Laboratory for Mathematical Modelling of Wave Phenomena

Wave processes in a loaded elastic shell are investigated. Such shells are actively investigated due to rich applications in engineering and construction. This is one of the most important elements in the modeling of acoustic waveguides, various pipelines, supports of offshore drilling rigs and other hydraulic structures. Waves propagating in this type of systems cause oscillations and vibrations of both the shells themselves and their joints, which may affect the strength properties of the entire system. The presence of different kinds of loads significantly complicates the nature of wave processes. These loads may be the environment surrounding the shell or the liquid inside the shell. As well as various local loads, such as stiffening ribs on the shell, massive elements on the shell, etc. All this requires research not only classical, but also more complex models of the shell, which is caused, in particular, by the relevance of the problem of hydrogen embrittlement of oil and gas pipelines. Hydrogen impact leads to weakening of the structural material, which is not taken into account by the existing calculation norms, which can lead to accidents. The mentioned technical problems require prediction of the shell behavior and possibility of occurrence of such modes, as well as derivation of formulas for reconstruction of the shell parameters according to the experimental data.

Areas of research:

• Boundary Contact Problems (BCP) of mathematical acoustics. Exact, asymptotic and numerical solutions of BCP (including shell-media systems)

• Study of energy fluxes in the shell-media systems.

• Periodically loaded shells

• Inhomogeneous shells, including those weakened by hydrogen

Main results:

• The BCP of interaction of the medium and shells of Timoshenko and Kirchhoff–Love type based on V.V. Eliseev shell models are proposed. The invariant expressions for energy fluxes in the shell-medium systems, consistent with the law of energy conservation in the system, are obtained. [⤤]

• Exact analytical solutions have been obtained and simplified models have been proposed for a number of boundary-contact problems of mathematical acoustics with harmonic dependence on time. In particular, for free vibrations of membranes, plates, cylindrical shells, as well as concentric cylindrical shells partially filled with liquid and surrounded by a medium, immersed in a liquid waveguide. [⤤], [⤤], [⤤]

• The exact analytical representation of the Green's function for the problem of harmonic vibrations of a cylindrical shell partially immersed in a liquid has been found. This allows us to calculate the forced vibrations of a cylindrical shell under the action of forces acting both in the underwater and above-water part of the shell, as well as at the liquid level (for example, the effect of ice). [⤤]

• The analysis of the structure of both energy flows and kinematic and dynamic characteristics of the vibration field in the shell-environment system in the vicinity of the special points of the dispersion curves has been carried out. The asymptotics of the energy flows were found. In particular, the rearrangement of the energy flux components in the vicinity of the viring points for the Kirchhoff–Love type shells, both "dry" and loaded by the medium, is investigated. The efficiency of energy fluxes is shown, since not only the kinematic and dynamic characteristics of the vibration field are taken into account, but also the phase shift between them. This allows predicting dangerous modes of vibrations of shells and shell-media systems in their design. [⤤], [⤤], [⤤]

• The effect of negative group velocity for "dry" shells of Kirchhoff–Love type was found. This effect was investigated both for "dry" shells and for shells loaded with liquid or Winkler base. Asymptotic estimates of the frequency ranges and wave numbers where the negative group velocity effect is observed are obtained, as well as estimates of the maximum negative group velocity in the system. These results can be useful for predicting the occurrence of dangerous modes of operation of pipelines and other similar systems. [⤤], [⤤], [⤤], [⤤]

• The effect of factorization of the equations of passband and stopband boundaries in infinite periodic shells is found. Explicit expressions of factorization of these equations in the case of rods and beams loaded with periodic masses have been obtained. Energy flows and vibration fields in such systems are investigated. [⤤], [⤤], [⤤]

• Formulas linking the frequency shift caused by hydrogen embrittlement of the shell with the parameters of the shell, the attenuation coefficients and the degree of wear with time were obtained. This, in turn, makes it possible to predict the frequency shift and, thus, to avoid the undesirable shell oscillations dangerous for its strength. [⤤], [⤤]

Oscillations of the cylindrical shell on Winkler base loaded with a mass belt: (a) physical model, (b) visualization of displacements.

