Papers by Yury Stepanyants
Physics of Fluids, 2021
In this paper, we study the transformation of surface envelope solitons traveling over a bottom s... more In this paper, we study the transformation of surface envelope solitons traveling over a bottom step in water of a finite depth. Using the transformation coefficients earlier derived in the linear approximation, we find the parameters of transmitted pulses and subsequent evolution of the pulses in the course of propagation. Relying on the weakly nonlinear theory, the analytic formulas are derived which describe the maximum attainable wave amplitude in the neighborhood of the step and in the far zone. Solitary waves may be greatly amplified (within the weakly nonlinear theory formally, even without a limit) when propagating from relatively shallow water to the deeper domain due to the constructive interference between the newly emerging envelope solitons and the residual quasi-linear waves. The theoretical results are in good agreement with the data of direct numerical modeling of soliton transformation. In particular, more than double wave amplification is demonstrated in the perfor...
Applied Mathematical Modelling, Jul 1, 2002
Basic mathematical equations which describe the processes of sulphide oxidation and gas, heat and... more Basic mathematical equations which describe the processes of sulphide oxidation and gas, heat and water transport in waste rock dumps are presented and discussed. The governing equations account for gas and water flow, vapourisation and condensation with latent heat effects, heat transport and mass balance. Gas, water and solid phases are assumed to be in local thermal equilibrium at all times. Air is approximated as an ideal three-component gas. Different semi-empirical relationships between physical values are used: Darcy's law for fluid flow, ideal gas law, the Van Genuchten formula for the relationship between degree of water saturation and pressure head, Mualem's formula for the relative hydraulic conductivity as a function of pressure head, etc. Some important global quantities, such as the fraction of sulphide sulphur oxidised and the global oxidation rate, are defined and considered as functions of time.
Chaos, Jul 1, 1993
Advection of small particles with inertia in two-dimensional ideal flows is studied both numerica... more Advection of small particles with inertia in two-dimensional ideal flows is studied both numerically and analytically. It is assumed that the flow disturbance around the particle corresponds to a potential dipole, so that the motion is driven by pressure gradient, inertial, and added-mass forces. It is found that in general the motion is nonintegrable, but particular exact solutions can be obtained. The problem is then studied for the cases of axisymmetric flow, when the motion proves to be completely integrable, and of a cellular flow, for which both regular and stochastic (bounded and unbounded) trajectories are calculated. In the latter case, the unbounded stochastic motion is of Brownian-like character, and the results derived show that the particle dispersion process is generally anomalous.
Journal of Fluid Mechanics, Nov 24, 2021
In the linear approximation, we study a one-dimensional problem of the reflectionless wave propag... more In the linear approximation, we study a one-dimensional problem of the reflectionless wave propagation on a surface of a shallow duct with the spatially varying water depth, duct width, and current. We show that both global and bounded exact solutions describing reflectionless propagation in opposite directions of long waves of arbitrary shape exist for the particular variations of these parameters. A general analysis of the problem is illustrated by a few solutions constructed for the specific cases of spatial profiles of the flow velocity. The results obtained can be of interest to mitigate the possible impact of waves on ships, marine engineering constructions, and human activity in the coastal zones.
Wave Motion, Mar 1, 2018
The interactions of multi-lumps within the Kadomtsev-Petviashvili-1 (KP1) equation are studied an... more The interactions of multi-lumps within the Kadomtsev-Petviashvili-1 (KP1) equation are studied analytically and numerically. The dependence of stationary multi-lump structures on free parameters is discussed. The interactions of single lumps with each other and with a more complex objects such as bi-lumps, as well as the interactions of bi-lumps with each other are studied numerically. The results obtained are described and illustrated graphically (the videos are also available).
Nonlinear spectra of shallow water waves
HAL (Le Centre pour la Communication Scientifique Directe), 2013
The problem of spectra interpretation of nonlinear shallow water waves is studied in terms of int... more The problem of spectra interpretation of nonlinear shallow water waves is studied in terms of interacting Korteweg-de Vries (KdV) solitons and quasi-linear wavetrains. The method of data processing of random wave field is suggested and illustrated by an example. The soliton component obscured in the random wave field can be determined either on the basis of the inverse scattering method or by direct numerical solution of the KdV equation. The distribution function of number of solitons on amplitudes was constructed for the illustrative example. The relevance of the suggested approach to field measurements is discussed.
