Two-dimensional Nonlinear Evolution Research Articles

A novel (2+1)-dimensional nonlinear evolution equation for weakly stratified free-surface boundary layers

To get an insight into the dynamics of the oceanic surface boundary layer we develop an asymptotic model of the nonlinear dynamics of linearly decaying three-dimensional long-wave perturbations in weakly stratified boundary-layer flows. Although in nature the free-surface boundary layers in the ocean are often weakly stratified due to solar radiation and air entrainment caused by wave breaking, weak stratification has been invariably ignored. Here, we consider an idealized hydrodynamic model, where finite-amplitude three-dimensional perturbations propagate in a horizontally uniform unidirectional weakly stratified shear flow confined to a boundary layer adjacent to the water surface. Perturbations satisfy the no-stress boundary condition at the surface. They are assumed to be long compared with the boundary-layer thickness. Such perturbations have not been studied even in a linear setting. By exploiting the assumed smallness of nonlinearity, wavenumber, viscosity and the Richardson number, on applying triple-deck asymptotic scheme and multiple-scale expansion, we derive in the distinguished limit a novel essentially two-dimensional nonlinear evolution equation, which is the main result of the work. The equation represents a generalization of the two-dimensional Benjamin–Ono equation modified by the explicit account of viscous effects and new dispersion due to weak stratification. It describes perturbation dependence on horizontal coordinates and time, while its vertical structure, to leading order, is given by an explicit analytical solution of the linear boundary value problem. It shows the principal importance of weak stratification for three-dimensional perturbations.

Interfacial stability and self-similar rupture of evaporating liquid layers under vapor recoil

We investigate interfacial stability of an evaporating viscous liquid layer above/below a horizontal heated substrate in the framework of a long-wave model that accounts for surface tension, positive/negative gravity, and evaporation effects of mass loss and vapor recoil. With the time-dependent linear stability analysis, it is found that the interface instability is enhanced by vapor recoil with time using an effective growth rate. The destabilizing mechanism of vapor thrust competes with the stabilizing surface tension, and the effects of the latter are not asymptotically negligible near rupture, reflected by a rescaled effective interfacial pressure. A two-dimensional nonlinear evolution is investigated for the quasi-equilibrium evaporating layers with different evaporative conditions for Rayleigh–Taylor unstable and sessile layers. For weak mass loss and strong vapor recoil, the well-defined capillary ridges emerge around a deepening narrow valley with increasing wavelength under a positive gravity, while, on the basis of initial condition, main and secondary droplets are either coalesced partially or separated by a sharp dry-out point under a negative gravity. The rupture location depends strongly on the characteristics of a given initial condition, except for the random perturbation. For both the cases, an increase in the modified evaporation number tends to reduce the rupture time tr and droplet thickness remarkably. Similarity analysis along with numerical strategy is presented for the final stage of touch-down dynamics, determined by a physical balance between the vapor recoil and capillary force. The evaporation-driven rupture with a significant vapor recoil and negligible mass loss is shown to contain a countably infinite number of similarity solutions whose horizontal and vertical length scales behave as (tr − t)1/2 and (tr − t)1/3. The first similarity solution represents a stable single-point rupture.

Nonlinear development of two-layer Couette–Poiseuille flow in the presence of surfactant

The two-dimensional nonlinear evolution of the interface between two superposed layers of viscous fluid moving in a channel in the presence of an insoluble surfactant is examined. A pair of coupled weakly nonlinear equations is derived describing the interfacial and surfactant dynamics when one of the two fluid layers is very thin in comparison to the other. In contrast to previous work, the dynamics in the thin film are coupled to the dynamics in the thicker layer through a nonlocal integral term. For asymptotically small Reynolds number, the flow in the thicker layer is governed by the Stokes equations. A linearized analysis confirms the linear instability identified by previous workers and it is proven that the film flow is linearly unstable if the undisturbed surfactant concentration exceeds a threshold value. Numerical simulations of the weakly nonlinear equations reveal the existence of finite amplitude traveling-wave solutions. For order one Reynolds number, the flow in the thicker layer is governed by the linearized Navier–Stokes equations. In this case the weakly nonlinear film dynamics are more complex and include the possibility of periodic traveling-waves and chaotic flow.

Dynamics and stability of flow down a flexible incline

The flow of a thin liquid film down a flexible inclined wall is examined. Two configurations are studied: constant flux (CF) and constant volume (CV). The former configuration involves constant feeding of the film from an infinite reservoir of liquid. The latter involves the spreading of a drop of constant volume down the wall. Lubrication theory is used to derive a pair of coupled two-dimensional nonlinear evolution equations for the film thickness and wall deflection. The contact-line singularity is relieved by assuming that the underlying wall is pre-wetted with a precursor layer of uniform thickness. Solution of the one-dimensional evolution equations demonstrates the existence of travelling-wave solutions in the CF case and self-similar solutions in the CV case. The effect of varying the wall tension and damping coefficient on the structure of these solutions is elucidated. The linear stability of the flow to transverse perturbations is also examined in the CF case only. The results indicate that the flow, which is already unstable in the rigid-wall limit, is further destabilized as a result of the coupling between the fluid and underlying flexible wall.

