Table of contents

The present volume of Journal of Physics: Conference Series represents contributions from participants of the Russian conference "Nonlinear waves: theory and new applications" held at the Technopark of Novosibirsk Akademgorodok, Russia, 29 February - 2 March 2016, organised by the Lavrentyev Institute of Hydrodynamics and the Novosibirsk State University.

The conference is longstanding and attracts the attention of mathematicians and physicists. This year the conference was dedicated to the 70^th anniversary of the birth of an outstanding scientist in the field of mechanics, Professor Vladimir M. Teshukov (2 March 1946 - 22 April 2008), who was a director of the Lavrentyev Institute of Hydrodynamics from 2004-2008.

Scientific activities of the conference included a discussion of a wide range of problems, mainly related to mathematical theory of nonlinear wave processes in heterogeneous fluids, construction of new models of multiphase fluids and development of analytical and numerical methods for solving problems of mechanics. The conference programme included oral presentations (20 minutes) given by more than 100 participants from 15 cities of Russia (including Moscow, Novosibirsk, Kazan, Vladivostok, Ufa, Yekaterinburg, Krasnoyarsk and Irkutsk), and France (Marseille, Toulouse). A significant part of the talks (about 30%) were presented by young researchers and postgraduate students.

We would like to thank the speakers for their significant contributions to the conference. We also would like to thank the members of the Organizing Committee. We cannot end without expressing our many thanks to our sponsors (Russian Foundation for Basic Research and the technology company Schlumberger) for their financial support.

All papers published in this volume of Journal of Physics: Conference Series have been peer reviewed through processes administered by the proceedings Editors. Reviews were conducted by expert referees to the professional and scientific standards expected of a proceedings journal published by IOP Publishing.

The system of one-dimensional shallow water equations above the rough bottom is considered. All its hydrodynamic conservation laws are found, and a group classification is performed. A new conservation law additional to the two basic conservation laws is found. It is shown that the system of shallow water equations can be linearized by a point change of variables only in cases of constant and linear bottom profiles.

Control problems for the 3D model of acoustic scattering which describes scattering acoustic waves by a permeable obstacle with the form of a spherical layer are considered. These problems arise while developing the design technologies of acoustic cloaking devices using the wave flow method. The solvability of direct and control problems for the acoustic scattering model under study is proved. The sufficient conditions which provide local uniqueness and stability of optimal solutions are established.

In the paper a model for description of a hydraulic fracture propagation in inhomogeneous poroelastic medium is proposed. Among advantages of the presented numerical algorithm, there are incorporation of the near-tip analysis into the general computational scheme, account for the rock failure criterion on the base of the cohesive zone model, possibility for analysis of fracture propagation in inhomogeneous reservoirs. The numerical convergence of the algorithm is verified and the agreement of our numerical results with known solutions is established. The influence of the inhomogeneity of the reservoir permeability to the fracture time evolution is also demonstrated.

The torsion problems for aluminum alloy rods with circular cross section are solved. The rods are cut out from a transversely-isotropic plate with reduced resistance to creep strain in the direction of 45° to the normal direction of plate. Approximate estimates and finite element analysis show that the warping of the cross-section occurs when the rods are cut out in the longitudinal direction. In this case the value of torsion angle exceeds more than three times the value of torsion angle calculated in solving the isotropic problem. The analytical solution and finite element analysis for the rods cut out in the normal direction of plate show that the warping of the cross-section is missing, but the value of torsion angle s greater by an order of the value of torsion angle calculated in solving of the isotropic problem.

This article considers method of describing the behaviour of hemodynamic parameters near vascular pathologies. We study the influence of arterial aneurysms and arteriovenous malformations on the vascular system. The proposed method involves using generalized model of Van der Pol-Duffing to find out the characteristic behaviour of blood flow parameters. These parameters are blood velocity and pressure in the vessel. The velocity and pressure are obtained during the neurosurgery measurements. It is noted that substituting velocity on the right side of the equation gives good pressure approximation. Thus, the model reproduces clinical data well enough. In regard to the right side of the equation, it means external impact on the system. The harmonic functions with various frequencies and amplitudes are substituted on the right side of the equation to investigate its properties. Besides, variation of the right side parameters provides additional information about pressure. Non-linear analogue of Nyquist diagrams is used to find out how the properties of solution depend on the parameter values. We have analysed 60 cases with aneurysms and 14 cases with arteriovenous malformations. It is shown that the diagrams are divided into classes. Also, the classes are replaced by another one in the definite order with increasing of the right side amplitude.

