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

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.

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BÉNARD-MARANGONI FERROCONVECTION WITH FEEDBACK CONTROL

International Journal of Modern Physics Conference Series ◽

10.1142/s201019451200565x ◽

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.

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Fingering Instability of a Gravity-Driven Thin Film Flowing Down a Vertical Tube with Wall Slippage

Applied Sciences ◽

10.3390/app10010076 ◽

2019 ◽

Vol 10 (1) ◽

pp. 76

Author(s):

Chicheng Ma ◽

Shuaizhao Hu ◽

Guangxu Dong ◽

Bo Li

Keyword(s):

Thin Film ◽

Stability Analysis ◽

Linear Stability ◽

Linear Stability Analysis ◽

Traveling Wave ◽

Flow Instability ◽

Vertical Tube ◽

Petroleum Engineering ◽

Gravity Driven ◽

Wall Slippage

Inspired by the antiwetting property of pitcher plants, specialists have designed different functional material with slippery surfaces, and a directional slippery surface has been fabricated. This paper considers a gravity-driven liquid film coating the interior surface of a vertical tube, and different slippery lengths in the azimuthal direction and the axial direction are taken into account. The evolution equation of coating flow is derived using the thin film model, and time responses for two dimensional flow are calculated. Linear stability analysis (LSA) is given based on the traveling wave solutions, demonstrating that the axial slippery effect suppresses the flow instability and causes a larger traveling wave speed. Simultaneously, the azimuthal slippery effect plays a destabilizing role for perturbations with small wavenumbers and it is stabilizing for large wavenumbers. Direct simulations of the fingering flow patterns agree well with the linear stability analysis. Our results offer insight into the influence of wall slippage on the flow stability of liquid in petroleum engineering.

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

Journal of Dynamic Systems Measurement and Control ◽

10.1115/1.1515325 ◽

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.

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