Long-wave Perturbations Research Articles

Thin film development on a double layer of fluids over a stretching sheet

Abstract This research investigates two layers of immiscible fluids over a stretching sheet, where the primary layer is a Newtonian fluid and the secondary layer is a non-Newtonian second-grade fluid. The governing equations describing the flow are the two-dimensional mass and momentum equations supported by the interface and boundary conditions. The small aspect ratio of the fluid domain allows the application of long-wave theory and thereby two partial differential equations for the dynamics of thin liquid film for each layer are obtained. The coupled partial differential equations thus obtained are solved numerically by the finite volume approach where the spatial derivatives are approximated using the upwind difference scheme and the time derivatives are by forward difference. The study focuses on analyzing the impact of fluid processing parameters like Reynolds number, viscosity ratio, non-Newtonian parameter, and surface tension on the flow dynamics. The study reveals that the non-Newtonian parameter delays the thinning effect of the fluids across the fluid layers and may have possible applications in coating technologies, biomedical devices, and advanced manufacturing.

Rogue waves on the periodic background of the Kuralay-II equation

We derive the rogue wave solutions of the Kuralay-II equation by applying the Darboux transformation method with the Lax pair on the periodic background. These solutions are represented using Jacobian elliptic functions: dnoidal and cnoidal. The rogue wave solutions can be obtained on the periodic background while the dnoidal travelling periodic wave and cnoidal travelling periodic wave are modulationally unstable with respect to the long-wave perturbations.

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.

Local linear stability of plumes generated along vertical heated cylinders in stratified environments

The linear temporal and absolute/convective stability characteristics of a thermal plume generated along a heated vertical cylinder are investigated theoretically under the Boussinesq approximation. Special focus is given to the uniform-wall-buoyancy-flux case whereby the cylinder surface sustains the same linear temperature gradient as the environment. A competition between the axisymmetric and helical modes is a remarkable feature of the instability, distinguishing these ‘annular plumes’ from free plumes/jets for which the helical mode is generally dominant. It is found that higher surface curvature stabilises the temporal axisymmetric mode significantly, but only has moderate effects on the helical mode. The most temporally unstable perturbation mode switches from a helical into an axisymmetric mode when the Prandtl number increases beyond a critical value. Both the roles of shear and buoyancy during the destabilisation are identified through an energy analysis which indicates that, while the shear work is usually a major source of perturbation energy, the buoyancy work manifests for long-wave axisymmetric perturbation modes, and for thin cylinders and high Prandtl numbers. For the specific temperature configuration considered herein, an annular plume is always convectively unstable whereas decreasing the cylinder radius from the planar limiting case first decreases and then increases the tendency of the flow towards being absolutely unstable. The helical mode is especially susceptible to being absolutely unstable on very thin cylinders. Several conditions for the onset of cellular thermal convection and plume detrainment are proposed based on our results and a hypothesis which connects the absolute instability to the detrainment phenomenon.

Benjamin–Feir Instability of Stokes Waves in Finite Depth

Whitham and Benjamin predicted in 1967 that small-amplitude periodic traveling Stokes waves of the 2d-gravity water waves equations are linearly unstable with respect to long-wave perturbations, if the depth {mathtt h} is larger than a critical threshold texttt{h}_{scriptscriptstyle {textsc {WB}}}approx 1.363 . In this paper, we completely describe, for any finite value of mathtt h >0 , the four eigenvalues close to zero of the linearized equations at the Stokes wave, as the Floquet exponent mu is turned on. We prove, in particular, the existence of a unique depth texttt{h}_{scriptscriptstyle {textsc {WB}}}, which coincides with the one predicted by Whitham and Benjamin, such that, for any 0< mathtt h < texttt{h}_{scriptscriptstyle {textsc {WB}}}, the eigenvalues close to zero are purely imaginary and, for any mathtt h > texttt{h}_{scriptscriptstyle {textsc {WB}}}, a pair of non-purely imaginary eigenvalues depicts a closed figure “8”, parameterized by the Floquet exponent. As {mathtt h} rightarrow texttt{h}_{scriptscriptstyle {textsc {WB}}}^{, +} the “8” collapses to the origin of the complex plane. The complete bifurcation diagram of the spectrum is not deduced as in deep water, since the limits texttt{h}rightarrow +infty (deep water) and mu rightarrow 0 (long waves) do not commute. In finite depth, the four eigenvalues have all the same size mathcal {O}(mu ), unlike in deep water, and the analysis of their splitting is much more delicate, requiring, as a new ingredient, a non-perturbative step of block-diagonalization. Along the whole proof, the explicit dependence of the matrix entries with respect to the depth texttt{h} is carefully tracked.

Natural Optical Activity from Density-Functional Perturbation Theory.

