Stabilising falling liquid film flows using feedback control

Falling liquid films become unstable due to inertial effects when the fluid layer is sufficiently thick or the slope sufficiently steep. This free surface flow of a single fluid layer has industrial applications including coating and heat transfer, which benefit from smooth and wavy interfaces, respectively. Here, we discuss how the dynamics of the system are altered by feedback controls based on observations of the interface height, and supplied to the system via the perpendicular injection and suction of fluid through the wall. In this study, we model the system using both Benney and weighted-residual models that account for the fluid injection through the wall. We find that feedback using injection and suction is a remarkably effective control mechanism: the controls can be used to drive the system towards arbitrary steady states and travelling waves, and the qualitative effects are independent of the details of the flow modelling. Furthermore, we show that the system can still be successfully controlled when the feedback is applied via a set of localised actuators and only a small number of system observations are available, and that this is possible using both static (where the controls are based on only the most recent set of observations) and dynamic (where the controls are based on an approximation of the system which evolves over time) control schemes. This study thus provides a solid theoretical foundation for future experimental realisations of the active feedback control of falling liquid films.

[1]  S. Sun,et al.  ON STABILITY OF LIQUID FLOW DOWN AN INCLINED PLANE , 2016 .

[2]  G. Pavliotis,et al.  Controlling spatiotemporal chaos in active dissipative-dispersive nonlinear systems. , 2015, Physical review. E, Statistical, nonlinear, and soft matter physics.

[3]  A. Thompson,et al.  Falling liquid films with blowing and suction , 2015, Journal of Fluid Mechanics.

[4]  G. Pavliotis,et al.  Stabilising nontrivial solutions of the generalised Kuramoto-Sivashinsky equation using feedback and optimal control , 2015, 1505.06086.

[5]  Edriss S. Titi,et al.  Abridged Continuous Data Assimilation for the 2D Navier–Stokes Equations Utilizing Measurements of Only One Component of the Velocity Field , 2015, 1504.05978.

[6]  Daniel H. Reck,et al.  Does the topography’s specific shape matter in general for the stability of film flows? , 2015 .

[7]  S. B. Islami,et al.  An experimental investigation on the developing wavy falling film in the presence of electrohydrodynamic conduction phenomenon , 2015 .

[8]  Sreeram K. Kalpathy,et al.  Thermally induced delay and reversal of liquid film dewetting on chemically patterned surfaces. , 2013, Journal of colloid and interface science.

[9]  M. Amaouche,et al.  Hydromagnetic thin film flow: linear stability. , 2013, Physical review. E, Statistical, nonlinear, and soft matter physics.

[10]  D. Tseluiko,et al.  Stability of film flow over inclined topography based on a long-wave nonlinear model , 2013, Journal of Fluid Mechanics.

[11]  M. Sellier,et al.  Flow domain identification from free surface velocity in thin inertial films , 2013, Journal of Fluid Mechanics.

[12]  N. Aksel,et al.  Crucial flow stabilization and multiple instability branches of gravity-driven films over topography , 2013 .

[13]  A. Bassom,et al.  Flow of a liquid layer over heated topography , 2012, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[14]  Y. C. Lee,et al.  Electrified thin film flow at finite Reynolds number on planar substrates featuring topography , 2012 .

[15]  J. Pascal,et al.  Gravity-driven flow over heated, porous, wavy surfaces , 2011 .

[16]  Serafim Kalliadasis,et al.  Falling Liquid Films , 2011 .

[17]  N. Aksel,et al.  Side wall effects on the instability of thin gravity-driven films—From long-wave to short-wave instability , 2011 .

[18]  E. Momoniat,et al.  The influence of slot injection/suction on the spreading of a thin film under gravity and surface tension , 2010 .

[19]  Vasilis Bontozoglou,et al.  Effect of channel width on the primary instability of inclined film flow , 2010 .

[20]  Felix Otto,et al.  Optimal bounds on the Kuramoto–Sivashinsky equation , 2009 .

[21]  Nuri Aksel,et al.  Bottom reconstruction in thin-film flow over topography: Steady solution and linear stability , 2009 .

