Long-wave Theory Research Articles

Identification of a source of oceanic ring waves near Darwin’s Island based on space photos

A system of ring surface waves in the Pacific Ocean is identified, which was registered near Darwin's Island in one of the photos taken according to the program of space experiment Uragan. Within the scope the hypothesis that the waves were generated as a result of a harmonic action on the ocean bed, a mathematical model, describing the excitation and propagation of waves in the approximation of the theory of long waves, was employed in image processing. As a result of calculations, the simulated wave system was approached to the one registered in the photo, which made it possible to estimate the physical parameters of the source of waves.

The Fracturing of LGBT Identities under Neoliberal Capitalism

AbstractHistorians have linked the emergence of contemporary lesbian/gay identities to the development of capitalism. A materialist approach should also look atdifferentforms of sexual identity, and their connections with specific phases of capitalist development. Marxist long-wave theory can help us understand how the decline of Fordism contributed to shifts in LGBT identities, speeding the consolidation of gay identity while fostering the rise of alternative sexual identities. These alternative identities, sometimes defined as ‘queer’, characterised by sexual practices that are still stigmatised, by explicit power-differentials and above all by gender-nonconformity, are particularly common among young and disadvantaged working-class strata. The growing diversity of identities is a challenge to any gay universalism that neglects class, gender, sexual, racial/ethnic and other differences, to the currently dominant forms of lesbian/gay organising, and ultimately to the prevailing division of human beings into gay and straight.1

Thin film flow over a heated nonlinear stretching sheet in presence of uniform transverse magnetic field

The unsteady flow of a thin viscous liquid film over a heated horizontal stretching surface permitted by uniform transverse magnetic field is studied by considering the stretching velocity and the temperature distribution in their general functional forms. An evolution equation for the film thickness is derived using long-wave approximation theory of thin liquid film and this nonlinear PDE is solved numerically for some representative values of non-dimensional parameters. It is observed that the magnetic field resists the film thinning process for all types of velocity and temperature distribution. But thermocapillarity enhances the film thinning rate even in presence of magnetic field. Further, effect of Marangoni and Prandtl numbers are explored in presence of magnetic field. Physical explanations are provided to understand the different effects on film thinning.

Generation of long waves in ice-covered lakes by moving disturbances of atmospheric pressure

The effect of ice cover on the oscillations of water in a two-dimensional constant-depth basin within the linear theory of long waves is considered. The ice cover is treated as thin elastic plate in the presence of compressive force. The ice is fastened hardly to a shore. The eigenfrequencies and eigenfunctions of free oscillations (seishes) are obtained. The forced oscillations of fluid and ice cover under the action of a moving disturbance of atmospheric pressure are investigated. The input data are taken to have values typical of Lake Baikal. The time dependence of total mechanical energy of the fluid is calculated for different velocities of pressure motion. The variations in ice-bending stresses are determined and the causes of shore-ice breaking are explained.

Influence of straits and bottom topography on the structure of barotropic currents in a flow-through basin

The development of currents and eddies in a basin of variable depth with two straits is studied within the framework of the nonlinear theory of long waves taking into account turbulent viscosity and the Coriolis parameter. The problem is solved numerically. We perform the comparative analysis of the results of modeling of currents in a homogeneous liquid. It is shown that these results depend on the location of the straits and bottom topography. Only jet currents are formed in basins of constant depth with symmetric straits. Eddy structures periodically appear in the presence of asymmetric straits.

Effect of permeability on the instability of a non-Newtonian film down a porous inclined plane

A thin film of a power–law fluid flowing down a porous inclined plane is considered. It is assumed that the flow through the porous medium is governed by the modified Darcy’s law together with Beavers–Joseph boundary condition for a general power–law fluid. Under the assumption of small permeability relative to the thickness of the overlying fluid layer, the flow is decoupled from the filtration flow through the porous medium and a slip condition at the bottom is used to incorporate the effects of the permeability of the porous substrate. Applying the long-wave theory, a nonlinear evolution equation for the thickness of the film is obtained. A linear stability analysis of the base flow is performed and the critical condition for the onset of instability is obtained. The results show that the substrate porosity in general destabilizes the film flow system and the shear-thinning rheology enhances this destabilizing effect. A weakly nonlinear stability analysis reveals the existence of supercritical stable and subcritical unstable regions in the wave number versus Reynolds number parameter space. The numerical solution of the nonlinear evolution equation in a periodic domain shows that the fully developed nonlinear solutions are either time-dependent modes that oscillate slightly in the amplitude or time independent stable two-dimensional nonlinear waves with large amplitude referred to as ‘permanent waves’. The results show that the shape and the amplitude of the nonlinear waves are strongly influenced by the permeability of the porous medium and the shear-thinning rheology.

