Home – Home – Jornals – Journal of Applied Mechanics and Technical Physics 2024 number 5

A boundary value problem is considered for a nonlinear mass transfer model that generalizes the classical Boussinesq approximation under inhomogeneous Dirichlet boundary conditions for velocity and mixed boundary conditions conditions for the concentration of the substance. It is assumed that the viscosity and diffusion coefficients and the buoyancy force in the model equations depend on the concentration. A mathematical apparatus for studying the problem is developed and used. to prove the theorem on the global existence of a weak solution. Sufficient conditions for similar problems that ensure the local uniqueness of weak solutions are given.

A problem is posed on a joint unsteady unidirectional motion of two immiscible fluids in a cylindrical tube with a constant temperature difference on the solid surface of the tube. From the mathematical viewpoint, this is an adjoint and inverse problem with respect to the pressure gradient of one of the fluids along the tube. The condition of problem overdetermination is a specified unsteady total flow rate of both fluids. A steady solution is found. A priori estimates of the solution of the unsteady problem in a uniform metric are obtained. Based on these estimates, sufficient conditions for input data are formulated at which the steady solution is exponentially stable.

This paper describes a multidimensional initial-boundary-value problem for Kelvin-Voigt equations for a viscoelastic fluid with a nonlinear convective term and a linear impulse term, which is a regular junior term describing impulsive phenomena. The impulse term depends on an integer positive parameter n , and, as n → +∞, weakly converges to an expression that includes the Dirac delta function that simulates impulse phenomena at the initial time. It is proven that, as n → +∞ an infinitesimal initial layer associated with the Dirac delta function is formed and the family of regular weak solutions of the initial-boundary value problem converges to a strong solution of a two-scale micro- and macroscopic model.

The characteristics of combined flows of evaporating liquid and laminar gas flow in a flat horizontal channel are studied based on the exact partially invariant solution of thermoconcentration convection equations. The influence of the liquid layer thickness and the conditions for the temperature function on the upper wall of the channel on the rate of evaporation caused by gas pumping is investigated. The exact solution is verified by comparison with experimental data. The linear stability of the exact solutions is studied. It is established that regardless of the type of boundary thermal regime, the system always experiences oscillatory instability in the in the form of cellular convection. Thermal insulation of the upper wall does not lead to a change in the structure of the most dangerous perturbations, slightly destabilizes the flow in the case of long-wave perturbations, and has a stabilizing effect in the case of short-wave perturbations.

One possible formulation of the inverse problem of identification of the vortex structure based on the flow velocity vectors at a set of points is considered, and an algorithmic method of its solution is proposed. The approach is based on the vortex structure presentation by a combination of the Rankine vortices. Identification is understood as determination of the number of model vortices, their intensities, centers, and radii. The method implies minimization in space of the parameters of the model system of the objective functional estimating the closeness of the known and modeled velocity vectors. The algorithm includes the following stages: search for the initial approximation for the vortex structure, refinement of the model vortex parameters, and correction of the resultant structure. Solving the direct problem of the flow development prediction is based on solving the initial-boundary value problem for the Euler equation for the ideal fluid dynamics by the spectral-vortex method. Results of test computations performed by the proposed approach are presented. It is demonstrated that the model system in all test computations ensures a sufficiently accurate description of the topology of streamlines during identification. Predictions at times corresponding to changes in the flow topology are obtained.

A nonlinear problem of unsteady motion of a circular cylinder in an ideal infinitely deep fluid under the action of arising hydrodynamic loads is considered. A method of reducing the solution of the initial mathematical problem to the solution of an equivalent integrodifferential system of equations for the function determining the shape of the sought free surface for the normal and tangential components of the fluid velocity on the free surface, and for the unknown trajectory of cylinder motion is used. The initial (in terms of time) asymptotic curve of the solution, which describes the cylinder motion from the state at rest is constructed.

This paper describes a study based on a three-dimensional Ostroumov-Birikha solution and pertaining to two-layer flows of liquid and gas-vapor mixture with account for diffusion-type evaporation on a thermocapillary interface. The results of analytical and numerical simulation of convective flows in a channel with solid impermeable walls arising under different temperature conditions are presented. The values of the mass evaporation rate and thermocapillary stresses calculated on the basis of the exact solution and obtained experimentally are compared.

Linear model equations in partial derivatives with two independent variables are considered. The highest operator symmetries and general solutions for a series of hyperbolic equations are found. Equivalence transformations are constructed for some equations.

Heat exchange in a liquid film moving along the bottom of a microchannel is calculated on the basis of a developed three-dimensional nonstationary model of motion. The liquid moves under the action of a cocurrent gas flow in the channel, with a square heater located at its bottom. In this case, the action of all the main physical factors during their interaction is taken into account: diffusive and convective heat transfer, dependence of liquid properties on temperature, thermocapillary effect, occurrence and evolution of surface deformations, and evaporation and condensation of liquid. It is revealed that the heater size significantly affects temperature fields, surface deformations, and temperature extremes. A formula for calculating the greatest excess of the average temperature achieved on the substrate is derived.

This paper considers hyperbolic systems of equations of a certain type that describe one-dimensional nonlinear waves propagating in the same way in both x directions. Each system of this type can be set in correspondence to a hyperbolic system of equations with a halved order, constructed on the basis of the initial system of equations. The similarity of the solutions of this system of equations and the original one is studied.

