Ulyanov, Oleg Nikolaevich

Ul’yanov O.N., Rubina L.I.

A system of equations of unsteady spatial free convection of an incompressible viscous fluid in the Boussinesq approximation is considered. The analysis is based on the methods of reduction of linear and nonlinear partial differential equations (PDEs) and systems of PDEs to ordinary differential equations (ODEs) and systems of ODEs. These methods were proposed by the authors earlier, and their general principles are given in the paper. The methods are based on the construction of a system of equations of characteristics for a first-order PDE (the basic equation). This equation is constructed in a certain way by analyzing the original system of equations. The reductions lead to ODEs or systems of ODEs in which an independent variable  $$\psi$$ is such that the equation $$\psi(x,y,z,t)=\mathrm{const}$$ defines a level surface for all unknown functions of the original system of PDEs. The methods are applicable to PDEs and systems of PDEs regardless of their type. The Oberbeck–Boussinesq equations are reduced to a system of ODEs with a functional arbitrariness, and an exact solution with a constant arbitrariness is found for the original system. The functional arbitrariness in the constructed reduction also yielded a system of ODEs in which the temperature  $$T$$ is an independent variable. For this system exact solutions are found. A possible (vortex or vortex-free) motion of an incompressible fluid with free convection is analyzed. The cases of vortex and vortex-free motion of the fluid are identified. An exact solution defining a vortex-free motion of the fluid is written as a result of reductions for the original system of PDEs.

Ульянов О.Н., Рубина Л.И.

Рубина Л.И., Ульянов О.Н.

Рассматриваются безвихревые векторные поля на поверхности, заданной уравнением $a=z+\alpha(x,y,t)=0$. Изучаются условия, при выполнении которых векторные линии таких полей располагаются на этой поверхности. Получены достаточные условия существования гармонического векторного поля с такими векторными линиями. Изучена переопределенная система уравнений в частных производных, решение которой обеспечивает получение гармонического поля, векторные линии которого лежат на заданной поверхности рассматриваемого вида. Выписано уравнение поверхности, для которой можно найти гармоническое векторное поле с векторными линиями расположенными на этой поверхности. Показано, что для любых поверхностей рассматриваемого вида можно найти безвихревые негармонические векторные поля с векторными линиями, расположенными на заданной поверхности. Приведен ряд поверхностей, для которых указаны гармонические или негармонические безвихревые векторные поля с векторными линиями расположенными на этих поверхностях. Рассмотрена система уравнений Навье - Стокса для вязкой несжимаемой жидкости в безразмерном виде. Для этой системы в предположении потенциальности поля скоростей выписано частное решение, обеспечивающее расположение векторных линии поля скоростей на параболоиде вращения. Irrotational vector fields are considered on the surface given by the equation $a=z+\alpha(x,y,t)=0$. The conditions under which the vector lines of such fields are located on this surface are studied. Sufficient conditions for the existence of a harmonic vector field with such vector lines are obtained. An overdetermined system of partial differential equations is studied, the solution of which provides a harmonic field, the vector lines of which lie on a given surface of the considered type. The equation of the surface is written, for which it is possible to find a harmonic vector field with vector lines located on this surface. It is shown that for any surfaces of the type under consideration, one can find irrotational nonharmonic vector fields with vector lines located on a given surface. A number of surfaces are given for which harmonic or nonharmonic irrotational vector fields with vector lines located on these surfaces are obtained. Navier - Stokes equations for a viscous incompressible fluid in nondimensional form are considered. For this system, under the assumption that the velocity field is potential, a particular solution is written that ensures the location of the vector lines of the velocity field on a paraboloid of revolution.

Ульянов О.Н., Рубина Л.И.

Rubina L.I., Ul'yanov O.N.