We present a new scheme for the Maxwell's equations computations in threedimensional domains, where size in one direction is much smaller than the other sizes. The scheme is based on the Langdon-Lasinski scheme, which is standard for numerical experiments in plasma physics. Our study is devoted to analysis of correct wave propagation due to the effects of using a finite-difference approximation. To show the main dependencies we present numerical results in one-dimensional case. The results demonstrate, that the new scheme maintains the wave amplitude, the propagation speed and allows using of bigger time step in comparison with the Langdon-Lasinski scheme.

An inverse coefficient problem is considered for a stationary nonlinear convection- diffusion-reaction equation, in which reaction coefficient has a rather common dependence on substance concentration and on spacial variable. The solvability of the considered nonlinear boundary value problem is proved in a general case. The existence of solutions of the inverse problem is proved for the reaction coefficients, which are defined by the product of two functions. The first function depends on a spatial variable, the second one depends nonlinearly on the solution of the boundary value problem. The mentioned inverse problem consists in reconstructing the first function with the help of additional information provided by the solution of the boundary value problem.

In this paper, one provides a theoretical justification of the possibility of hydraulic fracture parameters determination by using a non-stationary fluid injection. It is assumed that the fluid is pumped into the fractured well with the time-periodic flow rate. It is shown that there is a phase shift between waves of fluid pressure and velocity. For the modelling purposes, the length and width of the fracture are assumed to be fixed. In the case of infinite fracture, one constructs an exact solution that ensures analytical determination of the phase shift in terms of the physical parameters of the problem. In the numerical calculation, the phase shift between pressure and velocity waves is found for a finite fracture. It is shown that the value of the phase shift depends on the physical parameters and on the fracture geometry. This makes it possible to determine parameters of hydraulic fracture, in particular its length, by the experimental measurement of the time shift and comparison with the numerical solution.

Arteriovenous malformation is a chaotic disordered interlacement of very small diameter vessels, performing reset of blood from the artery into the vein. In this regard it can be adequately modeled using porous medium. In this model process of embolization described as penetration of non-adhesive substance ONYX into the porous medium, filled with blood, both of these fluids are not mixed with each other. In one-dimensional approximation such processes are well described by Buckley-Leverett equation. In this paper Buckley-Leverett equation is solved numerically by using a new modification of Cabaret scheme. The results of numerical modeling process of embolization of AVM are shown.

Progress of technology and medicine dictates the ever-increasing requirements (heat resistance, corrosion resistance, strength properties, impregnating ability, etc.) for non-Newtonian fluids and materials produced on their basis (epoxy resin, coating materials, liquid crystals, etc.). Materials with improved properties obtaining is possible by modification of their physicochemical structure. One of the most promising approaches to the restructuring of non-Newtonian fluids is cavitation generated by high-frequency acoustic vibrations. The efficiency of cavitation in non-Newtonian fluid is determined by dynamics of gaseous bubble. Today, bubble dynamics in isotropic non-Newtonian fluids, in which cavitation bubble shape remains spherical, is most full investigated, because the problem reduces to ordinary differential equation for spherical bubble radius. However, gaseous bubble in anisotropic fluids which are most wide kind of non-Newtonian fluids (due to orientation of macromolecules) deviates from spherical shape due to viscosity dependence on shear rate direction. Therefore, the paper presents the mathematical model of gaseous bubble dynamics in anisotropic non-Newtonian fluids. The model is based on general equations for anisotropic non-Newtonian fluid flow. The equations are solved by asymptotic decomposition of fluid flow parameters. It allowed evaluating bubble size and shape evolution depending on rheological properties of liquid and acoustic field characteristics.

In this paper, we study stratified flows and internal waves in the fracture zones of the Mid Atlantic Ridge. The results of measurements carried out in the 39th and 40th cruises of RV Akademik Sergey Vavilov in the autumn of 2014 and 2015 are presented. Hydrophysical properties of the near-bottom flows are studied experimentally on the basis of CTD- and LADCP profiling. Theoretical analysis involves mathematical formulation of stratified fluid flow which uses CTD-data obtained from field observation in the Vema Fracture Zone region. Spectral properties and kinematic characteristics of internal waves are calculated by finite element method.