We present an accurate and computationally efficient first-principles methodology to calculate natural optical activity. Our approach is based on the long-wave density-functional perturbation theory and includes self-consistent field terms naturally in the formalism, which are found to be of crucial importance. The final result is expressed exclusively in terms of response functions to uniform field perturbations and avoids troublesome summations over empty states. Our strategy is validated by computing the natural optical activity tensor in representative chiral crystals (trigonal Se, α-HgS, and α-SiO_{2}) and molecules (C_{4}H_{4}O_{2}), finding excellent agreement with experiment and previous theoretical calculations.

On the interaction of a wind turbine wake with a conventionally neutral atmospheric boundary layer

In this work, we investigate the dynamics of wind turbine tip-vortex breakdown in a conventionally neutral atmospheric boundary layer (ABL). To this end, high-resolution data are collected from large-eddy simulations of a wind turbine operating within a neutral ABL and studied by means of proper orthogonal decomposition (POD) and Fourier analysis. The high resolution of the generated data in both space and time allows us to gain insight into the tip-vortex breakdown mechanisms by (i) capturing the energy modes of the coherent structures, (ii) studying their contribution to the tip-vortex breakdown through their power spectra functions and mean kinetic energy (MKE) flux, and (iii) analysing the growth rate of each contributing perturbation frequency along tip vortices. Our analysis shows that under a fully turbulent scenario, the growth rate of perturbations along the tip vortices is largest for low wave numbers, i.e. long-wave perturbations. Additionally, the MKE flux reaches its highest value at two diameters downstream of the rotor plane, a behaviour that can be attributed to the coexistence of multiple interacting POD modes, with the streamwise vortex roller mode being the primary contributor to the total MKE flux budget, contributing approximately 24%. Finally, comparisons with a laminar, uniform flow scenario subject to a single-frequency perturbation highlight the differences between the two ambient flow conditions. In the non-turbulent, uniform flow scenario, the growth rate attains its maximum value at a wave number corresponding to the out-of-phase mutual-inductance mechanism, whereas the MKE flux exhibits local minima and maxima along the wake and at different downstream locations depending on the perturbation frequency. Our analyses suggest that the breakdown of the wind turbine tip vortices under a fully turbulent neutral ABL inflow is due to complex interactions across a range of excitation frequencies, in which the mutual-inductance instability may not be the dominant one.

Nonlinear Modulational Instabililty of the Stokes Waves in 2D Full Water Waves

The well-known Stokes waves refer to periodic traveling waves under the gravity at the free surface of a two dimensional full water wave system. In this paper, we prove that small-amplitude Stokes waves with infinite depth are nonlinearly unstable under long-wave perturbations. Our approach is based on the modulational approximation of the water wave system and the instability mechanism of the focusing cubic nonlinear Schrödinger equation.

Evolution of Water Wave Groups in the Forced Benney–Roskes System

For weakly nonlinear waves in one space dimension, the nonlinear Schrödinger Equation is widely accepted as a canonical model for the evolution of wave groups described by modulation instability and its soliton and breather solutions. When there is forcing such as that due to wind blowing over the water surface, this can be supplemented with a linear growth term representing linear instability leading to the forced nonlinear Schrödinger Equation. For water waves in two horizontal space dimensions, this is replaced by a forced Benney–Roskes system. This is a two-dimensional nonlinear Schrödinger Equation with a nonlocal nonlinear term. In deep water, this becomes a local nonlinear term, and it reduces to a two-dimensional nonlinear Schrödinger Equation. In this paper, we numerically explore the evolution of wave groups in the forced Benney–Roskes system using four cases of initial conditions. In the one-dimensional unforced nonlinear Schrödinger equa tion, the first case would lead to a Peregrine breather and the second case to a line soliton; the third case is a long-wave perturbation, and the fourth case is designed to stimulate modulation instability. In deep water and for finite depth, when there is modulation instability in the one-dimensional nonlinear Schdrödinger Equation, the two-dimensional simulations show a similar pattern. However, in shallow water where there is no one-dimensional modulation instability, the extra horizontal dimension is significant in producing wave growth through modulation instability.

Full description of Benjamin-Feir instability of stokes waves in deep water

Small-amplitude, traveling, space periodic solutions –called Stokes waves– of the 2 dimensional gravity water waves equations in deep water are linearly unstable with respect to long-wave perturbations, as predicted by Benjamin and Feir in 1967. We completely describe the behavior of the four eigenvalues close to zero of the linearized equations at the Stokes wave, as the Floquet exponent is turned on. We prove in particular the conjecture that a pair of non-purely imaginary eigenvalues depicts a closed figure “8”, parameterized by the Floquet exponent, in full agreement with numerical simulations. Our new spectral approach to the Benjamin-Feir instability phenomenon uses a symplectic version of Kato’s theory of similarity transformation to reduce the problem to determine the eigenvalues of a 4 times 4 complex Hamiltonian and reversible matrix. Applying a procedure inspired by KAM theory, we block-diagonalize such matrix into a pair of 2 times 2 Hamiltonian and reversible matrices, thus obtaining the full description of its eigenvalues.

