Reseach Topics

Fluid Dynamics is a well established and exciting area of research. As an academic subject it has its very deep fundamental (mathematical and physical) core and enormous number of applications to a wide range of natural phenomena ranging from swimming of microorganisms to airplane design, from weather forecast to astrophysics. The main current research topics of the Fluid Dynamics Group are

- Asymptotic and Averaging methods;
- Vibrodynamics;
- Hydrodynamic stability theory;
- Solutions of Navier-Stokes equations and Stokes equations;
- Microhydrodynamics;
- Biological Fluid Dynamics;
- Magnetohydrodynamics (MHD).

Please see the homepages of members of the group for more details.

[Under development]

Current research of the Group is focused mostly on *Vibrodynamics*.

Vibrodynamics studies the behaviour of any system (mechanical, physical, biological, etc.), which undergoes vibrations or oscillations and allows description in terms of slow changing (averaged) parameters and oscillating parameters. Our current research in Vibrodynamics includes the following topics:

- Developing of the Vishik-Lyusternik asymptotic method;
- Developing of the Two-Timing asymptotic method;
- Developing of Averaging methods;
- Oscillating fluid flows (both viscous and inviscid);
- Vibrations/oscillations of complex systems (solid+fuid systems, MHD flows with alternating electromagnetic fields,
*etc.*); - Analytical Mechanics with the presence of vibrations.

What is Vibrodynamics?

Researchers in physics and mechanics know how useful, when studying any physical or mechanical system, to raise a question: what happens if we make it to vibrate? This question proved its productive and inspiring power in the long and impressive serious of discoveries, achieved in attempts to find the answer.

Michael Faraday (1830) put a glass of water on a vertically vibrating table. It led him to the renowned discoveries of the parametric resonance and the parametric excitation of surface waves.

Andrew Stephenson (1908) and Petr Kapitza (1951) applied the high frequency vibrations to the pivot of a pendulum. It led to the discovery of the dynamic stabilization of the `up side down’ pendulum. This result has been generalized to a broad class of systems. For instance, a similar idea was realized for the levitating of the charged particles in an oscillating electric field by Wolfgang Paul (1958) and Michail Miller (1958) (and was honoured with the Nobel Prize in Physics in 1989).

V.N.Chelomey (1956) considered a high frequency longitudinal force applied to a beam. He predicted the increasing of the effective stiffness of the beam (it is not well known in the West, that Chelomey was also a renown General Constructor of Russian missiles). This theoretical idea was tested in the laboratory experiments by David Acheson and Tom Mullin (1993).

Igor Simonenko and Svetlana Zenkovskaya (1966) raised a question what happens if a container with an inhomogeneous fluid (for instance, a pot of water heated from below) undergoes vibrations. The result was the discovery that the thermal convection can be controlled by vibrations. In particular, vertical vibrations can completely suppress all convective motions. Now the vibrational convection represents a wide and attractive research field with applications to the space technology and to microgravity phenomena.

John Littlewood and Dame Mary Cartwright (1957) considered the well known Van-der-Pol equations and analyzed their solutions for the periodically modulated coefficients. In physical terms it means that they considered a lamp generator with periodically modulated parameters. They found specific chaotic regimes in this system. It led to the discovery of a fundamental object of the modern dynamical system and chaos theory (the Smale horseshoe).

It should be emphasized, that from the mathematical viewpoint all listed effects and results are specific for the modulated systems and are impossible if equations do not contain oscillating factors (for the autonomous system). Theoretical developments of systems of equation containing oscillating factors form an active and important research area in mathematics. Simultaneously with the mathematical and theoretical developments, a very active process of the creation of various vibrational technical devices takes place. It is very important to emphasize here, that the attitude of engineers towards the vibrations has been under the drastic change: the old approach of treating vibrations as only a ‘bad’ and ‘harmful’ factor which should be avoided or suppressed has been changed to the productive use of vibrations (vibrational drilling and cutting, vibrational transport, etc.). It is true that the problem of suppressing of the already existing harmful vibrations is always urgent. However, we suppose that now it is timely to start the solving the problem how to use the existing (unavoidable) vibrations, like we use the energy of wind, rivers, and radiation of the Sun.

Studies of the vibrational effects in various (physical, mechanical, geophysical, biological, medical, technical, and economical) systems continue to bring new and impressive discoveries. These studies can be united under the name of *Vibrodynamics*. This term was coined in our group (V.A.Vladimirov, V.I.Yudovich, K.I.Ilin).

Vibrodynamics represents a really broad and interdisciplinary research area, since of any physical, mechanical, *etc.* system (after the exposing it to vibrations) becomes its subject. It is already well recognized that the organizing of the interdisciplinary efforts is the key for success in Vibrodynamics. This resulted for instance in the creation of the Laboratory for Vibrational Mechanics in St-Petersburg (Russia), and the Centre of Vibro-Impact Systems in Loughborough (UK).

Significant research in the theoretical Vibrodynamics has been undertaken at the Department of Mathematics, University of York under the leadership of Prof. V.A.Vladimirov, Prof V.I.Yudovich, and Dr.K.I.Ilin. During last few years the participants produced a number of sound results, which drastically simplified the theory in this area. A number of interesting problems, which were not accessible before, become solvable. Naturally, simplification of the theory brings a great deal of its generalization. What is even more important, this breakthrough in theory (as well as some pilot laboratory experiments) brought us the confidence that this area is very promising for the obtaining of interesting theoretical, experimental, and engineering results. This success allowed us to choose the most encouraging directions (we do understand now, what targets we should hit and able to hit!). As the result of all these activities Vibrodynamics has been identified among the best research direction to concentrate the efforts of specialists in mechanics, mathematics, physics, chemistry anf many other areas of Sciences and Engineering.