The asymptotic theory of neutral stability curve of the supersonic plane Couette flow of vibrationally excited gas is constructed. The system of two-temperature viscous gas dynamics equations was used as original mathematical model. Spectral problem for an eighth order linear system of ordinary differential equations was obtained from the system within framework of classical theory of linear stability. Transformations of the spectral problem universal for all shear flows were carried along the classical Dunn — Lin scheme. As a result the problem was reduced to secular algebraic equation with a characteristic division on "inviscid" and "viscous" parts which was solved numerically. The calculated neutral stability curves coincide in limits of 10% with corresponding results of direct numerical solution of original spectral problem.

In our experiments, we investigate a flow of a viscous fluid in the model of the common carotid artery bifurcation. The studies are carried out using three hardware equipments: two magnetic resonance scanners by Philips and Bruker, and intravascular guidewire ComboWire. The flux is generated by a special pump CompuFlow that is designed to reproduce a flow similar to the one in the blood vessels. A verification of the obtained data is carried out. Conducted research shows the capabilities of the measurement instruments and reflects the character of fluid flow inside the model.

We consider one-phase (formal) asymptotic solutions in the Kuzmak-Whitham form for evolutionary nonlinear equations. In this case, the leading asymptotic expansion term has the form X(S(x, t)/ɛ + ϕ(x, t), A(x, t), x, t) + O(ɛ), where ɛ is a small parameter and the phase S(x, t) and slowly changing parameters A(x, t) are to be found from the system of averaged Whitham equations. The equation for the phase shift ϕ(x, t) is appearing by studying the second-order correction to the leading term. The corresponding procedure for finding the phase shift is then nonuniform with respect to the transition to a linear (and weakly nonlinear) case. We formulate the general conjecture (checked for some examples), which essentially follows from papers by R. Haberman and collaborators, is that if one incorporates the phase shift ϕ(x, t) into the phase and adjust A by setting S → (x, t, ɛ) = S + ɛϕ + O(h2), A → Ã(x, t, ɛ) = A + ɛa + O(ɛ²), then the new functions (x, t, ɛ) and Ã(x, t, ɛ) become solutions of the Cauchy problem for the same Whitham system but with modified initial conditions. These functions completely determine the leading asymptotic term in the Whitham method.

The work is devoted to the mathematical modeling of the influence of transversal jet and the near-wall energy sources on the shock wave structure of supersonic flow in channel with variable cross section. Stable regimes with the region of transonic velocities are obtained. Their stability is confirmed by the width of the corridor of the input power in the region of existence of the regime modes.

A nonlinear second-order parabolic equation with two variables is considered. Under additional conditions, this equation can be interpreted as the porous medium equation in case of dependence of the unknown function on two variables: time and distance from the origin. The equation has a wide variety of applications in continuum mechanics, for example, it is applicable for mathematical modeling of filtration of ideal polytropic gas in porous media or heat conduction. The authors deal with a special solutions which are usually called heat waves. A special feature of such solution is that it consists of two continuously joined solutions. The first of them is trivial and the second one is nonnegative. The heat wave solution can have discontinuous derivatives on the line of joint which is called the front of heat wave, i.e. smoothness of the solution, generally speaking, is broken. The most natural problem which has such solutions is the so-called "the Sakharov problem of the initiation of a heat wave". New solutions of the problem in the form of multiple power series for physical variables are constructed. The coefficients of the series are obtained from tridiagonal systems of linear algebraic equations. Herewith, the elements of matrices of this systems depend on the matrix order and the condition of the diagonal dominance is not fulfilled. The recurrent formulas for the coefficients are suggested.

In the present study, we consider the system of two layers of the immiscible constant density fluids which are modeled by the full Euler equations. The domain of the flow is infinite in the horizontal directions and delimited above by a free surface. Bottom topography is taken into account. This is a simple model of the wave propagation in the ocean where the upper layer corresponds to the (thin) layer of fluid above the thermocline whereas the lower layer is under the thermocline. Though even this simple framework is computationally too expensive and mathematically too complicated to describe efficiently propagation of waves in the ocean. Modeling assumption such as shallowness, vanishing vorticity and hydrostatic pressure are usually made to get the bi-layer shallow water models that are mathematically more manageable. Though, they cannot describe correctly the propagation of both internal and free surface waves and dispersive/non hydrostatic must be added. Our goal is to consider the regime of medium to large vorticities in shallow water flow. We present the derivation of the model for internal and surface wave propagation in the case of constant and general vorticities in each layer. The model reduces to the classical Green-Naghdi equations in the case of vanishing vorticities.