Symmetries of shallow water equations on a rotating plane

We consider a system of nonlinear differential equations which describes the spatial motion of an ideal incompressible fluid on a rotating plane in the shallow water approximation and a more general system of the theory of long waves which takes into account the specifics of shear flows. Using the group analysis methods, we calculate the 9-dimensional Lie algebras of infinitesimal operators admissible by the models. We establish an isomorphism of these Lie algebras with a known Lie algebra of operators admissible by the system of equations for the two-dimensional isentropic motions of a polytropic gas with the adiabatic exponent γ = 2. The nontrivial symmetries of the models under consideration enable us to carry out the group generation of the solutions. The class of stationary solutions to the equations of rotating shallow water transforms into a new class of periodic solutions.

Solitary wave transformation on the underwater step: Asymptotic theory and numerical experiments

The transformation of a solitary wave on an underwater step is studied analytically and numerically. The theoretical model includes the linear potential description of the wave transformation on a step and the weakly nonlinear theory of long waves based on the Korteweg–de Vries equation for reflected and transmitted waves far from a step. Numerical simulation of solitary wave transformation on an underwater step is performed in the framework of an extended 1D Boussinesq-like system and fully nonlinear fully dispersive 2D Navier–Stokes equations. The results of numerical simulations for the incident solitary wave of weak amplitude are in agreement with the theoretical predictions for the wave shapes of the secondary solitons, but not with the predictions for travel times.

Marangoni convection, evaporation and interface deformation in liquid films on heated substrates with non-uniform thermal conductivity

Marangoni convection plays an important role in hydrodynamics of liquid films on heated or cooled substrates. In this paper a model describing Marangoni convection, interface dynamics and evaporation in liquid films on composite substrates or substrates of functionally graded materials is developed. Non-uniform thermal conductivity of the substrate causes non-uniformity of temperature distribution at the liquid–gas interface which leads to appearance of Marangoni stresses, convective vortices and film deformation. The film dynamics is described in the framework of long-wave theory. The substrate thermal conductivity non-uniformity has a pronounced effect on transport processes in the liquid film.

Nonmodal and nonlinear dynamics of a volatile liquid film flowing over a locally heated surface

The stability of a thin, volatile liquid film falling under the influence of gravity over a locally heated, vertical plate is analyzed in the noninertial regime using a model based on long-wave theory. The model is formulated to account for evaporation that is either governed by thermodynamic considerations at the interface in the one-sided limit or limited by the rate of mass transfer of the vapor from the interface. The temperature gradient near the upstream edge of the heater induces a gradient in surface tension that opposes the gravity-driven flow, and a pronounced thermocapillary ridge develops in the streamwise direction. Recent theoretical analyses predict that the ridge becomes unstable above a critical value of the Marangoni parameter, leading to the experimentally observed rivulet structure that is periodic in the direction transverse to the bulk flow. An oscillatory, thermocapillary instability in the streamwise direction above the heater is also predicted for films with sufficiently large heat loss at the free surface due to either evaporation or strong convection in the adjoining gas. This present work extends the recent linear stability analysis of such flows by Tiwari and Davis [Phys. Fluids 21, 022105 (2009)] to a nonmodal analysis of the governing non-self-adjoint operator and computations of the nonlinear dynamics. The nonmodal analysis identifies the most destabilizing perturbations to the film and their maximum amplification. Computations of the nonlinear dynamics reveal that small perturbations can be sufficient to destabilize a linearly stable film for a narrow band of wave numbers predicted by the nonmodal, linearized analysis. This destabilization is linked to the presence of stable, discrete modes that appear as the Marangoni parameter approaches the critical value at which the film becomes linearly unstable. Furthermore, the thermocapillary instability leads to a new, time-periodic base state. This transition corresponds to a Hopf bifurcation with increasing Marangoni parameter. A linear stability analysis of this time-periodic state reveals further instability to transverse perturbations, with the wave number of the most unstable mode about 50% smaller than for the rivulet instability of the steady base state and exponential growth rate about three times larger. The resulting film behavior is reminiscent of inertial waves on locally heated films, although the wave amplitude is larger in the present case near the heater and decays downstream where the Marangoni stress vanishes. The film’s heat transfer coefficient is found to increase significantly upon the transition to the time-periodic flow.