Stability Analysis of Thin Power-Law Fluid Film Flowing down a Moving Plane in a Vertical Direction

This paper analyses the stability of thin power-law fluid flowing down a moving plane in a vertical direction by using the long-wave perturbation method. Linear and nonlinear stability are discussed. The linear stable region increases as the downward speed increases and the power-law index increases. More accurate results are obtained on the coefficients of the nonlinear generalized kinematic equation in the power-law part. The regions of sub-critical instability and absolute stability are expanded when the downward movement of plane increases, or the power-law index increases, and meanwhile the parts of supercritical stability and explosive supercritical instability are compressed. By substituting the power-law index and moving speed into the generalized nonlinear kinematic equation of the power-law film on the free surface, the results can be applied to estimate the stability of the thin film flow in the field of engineering.

Effect of odd-viscosity on the dynamics and stability of a thin liquid film flowing down on a vertical moving plate

In this study, we focus on the problem of hydrodynamic instability of a thin, viscous, Newtonian liquid film with broken time-reversal-symmetry flowing down along the surface of a vertical moving plate. The presence of odd viscosity gives rise to new terms in the pressure gradient of the flow. Utilizing the classical long-wave perturbation method, we obtain the analytical solutions as well as derive the nonlinear evolution equation of Benney-type in terms of film thickness h(x,t) which is significantly modified due to the presence of odd viscosity in the liquid. We solve the linear model by using the normal mode approach and for three different conditions, namely, the quiescent, up-moving and down-moving plate velocity. The linear study shows that the effect of the down-moving motion of the vertical plate is to enhance the stability of the film flow whereas the up-moving motion of the vertical plate tends to reduce it. Further, the study shows that odd viscosity always has a stabilizing effect on the flow field. In addition, the Orr–Sommerfeld equation is also derived and solved analytically to obtain the critical Reynolds number. Finally, we show the numerical solution of the evolution equation in a periodic domain which clearly demonstrates the role of odd-viscosity on the dynamics of the plate motions of thin film flows coating in isothermal environments. Our study clearly shows how odd viscosity influences the stability of the flow.

Odd-viscosity-induced instability of a thin film with variable density

The stability of a two-dimensional gravity-driven thin viscous Newtonian fluid with broken time-reversal-symmetry draining down a uniformly heated inclined plane is discussed. The presence of the odd part of the Cauchy stress tensor with an odd viscosity coefficient brings new characteristics in fluid flow. A theoretical model is implemented, which captures the dependence of the surface tension on temperature, and the model also allows for variation in the density of the liquid with a thermal difference. The coupled effect of odd viscosity, variable density, and surface tension has been investigated both analytically and numerically. A nonlinear evolution equation of the free surface is derived by the method of systematic asymptotic expansion. A linear stability analysis is carried out, which yields the critical conditions for the onset of instability in long-wave perturbations. New interesting results illustrating how the critical Reynolds number depends on the odd viscosity as well as other various dimensionless parameters have been obtained. In addition, a weakly nonlinear stability analysis is performed based on the method of multiple scales from which a complex Ginzburg–Landau equation is obtained. It is observed that the film not only has supercritical stable and subcritical unstable zones, but also unconditional stable and explosive zones. It has been also shown that the spatial uniform solution corresponding to the sideband disturbance may be stable in the unstable region. Employing the Crank–Nicolson method in a periodic domain, the spatiotemporal evolution of the model has been analyzed numerically for different values of the odd viscosity as well as other flow parameters. Nonlinear simulations are found to be in good agreement with the linear and weakly nonlinear stability analysis. The results are conducive to the further development of related experiments.

A new linearly unstable mode in the core-annular flow of two immiscible fluids

Abstract

Rogue waves for the fourth-order nonlinear Schrödinger equation on the periodic background.

In this paper, we construct rogue wave solutions on the periodic background for the fourth-order nonlinear Schrödinger (NLS) equation. First, we consider two types of the Jacobi elliptic function solutions, i.e., dn- and cn-function solutions. Both dn- and cn-periodic waves are modulationally unstable with respect to the long-wave perturbations. Second, on the background of both periodic waves, we derive rogue wave solutions by combining the method of nonlinearization of spectral problem with the Darboux transformation method. Furthermore, by the study of the dynamics of rogue waves, we find that they have the analogs in the standard NLS equation, and the higher-order effects do not have effect on the magnification factor of rogue waves. In addition, when the elliptic modulus approaches 1, rogue wave solutions can reduce to multi-pole soliton solutions in which the interacting solitons form weakly bound states.