Generalized Korteweg–de Vries equation for internal waves in two-layer fluid
Zeitschrift für Angewandte Mathematik und Physik, May 31, 2008
A modified Lundgren's model (the LABSRL model) accounting for the effect of surface tension is ap... more A modified Lundgren's model (the LABSRL model) accounting for the effect of surface tension is applied for the description of stationary bathtub vortices in a viscous liquid with a free surface. Laminar liquid flow through the circular bottom orifice is considered in a horizontally unbounded domain with the liquid being assumed to be undisturbed at infinity and approaching to a constant depth. An approximate analytical solution of the LABSRL model is obtained for small-dent vortices. Good agreement is achieved between the constructed analytical and numerical solutions for the same set of parameters.
Journal of Fluid Mechanics, May 14, 2008
A modified Lundgren model is applied for the description of stationary bathtub vortices in a visc... more A modified Lundgren model is applied for the description of stationary bathtub vortices in a viscous liquid with a free surface. Laminar liquid flow through the circular bottom orifice is considered in the horizontally unbounded domain. The liquid is assumed to be undisturbed at infinity and its depth is taken to be constant. Three different drainage regimes are studied: (i) subcritical, where whirlpool dents are less than the fluid depth; (ii) critical, where the whirlpool tips touch the outlet orifice; and (iii) supercritical, where surface vortices entrain air into the intake pipe. Particular attention is paid to critical vortices; the condition for their existence is determined and analysed. The influence of surface tension on subcritical whirlpools is investigated. Comparison of results with known experimental data is discussed.
Nonlinear Systems and Complexity provides a place to systematically summarize recent developments... more Nonlinear Systems and Complexity provides a place to systematically summarize recent developments, applications, and overall advance in all aspects of nonlinearity, chaos, and complexity as part of the established research literature, beyond the novel and recent findings published in primary journals. The aims of the book series are to publish theories and techniques in nonlinear systems and complexity; stimulate more research interest on nonlinearity, synchronization, and complexity in nonlinear science; and fast-scatter the new knowledge to scientists, engineers, and students in the corresponding fields. Books in this series will focus on the recent developments, findings and progress on theories, principles, methodology, computational techniques in nonlinear systems and mathematics with engineering applications. The Series establishes highly relevant monographs on wide ranging topics covering fundamental advances and new applications in the field. Topical areas include, but are not limited to: Nonlinear dynamics Complexity, nonlinearity, and chaos Computational methods for nonlinear systems Stability, bifurcation, chaos and fractals in engineering Nonlinear chemical and biological phenomena Fractional dynamics and applications Discontinuity, synchronization and control.
Two analytical techniques for the generation of wide classes of exact solutions of the nonlinear ... more Two analytical techniques for the generation of wide classes of exact solutions of the nonlinear Schrödinger equation (NLSE) containing an external potential are proposed. Both methods are illustrated by a variety of localized solutions, including solitary optical vortices, for both the self-focusing and self-defocusing nonlinearities. The stability of solutions was tested by direct numerical simulations of the NLSE; the existence of stable localized modes was confirmed through the simulation.
Studies in Applied Mathematics, Dec 16, 2011
A new approach to the description of stationary plane waves in ideal density stratified incompres... more A new approach to the description of stationary plane waves in ideal density stratified incompressible fluid is considered without the application of Boussinesq approximation. The approach is based on the equation derived by Dubreil-Jacotin [4, 5, 6] and Long [7] with the additional assumption that the mean vorticity of the flow is zero. It is shown that in the linear approximation the spectrum of eigenmodes and dispersion equations corresponding to these eigenmodes can be found in the closed analytical forms for many particular relationships between the fluid density and stream function. Examples are presented for waves of infinitesimal amplitude. Exact expression for the velocity of solitary wave of any amplitude is derived.
