arXiv subjects

Vladimir A. Vladimirov

Publications and source records attributed to Vladimir A. Vladimirov.

15 recordsLinked to original sources

arXiv2026

Continuity equations in the Generalised Lagrangian Mean theory

Generalised Lagrangian Mean (GLM) theory describes the joint evolution of the mean flow and its perturbations. This paper analyses the exact GLM continuity equations (GLM-CEs), presents them in new forms, and adds new equations to complete them. Our three-dimensional results include: (i) A modified approach to GLM theory that uses only the basic notions of classical fluid dynamics. The only tools used are Lagrangian X, Eulerian x, averaged (mean) Eulerian x' coordinates of fluid particles, and an averaging parameter a. The targeted forms of GLM-CEs are expressed with functions x(x',t, a) and r(x',t, a), where r is the fluid density. (ii) Actual velocity divergence div u expressed through div'u', where u and u' are the actual and mean fluid velocities. (iii) Three new forms of GLM-CEs. Each version represents a sum of a mean and an a-dependent (tilde) part. The latter addition makes each form mathematically complete. (iv) The original Andrews-McIntyre transformation (AMT) is also complemented by its tilde part. Then it also gives the full version of the exact GLM-CEs. However, it works only for a special class of fluid flows. Its structure explains why the tilde equations have been overlooked. (v) Our generalisation of AMT, which repeats the results (i)-(iii) but requires a more complex derivation. (vi) Examples of flows with small but finite Lagrangian perturbations, linking our exposition to the classical GLM theory. Keywords: continuity equations, general Lagrangian mean theory, average (mean) coordinates, Eulerian description, tilde equations, Andrews-McIntyre transformation.

physics.flu-dyn↗

arXiv2026

Mikhail Alekseevich Lavrentiev and Fluid Dynamics. On the 125th anniversary of his birth

The purpose of the present paper is to advertise and illuminate the applied research of Mikhail Alekseyevich Lavrentiev, a distinguished scientist who generated new ideas and obtained many outstanding results in mechanics and mathematics. He also initiated and maintained numerous flourishing activities in mechanics and applied mathematics in the former Soviet Union. He was notably an outstanding researcher in pure and applied mathematics as well as in engineering. On the applied side, his research spanned fluid dynamics, explosion physics, mechanics, technology, and several branches of engineering. It was always aimed at the practical challenges of his time and achieved impressive successes. It has been fascinating to uncover all of this. Much of his biographical material and a review of his activities in pure mathematics have been published, but no review of his applied achievements is available. This paper aims to give Lavrentiev's applied research the recognition it deserves in the communities of fluid dynamics, applied mathematics, engineering, and the history of science.

math.HO↗

arXiv2021

Distinguished Limits and Drifts: between Nonuniqueness and Universality

This paper deals with a version of the two-timing method which describes various `slow' effects caused by externally imposed `fast' oscillations. Such small oscillations are often called \emph{vibrations} and the research area can be referred as \emph{vibrodynamics}. The governing equations represent a generic system of first-order ODEs containing a prescribed oscillating velocity u, given in a general form. Two basic small parameters stand in for the inverse frequency and the ratio of two time-scales; they appear in equations as regular perturbations. The proper connections between these parameters yield the \emph{distinguished limits}, leading to the existence of closed systems of asymptotic equations. The aim of this paper is twofold: (i) to clarify (or to demystify) the choices of a slow variable, and (ii) to give a coherent exposition which is accessible for practical users in applied mathematics, sciences and engineering. We focus our study on the usually hidden aspects of the two-timing method such as the \emph{uniqueness or multiplicity of distinguished limits} and \emph{universal structures of averaged equations}. The main result is the demonstration that there are two (and only two) different distinguished limits. The explicit instruction for practically solving ODEs for different classes of u is presented. The key roles of drift velocity and the qualitatively new appearance of the linearized equations are discussed. To illustrate the broadness of our approach, two examples from mathematical biology are shown.

