The dynamics and stability of thin liquid films have fascinated scientists over many decades: the observations of regular wave patterns in film flows down a windowpane or along guttering, the patterning of dewetting droplets, and the fingering of viscous flows down a slope are all examples that are familiar in daily life. Thin film flows occur over a wide range of length scales and are central to numerous areas of engineering, geophysics, and biophysics; these include nanofluidics and microfluidics, coating flows, intensive processing, lava flows, dynamics of continental ice sheets, tear-film rupture, and surfactant replacement therapy. These flows have attracted considerable attention in the literature, which have resulted in many significant developments in experimental, analytical, and numerical research in this area. These include advances in understanding dewetting, thermocapillary- and surfactant-driven films, falling films and films flowing over structured, compliant, and rapidly rotating substrates, and evaporating films as well as those manipulated via use of electric fields to produce nanoscale patterns. These developments are reviewed in this paper and open problems and exciting research avenues in this thriving area of fluid mechanics are also highlighted.

Dynamics and stability of thin liquid films

Notes on numerical fluid mechanics

Hydrodynamic and hydromagnetic stability

We consider the gravity-driven laminar flow of a shallow fluid layer down an uneven incline with the principal objective of investigating the effect of bottom topography and surface tension on the stability of the flow. The equations of motion are approximations to the Navier–Stokes equations which exploit the assumed relative shallowness of the fluid layer. Included in these equations are diffusive terms that are second order relative to the shallowness parameter. These terms, while small in magnitude, represent an important dependence of the flow dynamics on the variation in bottom topography and play a significant role in theoretically capturing important aspects of the flow. Some of the second-order terms include normal shear contributions, while others lead to a nonhydrostatic pressure distribution. The explicit dependence on the cross-stream coordinate is eliminated from the equations of motion by means of a weighted residual approach. The resulting mathematical formulation constitutes an extension ...

Instability in gravity-driven flow over uneven surfaces

Here, the nonlinear flow instability of a thin film flowing down an inclined wall with smooth deformations is investigated. A nonlinear evolution equation is obtained under the small wavenumber approximation. This equation describes the film free surface deformation including the effects of inertia, viscosity, surface tension, and deformation of the wall. The equation has a forcing term that corresponds to periodic time-dependent perturbations hitting on the free surface. These time-dependent perturbations are tested against the presence of the free surface waves due to wall deformations. In particular, a wall sinusoidal profile is investigated. It is found that when the wall wavelength is small enough, the valleys of the surface response to the wall profile become very deep, showing a kind of resonance, which has an important stabilizing influence in the evolution of the superposed time-dependent perturbations.

Nonlinear instability of a thin film flowing down a smoothly deformed surface

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.

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

Dynamics of a thin viscoelastic film on an inclined plane

Interfacial phase change effects on the stability characteristics of thin viscoelastic liquid film down a vertical wall

Long waves on a viscoelastic film flow down a wavy inclined plane is investigated. The analysis is performed to see how long non-linear waves on viscoelastic film down an uneven inclined wall are deformed due to the non-uniformity of the basic flow. The results are then compared with those corresponding to Newtonian film down a wavy inclined wall as well as viscoelastic film down a plane inclined wall.

Long waves on a viscoelastic film flow down a wavy incline

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.

B. Uma

Papers

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

Dynamics of a thin viscoelastic film on an inclined plane

Interfacial phase change effects on the stability characteristics of thin viscoelastic liquid film down a vertical wall

Long waves on a viscoelastic film flow down a wavy incline

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