Solitary wave instability in the positive-dispersion media described by the two-dimensional Boussinesq equations
Journal of Experimental and Theoretical Physics, Jul 1, 1994
Nanoparticle Dynamics in a Viscous Fluid at Small Reynolds Numbers
Proceedings of the 6th Australasian Congress on Applied Mechanics, Dec 1, 2010
Dynamics of small size solid particles in viscous density stratified fluid is studied analyticall... more Dynamics of small size solid particles in viscous density stratified fluid is studied analytically and numerically within the framework of the creeping flow approximation corresponding to very small Reynolds numbers. The equation of motion for a particle includes a consideration of the gravity/buoyancy force, Stokes drag force and the Bossinesq-Basset drag (BBD) force. The problem studied is applicable to many practical situations where particle motion may be experienced in viscous fluid of variable density. Exact analytical solutions describing particle motion are obtained both for the buoyant and heavy particles. It is shown that for a non-stationary motion of a particle, the consideration of the BBD force is principally important, resulting in much slower decay of the particle velocity compared to the case when only the Stokes drag force is accounted. A heavy particle motion in piece-homogeneous multilayered fluid and in smoothly stratified fluid is also studied. The results obtained are relevant in particular to the physical processes occurring in the cooling systems of nuclear reactors such as the Open Pool Australian Light-water research reactor OPAL at Lucas Heights, Sydney.
Communications in Nonlinear Science and Numerical Simulation, Apr 1, 2020
We study weakly nonlinear wave perturbations propagating in a cold nonrelativistic and magnetized... more We study weakly nonlinear wave perturbations propagating in a cold nonrelativistic and magnetized ideal quark-gluon plasma. We show that such perturbations can be described by the Ostrovsky equation. The derivation of this equation is presented for the baryon density perturbations. Then we show that the generalized nonlinear Schrödinger (NLS) equation can be derived from the Ostrovsky equation for the description of quasi-harmonic wave trains. This equation is modulationally stable for the wave number k < k m and unstable for k > k m , where k m is the wave number where the group velocity has a maximum. We study numerically the dynamics of initial wave packets with the different carrier wave numbers and demonstrate that depending on the initial parameters they can evolve either into the NLS envelope solitons or into dispersive wave trains.
Radiophysics and Quantum Electronics, Oct 1, 1987
The structure of steady-state two-dimensional solutions of the soliton type with quadratic and cu... more The structure of steady-state two-dimensional solutions of the soliton type with quadratic and cubic nonlinearities and power-law dispersion is analyzed numerically. It is shown that steadily coupled two-dimensional multisolitons can exist for positive dispersion in a broad class of equations, which generalize the Kadomtsev-Petviashvili equation. i. Kadomtsev and Petviashvili [i] have derived an equation that generalizes the wellknown Korteweg-de Vries equation and describes quasiplanar disturbances in a quadratically nonlinear medium with weak dispersion. The basic approximation used in [i] is the assumption that the scale of the wave field in the direction of motion is much smaller than the scale in the transverse direction. Clearly, the same approximation can also be used to describe disturbances in other media having different types of nonlinearity and dispersion (see, e.g.,
On the connections between solutions of one-dimensional and quasi-one-dimensional evolution equations
Russian Mathematical Surveys, Feb 28, 1989
Journal of Fluid Mechanics, Mar 24, 2022
We show that in the linear approximation there are three classes of reflectionless wave propagati... more We show that in the linear approximation there are three classes of reflectionless wave propagation on a surface of shallow water in the channel with spatially varying depth, width and current speed. Two of these classes have been described in our previous paper (Churilov & Stepanyants, J. Fluid Mech., vol. 931, 2022, A15), and the third one was discovered recently and is described here. The general analysis of the problem shows that, within the approach used in both of our papers, these three classes apparently exhaust all possible cases of exact solutions of the problem considered. We show that the reflectionless flow can be global for certain conditions, i.e. it can exist on the entire x-axis. There are also reflectionless flows which exist only on limited intervals of the x-axis.
Mathematical Modelling of Natural Phenomena, 2014
The problem of transformation of quasimonochromatic wavetrains of surface gravity waves with narr... more The problem of transformation of quasimonochromatic wavetrains of surface gravity waves with narrow spatial-temporal spectra on the bottom shelf is considered in the linear approximation. By means of numerical modeling, the transmission and reflection coefficients are determined as functions of the depth ratio and wave number (frequency) of an incident wave. The approximation formulae are proposed for the coefficients of wave transformation. The characteristic features of these formulae are analyzed. It is shown that the numerical results agree quite satisfactorily with the proposed approximation formulae.
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Papers by Yury Stepanyants