Waves on the surface of a fluid are a common physical phenomenon; mathematical problems of wave theory arise in connection with important practical applications (first of all, calculation of hydrodynamic loads on submerged bodies). Problems are studied that describe in a linear approximation the steady motion of an ideal, heavy fluid with a free surface in the presence of obstacles. Finding uniquely solvable statements is one of the key issues [⤤] , often requiring development of original mathematical methods; for example, consideration of localized (trapped) modes is often reduced to studying point eigenvalues embedded into continuous spectrum.

Main results:

• Conditions for the uniqueness and solvability of a number of problems are found, including the plane problem of fluid flow over a bottom obstacle [⤤] , the problem of forward motion of bodies in a homogeneous [⤤] and two-layer fluid [⤤] , the problem of time-harmonic oscillations of a fluid in the presence of fixed bodies [⤤] , [⤤] and obstacles [⤤] , taking into account surface tension [⤤] , as well as freely floating bodies [⤤] , [⤤] , [⤤] .

• The question of supplementary conditions that turn the Neumann–Kelvin problem into a uniquely solvable one is studied for bodies that either cross the free surface or the interface between two layers of different density [⤤] , [⤤] , [⤤] .

• For the problem of the forward motion of bodies and the problem of fluid oscillations in the presence of bodies, criteria for unique solvability are found. Numerical algorithms are developed and, for the first time, examples of non-uniqueness with completely submerged bodies [⤤] , [⤤] , [⤤] , [⤤] , [⤤] , [⤤] are found (including taking into account surface tension and in a two-layer liquid). The connection of examples of non-uniqueness with anomalously large values of wave resistance is shown. The figure shows a body for which there is an example of non-uniqueness, and a picture of streamlines [⤤].

• A number of examples of non-uniqueness for bodies partially submerged in a fluid or crossing the interface are obtained. The inverse procedure is used: for a given velocity potential, a geometry is sought for which it is a solution to the homogeneous problem [⤤] , [⤤]. Significant progress has been made in problems of coupled oscillations of a fluid and freely floating bodies [⤤] , [⤤] and [⤤]. In the constructed examples of mode localization [⤤] , the submerged part of the bodies is axisymmetric, and the bodies are motionless, although they float freely — the figure shows an example of such a structure.

• The problem of fluid oscillations in the presence of partially submerged bodies is considered in the case when the fluid is bounded from above by a floating brash ice. The existence of localized oscillation modes is established, and examples of bodies supporting such modes with finite energy are constructed. On the other hand, we find conditions for the absence of localized modes [⤤] , [⤤].

• The problems of forward motion of bodies and vibrations of bodies in a two-layer fluid are considered. An important feature is a more complex picture of hydrodynamic loads due to the presence of internal waves associated with the phenomenon of "dead water". Question of unique solvability is studied, examples of non-uniqueness are found, properties of solutions are investigated , including hydrodynamic loads and their possible features , [⤤] , [⤤] (the problem of oscillations).

• For the three-dimensional problem of the motion of bodies in a fluid with a free surface (the problem of ship waves) , algorithms for fast calculation of the Green's function and derivatives have been developed. For this purpose, methods have been developed for calculating integrals of rapidly and non-harmonically oscillating functions over a semi-infinite interval [⤤] .

• Eigenmodes of fluid oscillations with a free surface in vertical circular and annular cylindrical containers are studied. The effect of violation of axial symmetry due to the presence of radial baffles going vertically from the free surface to the bottom was analyzed. It is shown that the presence of a baffles results in a significant change in the properties of eigenvalues and eigenfunctions [⤤].

Prof. A.P. Kiselev is a member of the organizing committee, the head of the program committee of the annual international conference Days of Diffraction. D.Sc. O.V. Motygin is a member of the program committee of the conference Days of Diffraction. A.P. Kiselev and O.V. Motygin are editors of Proceedings of the International Conference Days on Diffraction. Links: [⤤] and [⤤].