Bent Marshak waves

Radiation-driven heat waves (Marshak waves) are ubiquitous in astrophysics and terrestrial laser-driven high-energy density plasma physics experiments. Generally, the equations describing Marshak waves are so nonlinear, that solutions involving more than one spatial dimension require simulation. However, in this paper it is shown that one may analytically solve the problem of the two-dimensional nonlinear evolution of a Marshak wave, bounded by lossy walls, using an asymptotic expansion in a parameter related to the wall albedo and a simplification of the heat front equation of motion. Three parameters determine the nonlinear evolution: a modified Markshak diffusion constant, a smallness parameter related to the wall albedo, and the spacing of the walls. The final nonlinear solution shows that the Marshak wave will be both slowed and bent by the nonideal boundary. In the limit of a perfect boundary, the solution recovers the original diffusion-like solution of Marshak. The analytic solution will be compared to a limited set of simulation results and experimental data.

Surfactant-induced fingering phenomena beyond the critical micelle concentration

We consider the spreading of a droplet of soluble surfactant, at concentrations beyond its critical micelle concentration (CMC), on a pre-existing thin liquid layer. Lubrication theory is used to derive a coupled system of four two-dimensional nonlinear evolution equations for the film thickness and surfactant concentration at the interface and in the bulk as both monomers and micelles. These equations are parameterized by a number of dimensionless groups that reflect the relative importance of Marangoni stresses, surface and bulk diffusion, capillarity, surfactant solubility, sorption kinetics and the nonlinearity of the equation of state. Our results for the base state indicate that two parameters in particular exert a significant influence on the flow profiles: the dimensionless mass of surfactant deposited, further, this protuberance separates from the drop to form a distinct secondary front that lies behind a leading front that usually accompanies the spreading process. Our examination of the linear and nonlinear stability of the system through a transient growth analysis and transient numerical simulations, respectively, indicates that these features are vulnerable to transverse perturbations, leading to the formation of fingers. The results obtained in the present work are in qualitative agreement with recently available experimental data.

Dynamics of density irregularities in the E-region of the ionosphere

A system of equations describing the two-dimensional nonlinear evolution of the two-stream and gradient-drift instabilities in the E-region of the Earth's ionosphere is derived, taking into account turbulent ion viscosity and the nonlinearity associated with the ion inertia. The linear instabilities are briefly re-examined. Regular nonlinear wave forms are found for both instabilities in the one-dimensional limit. Estimates of characteristic values of amplitudes and wavelengths are also presented.

Stable localized patterns in thin liquid films.

We study a two-dimensional nonlinear evolution equation which describes the three-dimensional spatiotemporal behavior of the air-liquid interface of a thin liquid film lying on the underside of a cooled horizontal plate. We show that the equation has a Liapunov functional and exploit this fact to demonstrate that the Marangoni effect can stabilize the destabilizing effect of gravity (the Rayleigh-Taylor instability) allowing for the existence of stable localized axisymmetric solutions for a wide range of parameter values. Various properties of these structures are discussed.

Spectral transform for a two spatial dimension extension of the dispersive long wave equation

A two-dimensional nonlinear evolution equation is solved in the inverse spectral transform scheme. It coincides, when reduced to one spatial dimension, with the dispersive long wave equation. The Backlund transformation, soliton solution and superposition formula are obtained. The spectral transform is explicitly defined and the corresponding linear evolution of the spectral data is given. The inverse spectral problem is formulated as a non-local Riemann-Hilbert boundary value problem and solved.

Bäcklund transformation for two-dimensional nonlinear evolution equations and the decomposition theorem

The simplest Bäcklund transformations are derived for matrix spectral problems associated with the Dirac equation. Decomposition theorems reducing equations of the nonlinear Schrödinger type to the common action of commuting generalized Toda lattice flows are presented.

Two-dimensional non-linear evolution equations: The derivation and the transient wave solution

The general theory of two-dimensional evolution equations describing transient wave propagation in non-linear continuous media is presented. The ray method is used and the two-dimensional evolution equations for plane and cylindrical wave-beams are obtained. The transient wave solutions are discussed briefly. A transformation of variables is proposed that permits the transformation of the two-dimensional evolution equation into a one-dimensional evolution equation with coordinate-dependent coefficients. A breakdown time analysis is carried out for this case. The dispersion relations for plane and cylindrical wave-beams are presented. The non-linear dispersion relation is obtained by making use of a series representation.