Unsteady one-dimensional shear flows of a viscoelastic medium are considered. A general approach is formulated for media with several relaxation times, which allows the known models of viscoelastic flows to be presented as evolutionary systems of first-order equations. Conditions of hyperbolicity of flow classes considered are found for the Johnson-Segalman, Giesekus, and Rolie-Poly models. The equations of motion of the viscoelastic medium are presented in the form of a full nonlinear system of conservation laws. A method of calculating unsteady discontinuous flows within the framework of the models under consideration is proposed. The class of unsteady Couette flows in the gap between the cylinders used in rheological tests is studied numerically. The process of shear lamination of its influence on the structure of steady flows are investigated. The numerical results obtained are compared with experimental data.

This paper presents a mathematical model of a medium consisting of active particles capable of adjusting their movement depending on so-called signals or stimuli. Such models are used, for example, in studying the growth of living tissues, colonies of microorganisms and more highly organized populations. The interaction between two types of particles, one of which (predator) pursues the other (prey) is investigated. The predator's movement is described by the Cattaneo heat equation, and the prey is only capable of diffusing. In view of the hyperbolicity of the Cattaneo model, in the case of sufficiently weak diffusion of preys, the presence of long-lived short-wave structures can be assumed. However, the mechanism of instability and failure of such structures is found. The relations for the transport coefficients of the predator that block this mechanism are derived explicitly.

The normal coordinates method is used in conservative mechanical systems to reduce two quadratic forms to a sum of squares. In this case, a system of differential equations is split into a system of independent oscillators. A linear dissipative mechanical system with a finite number of degrees of freedom is determined by three quadratic forms: kinetic and potential energy of the system, as well as the Rayleigh dissipative function, which, generally speaking, cannot be reduced to a sum of squares. Conditions are considered under which all three quadratic forms are reduced to a sum of squares by a single transformation exactly or approximately. It is shown that, for such systems, normal coordinates can be introduced in which the system is split into independent second-order systems. This allows one to construct exact or approximate analytical solutions in general form and with an estimated relative error in the case of an approximate solution. The advantages of this approach are shown for problems of theoretical mechanics and electrical engineering, in which analytical solutions are constructed and optimization analysis is carried out. In this case, traditional methods allow only numerical calculations to be performed for given parameter values.

Asymptotic behavior of solutions of initial-boundary-value problems arising in simulation of motion of incompressible viscoelastic fluids is studied in the case of various combinations of small relaxation parameters (stress relaxation time at constant strain and strain relaxation time at constant stress), one of which can be equal to zero.

Publications dealing with investigations of corkscrew fluid flows with collinear velocity and vortex vectors are reviewed. New solutions are presented for the Navier-Stokes equations for an incompressible fluid and second-order fluid equations, which are two-dimensional analogs of corkscrew flows.

The existence of a classical solution was established for a one-phase radial viscous fingering problem in a Hele-Shaw cell under surface tension (original problem) by means of parabolic regularization for a certain subsequence {ε_n}_n∈N, ε_n > 0. In this paper, we prove the uniqueness of the classical solution to the original problem with the use of parabolic regularization for the full sequence of the parameter {ε}, ε > 0.

A viscous fluid flow between two wavy horizontal surfaces unlimited in longitudinal and transverse directions is considered. The full Navier-Stokes equations are applied to study the linear stability of such a flow with respect to various three-dimensional disturbances. Two types of wall waviness are studied: longitudinal and transverse periodic corrugation. At the first stage, the main solution is obtained and the initial equations are linearized in the vicinity of this solution. At the second stage, the generalized problem of determining eigenvalues is solved and the entire possible spectrum of disturbances is analyzed. The varied parameters are the Reynolds number, amplitude, period, and shape of the corrugation. Disturbances of velocity and pressure fields are generally characterized by two wave numbers, which are additional parameters. The influence of the parameters and shape of the wall waviness on the region where the laminar-turbulent transition begins is investigated.

This paper examines two-dimensional flows near a free critical point of an incompressible viscoelastic Maxwell medium using the Johnson-Segalman convected derivative. The flow is assumed to be axisymmetric, and its velocity profile is linear along the axial coordinate. A general exact analytical solution is found for the problem of the distribution of the stress tensor components near the stagnation point.

Rotational sedimentation of neutrally buoyant particles in suspensions is studied by mathematical simulation in the case of two-dimensional circular flows between two cylinders. Particle separation in the absence of gravity is caused by rotation of the inner cylinder. It is revealed that sedimentation depends on particle rotation. Within the framework of the Cosserat continuum, the suspension is considered as a micropolar fluid. The effect of the eccentricity of noncoaxial cylinders on the sedimentation front is investigated.

The role of the hidden integral of motion for correct description of the far field of velocities and pressures is discussed based on the complete Navier-Stokes equations for the case of non-self-similar submerged jets of an incompressible viscous fluid when the source of motion has a non-zero characteristic size. It is shown that the emergence of the hidden conservation integral is due to the fact that the coordinates of the point of the effective momentum source and the point of the effective mass source may not coincide for real spatially extended sources of jet flow. Using special functions, an exact analytical solution for all terms of the asymptotic expansion of the far field of a non-self-similar submerged jet was obtained that is described by all integrals of motion: conservation of the total momentum flux, conservation of the total angular momentum flux, conservation of the total mass flux, and the additional hidden conservation integral associated with the conservation of the total angular momentum flux. It is shown that the hidden integral was actually first obtained by L. G. Loitsyansky in studying a non-self-similar solution for a submerged jets within the boundary layer approximation, but it was mistakenly interpreted as the integral of conservation of the mass flux flowing from source of the jet. The obtained exact solution was used to perform calculations of velocity and pressure fields at different Reynolds numbers and different values of the hidden integral for the model of the jet outflow from a circular tube of finite size. The influence of the hidden integral of motion on the flow pattern is analyzed.