Research directions:

Development of mathematical models and study of the nonlinear dynamics of nano- and microelectromechanical sensors based on the phenomenon of modal localization of oscillations in weakly coupled systems under electrostatic and laser thermo-optical excitation, taking into account the coupling of thermal, electrical and mechanical processes.

Main results:

• The dynamics and stability of elastic elements of nano- and microelectromechanical systems (N/MEMS) under laser thermo-optical effects have been studied. Mathematical modeling and development of algorithms for laser generation and control of oscillations of moving N/MEMS elements have been carried out.

• The architecture has been developed and mathematical modeling of the nonlinear dynamics of N/MEMS sensors based on the principle of modal localization of oscillations in systems of weakly coupled continuous elastic elements has been carried out.

• Models of highly sensitive N/MEMS sensors of physical quantities (accelerometers, gyroscopes, particle mass detectors, flow velocity sensors, etc.) based on nonlinear static and spectral characteristics of distributed elastic moving elements with projected initial geometric perturbations (multi-stable systems - systems with switches) have been developed.

Synchronization of oscillations of weakly coupled elastic elements of a differential resonant MEMS accelerometer in the dual-circuit autogenerator mode.

Research directions:

Investigation of the behavior of various liquids during the flow in nanotubes.

Three-dimensional modeling of the processes of molecular-dynamic interaction.

Main results:

On the basis of analytical and numerical modeling, the features of the flow of liquids in nanochannels were studied. For a number of types of liquid, a theory has been proposed that makes it possible to estimate the viscosity of a liquid depending on the force of interaction between liquid molecules and channel walls. The velocity profiles for the Poiseuille and Couette flows are obtained, and the processes occurring in the fluid are studied.

Filling the gap between nanotubes with a mixture of water (highlighted in red) and methane (highlighted in purple) in various proportions from left to right: water - 100% (methane 0%); water - 80%; water - 60%; water - 40%; water - 20%; water - 0% (methane 100%). (mixture temperature 300K).

Flow velocity profile of water mixed with argon.

Flow velocity profiles of a mixture of water and argon depending on the gap between the tubes.

Research directions:

Investigation of the dynamics of systems with time-varying stiffness and mass parameters.

Dynamics of elastic structures interacting with moving ice cover.

Detachment of the film from the substrate under external impact.

Main results:

The process of damage growth in the film substrate under impact and stationary impacts has been studied. The possibility of damage growth because of the process of wave localization in the area with initial damage is shown, and a program for calculating the behavior of the film depending on the type of impact is compiled.

The dependence of the film displacement under impact.

Analytical methods are proposed for studying the dynamics of weakly nonlinear elastic beams with time-varying mass, or mass and stiffness (including that which changes as a result of material aging). A criterion has been developed for how, when applying methods of the Galerkin type, it is necessary to choose the number of modes taken into account in the solution of the problem being constructed. The results obtained are applied to the study of the phenomenon of vibration caused by rain and wind.

Phase portraits of the oscillator when changing the position of the variable rain mass located on the surface of the suspension bridge cable.

A continuum model has been proposed to describe ice-induced vibrations for offshore structures anchored to the bottom (such as oil platforms, wind turbines, for example).

Methods and programs have been developed to describe the dynamics of offshore structures interacting with a moving ice cover based on a simple mechanical model. The main modes of vibration are described, including the most dangerous ones in terms of strength. A model is proposed that describes the presence of water and broken ice in the gap between the structure and the ice cover. Wave regimes in liquid and ice cover are also considered during the operation of mechanisms on a floating structure, taking into account the gap between the structure and ice.

