBIC AND QUINTIC NONLINEARITIES

2014 ◽  
Vol 28 (04) ◽  
pp. 1450003 ◽  
Author(s):  
DIDIER BELOBO BELOBO ◽  
GERMAIN HUBERT BEN-BOLIE ◽  
TIMOLÉON CRÉPIN KOFANÉ
Keyword(s):  
Stability Analysis ◽  
Linear Stability ◽  
Linear Stability Analysis ◽  
Numerical Experiments ◽  
Modulational Instability ◽  
Numerical Results ◽  
Matter Wave ◽  
Bright Solitons ◽  
Bose Einstein

The modulational instability (MI) of binary condensates with cubic-quintic nonlinearities is investigated. Using a linear stability analysis, a gain of instability is derived then, effects of the quintic nonlinearities on the instability gain are identified. To be precise, attractive intraspecie quintic nonlinearities enhance the instability, while repulsive quintic intraspecie nonlinearities soften the instability. Besides, small attractive and large repulsive quintic inter-species nonlinearities increase the instability. Numerical experiments quite well corroborate the analytical predictions. Further numerical results show effects of the cubic and the quintic nonlinearities on the propagation of trains of bright solitons generated.

Feedback control of Marangoni convection in a thin film heated from below

2019 ◽  
Vol 876 ◽  
pp. 573-590 ◽  
Author(s):  
Anna E. Samoilova ◽  
Alexander Nepomnyashchy
Keyword(s):  
Thin Film ◽  
Stability Analysis ◽  
Feedback Control ◽  
Linear Stability ◽  
Linear Stability Analysis ◽  
Control Strategies ◽  
Oscillatory Mode ◽  
Linear Feedback ◽  
Marangoni Instability ◽  
Strategy I

We use linear proportional control for the suppression of the Marangoni instability in a thin film heated from below. Our keen interest is focused on the recently revealed oscillatory mode caused by a coupling of two long-wave monotonic instabilities, the Pearson and deformational ones. Shklyaev et al. (Phys. Rev. E, vol. 85, 2012, 016328) showed that the oscillatory mode is critical in the case of a substrate of very low conductivity. To stabilize the no-motion state of the film, we apply two linear feedback control strategies based on the heat flux variation at the substrate. Strategy (I) uses the interfacial deflection from the mean position as the criterion of instability onset. Within strategy (II) the variable that describes the instability is the deviation of the measured temperatures from the desired, conductive values. We perform two types of calculations. The first one is the linear stability analysis of the nonlinear amplitude equations that are derived within the lubrication approximation. The second one is the linear stability analysis that is carried out within the Bénard–Marangoni problem for arbitrary wavelengths. Comparison of different control strategies reveals feedback control by the deviation of the free surface temperature as the most effective way to suppress the Marangoni instability.

scholarly journals BÉNARD-MARANGONI FERROCONVECTION WITH FEEDBACK CONTROL

2012 ◽  
Vol 09 ◽  
pp. 552-559
Author(s):  
NOR FADZILLAH MOHD MOKHTAR ◽  
NORIHAN MD ARIFIN
Keyword(s):  
Stability Analysis ◽  
Feedback Control ◽  
Free Boundary ◽  
Galerkin Method ◽  
Linear Stability ◽  
Linear Stability Analysis ◽  
Lower Boundary ◽  
Critical Stability ◽  
Stability Parameters ◽  
The Galerkin Method

The effect of feedback control on the onset of Bénard-Marangoni ferroconvection in a horizontal ferrofluid layer heated from below is investigated theoretically. The lower boundary is rigid and the upper free boundary is assumed to be flat and undeformable. A linear stability analysis is used and the Galerkin method is employed to find the critical stability parameters numerically. It is found that the onset of instability can be delayed through the use of feedback control.

Modal and non-modal linear stability of the plane Bingham–Poiseuille flow

2007 ◽  
Vol 577 ◽  
pp. 211-239 ◽  
Author(s):  
C. NOUAR ◽  
N. KABOUYA ◽  
J. DUSEK ◽  
M. MAMOU
Keyword(s):  
Stability Analysis ◽  
Eigenvalue Problem ◽  
Linear Stability ◽  
Linear Stability Analysis ◽  
Poiseuille Flow ◽  
Linear Analysis ◽  
Bingham Fluid ◽  
Streamwise Vortices ◽  
Bingham Number ◽  
Spectra And Pseudospectra

The receptivity problem of plane Bingham–Poiseuille flow with respect to weak perturbations is addressed. The relevance of this study is highlighted by the linear stability analysis results (spectra and pseudospectra). The first part of the present paper thus deals with the classical normal-mode approach in which the resulting eigenvalue problem is solved using the Chebychev collocation method. Within the range of parameters considered, the Poiseuille flow of Bingham fluid is found to be linearly stable. The second part investigates the most amplified perturbations using the non-modal approach. At a very low Bingham number (B ≪ 1), the optimal disturbance consists of almost streamwise vortices, whereas at moderate or large B the optimal disturbance becomes oblique. The evolution of the obliqueness as function of B is determined. The linear analysis presented also indicates, as a first stage of a theoretical investigation, the principal challenges of a more complete nonlinear study.

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