The following report was presented at plenary Session of the Final Scientific Conference'2000 of the Kazan Science Center RAS on January 30, 2001:

Aganin A.A. Simulation of bubble dynamics and sonoluminescence.
The discovery of the phenomenon of single bubble sonoluminescence (SBSL) introduced an essential contribution into development of models of spherical bubble dynamics. This contribution can completely be compared with that introduced in to the area of bubble dynamics by invention of computers. In particular, some regimes of bubble oscillations which before the discovery of SBSL can be obtained by only theoretically and in reality of which nearly nobody believed because of very high temperatures, pressures and densities of the gas in the bubble reached in computations turned out to be relatively simply derivable experimentally. To explain those regimes new bubble dynamics models were put forward, which in detail describe local states of the medium both inside the bubble and in its environment. To do so, significant joint efforts of researchers from different fields of science (hydrodynamics, physics, chemistry) had been required.

The following reports were presented at morning Session (chaired by Director of IME KazSC RAS Dr. Phys. & Math. Sc. D.A.Gubaidullin) on January 31, 2001:

Gubaidullin D.A., Laptev C.A., Nikiforov A.A. Dynamics of weak waves of different geometry in polydispersed gas-vapor-drop media.
Propagation of linear waves in mono- and polydispersed gas-vapor-drop systems within the framework of dynamics of heterogeneous media was investigated. The plane, spherical and cylindrical cases were considered. The general dispersion equation independent of the distance from axis or the center of the symmetry was obtained. Dynamics of a weak pressure pulse of different initial form in polydispersed air fogs with the use of the fast Fourier transform algorithm was studied. The influence of geometry of the processe, phase exchange, polydispersed composition and the basic parameters was investigated. The significantly strong attenuation and change in the form of spherical and cylindrical perturbations in contrast to the plane case were found.

Aganin A.A. Compressibility of incompressible liquid.
In problems of fluid dynamics such flow conditions may occur that it becomes reasonable to use the model of incompressible liquid in one temporal interval (spatial area), the model of linear acoustics in the other interval (area) and the complete model of compressible fluid dynamics in one other interval (area). An issue appears of where, when and what model should be applied. This is in fact an issue of interrelation of these models. From this point of view most unclear position has the model of incompressible liquid. It is shown that this model can be considered as one approximation of the model of compressible fluid dynamics.

Gil'manov A.N., Guseva T.S. Supersonic gas flow deceleration in a 2d channel with heat transfer through the wall.
The phenomenon of deceleration of a supersonic gas flow in a flat channel with heat transfer through the wall is simulated based on the Euler equations. Their solution is found by a method combining TVD scheme of the increased order of approximation and dynamically adaptive grids. Results of numerical calculations have shown that the flow structure essentially depends on the degree as well as the character of the wall heating (cooling). It has been found that at a certain temperature distribution on the wall nearly continuous deceleration of the flow is possible.

Badertdinova Ye.R., Khairullin M.Kh. Determination of filtration properties of stratified reservoir by the method of regularization.
A numerical algorithm for determination of filtration properties of stratified reservoir composed by horizontal layers is proposed in this report.

Gabidullina A.N., Elesin A.V., Kadyirova A.Sh., Mazurov P.A. On identification of the hydraulic conductivity in case of three-dimensional anisotropic confined aquifer.
The constructing of the effective numerical methods for the estimation of aquifer parameters allows one to reduce and simplify the expensive experimental investigations. One of such methods is the solution of inverse problems with using the information about the measured head values in observation points. These inverse problems are ill-posed. A few iterative regularizational algorithms of quasi-Newton type has been constructed for identification of the hydraulic conductivity in case of three-dimensional anisotropic confined aquifer. The regularizational properties of the proposed algorithms are demonstrated on the solution of several ill-posed problems.

Zaripov R.G., Davydov R.I., Sonin N.V. Nonlinear oscillations of a gas in tubes.
Results of experimental investigations of nonlinear oscillations of a gas in tubes are presented. Forced longitudinal gas oscillations near linear and nonlinear resonances in a wide range of frequency and excitation amplitudes are studied. New nonlinear effects in an oscillating gas are revealed. They include subharmonic resonances near the frequency three time as less as the first eigen one, large-scaled vortex formations near the open end etc. Investigation of liquid atomizing processes in a wave field in the industrial ecology problems is of particular interest. The program of experimental investigations of nonlinear oscillations of aerosols in tubes is developed.