[22]  R. Craster,et al.  Dynamics and stability of thin liquid films , 2009 .

[23]  M. Velarde,et al.  Stability analysis of thin film flow along a heated porous wall , 2009 .

[24]  J. Vanden-Broeck,et al.  Effect of an electric field on film flow down a corrugated wall at zero Reynolds number , 2008 .

[25]  J. Vanden-Broeck,et al.  Electrified viscous thin film flow over topography , 2007, Journal of Fluid Mechanics.

[26]  Gaurav,et al.  Stability of gravity-driven free-surface flow past a deformable solid at zero and finite Reynolds number , 2007 .

[27]  Dmitri Tseluiko,et al.  Wave evolution on electrified falling films , 2006, Journal of Fluid Mechanics.

[28]  C. Pozrikidis,et al.  Effect of surfactant on the stability of film flow down an inclined plane , 2004, Journal of Fluid Mechanics.

[29]  O. Gottlieb,et al.  Subcritical and supercritical bifurcations of the first- and second-order Benney equations , 2004 .

[30]  Peter K. Jimack,et al.  Gravity-driven flow of continuous thin liquid films on non-porous substrates with topography , 2004, Journal of Fluid Mechanics.

[31]  Uwe Thiele,et al.  Nonlinear evolution of nonuniformly heated falling liquid films , 2002 .

[32]  R. Grigoriev Contact line instability and pattern selection in thermally driven liquid films , 2002, nlin/0207024.

[33]  Panagiotis D. Christofides,et al.  Optimal actuator/sensor placement for nonlinear control of the Kuramoto-Sivashinsky equation , 2002, Proceedings of the 2002 American Control Conference (IEEE Cat. No.CH37301).

[34]  V. Bontozoglou,et al.  Experiments on laminar film flow along a periodic wall , 2002, Journal of Fluid Mechanics.

[35]  P. Manneville,et al.  Improved modeling of flows down inclined planes , 2000 .

[36]  Antonios Armaou,et al.  Wave suppression by nonlinear finite-dimensional control , 1999, Proceedings of the 1999 American Control Conference (Cat. No. 99CH36251).

[37]  Panagiotis D. Christofides,et al.  Feedback control of the Kuramoto-Sivashinsky equation , 1998, Proceedings of the 37th IEEE Conference on Decision and Control (Cat. No.98CH36171).

[38]  Stephen H. Davis,et al.  Spreading and imbibition of viscous liquid on a porous base , 1998 .

[39]  Liu,et al.  Onset of spatially chaotic waves on flowing films. , 1993, Physical review letters.

[40]  R. Téman,et al.  Integral Manifolds and Inertial Manifolds for Dissipative Partial Differential Equations , 1988 .

[41]  E. Michaelides,et al.  Gravity flow of a viscous liquid down a slope with injection , 1988 .

[42]  C. Pozrikidis,et al.  The flow of a liquid film along a periodic wall , 1988, Journal of Fluid Mechanics.

[43]  J. M. Floryan,et al.  Instabilities of a liquid film flowing down a slightly inclined plane , 1987 .

[44]  Eitan Tadmor,et al.  The well-posedness of the Kuramoto-Sivashinsky equation , 1986 .

[45]  Hassan K. Khalil,et al.  Singular perturbation methods in control : analysis and design , 1986 .

[46]  Y. Pomeau,et al.  On solitary waves running down an inclined plane , 1983, Journal of Fluid Mechanics.

[47]  G. Sivashinsky,et al.  Irregular flow of a liquid film down a vertical column , 1982 .

[48]  G. Sivashinsky,et al.  On Irregular Wavy Flow of a Liquid Film Down a Vertical Plane , 1980 .

[49]  S. P. Lin Stability of liquid flow down a heated inclined plane , 1975 .

[50]  D. J. Benney Long Waves on Liquid Films , 1966 .

[51]  T. Brooke Benjamin,et al.  Wave formation in laminar flow down an inclined plane , 1957, Journal of Fluid Mechanics.