Nanofluidics (field of nanoliter liquids physics: 1 nL = 10⁻¹² m³) is of great interest to researchers, because of the promising applications of these systems in biology, optoelectronics, and various sensors and actuators based on anisotropic molecular liquids and liquid crystal (LC) materials. Manipulations with these nanosized anisotropic molecular systems in ultrathin capillaries and channels are often performed using an external electric field.This method of transport of nanoliter volumes is equally applicable both for molecular liquids and LC materials. A distinctive feature of LC systems in comparison with anisotropic molecular liquids is that orientation ordering of molecules described by director field n is formed in LC systems under certain thermodynamic conditions.

Microfluidics often uses micro-sized liquid crystal droplets to control the concentration of molecules in biomedical applications . Central to the success of microfluidic liquid crystal systems is the development of innovative methods for manipulating liquid crystal systems in microchannels.

Pressure gradients and an external electric field for controlling fluid motion are traditionally generated by mechanical, thermomechanical, or electrical drives. On the other hand, the liquid crystals are extremely sensitive to external stimuli and therefore can be used for the construction of stimuli-responsive devices, such as sensors or actuators. They have various advantages in comparison with other types of microsensors and microactuators; simple structure, high shape adaptability, easy downsizing, and low driving voltages. Nematic droplets of the appropriate size, confined in a cylindrical capillary or channel, are micro-devices, the orientation of the molecules of which can be manipulated not only by the electric field or pressure, but also by the temperature gradient ∇T. This gradient can be generated, for instance, by a laser beam focused both in the microfluidic volume and at the boundary of the LC channel.

One of the principles of liquid crystal pumping is based on the coupling, on the one hand, between the gradient of director field ∇n and the temperature gradient ∇T, excited, for instance, by a laser beam, focused both at the boundary of the LC channel and inside the LC microvolume, and, on the other hand, between ∇n and the velocity field v, excited in the microfluidic channel by the laser irradiation.

Main results

The complex dynamics of dissipation processes was revealed during the study of the reaction of liquid crystal material confined in thin and ultrathin channels due to locally formed temperature gradients. Several scenarios for the formation of hydrodynamic flows in microsized hybrid aligned nematic (HAN) channels, based on the appropriate nonlinear extension of the classical Ericksen–Leslie theory, supplemented by thermomechanical correction of the shear stress and Rayleigh dissipation function, as well as taking into account the entropy balance equation, are analyzed. Detailed numerical simulations were performed to elucidate the role of the heat flux q caused by laser radiation focused on the lower boundary of the equally warmed up the HAN channel containing a monolayer of azobenzene with the possibility of a trans-cis and cis-trans conformational changes in formation of the vortex flow v.

It is shown that a thermally excited vortex flow is maintained with motion in a positive sense (clockwise) in the vicinity of the orientation defect at the lower boundary of the HAN channel caused by the trans-cis and cis-trans conformational changes.

We have described a novel mechanism of a kink-like and double π- forms of distortions of the director field n in a microsized nematic volume, confined between two infinitely long horizontal cylinders with a radial temperature gradient ∇T, under the effect of a voltage U applied between cylinders. It is found that, on the one hand, under certain conditions, in terms of curvature of cylinders κ and the voltage U, the torques and forces acting on the director n may excite the kink-like distortion wave spreading along the normal to both cylindrical boundaries, whose resemblance to a kink-like distortion wave depends on the value of applied voltage U and the curvature of the inner cylinder. It has been worked out, on the other hand, the conditions, in terms of κ and U, producing the distortion mechanism of the n in the double π-form, with the intermediate relaxation wall.

Direction of research:

A study of non-stationary problems with free boundaries, governing the evolution of two viscous immiscible liquids with an unknown interface, on which surface tension is taken into account, in the framework of the Navier-Stokes model. Investigation of global unique solvability of the problems in the Sobolev–Slobodetskiǐ and Hölder spaces, a priori estimates of solutions, a study of motion stability over time.

