A theory for two dimensional long and stationary waves of finite amplitude on a thin viscous liquid film down an inclined plane in the presence of uniform electric field at infinity is investigated. A set of exact averaged equations for the film flow system is described and linearized stability analysis of the uniform flow is performed using normal-mode formulation and the critical condition for linear instability is obtained. The linearized instability for the permanent wave equation, consistent to the second order in ε, is examined and the eigenvalue properties of the fixed points are classified in various parametric regimes. Numerical integration of the permanent wave equation as a third-order dynamical system is carried out. Different bifurcation scenarios leading to multiple-hump solitary waves or leading to chaos are exhibited in the parametric space. © 2008 American Institute of Physics.

Usha R

Department of Mathematics

Bifurcation (mathematics)

Conductive films

Eigenvalues and eigenfunctions

Electric field effects

Linear stability analysis

SOLITONS

Thin films

Wave equations

FILM FLOW

PERMANENT WAVE EQUATION

THIRD-ORDER DYNAMICAL SYSTEM

Uniform flow

Viscous flow

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IIT Madras

A thin conducting viscous film on an inclined plane in the presence of a uniform normal electric field: Bifurcation scenarios

Physics of Fluids

The time evolution of a thin conducting liquid film flowing down a porous inclined substrate is investigated when an electric field acts normal to the substrate. It is assumed that the flow through the porous medium is governed by Darcy's law together with Beavers-Joseph condition. Under the assumption of small permeability relative to the thickness of the overlying fluid layer, the flow is decoupled from the filtration flow through the porous medium. A slip condition at the bottom is used to incorporate the effects of the permeability of the substrate. From the set of exact averaged equations derived using integral boundary method for the film thickness and for the flow rate, a nonlinear evolution equation for the film thickness is derived through a long-wave approximation. A linear stability analysis of the base flow is performed and the critical Reynolds number is obtained. The results reveal that the substrate porosity in general destabilizes the liquid film flow and the presence of the electric field enhances this destabilizing effect. A weakly nonlinear stability analysis divulges the existence of supercritical stable and subcritical unstable zones in the wave number/Reynolds number parameter space and the results demonstrate how the neutral curves change as the intensity of the electric filed or the permeability of the porous medium is varied. The numerical solution of the nonlinear evolution equation in a periodic domain reveals that the base flow yields to surface structures that are either time independent waves of permanent form that propagate or time-dependent modes that oscillate slightly in the amplitude. Further, it is observed that the shape and amplitude of long-time waveforms are influenced by the permeability of the porous medium as well as by the applied electric field. The results reveal that the destabilization induced by the electric field in an otherwise stable film over a porous medium is exhibited in the form of traveling waves of finite amplitude. The presence of the porous substrate promotes the oscillatory behavior of the long-time waveform; however, the electric field has a tendency to suppress this oscillatory behavior. © 2010 The American Physical Society.

Physical Review E - Statistical, Nonlinear, and Soft Matter Physics

Electrified film on a porous inclined plane: Dynamics and stability

The theory describing the nonlinear stationary waves of finite amplitude and long wavelength on a thin viscous Newtonian film at high Reynolds numbers and moderate Weber numbers has been developed using the energy integral method (EIM). The linear instability of the uniform flow by EIM has been analyzed and the linear instability threshold has been obtained as cot θ/Re=6/5, which agrees with the classical results of the Orr-Sommerfeld analysis by Benjamin [J. Fluid Mech. 2, 554 (1957)] and Yih [Phys. Fluids 6, 321 (1963)] and verified experimentally by Liu and Gollub [Phys. Rev. Lett. 70, 2289 (1993)]. Further, in the frame of reference moving with the steady wave speed, the second order approximate equations reduce to a third order dynamical system. While wave transitions in real life involve complex spatio-temporal dynamics and many of these transitions lead to chaotic waves that are not stationary traveling waves, bifurcation of stationary traveling waves has been examined as a preliminary study of the more complex transitions. Stability of the fixed points of the dynamical system, parametric regimes of heteroclinic orbits and Hopf bifurcations are delineated. Numerical integration has been carried out in order to study the different bifurcation scenarios as the phase speed deviates from the Hopf-bifurcation thresholds. Four different bifurcation scenarios have been observed and the dependence of bifurcation scenarios on the inclination angle, Reynolds numbers and Weber numbers have been discussed. Although the results obtained by the momentum integral method and EIM exhibit similar bifurcation scenarios, there are quantitative differences which shows that the modeling differences exist in the literature . © 2004 American Institute of Physics.

Fulltext

Modeling of stationary waves on a thin viscous film down an inclined plane at high Reynolds numbers and moderate Weber numbers using energy integral method

Journal	Data powered by TypesetPhysics of Fluids
Publisher	Data powered by TypesetAmerican Institute of Physics Inc.
ISSN	10706631
Open Access	No