The work is devoted to the mathematical modeling of the influence of forced vibrations of a surface element on one side of the airfoil on the shock-wave structure of transonic flow around. The influence of parameters of oscillations on the airfoil wave drag and the lift force were qualitatively and quantitatively investigated for constant maximum velocity amplitude, which is close in magnitude to the sound velocity in the incoming flow, and for a wide range of frequencies. The arising of additional lift force is shown.

A numerical and experimental study of the spectral distribution of disturbances hotshot wind tunnel was performed. It was found that disturbances are the superposition of fast acoustic waves propagating along axial direction and oblique slow acoustic waves which have less amplitude. Numerical simulation of disturbances evolution in the nozzle of the wind tunnel was carried out by solving 2D Navier-Stokes equations using the ANSYS Fluent program package. Experiments were carried out in the IT-302M high-enthalpy hotshot wind tunnel ITAM SB RAS.

A shear flow of a viscosity-stratified fluid in a Hele-Shaw cell is considered. The long-wave approximation is applied to the governing equations. To describe the evolution of the mixing layer, a special flow with a three-layered structure is considered. A one-dimensional model is derived by averaging the motion equations over the cell width, taking into account the flow structure. For a stationary flow, solutions of motion equations are constructed. The influence of viscosity on the mixing layer evolution is investigated by performing a numerical experiment for a flow with different viscosities in the layers and for a flow with always zero viscosity. It is shown that viscosity has a significant influence on the flow evolution.

A problem on generation of unsteady nonlinear waves on the surface of an infinitely deep ideal fluid due to the motion of a submerged elliptical cylinder is considered. It is supposed that the cylinder can rotate in addition to translational two-dimensional motion. The initial formulation of the problem is reduced to an integrodifferential system of equations for the functions defining the free surface shape, the normal and tangential components of velocity on the free boundary. The small-time asymptotics of the solution is constructed in the case of the cylinder that moves with a constant acceleration from rest.

The role of linear and nonlinear effects in the process of formation of detonation wave structure is investigated using linear stability analysis and direct numerical simulation. A simple model with a one-step irreversible chemical reaction is considered. For linear stability computations, both the local iterative shooting procedure and the global Chebyshev pseudospectral method are employed. Numerical simulations of 1D pulsating instability are performed using a shock fitting approach based on a 5th order upwind-biased compact-difference discretization and a shock acceleration equation deduced from the Rankine-Hugoniot conditions. A shock capturing WENO scheme of the 5th order is used to simulate propagation of detonation wave in a plane channel. It is shown that the linear analysis predicts correctly the mode dominating on early stages of flow evolution and the size of detonation cells which emerge during these stages. Later, however, when a developed self-reproducing cellular structure forms, the cell size is approximately doubled due to nonlinear effects.

The results of mathematical modeling of formation of galaxies in cosmological context with using of multiphase hydrodynamical model are presented in the paper. Mathematical model of the problem of cosmological modeling, numerical methods for solving the hyperbolical equations and brief description of parallel implementation of the software complex CosmoPhi are described in details. The results of numerical experiments of large-scale cosmological simulations are presented.

The deformation behavior of high-chromium stainless steel of sorbitic structure upon high-temperature tempering and of electrically saturated with hydrogen in the electrochemical cell during 12 hours is investigated. The stress-strain curves for each state were obtained. From the stress-strain curves, one can conclude that hydrogen markedly reduces the elongation to the fracture of specimen. Using double-exposed speckle photography method it was found that the plastic flow of the material is of a localized character. The pattern distribution of localized plastic flow domains at the linear hardening stage was investigated. Comparative study of autowave parameters was carried out for the tempered steel as well as the electrically saturated with hydrogen steel.

