Evgeny A. Demekhin

NONLINEAR EVOLUTION OF WAVES ON A VERTICALLY FALLING FILM

Wave formation on a falling film is an intriguing hydrodynamic phenomenon involving transitions among a rich variety of spatial and temporal structures. Immediately beyond an inception region, short, near-sinusoidal capillary waves are observed. Further downstream, long, near-solitary waves with large tear-drop humps preceded by short, front-running capillary waves appear. Both kinds of waves evolve slowly downstream such that over about ten wavelengths, they resemble stationary waves which propagate at constant speeds and shapes. We exploit this quasi-steady property here to study wave evolution and selection on a vertically falling film. All finite-amplitude stationary waves with the same average thickness as the Nusselt flat film are constructed numerically from a boundary-layer approximation of the equations of motion. As is consistent with earlier near-critical analyses, two travelling wave families are found, each parameterized by the wavelength or the speed. One family γ 1 travels slower than infinitesimally small waves of the same wavelength while the other family γ 2 and its hybrids travel faster. Stability analyses of these waves involving three-dimensional disturbances of arbitrary wavelength indicate that there exists a unique nearly sinusoidal wave on the slow family γ 1 with wavenumber α s (or α 2 ) that has the lowest growth rate. This wave is slightly shorter than the fastest growing linear mode with wavenumber α m and approaches the wave on γ 1 with the highest flow rate at low Reynolds numbers. On the fast γ 2 family, however, multiple bands of near-solitary waves bounded below by α f are found to be stable to two-dimensional disturbances. This multiplicity of stable bands can be interpreted as a result of favourable interaction among solitary-wave-like coherent structures to form a periodic train. (All waves are unstable to three-dimensional disturbances with small growth rates.) The suggested selection mechanism is consistent with literature data and our numerical experiments that indicate waves slow down immediately beyond inception as they approach the short capillary wave with wavenumber α 2 of the slow γ 1 family. They then approach the long stable waves on the γ 2 family further downstream and hence accelerate and develop into the unique solitary wave shapes, before they succumb to the slowly evolving transverse disturbances.

Iterated stretching of viscoelastic jets

We examine, with asymptotic analysis and numerical simulation, the iterated stretching dynamics of FENE and Oldroyd-B jets of initial radius r0, shear viscosity ν, Weissenberg number We, retardation number S, and capillary number Ca. The usual Rayleigh instability stretches the local uniaxial extensional flow region near a minimum in jet radius into a primary filament of radius [Ca(1−S)/We]1/2r0 between two beads. The strain-rate within the filament remains constant while its radius (elastic stress) decreases (increases) exponentially in time with a long elastic relaxation time 3We(r02/ν). Instabilities convected from the bead relieve the tension at the necks during this slow elastic drainage and trigger a filament recoil. Secondary filaments then form at the necks from the resulting stretching. This iterated stretching is predicted to occur successively to generate high-generation filaments of radius rn, (rn/r0)=√(rn−1/r0)3/2 until finite-extensibility effects set in.

Thermocapillary instability and wave formation on a film falling down a uniformly heated plane

We consider a thin layer of a viscous fluid flowing down a uniformly heated planar wall. The heating generates a temperature distribution on the free surface which in turn induces surface tension gradients. We model this thermocapillary flow by using the Shkadov integral-boundary-layer (IBL) approximation of the Navier–Stokes/energy equations and associated free-surface boundary conditions. Our linear stability analysis of the flat-film solution is in good agreement with the Goussis & Kelly (1991) stability results from the Orr–Sommerfeld eigenvalue problem of the full Navier–Stokes/energy equations. We numerically construct nonlinear solutions of the solitary wave type for the IBL approximation and the Benney-type equation developed by Joo et al. (1991) using the usual long-wave approximation. The two approaches give similar solitary wave solutions up to an

Laminarizing effects of dispersion in an active-dissipative nonlinear medium

O(1)

A spectral theory for small-amplitude miscible fingering

Reynolds number above which the solitary wave solution branch obtained by the Joo et al. equation is unrealistic, with branch multiplicity and limit points. The IBL approximation on the other hand has no limit points and predicts the existence of solitary waves for all Reynolds numbers. Finally, in the region of small film thicknesses where the Marangoni forces dominate inertia forces, our IBL system reduces to a single equation for the film thickness that contains only one parameter. When this parameter tends to zero, both the solitary wave speed and the maximum amplitude tend to infinity.