An algorithm is proposed for obtaining solutions of partial differential equations with right-hand side defined on the grid $\{ x_{1}^{\mu}, x_{2}^{\mu}, \ldots, x_{n}^{\mu}\},\ (\mu=1,2,\ldots,N)\colon f_{\mu}=f(x_{1}^{\mu}, x_{2}^{\mu}, \ldots, x_{n}^{\mu}).$ Here $n$ is the number of independent variables in the original partial differential equation, $N$ is the number of rows in the grid for the right-hand side, $f=f( x_{1}, x_{2}, \ldots, x_{n})$ is the right-hand of the original equation. The algorithm implements a reduction of the original equation to a system of ordinary differential equations (ODE system) with initial conditions at each grid point and includes the following sequence of actions. We seek a solution to the original equation, depending on one independent variable. The original equation is associated with a certain system of relations containing arbitrary functions and including the partial differential equation of the first order. For an equation of the first order, an extended system of equations of characteristics is written. Adding to it the remaining relations containing arbitrary functions, and demanding that these relations be the first integrals of the extended system of equations of characteristics, we arrive at the desired ODE system, completing the reduction. The proposed algorithm allows at each grid point to find a solution of the original partial differential equation that satisfies the given initial and boundary conditions. The algorithm is used to obtain solutions of the Poisson equation and the equation of unsteady axisymmetric filtering at the points of the grid on which the right-hand sides of the corresponding equations are given.

Ульянов О.Н., Рубина Л.И.

Ульянов О.Н., Рубина Л.И.

Rubina L.I., Ul’yanov O.N.

A system of equations for the motion of an ionized ideal gas is considered. An algorithm for the reduction of this system of nonlinear partial differential equations (PDEs) to systems of ordinary differential equations (ODEs) is presented. It is shown that the independent variable ψ in the systems of ODEs is determined from the relation ψ = t + xf1(ψ) + yf2(ψ) + zf3(ψ) after choosing (setting or finding) the functions fi(ψ), i = 1, 2, 3. These functions are either found from the conditions of the problem posed for the original system of PDEs or are given arbitrarily to obtain a specific system of ODEs. For the problem on the motion of an ionized gas near a body, we write a system of ODEs and discuss the issue of instability, which is observed in a number of cases. We also consider a problem of the motion of flows (particles) in a given direction, which is of significant interest in some areas of physics. We find the functions fi(ψ), i = 1, 2, 3, that provide the motion of a flow of the ionized gas in a given direction and reduce the system of PDEs to a system of ODEs.

Rubina L.I., Ulyanov O.N.

We study the potential double wave equation and the system of spatial double wave equations. In the class of solutions of multiple wave type, these equations are reduced to an ODE and the system of ODEs respectively. We find some exact solutions and obtain formulas for the contact lines of the corresponding double waves with a simple wave, show that in a neighborhood of an arbitrary point in the plane of self-similar variables there exists a special flow of potential double wave type, and construct a spatial double wave type flow around a specified smooth body.

Рубина Л.И., Ульянов О.Н.

Rubina L.I., Ul’yanov O.N.

For Euler equations describing a steady motion of an ideal polytropic gas, we consider the problem of a flow around a body with known surface in the class of twice continuously differentiable functions. We use approaches of the geometric method developed by the authors. In the first part of the paper, the problem of a flow around a given body is solved in a special class of flows for which the continuity equation holds identically. We show that the class of solutions is nonempty and obtain one exact solution. In the second part of the paper, we consider the general case of stationary flows of an ideal polytropic gas. The Euler equations are reduced to a system of ordinary differential equations, for which we obtain an exact solution for a given pressure on the body. Examples illustrating the properties of the obtained exact solutions are considered. It is shown that such solutions make it possible to find points of a smooth surface of a body where blowups or strong or weak discontinuities occur.

Rubina L.I., Ul’yanov O.N.

Rubina L.I., Ul’yanov O.N.

We discuss the initial and boundary value problems for the system of dimensionless Navier–Stokes equations describing the dynamics of a viscous incompressible fluid using the method of characteristics and the geometric method developed by the authors. Some properties of the formulation of these problems are considered. We study the effect of the Reynolds number on the flow of a viscous fluid near the surface of a body.

Рубина Л.И., Ульянов О.Н.

Rubina L.I., Ul’yanov O.N.

Previously the authors developed a geometric method for studying and solving nonlinear equations and systems of equations with partial derivatives. This method is used in this paper to obtain a series of exact solutions to certain nonlinear acoustics equations, as well as to reduce the system of Euler equations to systems of common differential equations.