Tukmakov A.L. Nonlinear effects of gas oscillations in closed tubes and canals.
Gas oscillations in closed tubes and canals are numerically investigated on basis of viscous compressible termoconductive gas model. The gas motion is described by Navier-Stokes equations for laminar flows and Reynolds equations at turbulence regimes. Navier-Stokes or Reynolds systems was solved by the explicit MacCormack second-order difference splitting schemes with FCT correction procedure and algebraic turbulence model for Reynolds equations.

Zakirov U.N. Some scientific problems of mechanics in the twenty first century.
The underground (underplanetary) hydromechanics of the Sun Sistem, including Mars, aerosol's mechanics of the planet's atmosphere, mechanics of shells long time staying cosmic spase, nonlinear mechanics of satellites with connections, relativity mechanics of continuous mediumin of variable masses should be some scientific problems of mechanics in the 21-st century.

The following reports were presented at evening Session (chaired by Deputy Director of IME KazSC RAS Dr. Phys. & Math. Sc. R.G.Zaripov) on January 31, 2001:

Yakupov N.M. Laboratory of a nonlinear mechanics of shells during last ten years.
This report presents development of spline version of FEM for calculation of shells of complicated geometry: an estimation of convergence, technique of local refinement of a grid and prediction of durability of corrosive wear, ratio for matching intersected shells of composite geometry, algorithm of calculation of shells of mean thickness, registration of a non-uniformity of pattern of a material. Development of spline version FEM for calculation of three-dimensional objects. Creation of experimental - theoretical method for testing of polymer films. The solution of a number of particular applied problems.

Yakupov N.M., Galimov N.K., Galimov Sh.K., Leontiev A.A., Nurgaliev A.R. Experimental - theoretical method for testing of strength of polymer films.
The experimental-theoretical method for testing of strength of polymer films is designed. The specimen as a round diaphragm is restricted between a matrix and punch of the special plant through a hydraulic press. The specimen is swelt under loading by air. Arresting given pressure and hoisting height of a diaphragm in its center and using ratio of a non-linear shell theory and the plasticities, curves of deforming are plotted.

Bocharov N.V. Transient processes in problems of shock loading of elastic structural components.
In the work numerical models of temporal forms of outside force pulse are investigated, described by impulse functions with discontinuity of the first kind. Mathematical conditions guaranteeing the best convergence to the solution of the problem of shock loading of elastic structural components are determined for those models. Convergence is achieved in the method of expansion of complete dynamic solution by natural oscillation forms using separating quasistatic component method.

Fedyaev V.L. On the use of modelling methods for improvement of technological equipment.
In thermal power engineering, chemical, petrochemical and other branches of industry there, the heat-exchange equipment is widely used. The basic objects of investigations in the present work are shell-and-tube heat exchangers and graduation towers. The mathematical models describe all the processes taken place in their operation: deformation of elements, flows of liquid and gas, heating and heat exchange between media. Corresponding problems are solved with the help of analytical as well as numerical methods. Some original technical solutions are proposed based on the results of the investigations conducted.

Mazo A.B., Fedyaev V.L., Snigerev B.A. Mathematical modelling of filters for crude cleaning of circulating water of industrial plants.
Effective work of heat-exchange equipment is to a large extent dependent on the property of cooling water. Partial solution of this problem is that of crude cleaning of circulating water with some special filters. A mathematical model of computation of fluid flows and particle transfer in filters is proposed in this work.

Mazo A.B., Alimov M.M. (KSU). Modelling of separating flows with stationary vortex zones.

Yakupov N.M.., Hisamov R.Z. Allowing for non-uniformity of material in calculation of elements of constructions.
A spline version of the finite element method for calculation of stress - strained state of three-dimensional elements of constructions of complicated geometry is developed. The problem of parametrization of a curved area is solved. The solution in each element is presented in the form of Hermite's cubic spline of three variables. The fibers with different mechanical properties are taken into account using an assumption of a continuity of deformation inside each element. Test problems for one element with fibers permeating it are given.

Aksenov I.B. (KSTU of A.N.Tupolev's name), Tukmakov A.L. Fractal dimensionality of acoustic signal in tubes with roughness.
Samples of steel tubes with internal corrosion layer are experimentally investigated. Fractal nature of acoustic oscillations under flick shock excitation of the surface is found. The results show that Hausdorff dimensionality of acoustic signal is variating with thickness of corrosion layer and tends to integer number value for shooth-walled tubes.