Charge leakage of the weakly conducting liquid layers in a thin bilayer can engender interesting interfacial instabilities when exposed to an external electrostatic field. A general linear stability analysis including the full descriptions of the Maxwell stresses uncovers the key short to long-wave features of the instabilities of the bilayers composed of purely dielectric films, leaky dielectric films, and a combination of leaky and dielectric films. The study highlights that for the leaky bilayers the additional electrostatic stress due to the presence of free charges at the interface(s) can significantly reduce the length scale to enforce pattern miniaturization. Unlike a purely dielectric bilayer where the dielectric-contrast across the interfaces dictates the direction of the interfacial deformations, for leaky bilayers the nature of the charge (positive or negative) at the interface can also contribute to the deformation towards or away from the electrodes (anode or cathode). Nonlinear simulations uncover that the interfaces can develop unique morphologies when the spatiotemporal variation of the attractive or repulsive force at the charged interface act together or against the electrical stress due to the induced charge separation across the interface. Exploiting these features a host of periodic interfacial patterns such as core-shell columns, a hole encapsulated by a column, a bundle of columns embedded inside a single column, a collection of holes embedded under a column, and “caged” columns are obtained, which are rather difficult to assemble using other conventional patterning techniques. The results reported can be of importance in the diverse areas of micro/nanotechnology.

Electric field induced instabilities of thin leaky bilayers: pathways to unique morphologies and miniaturization.

A gravity-driven, thin, incompressible liquid film flow on a non-uniformly heated, slippery inclined plane is considered within the framework of the long-wave approximation method. A mathematical model incorporating variation in surface tension with temperature has been formulated by coupling the Navier–Stokes equation, governing the flow, with the equation of energy. For the slippery substrate, the Navier slip boundary condition is used at the solid–liquid interface. An evolution equation is formed in terms of the free surface, which includes the effects of inertia, thermocapillary as well as slip length. Using the normal mode approach, linear stability analysis is carried out and a critical Reynolds number is obtained, which reflects its dependence on the Marangoni number M n as well as slip length δ . This depicts that δ and M n both have the destabilization effect on the flow field. The linear study also reveals that the inertia force has a negligible effect compare to the thermocapillary or slip. In addition, the study highlights a critical Marangoni number at which the instability sets in when the thermocapillary stress attains a critical value. The method of multiple scales is used to investigate the weakly nonlinear stability analysis of the flow. The study interprets that the variation of M n and δ have substantial effects on different stable/unstable zones. It also shows that within a considered parametric domain, the unconditional stable zone completely vanishes for any value of M n , when the slip length δ attains a critical value. The study also divulges that in the subcritical unstable (supercritical stable) zone the threshold amplitude ( ζ a ) decreases (increases) with the increment of M n and δ . Further, we discussed the spatial uniform solution of the complex Ginzburg–Landau equation for sideband disturbances. Employing the Crank–Nicolson method, the nonlinear evolution equation of the free surface is solved numerically in a periodic domain, considering the sinusoidal initial perturbation of small amplitude. The nonlinear simulations are found to be in good agreement with the linear and weakly nonlinear stability analysis. The evolution of the maximum h max and minimum h min thickness amplifies, for small change of M n and δ . Further, it shows that the influence of the thermocapillary force amplifies the destabilizing nature of δ . The traveling wave solution confirms the existence of a fixed point for the considered parametric domain, chosen from the experimental data. Finally, the Hopf bifurcation of the fixed point exhibits that the nonlinear wave speed at the transcritical point increases as δ increases but decreases as M n increases. The noteworthy result which arises from the study is that a transcritical Hopf bifurcation exists if the slip length δ > max 1 6 M n − 1 3 , 1 2 M n − 2 3 − M n .

Thermocapillary instability on a film falling down a non-uniformly heated slippery incline

In this paper, liquid films flowing down a porous vertical cylinder were investigated by an integral boundary layer model. Linear stability and nonlinear evolution were studied. Linear stability results of the integral boundary layer model were in good agreement with the linearized Navier-Stokes equations which indicated that the permeability of the porous medium enhanced the instability of the flow system. The growth rate and cut-off wave number increased with increasing the permeability and the Reynolds number. Linear stability analysis showed that the system was more unstable for a larger Reynolds number Re. Nonlinear studies showed that, for a very small Re, the film evolved with time while a saturated state was not observed. In addition, it was observed that the film ruptured when the permeability parameter beta > 0, and the rupture time decreased with increasing beta. However, for a moderate Reynolds number, a small finite harmonic disturbance evolved to a saturated traveling wave. Further investigation was conducted on the droplet-like wave solution. Results showed that the wave speed increased as the permeability parameter increased.

/pdf/viscous-liquid-films-on-a-porous-vertical-cylinder-dynamics-57vk974muo.pdf

Viscous liquid films on a porous vertical cylinder: Dynamics and stability

Falling thin liquid film on a substrate with complex topography is modeled using a three equation integral boundary layer system. Linear stability and nonlinear dynamics of the film in the framework of this model are studied on a topography with sinusoidal longitudinal grooves aligned parallel in the direction of the main flow. The linear stability theory reveals the stabilizing nature of the surface tension force and the groove measure on the film, and the pronounced destabilizing effects of inertia. The evolution of the film thickness is tracked numerically for a vertically falling film on a grooved geometry by choosing wavenumbers corresponding to the unstable mode where the growth rate of instability is maximum. The effect of surface geometry on the temporal evolution of the film dynamics is analyzed on a periodic domain. Numerical investigations agree with the linear stability predictions and show that the longitudinal grooves exert a stabilizing effect on the film and the waviness is suppressed when the steepness of the longitudinal groove measure increases.

Falling liquid films on longitudinal grooved geometries: Integral boundary layer approach

In this paper, we investigated the stability of a two coaxial leaky dielectric fluid system flowing in an annulus duct. A constant pressure gradient was applied to drive the flow in the duct. A radial electric field was imposed between the outer and inner surfaces of the duct. Linear stability analysis was employed to discuss the influences of electric field on the capillary and interface wave instabilities. The former instability is caused by surface tension and the latter is caused by viscosity stratification at the interface. It was found that, depending on the electrical permittivities and conductivities of the two liquids, the electric field either stabilized or destabilized the flow system. Apart from that, it was found that an external electric field could impede the capillary and interface wave instabilities. Influences of the inner radius of the duct, viscosity ratio, thickness ratio, and Reynolds number on the stability of the system were discussed as well.

/pdf/stability-of-two-immiscible-leaky-dielectric-liquids-tzkbu6spl7.pdf

Stability of two immiscible leaky-dielectric liquids subjected to a radial electric field in an annulus duct

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.

B. Uma

Papers

Electrified film on a porous inclined plane: dynamics and stability.