Alexander L. Frenkel

Nonlinear saturation of Rayleigh–Taylor instability in thin films

A general mechanism of nonlinear saturation of instabiities in flowing films is described using the Rayleigh–Taylor instability as an example. The combined action of flow shear and surface tension is the essence of the saturation mechanism. As a result, the streamwise perturbations of the interface that would rupture a stagnant film do not rupture a film flowing in a certain range of shear rates.

Annular flows can keep unstable films from breakup: Nonlinear saturation of capillary instability

Abstract We consider a fluid film on the inner walls of a capillary. The film surrounds another fluid in the core. It is known that the capillary instability, driven by the surface tension at the fluid-fluid interface, breaks up the film if it is primarily stagnant. In contrast, as we show, a primary flow, in a certain range of parameters, can keep the linearly unstable film from rupturing. This is a result of the nonlinear low-level saturation of the interface instability. This saturation is due to the coordinated action of the destabilizing factors, the shear of flow, and the surface tension at the interface. The resulting state of the interface is, in general, chaotic oscillations, with their amplitude being much less than the unperturbed film thickness. The approximate equation of interface evolution is derived. The saturation mechanism is explained. The characteristic scales of the developed oscillations are found, and the parameter range of the theory applicability is discussed.

Stokes-flow instability due to interfacial surfactant

The linear stability of a two-fluid shear flow with an insoluble surfactant on the flat interface is investigated in the Stokes approximation. Gravity is neglected in order to isolate the Marangoni effect of the surfactant. In contrast to all earlier studies of related fluid systems, we encounter (i) the destabilization (here, of a shear flow) caused solely by the introduction of an interfacial surfactant and (ii) the destabilization (here, of a system with a surfactant) caused solely by the imposition of a Stokes flow. Asymptotic long-wave expressions for the growth rates are obtained.

Nonlinear theory of strongly undulating thin films flowing down vertical cylinders

The nonlinear system of approximate equations is obtained for thin annular films flowing down vertical fibers. The multiparameter perturbation approach is used which justifies the theory for a much wider range of basic flow parameters than the more conventional methods. From the evolution equation that is applicable to the strongly undulating films which are perpetually on the verge of break up, certain dependences follow from observable quantities, in excellent agreement with experiments.

Destabilization of a creeping flow by interfacial surfactant: linear theory extended to all wavenumbers

Creeping flow of a two-layer system with a monolayer of an insoluble surfactant on the interface is considered. The linear-stability theory of plane Couette-Poiseuille flow is developed in the Stokes approximation. To isolate the Marangoni effect, gravity is excluded. The shear-flow instability due to the interfacial surfactant, uncovered earlier for long waves only (Frenkel & Halpern), is studied with inclusion of all wavelengths, and over the entire parameter space of the Marangoni number M, the viscosity ratio m, the interfacial velocity shear s, and the thickness ratio n (≥ 1). The complex wave speed of normal modes solves a quadratic equation, and the growth rate function is continuous at all wavenumbers and all parameter values. If M > 0, s ¬= 0, m 1, the small disturbances grow provided they are sufficiently long wave. However, the instability is not long wave in the following sense: the unstable waves are not necessarily much longer than the smaller of the two layer thicknesses. On the other hand, there are parametric regimes for which the instability has a mid-wave character, the flow being stable at both sufficiently large and small wavelengths and unstable in between

Interactions of coherent structures in a film flow: Simulations of a highly nonlinear evolution equation

Numerical simulations of the evolution equation [14] for thickness of a film flowing down a vertical fiber are presented. Solutions with periodic boundary conditions on extended axial intervals develop trains of pulse-like structures. Typically, a group of several interacting pulses (or a solitary pulse) is bracketed by spans of nearly uniform thinned film and is virtually isolated: The evolution of such a “section” is modeled as a solution with periodic boundary conditions on the corresponding, comparatively short, interval. Single-pulse sections are steady-shape traveling waveforms (“cells” of shorter-period solutions). The collision of two pulses can be either a particle-like “elastic” rebound, or—and only if a control parameter S (proportional to the average thickness) exceeds a certain critical value, Sc ≈ 1—a “deeply inelastic” coalescence. A pulse which grows by a cascade of coalescences is associated with large drops observed in experiments by Quéré [39] and our Sc is in excellent agreement with its laboratory value.

On the convection of a passive scalar by a turbulent Gaussian velocity field

Through the use of the Novikov-Furutsu formula for Gaussian processes an equation is obtained for the diffusion of the ensemble average of a passive scalar in an incompressible turbulent velocity field in terms of the two-point, two-time correlator of this field. The equation is valid for turbulence which is not necessarily homogeneous or stationary and thus generalizes previous work.

Stability of an oscillating Kolmogorov flow

The linear stability of unidirectional flow sinusoidal both in a transverse direction and in time is considered. The problem reduces to an infinite algebraic eigenvalue problem. By using continued fractions, it is proved rigorously that the time‐independent flow is unstable to perturbation modes which do not have the periodicity of the basic flow in the transverse direction. Also, instability is proved for the inviscid case, for which the proofs known before do not work even when perturbation modes have the same periodicity as the basic flow. In the case of time‐dependent inviscid flow, exact solutions of a generalized Orr–Sommerfeld equation are found by the separation of continuous variables. Comparison with Galerkin solutions of the eigenvalue problem obtained by the separation of a discrete ‘‘time’’ variable leads to insights into reliability of the Galerkin results for the original, two‐dimensional eigenvalue problem.

On negative eddy viscosity under conditions of isotropy

This study considers a large‐scale flow freely evolving through a periodic array of oscillating triangular eddies. The effective eddy viscosity induced by the system appears to be isotropic, i.e., insensitive to the spatial orientation of the underlying periodic flow field. In contrast to the previously studied situation with time‐independent periodic flow, the eddy viscosity generated by rapidly oscillating eddies may be negative, thereby reducing the total (molecular and eddy) viscosity of the system. At sufficiently high Reynolds numbers, the total viscosity might even become negative promoting spontaneous formation of large‐scale structures.

On evolution equations for thin films flowing down solid surfaces

A wavy free‐surface flow of a viscous film down a cylinder is considered. It is shown that if the cylinder radius is large, as compared to the film thickness, the long‐wave perturbation approach yields a rather simple evolution equation. This nonlinear equation is similar to the well‐known Benney equation of planar films, and becomes exactly the latter in the limit of infinite radius. Thus it is the annular‐case analog—which was missing in the literature—of the Benney equation. It is argued that under conditions implicitly implied in their derivation, the Benney‐type equations are not uniformly valid for large times. However, both the new and Benney equations are important heuristically—as sources of other, simpler, equations which, in certain domains of system parameters, are valid for all time. Also, the new equation of annular films is important as a qualitative model incorporating all significant physical factors.