Using numerical experiment the one-dimensional unsteady process of heterogeneous combustion in porous object under free convection is considered when the dependence of permeability on porosity is taken into account. The combustion is due to exothermic reaction between the fuel in the solid porous medium and oxidizer contained in the gas flowing through the porous object. In the present work the process is considered under natural convection, i.e. when the flow rate and velocity of the gas at the inlet to the porous objects are unknown, but the gas pressure at object boundaries is known. The influence of changing of permeability due to the changing of porosity on the solution is investigated using original numerical method, which is based on a combination of explicit and implicit finite-difference schemes. It was shown that taking into account the dependence of permeability on porosity, which is described by some known equations, can significantly change the solution in one-dimensional case. The changing of permeability due to the changing of porosity leads to the speed increasing of both cocurrent and the countercurrent combustion waves, and to the temperature increasing in the combustion zone of countercurrent combustion wave.

The purpose of this paper is to develop a numerical method of solving the problem of evolution of the finite gas volume that entered in a liquid flow at a set flow rate. The drift- flux model is used as gas-liquid mixture equations. The velocities of both phases, mixture and gas, are related by the Zuber-Findlay equation which coefficients depend on flow regime and gas void fraction. Lagrangian coordinates are used to simplify the initial equations. The numerical solution scheme is proposed. The numerical solution of the Riemann problem is verified by comparison with the exact self-similar solution. The model and numerical method efficiency is illustrated by examples of gas kick calculations in a vertical well.

A non-linear problem on steady non-homogeneous flows over an uneven bottom is considered. A semi-analytical model deals with asymptotic solutions of the Dubreil-Jacotin — Long equation of uniformly stratified fluid. Approximate solutions are constructed by the perturbation procedure combined with the Fourier method of modal expansion. Stationary wave patterns forced above the rough terrain of finite horizontal extension are calculated and analyzed.

The system of weakly coupled differential equations describing traveling waves in dispersive media is considered. The Lyapunov — Schmidt construction is used to study the branching of cnoidal-type periodic solutions. The analysis of bifurcation equations uses the group symmetry and cosymmetry of original equations. Sufficient condition for existence of the phase-shifted modes of cnoidal waves is formulated in terms of the Pontryagin's function determined by the nonlinear perturbation terms

The blood realizes the transport of substances, which are necessary for livelihoods, throughout the body. The assumption about the relationship genotype and structure of vasculature (in particular of brain) is natural. In the paper we consider models of vessel net for two genetic lines of laboratory mice. Vascular net obtained as a result of preprocessing MRI data. MRI scanning is realized using the method of variation of slope of scanning plane, i.e. by several sets of parallel planes specified by different normal vectors. The following special processing allowed to construct models of vessel nets without fragmentation. The purpose of the work is to compare the vascular network models of two different genetic lines of laboratory mice.

The present paper discusses the method of identification (diseased/healthy) human cerebral vessels by using of mathematical model. Human cerebral circulation as a single tuned circuit, which consists of blood flow, elastic vessels and elastic brain gel tissue is under consideration. Non linear Van der Pol-Duffing equation is assumed as mathematical model of cerebrovascular circulation. Hypothesis of vascular pathology existence in some position of blood vessel, based on mathematical model properties for this position is formulated. Good reliability of hypothesis is proved statistically for 7 patients with arterial aneurysms.

A theoretical model of internal solitary waves of large amplitude in a weakly stratified fluid is considered. It is assumed that the background density profile depends linearly or exponentially on the fluid depth. It is demonstrated that inverse problem on determining fine-scale structure of the density profile by a known curve of amplitude dispersion is reduced to solving a linear Fredholm integral equation of the first kind having special form of the kernel. The one-to-one correspondence between the density coefficient and the dispersion function is established in the case of analytical stratification.

A one-dimensional free boundary problem on a motion of a heavy piston in a tube filled with viscous gas is considered. The system of governing equations and boundary conditions is derived. The obtained system of differential equations can be regarded as a mathematical model of an exterior combustion engine. The existence of a weak solution to this model is proved. The problem of maximization of the total work of the engine is considered.

The paper deals with the numerical solution of an equilibrium problem for an elastic membrane with a thin rigid inclusion. The thin inclusion is supposed to delaminate, therefore a crack between the inclusion and the membrane is considered. The boundary conditions for nonpenetration of the crack faces are fulfilled. We provide the relaxation of the problem and propose an iterative method for the numerical solution of the approximated problem. The method is based on a domain decomposition and the Uzawa algorithm for finding a saddle point of the Lagrangian. Examples of the numerical solution of the initial problem are presented.