Mechanism for drop formation on a coated vertical fibre

Abstract The Kuramoto-Sivashinsky equation has become a popular prototype for systems which exhibit spatial-temporal chaos. We show here that a linear dispersion term δ h xxx tends to arrest such irregular behavior in favor of spatially periodic cellular structures, as is consistent with prior numerical and experimental observations. The study includes a normal form analysis and a numerical investigation of periodic and solitary stationary wave solutions of the equation. It is shown that by δ > 1.1, the infinite families of stationary periodic waves of the Kuramoto-Sivashinsky equation, each ending in a solitary wave, have been annihilated successively such that only a lone family of periodic waves, consisting of one-hump KdV pulses for δ > 3.7, remains as the only periodic stationary wave attractors of the system. These periodic waves have much larger domains of attraction than the strange attractors and hence tend to dominate spatial-temporal chaos in an extended domain with significant dispersion.

Two-dimensional wave dynamics in thin films. I. Stationary solitary pulses

Using the self-similar symmetry of a diffusing front, we develop a linear spectral theory for miscible fingering at inception that accurately captures the destabilization of localized disturbances (with large transverse wavelengths compared to the front width) by the unsteady front. Our theory predicts a generic selected wavelength (4πηD/U0 for gravity fingering, where η is the transverse to longitudinal dispersion ratio, and an additional factor proportional to the logarithm of the mobility ratio for viscous fingering) at the small time of O(D/U02), where D is the dispersion coefficient or diffusivity in the flow direction and U0 is the displacement velocity. This wavelength then grows in time and approaches a universal asymptotic wavelength coarsening dynamics of (η2D5/U02)1/8(t′)3/8, where t′ is the dimensional time, for all small-amplitude miscible fingering phenomena in a slot or in porous media. The 38 exponent in time is due to a unique long-wave stabilization mechanism due to transverse convection...

Generation and suppression of radiation by solitary pulses

Rayleigh instability on an axisymmetric viscous film around a vertical fibre produces localized wave structures (pulses) on a flat substrate film that can grow by an order of magnitude to form large capillary drops. We show that this drop formation process is driven by a unique mechanism in the form of an ever-growing pulse which leaves behind a trailing film thinner than the one it advances into. In addition to accumulating liquid from the film, the growing pulse also captures smaller and slower pulses in coalescence cascades. We construct this supercritical growing pulse by matched asymptotics

Local stability theory of solitary pulses in an active medium

We consider two-dimensional stationary solitary pulses in a falling film by using the two-dimensional generalized Kuramoto-Sivashinsky equation as a model system. We numerically construct solitary wave solutions of this equation as a function of the dispersion parameter. We obtain an analytical estimate for the speed of these waves in the strongly dispersive case by using a perturbation from the Korteweg-de Vries limit. An impulse response analysis in which the nonlinearity is replaced with a delta function leads to an approximate analytical solution for the shape of two-dimensional solitary waves. The analytical predictions are in excellent agreement with numerical results for the speed and shape of these waves.

Coherent structures, self-similarity, and universal roll wave coarsening dynamics

The description of the spatio-temporal dynamics of an extended active/dispersive medium, such as wave dynamics on a falling film, can be simplified considerably if the dynamics is dominated by a fixed number of solitary pulses separated by radiation-free flat substrates. Radiation is generated in a noise-free environment when excess mass drains out of a nonequilibrium (excited) pulse in the form of a spreading shelf and the radiation grows rapidly by feeding on the active substrate. However, this growth can be suppressed if the localized radiation packet is absorbed by a second pulse. It is shown that both the generation and suppression mechanisms can be quantitatively understood by analyzing the essential spectrum of the equilibrium pulse that determines how it attenuates an absorbed radiation wavepacket and a peculiar resonance pole that captures the drainage dynamics of an excited pulse. The instantaneous speed of a decaying pulse is shown to scale linearly with respect to its instantaneous amplitude a...