math.DS↗

arXiv2018

A theory of flying/swimming saucers. Exact solutions for rectilinear locomotion

We study self-propulsion (or locomotion) of a robot (or an underwater vehicle) in an inviscid incompressible fluid. The robot's body is rigid, while its locomotion ability is due to an internal actuator, which can perform controlled translational and rotational oscillations. Our attention is focused on two classes of the plane analytic exact solutions, describing rectilinear locomotion. Solutions of the first class describe the \emph{tumbling locomotion}, while the second class corresponds to the \emph{zigzag locomotion} without tumbling. We show, that tumbling locomotion is more efficient. Both classes of solutions show, that the use of actuator allows to choose any desired direction and any speed of locomotion. As a special case, we consider the self-propulsion caused by small-amplitude and high-frequency actuator oscillations. The exact and elementary character of our solutions makes the results potentially useful as the tests to verify physical and engineering models, as well as numerical and asymptotic results. In contrary to many recent publications in this area, the material is accessible to UG students in Engineering, Physics, and Applied Mathematics.

cond-mat.soft↗

arXiv2016

Many Faces of Boussinesq Approximations

The \emph{equations of Boussinesq approximation} (EBA) for an incompressible and inhomogeneous in density fluid are analyzed from a viewpoint of the asymptotic theory. A systematic scaling shows that there is an infinite number of related asymptotic models. We have divided them into three classes: `poor', `reasonable' and `good' Boussinesq approximations. Each model can be characterized by two parameters $q$ and $k$, where $q =1, 2, 3, \dots$ and $k=0, \pm 1, \pm 2,\dots$. Parameter $q$ is related to the `quality' of approximation, while $k$ gives us an infinite set of possible scales of velocity, time, viscosity, \emph{etc.} Increasing $q$ improves the quality of a model, but narrows the limits of its applicability. Parameter $k$ allows us to vary the scales of time, velocity and viscosity and gives us the possibility to consider any initial and boundary conditions. In general, we discover and classify a rich variety of possibilities and restrictions, which are hidden behind the routine use of the Boussinesq approximation. The paper is devoted to the multiplicity of scalings and related restrictions. We do not study any particular solutions and particular failures of EBA.

physics.flu-dyn↗

arXiv2015

Homogenization Method, uncovered on 2 pages

This note gives a brief and `crash' introduction to the method of Homogenization with the use of wave equation and diffusion equation with periodic in space coefficients as instructive examples. We expose the method with the use of an approach, which appears in Vibrodynamics. The note can be interesting for people who want to use the Homogenization method immediately.

math-ph↗

arXiv2015

Hierarchy of Distinguished Limits and Drifts for Oscillating Flows

Lagrangian motions of fluid particles in a general velocity field oscillating in time are studied with the use of the two-timing method. Our aims are: (i) to calculate systematically the most general and practically usable asymptotic solutions; (ii) to check the limits of applicability of the two-timing method by calculating the averaged motion without making any assumptions; (iii) to classify various drift motions and find their limits of applicability; (iv) to introduce a logical order into the area under consideration; (v) to open the gate for application of the same ideas to the studying more complex systems. Our approach to study a drift is rather unusual: instead of solving the ODE for trajectories we consider a hyperbolic PDE for a scalar lagrangian field $a(\vx,t)$, trajectories represent the characteristics curves for this PDE. It leads us to purely eulerian description of lagrangian motion, that greatly simplifies the calculations. There are two small scaling parameters in the problem: a ratio of two time-scales and a dimensionless vibrational amplitude. It leads us to the sequence of problems for distinguished limits. We have considered four distinguished limits which differ from each other by the scale of slow time. We have shown that each distinguished limit produces an infinite number of solutions for $a(\vx,t)$. A classical drift appears in the case of a purely oscillating flow. At the same time, we have shown that the concept of drift motion is not sufficient for the description of lagrangian dynamics and some `diffusion' terms do appear. Five examples of different options of drifts and `diffusion' are given.

physics.flu-dyn↗

arXiv2013

An asymptotic model in acoustics: acoustic drift equations

A rigorous asymptotic procedure with the Mach number as a small parameter is used to derive the equations of mean flows which coexist and are affected by the background acoustic waves in the limit of very high Reynolds number.

physics.flu-dyn↗

arXiv2012

Dumbbell micro-robot driven by flow oscillations

In this paper we study the self-propulsion of a dumbbell micro-robot submerged in a viscous fluid. The micro-robot consists of two rigid spherical beads connected by a rod or a spring; the rod's/spring's length is changing periodically. The constant density of each sphere differs from the density of a fluid, while the whole micro-robot has neutral buoyancy. An effective oscillating gravity field is created via rigid-body oscillations of the fluid. Our calculations show that the micro-robot undertakes both translational and rotational motion. Using an asymptotic procedure containing a two-timing method and a distinguished limit, we obtain analytic expressions for the averaged self-propulsion velocity and averaged angular velocity. The important special case of zero angular velocity represents rectilinear self-propulsion with constant velocity.