INSTABILITY OF A GRAVITY-DRIVEN LIQUID FILM CONFINED IN AN INCLINED RECTANGULAR DUCT

This study deals with the confinement effects due to top and side walls on the stability of a falling liquid film flowing down an incline. To this aim, we consider a liquid film flowing between two infinite plates and in rectangular ducts in the presence of an adjacent gas phase, focusing on the case of zero net gas flow rate. A rigorous linear stability analysis is conducted for three gas-liquid systems of particular interest in applications. This enables us to demonstrate the effect of the gas-liquid density ratio and viscosity ratio in small channels/ducts and shallow downward inclinations. Although the analysis considers all wave numbers, the neutral stability boundary is found to be defined mainly by long-wave perturbations in the two-plate (TP) geometry, as well as in the rectangular ducts. The decrease of the channel height for the considered liquid-gas systems always stabilizes the liquid film. However, the Kapitza criterion for the onset of Kapitza waves on the surface of a falling liquid film can either overpredict or underpredict the stability limits of the channel/duct flows, depending on the channel height, its aspect ratio, and the viscosity and density ratios of the phases. The falling liquid film, bounded by the side walls, becomes unstable at larger flow rates with the decrease of the duct aspect ratio. The presence of recirculating gas in the channel/duct is found to affect the stability through the dynamic shear stress components at the gas-liquid interface. Therefore, its presence must be taken into account.

Influence of the interface internal energy on monotone disturbances of a creeping stationary flow with a velocity field of the Hiemenz type

The stability of a creeping stationary flow, which arises as a result of the interface internal energy, is studied. It is shown that the system of amplitude equations allows exact integration. Complex decrement as a solution to the transcendental equation is found. It was found that long-wave perturbations decay with increasing time. Neutral curves for the transformer oil - formic acid system were constructed.

Restructuring and breakup of nanowires with the diamond cubic crystal structure into nanoparticles

A kinetic Monte Carlo approach is applied to study physical mechanisms responsible for the breakup of nanowires with the diamond cubic crystal structure into a chain of nanoparticles discovered in preceding experiments on Silicon nanowires. We show that this process is based on the well-known mechanism of roughening transition, which specifically manifests itself in quasi-one-dimensional systems/nanowires with a pronounced anisotropy of the surface energy density. Depending on the temperature and orientation of the nanowire relative to its internal crystal structure, the wavelengths of substantial cross-sectional modulations exceed its initial radius by 4–18 times. For certain orientations, a straight nanowire at the initial stage of evolution forms a serpentine/helical structure. The scenarios of the stage of nanowire ruptures into single nanoclusters are also diverse: either each spindle-shaped region of the nanowire transforms into a separate drop (by long-wave surface perturbations), or the adjacent short-scale beads absorb each other due to the Ostwald ripening effect, which can be accompanied by the formation of long-lived many-body dumbbells. The discovered features of the dynamics of quasi-one-dimensional systems expand our conceptions of the physical mechanisms involved in the breakup of nanowires (presented by Nichols and Mullins as a classical model for such instabilities) which could be useful in applications based on chains of ordered nanoparticles.

Modelling and nonlinear boundary stabilization of the modified generalized Korteweg\u2013de Vries\u2013Burgers equation

In this paper, we study the modelling and nonlinear boundary stabilization problem of the modified generalized Korteweg–de Vries–Burgers equation (MGKdVB) when the spatial domain is [0,1]. First, the MGKdVB equation is derived using the long-wave approximation and perturbation method. Then, two nonlinear boundary controllers are proposed for this equation and the L^{2} -global exponential stability of the solution is shown. Numerical simulations are given to illustrate the efficiency of the developed control schemes.

Marangoni instability in a heated viscoelastic liquid film: Long-wave versus short-wave perturbations.

We investigate the Marangoni instability in a thin layer of viscoelastic fluid, confined between its deformable free surface and a substrate of low thermal conductivity. Following a theoretical analysis, we study the stability of the present system for the case when the fluid layer is subjected to heating from below. Here, we use the Maxwell model to depict the rheology of the viscoelastic fluid. Linear stability analysis of the quiescent base state reveals that, in addition to the conventional short-wave mode, a long-wave instability can also emerge in this system. We demonstrate the appearance of both the long-wave monotonic and oscillatory instabilities in such a system. We study this long-wave mode analytically using the scaling k∼sqrt[Bi] (k is the wave number and Bi is the Biot number), whereas the short-wave mode is examined numerically. The influential role of elasticity of the fluid and the other involved parameters on the stability of the system is aptly discussed, and their ranges are identified within which a particular instability mode gets critical.