Midlatitude Baroclinic Rossby Waves in a High-Resolution OGCM Simulation

Abstract The effects of background baroclinic zonal flow and bottom pressure decoupling on midlatitude oceanic Rossby wave dynamics using a high-resolution OGCM simulation are investigated. To examine these effects, the phase speed and vertical structure of the simulated wave are compared with each of the different linear Rossby wave solutions obtained for two different circumstances (with or without background flow) and two different boundary conditions (a flat bottom or a bottom pressure decoupling condition). First, a frequency–wavenumber spectrum is examined for depth anomaly of the permanent thermocline (27.0σθ surface) along 32°S. Most of the energy is distributed along the theoretical dispersion curve including the effects of background flow and bottom pressure decoupling. The authors focus on a secondary dominant peak (appearing at a frequency greater than 1 cycle per year) at which the differences between the dispersion curves are large enough to discuss the relation between the spectral peak and the dispersion curves. The phase speed of this peak is nearly 1.5 times larger than that of the standard long-wave theory (flat bottom and no background flow), which is similar to results from previous observational studies. The extended long-wave theory including background flow and bottom pressure decoupling effects overestimates the phase speed. However, taking into account finite wavelength effects, this theory provides a phase speed much closer to that of the secondary dominant peak. The vertical structure corresponding to the wave of the secondary dominant peak extracted by composite analysis is intensified in the surface layer, a result similar to that from the theory including background flow and bottom pressure decoupling effects. The authors also compare the latitudinal distribution of midlatitude phase speed estimated by the frequency–wavenumber spectrum with theoretical results. The theory including background flow, bottom pressure decoupling, and finite wavelength effects reproduces the latitudinal distribution well, suggesting that these effects are important for explaining Rossby wave speed. The dominant factor enhancing the phase speed is bottom pressure decoupling related to rough bottom topography, while north of 30°N the background flow makes a strong contribution to the phase speed enhancement.

Weakly Nonlinear Stability Analysis of an Electro-Osmotic Thin Film Free Surface Flow

The flow of a thin incompressible fluid driven by an electro-osmotic force due to an applied electric field is analyzed. Linear stability of the film flow system is analyzed by solving the pertinent Orr-Sommerfeld eigenvalue problem. Applying long-wave theory, a nonlinear evolution equation for the local film thickness is obtained and the numerical investigation of the evolution equation in the supercritical stable region is carried to analyze the transient development of nonlinear waves incorporating various effects.

Long-Wave Instabilities in a Non-Newtonian Film on a Nonuniformly Heated Inclined Plane

A thin liquid layer of a non-Newtonian film falling down an inclined plane that is subjected to nonuniform heating has been considered. The temperature of the inclined plane is assumed to be linearly distributed and the case when the temperature gradient is positive or negative is investigated. The film flow is influenced by gravity, mean surface tension, and thermocapillary forces acting along the free surface. The coupling of thermocapillary instability and surface-wave instabilities is studied for two-dimensional disturbances. A nonlinear evolution equation is derived by applying the long-wave theory, and the equation governs the evolution of a power-law film flowing down a nonuniformly heated inclined plane. The linear stability analysis shows that the film flow system is stable when the plate temperature decreases in the downstream direction while it is less stable for increasing temperature along the plate. Weakly nonlinear stability analysis using the method of multiple scales has been investigated and this leads to a secular equation of the Ginzburg–Landau type. The analysis shows that both supercritical stability and subcritical instability are possible for the film flow system. The results indicate the existence of finite-amplitude waves, and the threshold amplitude and nonlinear speed of these waves are influenced by thermocapillarity. The nonlinear evolution equation for the film thickness is solved numerically in a periodic domain in the supercritical stable region, and the results show that the shape of the wave is influenced by the choice of wave number, non-Newtonian rheology, and nonuniform heating.

Identification of a system of oceanic waves based on space imagery

A system of circular surface oceanic waves that was registered on one of the images taken in accordance with the Uragan program of space experiment is identified. Within the framework of the hypothesis that the waves were generated as a result of an impulse effect on the oceanic surface, mathematical models that describe excitation and propagation of waves in the conditions of long-wave theory, as well as in liquid of infinite and finite depths were involved to image analysis. As a result of calculations, the simulated wave system was made close to the one registered in the image. This confirmed the hypothesis and allowed us to estimate the physical parameters of the initial impulse perturbation.

Unsteady thin-film flow over a heated stretching sheet

The unsteady flow in a thin viscous liquid film over a heated horizontal stretching surface are analyzed considering the stretching velocity and the temperature distribution in their general forms. An evolution equation for the film thickness, that retains the convective heat transport effects, is derived using long-wave theory of thin liquid film and is solved numerically for some representative values of non-dimensional parameters. It is observed that the thermocapillary effects are responsible in shaping the film thickness. Further the thermocapillary effects are more pronounced for lower values of Prandtl number and Biot number.

Trapped modes in 3D topographically varying plates

Trapped modes in 3D elastic plates are considered as a model of waves that are guided along, and localized to the vicinity of, welds. These waves propagate unattenuated along the weld and exponentially decay with distance transverse to it. In the direction of propagation (y), there is no change in geometry and we assume that waves have the form exp(iβy). An asymptotic long-wave theory provides numerical values of the trapped mode frequencies and gives conditions at which trapping can occur; these depend on the components of the wave number in different directions and variations of the plate thickness. The results of this long-wave theory are compared with a numerical solution of the full governing equations.