Баутин С.П., Вазиева И.А.

В работе рассматривается нелинейное уравнение теплопроводности в одномерном плоскосимметричном случае. Для него на отрезке [0; p] ставится задача Коши с непрерывными начальными данными. Эти данные четным образом продолжаются на отрезок [–p; 0], а затем с периодом 2p на всю числовую ось. Решение получившейся задачи Коши представляется в виде соответствующего тригонометрического ряда по косинусам от пространственной переменной. Коэффициенты ряда являются искомыми функциями от времени. Для этих коэффициентов приведена бесконечная система обыкновенных дифференциальных уравнений с соответствующими начальными условиями. Построены конечные отрезки тригонометрических сумм, приближенно передающие решения рассматриваемых задач Коши

Kazakov A., Lempert A.

The paper concerns a nonlinear second-order parabolic evolution equation, one of the well-known objects of mathematical physics, which describes the processes of high-temperature thermal conductivity, nonlinear diffusion, filtration of liquid in a porous medium and some other processes in continuum mechanics. A particular case of it is the well-known porous medium equation. Unlike previous studies, we consider the case of several spatial variables. We construct and study solutions that describe disturbances propagating over a zero background with a finite speed, usually called ‘diffusion-wave-type solutions’. Such effects are atypical for parabolic equations and appear since the equation degenerates on manifolds where the desired function vanishes. The paper pays special attention to exact solutions of the required type, which can be expressed as either explicit or implicit formulas, as well as a reduction of the partial differential equation to an ordinary differential equation that cannot be integrated in quadratures. In this connection, Cauchy problems for second-order ordinary differential equations arise, inheriting the singularities of the original formulation. We prove the existence of continuously differentiable solutions for them. A new example, an analog of the classic example by S.V. Kovalevskaya for the considered case, is constructed. We also proved a new existence and uniqueness theorem of heat-wave-type solutions in the class of piece-wise analytic functions, generalizing previous ones. During the proof, we transit to the hodograph plane, which allows us to overcome the analytical difficulties.

Stȩpień Ł.T.

Some exact solutions of boundary or initial conditions formulated for Bogomolny equations (derived by using the strong necessary conditions and associated with some ordinary equation and some partial differential equations) have been found. The solution obtained for the restricted baby Skyrme model, as well the density of energy for this solution, are localized. Moreover, it turns out that the densities of the ungauged Hamiltonian and the gauged Hamiltonian are correspondingly, non-zero and zero for the found solution of the Cauchy problem associated with the Bogomolny equation of the restricted baby Skyrme model. Hence, a degeneracy of the Hamiltonian for this model has been established. As such, one can see the breaking of some symmetry.

Ul’yanov O.N., Rubina L.I.

A system of equations of unsteady spatial free convection of an incompressible viscous fluid in the Boussinesq approximation is considered. The analysis is based on the methods of reduction of linear and nonlinear partial differential equations (PDEs) and systems of PDEs to ordinary differential equations (ODEs) and systems of ODEs. These methods were proposed by the authors earlier, and their general principles are given in the paper. The methods are based on the construction of a system of equations of characteristics for a first-order PDE (the basic equation). This equation is constructed in a certain way by analyzing the original system of equations. The reductions lead to ODEs or systems of ODEs in which an independent variable  $$\psi$$ is such that the equation $$\psi(x,y,z,t)=\mathrm{const}$$ defines a level surface for all unknown functions of the original system of PDEs. The methods are applicable to PDEs and systems of PDEs regardless of their type. The Oberbeck–Boussinesq equations are reduced to a system of ODEs with a functional arbitrariness, and an exact solution with a constant arbitrariness is found for the original system. The functional arbitrariness in the constructed reduction also yielded a system of ODEs in which the temperature  $$T$$ is an independent variable. For this system exact solutions are found. A possible (vortex or vortex-free) motion of an incompressible fluid with free convection is analyzed. The cases of vortex and vortex-free motion of the fluid are identified. An exact solution defining a vortex-free motion of the fluid is written as a result of reductions for the original system of PDEs.

