• Methods of Theoretical Physics in Continuous Media Mechanics
Interns will work on four projects with the staff of the Theoretical Physics Department. The projects are “Introduction to Perturbation Techniques”;“Instabilities in Hydrodynamics and Methods of Analysis”; “New Method of Analytical Solution of Nonlinear Second Order Differential Equations”;“The Mass Transport in Adsorptive Medium”.
Introduction to perturbation techniques
Perturbation techniques are a powerful tool for dealing with dynamics of nonlinear systems, both null-dimensional and distributed. It allows constructing a comprehensive picture of the possible dynamic regimes for the systems which cannot be solved analytically. We will consider techniques ranging from the simplest ones to the multiscale method in application to example physical problems. For auxiliary reading one can use the book "Introduction to perturbation techniques" by A.H. Nayfeh.
The new method of analytical solution of nonlinear second order differential equations
Many problems in modern physics can be formulated as nonlinear ordinary differential equations. The general method to find analytical solution of such equations is Lie group analysis (described in Handbook of Lie Group Analysis of Differential Equations by N.H. Ibragimov). This method is more easy to apply and the most effective for the solution of nonlinear second order equations, which will be demonstrated in our training.
The mass transport in adsorptive medium
The transport of solutes in porous media (e.g. soil, sandstone, fractured rocks, some filters etc.) in most important situations cannot be described within the classical Diffusion-Advection Theory (the Fick's law). The solute can deposit to solid matrix of media and detach from it. These features change the transport law and its description. Within our training we shall obtain the actual law for porous media and apply it for some simple problems.
Instabilities in Hydrodynamics and Methods of Analysis
First in the frames of the training course it is planned to derive the system of hydrodynamics equations (briefly). Then there will be the familiarization with the theory of hydrodynamics stability. The third step is to solve any simple problem of convective stability in a field of magnetohydrodynamics. The material on this topic hasmany applications and it can be important in a future for the students who are specialized in different branches of fundamental physics, for example, in astrophysics.