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V. A. Vladimirov

Publications and source records attributed to V. A. Vladimirov.

15 recordsLinked to original sources
arXiv2026

Discover the GLM and pseudo-Lagrangian equations of fluid dynamics on four pages

The General Lagrangian Mean (GLM) theory uses a set of averaged equations of fluid dynamics to describe interactions between mean flows and waves. These equations are formulated in coordinates that follow the fluid's average velocity and are often referred to as `pseudo-Lagrangian' or `semi-Lagrangian'. This paper focuses on the principles for deriving the pseudo-Lagrangian and GLM equations, using an inviscid, incompressible, homogeneous fluid as a demonstration case. Our exposition differs methodically from that of others and is aimed at the learners of the subject. Keywords: fluid flows, pseudo-Lagrangian description, GLM theory, inviscid incompressible fluid, Lagrangian displacements, mean flows, waves, averaged equations.

physics.flu-dyn↗
arXiv2020

Distinguished Limits and Vibrogenic Force revealed by Newton's Equation with Oscillating Force

In this paper, we analyse the basic ideas of Vibrodynamics and the two-timing method. To make our analysis most instructive, we have chosen the Newton's equation with a general oscillating force. We deal with its asymptotic solutions in the high frequency limit. Our treatment is simple but general. The targets of our study are \emph{the distinguished limits} and \emph{the universal vibrogenic force}. The aim of \emph{the distinguished limit procedure} is to identify how the small parameter can appear in an equation. The proper appearance of a small parameter leads to \emph{valid successive approximations}, and, in particular, to closed systems of averaged equations. We show, that there are only two distinguished limits. This means that Newton's equation, with high-frequency forcing, has two types of interesting asymptotic solutions. The key item in the averaged equations for all distinguished limits is \emph{the unique vibrogenic force}. The current state-of-the-art in this area is: a large number of particular examples are well-known, effective and advanced general methods (like the Krylov-Bogolyubov approach) are well developed. However, the presented general and simple analysis of distinguished limits and the vibrogenic force, formulated as a compact practical guide, is novel. An advantage of our treatment is the possibility of its straightforward use for various ODEs and PDEs with oscillating coefficients.

physics.gen-ph↗
arXiv2019

Variations of Saffman's robot: two mechanisms of locomotion?

We study two mechanisms of locomotion of a body in an inviscid fluid, which take place without the shedding of vorticity; we consider two simple examples of robots which are able to move along a straight trajectory. The first one consists of a sphere with an internal moving material point (actuator); this illustrates \emph{the recoil locomotion}. The second robot represents a sphere moving along a thin rigid rod, this is aimed at illustrating \emph{the deformational locomotion}. The latter appears since this motion of a sphere can be seen as a `soliton' deformation, moving along the rod. The equations of motion for both robots are the same, while the intervals of variables are different. The first robot was introduced by \cite{Saffman}, who also wrote that he was unable to find any exact solution for the deformational locomotion: our paper fills this gap. Some previous papers emphasize that, in the cases of locomotion, the deformations must be not axisymmetric. We consider only axisymmetric examples, which expands the range of the involved possibilities. The simple construction of presented robots allows us to operate with the exact solutions only, which can play a `reference' role. Our aim is to analyse the main notions and terminology, used in this high-impact research area.

physics.flu-dyn↗
arXiv2019

Chiral transfer of angular momentum

Suppose that viscous fluid is contained in the space between a fixed sphere $S_2$ and an interior sphere $S_1$ which moves with time-periodic velocity ${\bf U}(t)$ and angular velocity ${\bf Ω}(t)$, with $ \left<{\bf U}(t)\right> = \left<{\bf Ω}(t)\right> = 0$. It is shown that, provided this motion is chiral in character, it can drive a flow that exerts a non-zero torque on $S_2$. Thus angular momentum can be transferred through this mechanism.

physics.flu-dyn↗
arXiv2018

Generalized symmetry reduction of nonlinear differential equations

We study the application of generalized symmetry for reducing nonlinear partial differential equations. We construct the ansatzes for dependent variable $u$ which reduce the scalar partial differential equation with two independent variables to systems of ordinary differential equations. The operators of Lie-Bäcklund symmetry of the second order ordinary differential equation are used. We apply the method to nonlinear evolutionary equations and find solutions which cannot be obtained in the framework of classical Lie approach. The method is also applicable to partial differential equations which are not restricted to evolution type ones. We construct the solution of nonlinear hyperbolic equation depending on an arbitrary smooth function on one variable. We study also the correlation between the dimension of symmetry Lie algebra and possibility of constructing non-invariant solutions to the equation under study.

math.AP↗
arXiv2016

Fluid Flows driven by Oscillating Body Force

In this note we consider general formulation of Euler's equations for an inviscid incompressible homogeneous fluid with an oscillating body force. Our aim is to derive the averaged equations for these flows with the help of two-timing method. Our main result is the general and simple form of the equation describing the averaged flows, which are derived without making any additional assumptions. The presented results can have many interesting applications.