The process of internal erosion in a three-phase saturated soil is studied. The problem is described by the equations of mass conservation, Darcy's law and the equation of capillary pressure. The original system of equations is reduced to a system of two equations for porosity and water saturation. In general, the equation of water saturation is degenerate. The degenerate problem in a one-dimensional domain and one special case of the problem in a two-dimensional domain are solved numerically using a finite-difference method. Existence and uniqueness of a classical solution of a nondegenerate problem is proved.

A two-dimensional problem of a potential free-surface flow of an ideal incompressible fluid caused by a singular sink is considered. The sink is placed at the horizontal bottom of the channel. By employing a conformal map, the problem is equivalently rewritten in the unit circle. After that, it is investigated by the Levi — Civita technique with the extraction of the singular part of the flow that corresponds to the sink. We derive a Nekrasov type equation that describes exactly the form of the free boundary. This equation is studied at first numerically and then by an exact mathematical technique. It is shown that for the Froude number greater than some particular value, there exists a unique solution of the problem such that the free surface decreases monotonically when moving from the infinity to the sink. At the point over the sink, the free surface has a cusp.

In this paper we study the emergence and development of roll waves in two-layer fluid flow in a Hele-Shaw cell. We propose the mathematical model of such flow and define the conditions of transition from stable state to instability in the form of the roll waves. We find out the physical parameters of flows at which the roll waves exist. A linear stability analysis and the Whitham criterion of roll waves existence are used for solving the problem and arrive to identical conclusions on depths of upper and lower layers at which violation of flow stability occurs. The numerical calculations for the obtained mathematical model at found ratios of densities, viscosities and depths of layers are performed. They confirm development of the roll waves of finite amplitude from small oscillations of the interface.

The study is devoted to the mathematical model of fluid filtration in poroelastic media. The laws of conservation of mass for each phase, Darcy's law for fluid phase, the rheological law and the general equation of conservation of momentum for system describe this process. The local solvability of the problem is proved in this paper for the case in which the density of the mass forces is equal to zero and the fluid is compressible.

Exact solutions of the system of nonlinear shallow water equations on paraboloid are constructed by the method of group analysis. These solutions describe fast wave motion of the fluid layer and slow evolution of symmetric localized vortices. Explicit formulae are obtained for asymptotic solution related to the linear shallow water approximation. Numerical methods are used by the modeling the trajectory of the vortex center in the case of asymmetric vortices.

The researches of stability of cylindrical front of deflagration combustion in an annular combustion chamber were made using phenomenological model. The flame front is described as discontinuity of gasdynamic parameters. It is considered that the combustion products are under chemical equilibrium. The combustible mixture and the combustion products are ideal gases. The velocity of deflagration combustion is determined using the Chapman-Jouget theory. It depends on the temperature of combustible mixture only. It is found that the combustible flame front is unstable for several types of small disturbances in the system Mechanics of instabilities are examined using both the numeric and analytical methods. The cases of evolution of the unstable waves rotating in circular channel are presented.

One of the analytical methods of presenting solutions of nonlinear partial differential equations is the method of special series in powers of specially selected functions called basis functions. The coefficients of such series are found successively as solutions of linear differential equations. To find recurrence, the coefficient is achieved by the choice of basis functions, which may also contain arbitrary functions. By using such functional arbitrariness, it allows in some cases to prove the global convergence of the corresponding constructed series, as well as the solvability of the boundary value problem.

In this paper an endovascular measurement system used for intraoperative cerebral blood flow monitoring is described. The system is based on a Volcano ComboMap Pressure and Flow System extended with analogue-to-digital converter and PC laptop. A series of measurements performed in patients with cerebrovascular pathologies allows us to introduce "velocity-pressure" and "flow rate-energy flow rate" diagrams as important characteristics of the blood flow. The measurement system presented here can be used as an additional instrument in neurosurgery for assessment and monitoring of the operation procedure. Clinical data obtained with the system are used for construction of mathematical models and patient-specific simulations. The monitoring of the blood flow parameters during endovascular interventions was approved by the Ethics Committee at the Meshalkin Novosibirsk Research Institute of Circulation Pathology and included in certain surgical protocols for pre-, intra- and postoperative examinations.

