Alexander Oron

Some symmetries of the nonlinear heat and wave equations

Abstract The Lie-group formalism is applied to deduce the classical symmetries of the nonlinear heat equation, the diffusion-convection equation and the nonlinear wave equations. Some nonclassical symmetries are also presented.

Nonlinear evolution of nonuniformly heated falling liquid films

The present theoretical study focuses on the dynamics of a thin liquid film falling down a vertical plate with a nonuniform, sinusoidal temperature distribution. The results are compared to those obtained in the case of the uniform temperature distribution. The governing evolution equation for the film thickness profile based on long-wave theory accounts for two instability mechanisms related to thermocapillarity. The first mechanism is due to an inhomogeneity of the temperature at the liquid–gas interface induced by perturbations of the film thickness, when heat transfer to the gas phase is present, while the second one is due to the nonuniform heating imposed at the plate and leads to steady-state deformations of the liquid–gas interface. For a moderate nonuniform heating the traveling waves obtained in the case of a uniform heating are modulated by an envelope. When the temperature modulation along the plate increases the shape of the liquid–gas interface becomes “frozen” and the oscillatory traveling ...

Nonlinear dynamics of three-dimensional long-wave Marangoni instability in thin liquid films

The three-dimensional evolution of the long-wave Marangoni instability of thin liquid films is studied. According to earlier theoretical predictions no continuous steady states exist and the film ruptures. As in two dimensions, the mechanism of fingering is found to be a main route to rupture. A four-fold rotational symmetry of the film interface is retained, when a square periodic domain and the harmonic initial disturbance are used. A use of initial random disturbances in general eliminates the square symmetry of the solution. An increase of the domain size results in a growing complexity of the emerging patterns. In contrast with the dynamics in two dimensions the evolution of the interface in three dimensions and in particular the pattern emerging at rupture may strongly depend on the choice of the initial condition. The two-dimensional evolution of the film is found to be unstable to small three-dimensional random disturbances.

On a nonlinear thermocapillary effect in thin liquid layers

Dilute aqueous solutions of long alcohol chains were recently found to cause a quadratic dependence of surface tension on the temperature without affecting other bulk properties of the liquid: σ = σ 0 + α Q ( T − T 0 ) 2 , α Q > 0. The impact of such Marangoni instability on the behaviour of a thin liquid layer is studied in this work. We derive an equation describing a nonlinear spatiotemporal evolution of a thin film. The behaviour of the perturbed film in the absence of gravity, critically depends on whether the temperature T 0 , yielding a minimal surface tension, is attained on the surface of the film. When this is the case, a qualitatively new behaviour is observed: perturbations of the film interface may evolve into continuous steady patterns that do not rupture. Otherwise, the observed patterns due to the linear and quadratic Marangoni effects are qualitatively similar and result in the rupture of the film into separate drops.

Evolution and breaking of liquid film flowing on a vertical cylinder

An amplitude equation is derived, which describes the evolution of a disturbed film interface H(τ,Z,Y) flowing down an infinite vertical cylindrical column. Using a new approach, which accounts for fast spatial changes, the nonlinear evolution of the interface is shown to be governed by Hτ+βHHZ+αHZZ +γ∇2{N[(1/ω2)H+∇2H]}=0, where ω is the normalized cylinder radius and α, β, and γ are constants, ∇≡(∂Z, ∂Y), and N=[1+e4(∇H)2]−3/2. It is shown numerically that for some linearly unstable equilibria the evolving waves break in a finite time.

Bounded and unbounded patterns of the Benney equation

The boundedness of 2‐D liquid film flows on an inclined plane in the context of the regularized Benney, uτ+λu2ux+[(μu6−νu3)ux]x +σ{u3[uxx/(1+e2ux2)3/2]x}x=0, and the Benney (e=0) equation are studied. Here u, x, τ are the rescaled film thickness, the longitudinal coordinate, and time, respectively; λ, μ, and ν are non‐negative constants determined at equilibrium; and e is the parameter related to the film aspect ratio. For a vertical plane (ν=0) a critical curve λ=λc(μ) has been found bifurcating from the point (λ,μ)=(0,1) which divides the λ‐μ space into two domains. When λ≳λc(μ) the initial data evolves into modulating traveling waves similar to the solutions of the Kuramoto–Sivashinsky equation. However, when λ<λc(μ), either an infinite spike forms in the solution in finite time and the original Benney model breaks down or the solution of the regularized Benney equation forms an infinite slope when the wavelike solution attempts to become multivalued. In a tilted plane (ν≳0) the boundedness of the emer...

Long-wave Marangoni instability in a binary-liquid layer with deformable interface in the presence of Soret effect: Linear theory

We investigate the long-wave Marangoni instability in a binary-liquid layer in the limit of a small Biot number B. The surface deformation and the Soret effect are both taken into account. It is shown that the problem is characterized by two distinct asymptotic limits for the disturbance wave number k, k∼B1∕4 and k∼B1∕2, which are caused by the action of two instability mechanisms, namely, the thermocapillary and solutocapillary effects. The asymptotic limit of k∼B1∕2 is novel and is not known for pure liquids. A diversity of instability modes is revealed. Specifically, a new long-wave oscillatory mode is found for sufficiently small values of the Galileo number.

Dynamics of a condensing liquid film under conjoining'disjoining pressures

The dynamics of a condensing apolar ultrathin liquid film is studied in the framework of long-wave theory in the cases of both horizontal and slightly tilted solid coated surfaces. When condensation is slow, the film on a horizontal substrate passes through the stages of hole opening driven by the “reverse reservoir effect,” hole closing, eventual thickness equilibration and further spatially uniform growth of the condensate. When condensation is faster and the resistance to phase change is lower, secondary droplet(s) may emerge within the hole. During the film evolution the thickness of the microlayer covering the hole remains practically constant due to the “reverse reservoir effect.” The total heat flux across the condensate film is found to decrease with the absolute value of the condensation constant. When the solid substrate is tilted, the film dynamics exhibits the formation of multidrop structures and their coarsening along with the stages typical for the horizontal case. The increase of the tilt ...

Capillary instability of thin liquid film on a cylinder

The capillary instability of a thin liquid layer on a cylinder is studied. Using an integral approach and lubrication approximation, transport equations governing the spatiotemporal evolution of a film thickness and the temperature along the film are obtained. Evolution of the system under both isothermal and nonisothermal conditions is studied numerically. It is shown that nonlinear interaction of the linearly unstable modes begets an additional mode with a wavelength equal to that of the fastest growing wave. This, in turn, causes the formation of satellite drops along with the main ones. Application of these results in a possible continuous technology of high‐temperature superconductor wire fabrication is discussed.

Nonlinear dynamics of temporally excited falling liquid films

The two-dimensional spatiotemporal dynamics of falling thin liquid films on a solid vertical wall periodically oscillating in its own plane is studied within the framework of long-wave theory. A pertinent nonlinear evolution equation referred to as the temporally modulated Benney equation (TMBE) is derived and its solutions are investigated numerically. The bifurcation diagram of the Benney equation (BE) describing the film dynamics in the unforced regime is computed depicting the domains of linearly stable, linearly unstable bounded, and unbounded behaviors. The solutions obtained for film dynamics via the BE are compared to those documented for direct numerical simulations of the Navier–Stokes equations (NSE). The comparison demonstrates that the BE constitutes an accurate asymptotic reduction of the NSE in the domain preceding the transition to the regime of its unbounded solutions. It is found that periodic planar boundary excitation does not alter the fundamental unforced bifurcation structure and th...