physics.flu-dyn↗
arXiv2016

Advection equation analysed by two-timing method

The aim of this paper is to study and classify the multiplicity of distinguished limits and asymptotic solutions for the advection equation with a general oscillating velocity field with the systematic use of the two-timing method. Our results are: (i) the dimensionless advection equation contains two independent small parameters, which represent the ratio of two characteristic time-scales and the spatial amplitudes of oscillations; the scaling of the variables and parameters contains Strouhal number; (ii) an infinite sequence of distinguished limits has been identified; this sequence corresponds to the successive degenerations of a drift velocity; (iii) we have derived the averaged and oscillatory equations for the first four distinguished limits; derivations are performed up to the forth orders in small parameters; (v) we have shown, that each distinguish limit solution generates an infinite number of parametric solutions; these solutions differ from each other by the slow time-scale and the amplitude of prescribed velocity; (vi) we have discovered the inevitable appearance of pseudo-diffusion, which appears as a Lie derivative of the averaged tensor of quadratic displacements; we have clarified the meaning of pseudo-diffusion using a simple example; (vii) our main methodological result is the introduction of a logical order into the area and classification of an infinite number of asymptotic solutions; we hope that it can help to study the similar problems for more complex systems; (viii) since in our calculations we do not employ any additional assumptions, our study can be used as a test for the validity of the two-timing hypothesis; (ix) the averaged equations for five different types of oscillating velocity fields have been considered as the examples of different drifts and pseudo-diffusion.

physics.flu-dyn↗
arXiv2010

Admixture and Drift in Oscillating Fluid Flows

The motions of a passive scalar $\hat{a}$ in a general high-frequency oscillating flow are studied. Our aim is threefold: (i) to obtain different classes of general solutions; (ii) to identify, classify, and develop related asymptotic procedures; and (iii) to study the notion of drift motion and the limits of its applicability. The used mathematical approach combines a version of the two-timing method, the Eulerian averaging procedure, and several novel elements. Our main results are: (i) the scaling procedure produces two independent dimensionless scaling parameters: inverse frequency $1/ω$ and displacement amplitude $δ$; (ii) we propose the \emph{inspection procedure} that allows to find the natural functional forms of asymptotic solutions for $1/ω\to 0, δ\to 0$ and leads to the key notions of \emph{critical, subcritical, and supercritical asymptotic families} of solutions; (iii) we solve the asymptotic problems for an arbitrary given oscillating flow and any initial data for $\hat{a}$; (iv) these solutions show that there are at least three different drift velocities which correspond to different asymptotic paths on the plane $(1/ω,δ)$; each velocity has dimensionless magnitude ${O}(1)$; (v) the obtained solutions also show that the averaged motion of a scalar represents a pure drift for the zeroth and first approximations and a drift combined with \emph{pseudo-diffusion} for the second approximation; (vi) we have shown how the changing of a time-scale produces new classes of solutions; (vii) we develop the two-timing theories of a drift based on both the \emph{GLM}-theory and the dynamical systems approach; (viii) examples illustrating different options of drifts and pseudo-diffusion are presented.

physics.flu-dyn↗
arXiv2009

Rotating electrohydrodynamic flow in a suspended liquid film

The mathematical model of a rotating electrohydrodynamic flow in a thin suspended liquid film is proposed and studied. The motion is driven by the given difference of potentials in one direction and constant external electrical field $\vE_\text{out}$ in another direction in the plane of a film. To derive the model we employ the spatial averaging over the normal coordinate to a film that leads to the average Reynolds stress that is proportional to $|\vE_\text{out}|^3$. This stress generates tangential velocity in the vicinity of the edges of a film that, in turn, causes the rotational motion of a liquid. The proposed model is aimed to explain the experimental observations of the \emph{liquid film motor} (see arXiv:0805.0490v2).

physics.flu-dyn↗
arXiv2009

Modeling of zonal electrophoresis in plane channel of complex shape

The zonal electrophoresis in the channels of complex forms is considered mathematically with the use of computations. We show that for plane S-type rectangular channels stagnation regions can appear that cause the strong variations of the spatial distribution of an admixture. Besides, the shape of an admixture zone is strongly influenced by the effects of electromigration and by a convective mixing. Taking into account the zone spreading caused by electromigration, the influence of vertex points of cannel walls, and convection would explain the results of electrophoretic experiments, which are difficult to understand otherwise.

physics.flu-dyn↗
arXiv2009

Anomalous pH-gradient in Ampholyte Solution

A mathematical model describing a steady pH-gradient in the solution of ampholytes in water has been studied with the use of analytical, asymptotic, and numerical methods. We show that at the large values of an electric current a concentration distribution takes the form of a piecewise constant function that is drastically different from a classical Gaussian form. The correspondent pH-gradient takes a stepwise form, instead of being a linear function. A discovered anomalous pH-gradient can crucially affect the understanding of an isoelectric focusing process.

physics.chem-ph↗