In this paper a computer simulation of a blood flow in cerebral vessels with a giant saccular aneurysm at the bifurcation of the basilar artery is performed. The modelling is based on patient-specific clinical data (both flow domain geometry and boundary conditions for the inlets and outlets). The hydrodynamic and mechanical parameters are calculated in the frameworks of three models: rigid-wall assumption, one-way FSI approach, and full (two-way) hydroelastic model. A comparison of the numerical solutions shows that mutual fluid- solid interaction can result in qualitative changes in the structure of the fluid flow. Other characteristics of the flow (pressure, stress, strain and displacement) qualitatively agree with each other in different approaches. However, the quantitative comparison shows that accounting for the flow-vessel interaction, in general, decreases the absolute values of these parameters. Solving of the hydroelasticity problem gives a more detailed solution at a cost of highly increased computational time.

Equilibrium problems for a 2D elastic bodies with thin Euler-Bernoulli and Timoshenko elastic inclusions are considered. It is assumed that inclusions have a joint point, and a junction problem for these inclusions is analyzed. Existence of solutions is proved, and different equivalent formulations of problems are discussed. In particular, junction conditions at the joint point are found. A delamination of the elastic inclusions is also assumed. In this case, inequality type boundary conditions are imposed at the crack faces to prevent a mutual penetration between crack faces. A convergence to infinity of a rigidity parameter of the elastic inclusions is investigated. Limit problems are analyzed.

Structures and propagation of shock waves in high density particle suspensions in gas are investigated theoretically and numerically. A physical and mathematical model which takes into account integral collisions between the particles on the basis of molecular-kinetic approaches of theory of granular materials is applied. The possibility of different types of shock waves, including double front structures is revealed. The role of particle collisions in the dynamics of particle dense layer expansion under an influence of divergent shock wave and in processes of shock wave diffraction past a backward-facing step is analyzed.

Simulation of a blood flow under normality as well as under pathology is extremely complex problem of great current interest both from the point of view of fundamental hydrodynamics, and for medical applications. This paper proposes a model of Van der Pol - Duffing nonlinear oscillator equation describing relaxation oscillations of a blood flow in the cerebral vessels. The model is based on the patient-specific clinical experimental data flow obtained during the neurosurgical operations in Meshalkin Novosibirsk Research Institute of Circulation Pathology. The stability of the model is demonstrated through the variations of initial data and coefficients. It is universal and describes pressure and velocity fluctuations in different cerebral vessels (arteries, veins, sinuses), as well as in a laboratory model of carotid bifurcation. Derived equation describes the rheology of the "blood stream - elastic vessel wall gelatinous brain environment" composite system and represents the state equation of this complex environment.

In the paper, the asymptotic behavior of solutions of the Cauchy problem is described for the linearized Navier-Stokes and MHD equations with the initial condition localized in a neighborhood of a two-dimensional surface in three-dimensional space. In particular, conditions for the growth of the perturbation in plane-parallel, two-dimensional, and helical external flows are obtained.

An equilibrium problem for a two-dimensional homogeneous linear elastic body containing a thin elastic inclusion and an interfacial crack is considered. The thin inclusion is modeled within the framework of Euler-Bernoulli beam theory. An explicit formula for the first derivative of the energy functional with respect to the crack perturbation along the interface is presented. It is shown that the formulas for the derivative associated with translation and self-similar expansion of the crack are represented as path-independent integrals along smooth contour surrounding one or both crack tips. These path-independent integrals consist of regular and singular terms and are analogs of the well-known Eshelby-Cherepanov-Rice J-integral and Knowles-Sternberg M-integral.

Scenarios of the transition to turbulence in overturning lee waves generated by the two-dimensional obstacle in a stably stratified flow have been explored by visualization of velocity and scalar (density) fields, with analysis of spanwise spectra. The results are obtained by numerical solution of the continuity, Navier–Stokes and scalar equations for stratified fluid with the Boussinesq approximation, for varied Reynolds and Prandtl numbers relating to tank experiments, situations in atmosphere and oceans. Based on the computed data, the dependence of the most unstable perturbation wavelength on Reynolds and Prandtl numbers is derived.