Kazakov A.L., Kuznetsov P.A., Spevak L.F.

The study of nonlinear singular parabolic equations occupies a key place in the scientific school of A. F. Sidorov. In particular, the problem on initiating a heat wave has been studied since the 1980s. The present study aims to extend the results of Sidorov and his followers, including the authors, to the case of systems of the corresponding type. We find that the heat (diffusion) wave for the system considered has a more complex (three-part) structure, which follows from the fact that the zero fronts are different for the unknown functions. A theorem on the existence and uniqueness of a piecewise analytical solution, which has the form of special series, is proved. We find an exact solution of the desired type, the construction of which is reduced to the integration of ordinary differential equations (ODEs). We managed to integrate the ODEs by quadratures. In addition, we propose an algorithm based on the collocation method, which allows us to effectively construct an approximate solution on a given time interval. Illustrative numerical calculations are performed. Since we have not managed to prove the convergence in this case (this is far from always possible for nonlinear singular equations and systems), exact solutions, both obtained in this paper and previously known, have been used to verify the calculation results.

Krasin G.K., Stsepuro N.G., Kovalev M.S., Danilov P.A., Kudryashov S.I.

Abstract The bulk mapping of natural diamond poses problems where it is required to characterize various defects and measure their optical properties in volume. The combination of photoluminescence spectroscopy methods and methods for detecting the state of polarization in the volume will expand the functionality for mapping natural and artificial diamonds. The implemented methods will be an effective tool for the structural description of diamond optical centers.

Stȩpień Ł.T.

Some new classes of exact solutions (so-called functionally-invariant solutions) of the elliptic and hyperbolic complex Monge-Amp$\grave{e}$re equations and of the second heavenly equation, mixed heavenly equation, asymmetric heavenly equation, evolution form of second heavenly equation, general heavenly equation, real general heavenly equation and one of the real sections of general heavenly equation, are found. Besides non-invariance of these found classes of solutions has been investigated. These classes of solutions determine the new classes of metrics without Killing vectors. A criterion of non-invariance of the solutions belonging to found classes, has been also formulated.

Rubina L.I., Ul’yanov O.N.

A system of equations for the motion of an ionized ideal gas is considered. An algorithm for the reduction of this system of nonlinear partial differential equations (PDEs) to systems of ordinary differential equations (ODEs) is presented. It is shown that the independent variable ψ in the systems of ODEs is determined from the relation ψ = t + xf1(ψ) + yf2(ψ) + zf3(ψ) after choosing (setting or finding) the functions fi(ψ), i = 1, 2, 3. These functions are either found from the conditions of the problem posed for the original system of PDEs or are given arbitrarily to obtain a specific system of ODEs. For the problem on the motion of an ionized gas near a body, we write a system of ODEs and discuss the issue of instability, which is observed in a number of cases. We also consider a problem of the motion of flows (particles) in a given direction, which is of significant interest in some areas of physics. We find the functions fi(ψ), i = 1, 2, 3, that provide the motion of a flow of the ionized gas in a given direction and reduce the system of PDEs to a system of ODEs.

Rubina L.I., Ulyanov O.N.

We study the potential double wave equation and the system of spatial double wave equations. In the class of solutions of multiple wave type, these equations are reduced to an ODE and the system of ODEs respectively. We find some exact solutions and obtain formulas for the contact lines of the corresponding double waves with a simple wave, show that in a neighborhood of an arbitrary point in the plane of self-similar variables there exists a special flow of potential double wave type, and construct a spatial double wave type flow around a specified smooth body.

Rubina L.I., Ul’yanov O.N.

For Euler equations describing a steady motion of an ideal polytropic gas, we consider the problem of a flow around a body with known surface in the class of twice continuously differentiable functions. We use approaches of the geometric method developed by the authors. In the first part of the paper, the problem of a flow around a given body is solved in a special class of flows for which the continuity equation holds identically. We show that the class of solutions is nonempty and obtain one exact solution. In the second part of the paper, we consider the general case of stationary flows of an ideal polytropic gas. The Euler equations are reduced to a system of ordinary differential equations, for which we obtain an exact solution for a given pressure on the body. Examples illustrating the properties of the obtained exact solutions are considered. It is shown that such solutions make it possible to find points of a smooth surface of a body where blowups or strong or weak discontinuities occur.