Main results:

A general picture of the smoothness of solutions to the problems on the simultaneous motion of two viscous fluids was obtained. In particular, a study was made of the unique solvability in the Sobolev–Slobodetskiǐ and Hölder spaces of initial-boundary value problems for the Stokes and Navier–Stokes equations with a closed interface between two media. Exponential estimates of the solution to the problem governing the motion of a two-phase incompressible capillary fluid in a bounded domain for small initial data are exact and guarantee the stability of the solution over time. The results were published in the monograph by I. V. Denisova and V. A. Solonnikov, “Motion of a Drop in an Incompressible Fluid”, St. Petersburg: Lan', 2020, 296 p. [⤤] (translation: I. V. Denisova, V. A. Solonnikov, “Motion of a Drop in an Incompressible Fluid”, Springer, 2021, 316 с. [⤤]).

The stability of a uniformly rotating drop consisting of a two-layer viscous self-gravitating capillary liquid is studied. This motion is governed by an interface problem, a global unique solvability of which is obtained for small initial data, external forces and rotation speed, initial surfaces being given close to axisymmetric equilibrium figures. It is proved that if the second variation of energy functional is positive, then a small perturbation of the axially symmetric two-phase equilibrium figure exponentially tends to zero, and the motion of the drop passes over time into the rotation of the liquid mass as a solid body[⤤].

A plane nonlinear problem with a free boundary is studied, which describes steady waves on the surface of an ideal, incompressible fluid of finite depth, taking into account the vortex nature of its motion. In the case of a solitary wave, the flow has the following properties: at both infinities it is a unidirectional, shear, supercritical flow of constant depth, the latter corresponding to the given Bernoulli constant of the original problem. This mathematical model of waves under the action of gravity is important for taking into account the vorticity of a given type in the interaction of waves with currents; It follows from observations that this behavior is very common in nature.

Main results:

1. Two integro-differential equations of the Babenko type are derived and investigated to describe bifurcations, as a result of which nonlinear Stokes waves arise on the surface of a horizontal irrotational flow of finite depth, and as a result of branching of the bifurcation curves, these waves change their profile. For the first time, the corresponding diagrams, including those with multiple branching, were obtained numerically, and wave profiles were found for each of the branches, including extreme-shaped waves with corner points on crests. See publications: [⤤] and [⤤].

2. The existence of a new type of solitary waves on the flow surface with constant negative vorticity, the main feature of which is the presence of a stagnation point (stagnation) inside the vorticity region of the Kelvin "cat's eye" type, is proved. This bottom eddy is symmetrical with respect to the vertical passing through the wave crest. The considered type of fluid motion is observed in nature. See publications: [⤤] and [⤤].

Illustration of cat's-eye vorticity streamlines.

An interesting phenomenon predicted within the framework of the model is the presence of countercurrents, which are separated from one another by the so-called critical layers, and the latter are characterized by the presence of stagnation points. It is shown that the counterflow necessarily takes place in the case when the minimum of the wave profile exceeds a certain critical value. In addition, a constraint on the admissible values of the Bernoulli constant is found. New boundaries are found for the maxima and minima of wave profiles, which are valid for all their types, including bulbous waves, for which stationary propagation is possible instead of breaking.

Direction of research.

Main direction of research is concerned with the investigation of non-stationary processes in mechanical systems with inhomogeneities of different types:

• Moving loads

• Non-stationary processes in phase-transforming materials

• Non-stationary localized oscillations

Main results:

• Stiff phase nucleation in a phase-transforming bar due to the collision of non-stationary waves is considered [⤤].

• The resolution of the classical Stokes’ paradox for inertial moving load is proposed [⤤].

• The analytical solutions describing non-stationary motions in mechanical systems possessing trapped modes of oscillation are obtained in the case of slowly varying in time system parameters [⤤], [⤤], [⤤].

• The problem on the passage through a resonance for a mechanical system, having time-varying parameters and possessing a single trapped mode, is solved [⤤].

• Non-stationary oscillation of a string on the Winkler foundation subjected to a discrete mass-spring system (an oscillator) non-uniformly moving at a sub-critical speed is considered [⤤].

Comparison of the analytical and numerical solutions for the accelerated motion of a moving oscillator (a) The internal force (b) The displacement.