physics.flu-dyn↗

arXiv2012

Theory of a triangular micro-robot

In this paper we study the self-propulsion of a triangular micro-robot (or triangle-robot) which consists of three spheres connected by three rods; the rods' lengths are changing independently and periodically. Using the asymptotic procedure containing the two-timing method and distinguished limit arguments, we obtain analytic expressions for self-propulsion velocity the angular velocity. Our calculations show that a triangle-robot rotates with constant angular velocity around its centroid, while the centroid moves in a circle. The important special case of zero angular velocity represents rectilinear translational self-propulsion with constant velocity.

physics.flu-dyn↗

arXiv2012

Self-propulsion of V-shape micro-robot

In this paper we study the self-propulsion of a symmetric V-shape micro-robot (or V-robot) which consists of three spheres connected by two arms with an angle between them; the arms' lengths and the angle are changing periodically. Using an asymptotic procedure containing two-timing method and a distinguished limit, we obtain analytic expressions for the self-propulsion velocity and Lighthill's efficiency. The calculations show that a version of V-robot, aligned perpendicularly to the direction of self-swimming, is both the fastest one and the most efficient one. We have also shown that such $V$-robot is faster and more efficient than a linear three-sphere micro-robot. At the same time the maximal self-propulsion velocity of V-robots is significantly smaller than that of comparable microorganisms.

physics.flu-dyn↗

arXiv2012

On self-propulsion of $N$-sphere micro-robot

The aim of this paper is to describe the self-propulsion of a micro-robot (or micro-swimmer) consisting of $N$ spheres moving along a fixed line. The spheres are linked to each other by arms with the lengths changing periodically. For the derivation, we use the asymptotic procedure containing the two-timing method and a distinguished limit. We show that in the main approximation, the self-propulsion velocity appears as a linear combination of velocities of all possible triplets of spheres. Velocities and efficiencies of three-, four-, and five-swimmers are calculated. The paper is devoted to H.K.Moffatt, who initiated the author's interests in low-Reynolds-number fluid dynamics.

physics.flu-dyn↗

arXiv2012

Acoustic-drift equation

The aim of this paper is to derive a new equation (the \emph{acoustic-drift equation} (ADE)) describing the generation of a flow by an acoustic wave. We consider acoustic waves of perfect barotropic gas as the zero-order solution and derive the equation for the averaged flow of the first order. The used small parameter of our asymptotic study is dimensionless inverse frequency, and the leading term for a velocity field is chosen to be a purely oscillating acoustic field. The employed mathematical approach combines the two-timing method and the notion of a distinguished limit. The properties of commutators are used to simplify calculations. The derived averaged equation is similar to the original vorticity equation, where the Reynolds stresses has been transformed to an additional advection with the drift velocity. Hence ADE can be seen as a compressible version of the Craik-Leibovich equation. In particular, ADE shows that if the averaged vorticity is absent in the initial data, then it will always be equal to zero. This property confirms that the acoustic streaming is of viscous nature. At the same time ADE can be useful for the description of secondary motions which take place for `strong' acoustic streaming flow.

physics.flu-dyn↗

arXiv2012

Self-propulsion velocity of N-sphere micro-robot

The aim of this paper is to derive an analytical expression for the self-propulsion velocity of a micro-swimmer that consists of N spheres moving along a fixed line. The spheres are linked to each other by the rods of the prescribed lengths changing periodically. For the derivation we use the asymptotic procedure containing the two-timing method and a distinguished limit. Our final formula shows that in the main approximation the self-propulsion velocity is determined by the interactions between all possible triplets of spheres.

physics.flu-dyn↗

arXiv2011

MHD-Drift Equations: from Langmuir circulations to MHD-dynamo?

We have derived the closed system of averaged MHD-equations for general oscillating flows, which are purely oscillating in the main approximation. We have used the mathematical approach which combines the two-timing method and the notion of the distinguished limit. Properties of the commutators are used to simplify calculations. The direct connection with a vortex dynamo (or the Langmuir circulations) has been demonstrated and a conjecture on the MHD-dynamo has been formulated.

physics.flu-dyn↗