Nonlinear radial oscillations of a fluid in a parabolic basin with regard for the external action

Within the framework of the theory of long waves, we find a class of exact analytic solutions of the problem of description of nonlinear axially symmetric oscillations of a fluid in a parabolic basin with regard for the action of stationary radial bulk forces. The radial projection of the velocity of these oscillations (seiches) is a linear function of the radial coordinate, whereas the azimuthal velocity and displacements of the free surface of the fluid are polynomials in the radial coordinate with time-dependent coefficients. The method of finding solutions is based on the exact replacement of the original problem by a system of ordinary differential and algebraic equations. The action of the bulk forces may result either in the increase in the frequency of oscillations of the fluid and in the decrease in this frequency and affect the motion of the water edge, the characteristics of waves, and the velocity field.

Effect of Longitudinal Minigrooves on Flow Stability and Wave Characteristics of Falling Liquid Films

Falling liquid films are used in many industrial apparatuses. In many cases the film flow along a wall with topography is considered advantageous for intensification of the heat and mass transport. One of the promising types of the wall topography for the heat transfer intensification is comprised of minigrooves aligned along the main flow direction. The wall topography affects the development of wavy patterns on the liquid-gas interface. Linear stability analysis of the falling film flow based on the long-wave theory predicts that longitudinal grooves lead to the decrease in the disturbance growth rate and therefore stabilize the film. The linear stability analysis also predicts that the frequency of the fastest growing disturbance mode and the wave propagation velocity decrease on a wall with longitudinal minigrooves in comparison with a smooth wall. In the present work the effect of the longitudinal minigrooves on the falling film flow is studied experimentally. We use the shadow method and the confocal chromatic sensoring technique to study the wavy structure of falling films on smooth walls and on walls with longitudinal minigrooves. The measured film thickness profiles are used to quantify the effect of the wall topography on wave characteristics. The experimental results confirm the theoretical predictions.

Fission of a weakly nonlinear interfacial solitary wave at a step

The transformation of a weakly nonlinear interfacial solitary wave in an ideal two-layer flow over a step is studied. In the vicinity of the step the wave transformation is described in the framework of the linear theory of long interfacial waves, and the coefficients of wave reflection and transmission are calculated. A strong transformation arises for propagation into shallower water, but a weak transformation for propagation into deeper water. Far from the step, the wave dynamics is described by the Korteweg-de Vries equation which is fully integrable. In the vicinity of the step, the reflected and transmitted waves have soliton-like shapes, but their parameters do not satisfy the steady-state soliton solutions. Using the inverse scattering technique it is shown that the reflected wave evolves into a single soliton and dispersing radiation if the wave propagates from deep to shallow water, and only dispersing radiation if the wave propagates from shallow to deep water. The dynamics of the transmitted wave is more complicated. In particular, if the coefficient of the nonlinear quadratic term in the Korteweg-de Vries equation is not changed in sign in the region after the step, the transmitted wave evolves into a group of solitons and radiation, a process similar to soliton fission for surface gravity waves at a step. But if the coefficient of the nonlinear term changes sign, the soliton is destroyed completely and transforms into radiation. The effects of cubic nonlinearity are studied in the framework of the extended Korteweg-de Vries (Gardner) equation which is also integrable. The higher-order nonlinear effects influence the amplitudes of the generated solitons if the amplitude of the transformed wave is comparable with the thickness of lower layer, but otherwise the process of soliton fission is qualitatively the same as in the framework of the Korteweg-de Vries equation.

Thin Newtonian film flow down a porous inclined plane: Stability analysis

The flow of a thin Newtonian fluid layer on a porous inclined plane is considered. Applying the long-wave theory, a nonlinear evolution equation for the thickness of the film is obtained. It is assumed that the flow through the porous medium is governed by Darcy’s law. The critical conditions for the onset of instability of a fluid layer flowing down an inclined porous wall, when the characteristic length scale of the pore space is much smaller than the depth of the fluid layer above, are obtained. The results of the linear stability analysis reveal that the film flow system on a porous inclined plane is more unstable than that on a rigid inclined plane and that increasing the permeability of the porous medium enhances the destabilizing effect. A weakly nonlinear stability analysis by the method of multiple scales shows that there is a range of wave numbers with a supercritical bifurcation, and a range of larger wave numbers with a subcritical bifurcation. Numerical solution of the evolution equation in a periodic domain indicates the existence of permanent finite-amplitude waves of different kinds in the supercritical stable region. The long-time waveforms are either time-independent waves of permanent form that propagate or time-dependent modes that oscillate slightly in the amplitude. The presence of the porous substrate promotes this oscillatory behavior. The results show that the shape and amplitude of the waves are influenced by the permeability of the porous medium.