A. V. Vyazmin

Interfacial Mass Transfer in the Liquid–Gas System: An Optical Study

Mass transfer across the liquid–gas interface was studied by a number of optical methods: holographic interferometry in real time, polarization microscopy, shadow imaging, and modified surface-fluctuation spectroscopy. These methods visualize the evolution of the surface layer, the in-depth profile of density, and mass fluxes. Absorption and desorption of various gases by perfluorodecalin is discussed. The origination and development of irregularities in the diffusion–reaction zone were observed during chemical absorption of carbon dioxide by monoethanolamine and KOH solutions.

Decomposition of three-dimensional linearized equations for Maxwell and Oldroyd viscoelastic fluids and their generalizations

A new exact method of solving general three-dimensional nonstationary linearized equations for viscoelastic fluids is described based on breaking these equations down into several simpler equations. Formulas are given that make it possible to express the solution in the respective systems (consisting of four connected equations) by solving two independent equations. The most widespread rheological models of viscoelastic fluids are considered to illustrate the powerful capabilities of the proposed method. A new differential-difference model for a viscous fluid with a constant relaxation time is proposed that gives a finite disturbance propagation rate and is in good agreement with the Maxwell and Oldroyd differential models of viscoelastic fluids. The axial flows of viscoelastic fluids are studied, and solutions to certain hydrodynamic problems are given.

Differential-difference heat-conduction and diffusion models and equations with a finite relaxation time

AbstractDifferential-difference heat conduction and diffusion models and equations with a finite relaxation time are described that give a finite disturbance propagation rate. The modified Biot-Fourier law with delay is used for the heat flux, which leads to the differential-difference heat-conduction equation

Peculiarities of diffusion in gels

\left. {\frac{{\partial T}} {{\partial t}}} \right|_{t + \tau } = a\Delta T,

Peculiarities of unsteady mass transfer in flat channels with liquid and gel

where the left-hand side is calculated at t + τ (τ is the relaxation time) and the right-hand side is calculated, as usual, at t (without a time shift). At τ = 0, the differential-difference heat-conduction equation turns into the classical parabolic heat-conduction equation; if the left-hand side is expanded into a series in τ and two main terms of expansion are retained, we obtain the Cattaneo-Vernotte hyperbolic heat-conduction equation. An exact solution to the differential-difference heat-conduction equation is derived for a one-dimensional problem without the initial conditions with an arbitrary periodic boundary condition. The approximate solution is constructed to the general three-dimensional initial-boundary value problem of heat propagation with a finite relaxation time in a bounded domain with arbitrary initial heat distribution and the boundary condition of the third kind. It is shown that the differential-difference model makes it possible to derive the Oldroyd-type differential heat-conduction model.

Exact solutions and qualitative features of nonlinear hyperbolic reaction—diffusion equations with delay

An optical method was applied to study the peculiarities of diffusion in gel: this method provides real-time visualization of spreading of solutes brought into the gel. It was shown that spectral characteristics of reflected light give additional information about nature of diffusive spreading of solutes and about state of the gel. Gels with different densities and lifetime were studied. These parameters have strong influence on the velocity of diffusion. The study demonstrated critical differences for diffusion process in gels with true solutions and with solutions with nanoparticles. Experiments discovered the anisotropy in 3D diffusion of solutes in gels; physical explanation of this phenomenon was proposed.

Decomposition and exact solutions of three-dimensional nonstationary linearized equations for a viscous fluid

The rate of the unsteady mass-transfer process in horizontal flat channels filled with viscous liquid (water) and in channels filled with gel has been measured experimentally. It has been established that the rate of propagation of a substance through a channel under the unsteady one-dimensional mass transfer conditions depends on its width and exceeds the value corresponding to the rate of diffusion. The measured mass transport rate in an open channel is less than in a closed channel. The reason for the increased mass-transfer rate with regard to the diffusion mechanism is the slow convective flow of liquid caused by the difference between the densities of a diffusing substance and water. The presence of the surface concentration gradient of liquid gives rise to the appearance of surface forces that significantly affect the mass transport rate. The similarity of the mass-transfer behavior in channels filled with pure liquid and gels is found experimentally, which allows one to talk about their presence, as well as of convective transport in them.

Reaction-Diffusion Models with Delay: Some Properties, Equations, Problems, and Solutions

New classes of exact solutions to nonlinear hyperbolic reaction—diffusion equations with delay are described. All of the equations under consideration depend on one or two arbitrary functions of one argument, and the derived solutions contain free parameters (in certain cases, there can be any number of these parameters). The following solutions are found: periodic solutions with respect to time and space variable, solutions that describe the nonlinear interaction between a standing wave and a traveling wave, and certain other solutions. Exact solutions are also presented for more complex nonlinear equations in which delay arbitrarily depends on time. Conditions for the global instability of solutions to a number of reaction—diffusion systems with delay are derived. The generalized Stokes problem subject to the periodic boundary condition, which is described by a linear diffusion equation with delay, is solved.

The heat and mass transfer modeling with time delay

A new exact method for solving three-dimensional linear systems of hydrodynamic equations is described based on decomposing these systems into three simpler equations. It is shown that the general solution to three-dimensional Stokes equations (when there are no mass forces) can be expressed by means of solutions to two independent equations: the heat conduction equation and the Laplace equation. A class of solutions with the linear dependence of velocity components on two space variables is studied, and their physical interpretation is given. Axial flows are considered, and certain hydrodynamic problems are solved. A general solution to three-dimensional Oseen equations is constructed. The linearized equations of motion for a viscoelastic Oldroyd fluid are studied

Agar Gels: Kinetics of Formation and Structure

The delay reaction-diffusion models used in thermal physics, chemistry, biochemistry, biology, ecology, biomedicine, and control theory were reviewed. New exact solutions were obtained for several classes of one- and three-dimensional nonlinear equations with distributed parameters, in which the kinetic functions involve a delay. The qualitative features of these equations related to nonsmoothness and potential instability of solutions (these features should be taken into account in the mathematical modeling of the corresponding processes) were discussed. The properties of delay reaction-diffusion equations were described, which allow exact solutions to be obtained and multiplied. The key principles of construction, selection, and use of the test problems of the reaction-diffusion type were formulated, which can be used for evaluating the accuracy of rough analytical and numerical methods for solving the delay equations.