Konstantin Ilin

The stability of steady magnetohydrodynamic flows with current-vortex sheets

The stability of steady magnetohydrodynamic flows of an inviscid incompressible fluid with current-vortex sheets to small three-dimensional perturbations is studied. The energy method of Frieman and Rotenberg is extended to the case of steady flows with surfaces of tangential discontinuities across which the tangent velocity or the tangent magnetic field or both of them have jump discontinuities. Sufficient conditions for linear stability of some classes of steady flows with parallel velocity and magnetic field are obtained. Also, a sufficient condition for instability of a tubular current-vortex sheet is given.

Energy principle for magnetohydrodynamic flows and Bogoyavlenskij's transformation

The stability of steady magnetohydrodynamic flows of an inviscid incompressible fluid is studied using the energy method. It is shown that certain symmetry transformations of steady solutions of the equations of ideal magnetohydrodynamics have an important property: if a given steady magnetohydrodynamic flow is stable by the energy method, then certain infinite-dimensional families of steady flows obtained from the given flow by these transformations are also stable. This result is used to obtain new sufficient conditions for linear stability. In particular, it is shown that certain classes of steady magnetohydrodynamic flows in which both the magnetic field and the velocity depend on all three spatial coordinates are stable.

Instability of an inviscid flow between porous cylinders with radial flow

The stability of a two-dimensional viscous flow between two rotating porous cylinders is studied. The basic steady flow is the most general rotationally-invariant solution of the Navier-Stokes equations in which the velocity has both radial and azimuthal components, and the azimuthal velocity profile depends on the Reynolds number. It is shown that for a wide range of the parameters of the problem, the basic flow is unstable to small two-dimensional perturbations. Neutral curves in the space of parameters of the problem are computed. Calculations show that the stability properties of this flow are determined by the azimuthal velocity at the inner cylinder when the direction of the radial flow is from the inner cylinder to the outer one and by the azimuthal velocity at the outer cylinder when the direction of the radial flow is reversed. This work is a continuation of our previous study of an inviscid instability in flows between rotating porous cylinders (see \cite{IM2013}).

Instability of a two-dimensional viscous flow in an annulus with permeable walls to two-dimensional perturbations

The stability of a two-dimensional viscous flow in an annulus with permeable walls with respect to small two-dimensional perturbations is studied. The basic steady flow is the most general rotationally invariant solution of the Navier-Stokes equations in which the velocity has both radial and azimuthal components, and the azimuthal velocity profile depends on the radial Reynolds number. It is shown that for a wide range of parameters of the problem, the basic flow is unstable to small two-dimensional perturbations. Neutral curves in the space of parameters of the problem are computed. Calculations show that the stability properties of this flow are determined by the azimuthal velocity at the inner cylinder when the direction of the radial flow is from the inner cylinder to the outer one and by the azimuthal velocity at the outer cylinder when the direction of the radial flow is reversed. This work is a continuation of our previous study of an inviscid instability in flows between rotating porous cylinders [K. Ilin and A. Morgulis, “Instability of an inviscid flow between porous cylinders with radial flow,” J. Fluid Mech. 730, 364–378 (2013)].

On the stability of Alfven discontinuity

The stability of Alfven discontinuities for the equations of ideal compressible magneto-hydrodynamics (MHD) is studied. The Alfven discontinuity is a characteristic discontinuity for the hyperbolic system of MHD equations but, as in the case of shock waves, there is a mass flux through its front. The Lopatinskii condition for a planar Alfven discontinuity is tested numerically, and the domain in the space of parameters of the discontinuity where it is unstable is determined. In fact, in this domain the Alfven discontinuity is not only unstable, but the initial-boundary-value problem for corresponding linearized equations is ill-posed in the sense of Hadamard.

ON THE STEADY STREAMING INDUCED BY VIBRATING WALLS

We consider incompressible viscous flows between two transversely vibrating solid walls and construct an asymptotic expansion of solutions of the Navier--Stokes equations in the limit when both the amplitude of the vibration and the thickness of the oscillatory boundary layers (the Stokes layers) are small and have the same order of magnitude. Our asymptotic expansion is valid up to the flow boundary. In particular, we derive equations and boundary conditions for the averaged flow. At leading order, the averaged flow is described by the stationary Navier--Stokes equations with an additional term which contains the leading-order Stokes drift velocity. The same equations had been derived by Craik and Leibovich in 1976 when they proposed their model of Langmuir circulations in the ocean [A. D. D. Craik and S. Leibovich, J. Fluid Mech., 73 (1976), pp. 401--426]. In the context of flows induced by an oscillating conservative body force, these equations had also been derived by Riley [Ann. Rev. Fluid Mech., 33 ...

Dynamics of a rolling robot

Significance The dynamics of a rolling ball activated by an internal battery mechanism are analyzed by theoretical and numerical techniques. The problem involves four independent dimensionless parameters and is governed by a six-dimensional nonholonomic nonautonomous dynamical system with cubic nonlinearity. It can serve as a prototype for rolling bodies activated by any internal mechanism and is relevant to robotic systems for which such an internal mechanism may be subject to remote control. This is believed to be a unique problem of its kind to have been solved by appeal to fundamental principles of classical dynamics. For this reason, it should be accessible to a wide readership. The numerical results provide clear evidence of both regular and chaotic behavior. Equations describing the rolling of a spherical ball on a horizontal surface are obtained, the motion being activated by an internal rotor driven by a battery mechanism. The rotor is modeled as a point mass mounted inside a spherical shell and caused to move in a prescribed circular orbit relative to the shell. The system is described in terms of four independent dimensionless parameters. The equations governing the angular momentum of the ball relative to the point of contact with the plane constitute a six-dimensional, nonholonomic, nonautonomous dynamical system with cubic nonlinearity. This system is decoupled from a subsidiary system that describes the trajectories of the center of the ball. Numerical integration of these equations for prescribed values of the parameters and initial conditions reveals a tendency toward chaotic behavior as the radius of the circular orbit of the point mass increases (other parameters being held constant). It is further shown that there is a range of values of the initial angular velocity of the shell for which chaotic trajectories are realized while contact between the shell and the plane is maintained. The predicted behavior has been observed in our experiments.

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.

Shallow-water models for a vibrating fluid

We consider a layer of an inviscid fluid with free surface which is subject to vertical high-frequency vibrations. We derive three asymptotic systems of equations that describe slowly evolving (in comparison with the vibration frequency) free-surface waves. The first set of equations is obtained without assuming that the waves are long. These equations are as difficult to solve as the exact equations for irrotational water waves in a non-vibrating fluid. The other two models describe long waves. These models are obtained under two different assumptions about the amplitude of the vibration. Surprisingly, the governing equations have exactly the same form in both cases (up to interpretation of some constants). These equations reduce to the standard dispersionless shallow-water equations if the vibration is absent, and the vibration manifests itself via an additional term which makes the equations dispersive and, for small-amplitude waves, is similar to the term that would appear if surface tension were taken into account. We show that our dispersive shallow water equations have both solitary and periodic travelling wave solutions and discuss an analogy between these solutions and travelling capillary-gravity waves in a non-vibrating fluid.

Steady streaming in a channel with permeable walls

We study steady streaming in a channel between two parallel permeable walls induced by oscillating (in time) blowing/suction at the walls. We obtain an asymptotic expansion of the solution of the Navier-Stokes equations in the limit when the amplitude of the normal displacements of fluid particles near the walls is much smaller that both the width of the channel and the thickness of the Stokes layer. It is demonstrated that the magnitude of the steady streaming is much bigger than the corresponding quantity in the case of the steady streaming produced by vibrations of impermeable boundaries.