Rubina L.I., Ul’yanov O.N.

We discuss the initial and boundary value problems for the system of dimensionless Navier–Stokes equations describing the dynamics of a viscous incompressible fluid using the method of characteristics and the geometric method developed by the authors. Some properties of the formulation of these problems are considered. We study the effect of the Reynolds number on the flow of a viscous fluid near the surface of a body.

Rubina L.I., Ul’yanov O.N.

Previously the authors developed a geometric method for studying and solving nonlinear equations and systems of equations with partial derivatives. This method is used in this paper to obtain a series of exact solutions to certain nonlinear acoustics equations, as well as to reduce the system of Euler equations to systems of common differential equations.

Rubina L.I., Ul’yanov O.N.

A method for reducing systems of partial differential equations to corresponding systems of ordinary differential equations is proposed. A system of equations describing two-dimensional, cylindrical, and spherical flows of a polytropic gas; a system of dimensionless Stokes equations for the dynamics of a viscous incompressible fluid; a system of Maxwell’s equations for vacuum; and a system of gas dynamics equations in cylindrical coordinates are studied. It is shown how this approach can be used for solving certain problems (shockless compression, turbulence, etc.).

Rubina L.I., Ul’yanov O.N.

Some exact solutions to a nonlinear heat equation are constructed. An initial-boundary value problem is examined for a nonlinear heat equation. To construct solutions, the problem for a partial differential equation of the second order is reduced to a similar problem for a first order partial differential equation.

Ульянов О.Н., Рубина Л.И.

Lappa M.

Burmasheva N.V., Prosviryakov E.Y.

The paper considers an exact solution to the equations of thermal diffusion of a viscous incompressible fluid in the Boussinesq approximation with neglect of the Dufour effect for a steady shear flow. It is shown that the reduced system of constitutive relations is nonlinear and overdetermined. A nontrivial exact solution of this system is sought in the Lin–Sidorov–Aristov class. The resulting family of exact solutions allows one to describe steady-state inhomogeneous shear flows. This class generalizes the classical Couette, Poiseuille, and Ostroumov–Birikh solutions. It is demonstrated that the system of ordinary differential equations reduced within this class retains the properties of nonlinearity and overdetermination. A theorem on solvability conditions for the overdetermined system is proved; it is reported that, when these conditions are met, the solution is unique. The overdetermined system is solvable owing to the algebraic identity relating the horizontal velocity gradients, which are linear functions of the vertical coordinate. The constructive proof of the computation of hydrodynamic fields consists in the successive integration of the polynomials, the polynomial degree being dependent on the values of the boundary parameters.

Mayeli P., Sheard G.J.

The well-known Boussinesq (also known as Oberbeck—Boussinesq) approximation is still the most common approach for the numerical simulation of natural convection problems. However, the accurate performance of this approximation is mainly restricted by small temperature differences. This encourages researchers and engineers to use other approaches beyond the range of validity of the Boussinesq approximation, especially when buoyancy-driven flows are generated by large temperature differences. This paper assembles and classifies the various approaches for numerical simulation of laminar natural convection, including Boussinesq and non-Boussinesq approximations for Newtonian fluids. These classifications reside under two overarching classes capturing compressible and incompressible approaches, respectively. This review elaborates on the different approaches and formulations adopted within each category.

Rubina L.I., Ul’yanov O.N.

Previously the authors developed a geometric method for studying and solving nonlinear equations and systems of equations with partial derivatives. This method is used in this paper to obtain a series of exact solutions to certain nonlinear acoustics equations, as well as to reduce the system of Euler equations to systems of common differential equations.

Barna I.F., Mátyás L.

In this article we will present pure two-dimensional analytic solutions for the coupled non-compressible Newtoniain Navier-Stokes --- with Boussinesq approximation --- and the heat conduction equation. The system was investigated from E.N. Lorenz half a century ago with Fourier series and pioneered the way to the paradigm of chaos. We present a novel analysis of the same system where the key idea is the two-dimensional generalization of the well-known self-similar Ansatz of Barenblatt which will be interpreted in a geometrical way. The results, the pressure, temperature and velocity fields are all analytic and can be expressed with the help of the error functions. The temperature field has a strongly damped oscillating behavior which is an interesting feature.

Perepelkin E.E., Repnikova N.P., Inozemtseva N.G.

We obtain an exact solution of the space charge problem for a uniformly charged ball moving in an exterior constant homogeneous electric field. The results obtained can be used as tests in the numerical simulation of the effect of a space charge of actual beams and for estimating the accuracy of the numerical methods used in the solution of the space charge problem.

Rubina L.I., Ul’yanov O.N.

A method for reducing systems of partial differential equations to corresponding systems of ordinary differential equations is proposed. A system of equations describing two-dimensional, cylindrical, and spherical flows of a polytropic gas; a system of dimensionless Stokes equations for the dynamics of a viscous incompressible fluid; a system of Maxwell’s equations for vacuum; and a system of gas dynamics equations in cylindrical coordinates are studied. It is shown how this approach can be used for solving certain problems (shockless compression, turbulence, etc.).

Andreev V.K., Stepanova I.V.

The thermodiffusion convection equations with nonlinear buoyancy force are studied. Some particular solutions for describing of thermoconcentration flows are found. Two boundary value problems are solved for thermal convection case. The comparison with the linear Oberbeck-Boussinesq model is given.

Kraiko A.N., P’yankov K.S., Yakovlev Y.A.

Rubina L.I., Ul’yanov O.N.

Some exact solutions to a nonlinear heat equation are constructed. An initial-boundary value problem is examined for a nonlinear heat equation. To construct solutions, the problem for a partial differential equation of the second order is reduced to a similar problem for a first order partial differential equation.

Andreev V.K., Gaponenko Y.A., Goncharova O.N., Pukhnachev V.V.

Pukhnachev V.V., Todorov M.D., Christov C.I.

Views Icon Views Article contents Figures & tables Video Audio Supplementary Data Peer Review Share Icon Share Twitter Facebook Reddit LinkedIn Tools Icon Tools Reprints and Permissions Cite Icon Cite Search Site Citation V. V. Pukhnachev; Group‐theoretical Methods in Convection Theory. AIP Conference Proceedings 29 November 2011; 1404 (1): 27–38. https://doi.org/10.1063/1.3659901 Download citation file: Ris (Zotero) Reference Manager EasyBib Bookends Mendeley Papers EndNote RefWorks BibTex toolbar search Search Dropdown Menu toolbar search search input Search input auto suggest filter your search All ContentAIP Publishing PortfolioAIP Conference Proceedings Search Advanced Search |Citation Search

Ibragimov N.H., Meleshko S.V., Rudenko O.V.

Kudryavtsev A.G., Sapozhnikov O.A.

A method of determining the exact solutions to the Burgers equation on the basis of the Darboux transformation is described. It is shown that a single application of the Darboux transformation to the homogeneous Burgers equation transforms the latter into the inhomogeneous equation describing acoustic wave propagation against transonic flow in the de Laval nozzle. In this case, the contraction ratio of the nozzle is fixed and determined by the viscosity coefficient of the medium. Based on the exact solution of the homogeneous Burgers equation, for the aforementioned problem of the flow in the nozzle, all the possible regular steady-state solutions are presented and the evolution of nonstationary solutions is investigated. The algorithm of a multiple Darboux transformation, which allows an increase in the strength of inhomogeneity, i.e., in the contraction ratio of the nozzle, is determined. This approach leads to a discrete set of possible contraction ratios at which exact solutions can be obtained. The Crum’s theorem is used to derive a formula that allows determination of the exact solutions to the inhomogeneous Burgers equation from the solutions to the homogeneous heat transfer equation. It is noted that, in fact, the proposed algorithm of the multiple Darboux transformation makes it possible to decrease the viscosity coefficient of the medium in a discrete way.