METHOD OF NUMERICAL ANALYSIS OF THE PROBLEM OF MASS TRANSFER OF A CYLINDRICAL BODY WITH THE UNIFORM TRANSLATIONAL FLOW

1Ph.D, Associate Professor of the Department of Advanced Mathematics of O. M. Beketov National University of Urban Economy in Kharkiv, Kharkiv, Ukraine 2Ph.D, Associate Professor, Associate Professor of the Department of Applied Mathematics of Kharkiv National University of Radio Electronics, Kharkiv, Ukraine 3Ph.D, Associate Professor of the Department of Advanced Mathematics of O. M. Beketov National University of Urban Economy in Kharkiv, Kharkiv, Ukraine

)} , ( { ϕ φ r i -a sequence of functions that is complete with respect to the whole plane; 1 Φ , 2 Φ -undefined components of the solution structure; ) , ( ϕ ψ = ψ r -the stream function; ω -a sufficiently smooth function built by using the constructive apparatus of the R-functions theory and describing the geometry of the domain Ω ; 0 = ω -a normalized equation of Ω ∂ ; Ω -the flow domain;

INTRODUCTION
The problems of calculating viscous flows, which are complicated by mass transfer, are widely used in heat power engineering, chemical and food technologies, geo-and astrophysical studies, and environmental protection. Many processes of chemical technology are associated with fluid movement in process equipment. In the preparation of reagents and in the isolation of reaction products, such operations as leaching, absorption, extraction and distillation play an important role. The laws of hydrodynamics, heat and mass transfer are essential for the entire technological process. Processes of heat and mass transfer are also ones of the most important in the energy sector, as well as in a number of the technological processes in the metallurgical and other industries. In addition, the problems of mass transfer of bodies with a uniform viscous flow underlie the calculation of many technological processes, which are associated with dissolution, extraction, evaporation, precipitation of colloids, etc. Therefore, the development of new, as well as the improvement of existing methods of mathematical modeling and numerical analysis of external stationary problems of hydrodynamics of a viscous incompressible fluid, which take into consideration the mass transfer, is an actual scientific issue.
The object of this study is the stationary hydrodynamic process of flow past bodies by a viscous incompressible fluid, complicated by mass transfer and described by a system of equations with respect to the stream function and concentration.
The subject of this study is a mathematical model of the stationary task of flow past body by a viscous incompressible fluid with allowance for mass transfer, and a method for its numerical analysis.
The purpose of the work is to develop a new method for mathematical modeling of the mass transfer of a cylindrical body with a uniform translational flow which based on the R-functions method and Galerkin method.

PROBLEM STATEMENT
At small Peclet numbers, to describe the mass transfer process the Oseen approximation (equation 1) is used [1]: In the general case, the mass transfer process is described by the equation for the concentration (equation 2) in the form [1,2]: In equations (1) and (2) Equations (1) and (2) in a rectangular coordinate system have the following form: Equations (1) and (2) should be supplemented by the boundary condition on Ω ∂ and the condition at infinity [1]: The stream function ) , ( ϕ ψ r can be found, for example, as the solution of the following nonlinear task of flow past a cylindrical body by a viscous incompressible fluid [3][4][5][6]: The task (1), (5), (6) does not depend on the stream function ) , ( ϕ ψ r , and the solution of the task (2), (5), (6) consists of two steps: а) the search of the stream function as a solution of the task (7)-(9); b) the solution of the task for concentration.

REVIEW OF THE LITERATURE
Various tasks which are arising in the study of external viscous fluid flows can be investigated theoretically or by means of physical experiment. At present, mathematical modeling and computational experiment are increasingly МАТЕМАТИЧНЕ ТА КОМП'ЮТЕРНЕ МОДЕЛЮВАННЯ being used in the study of hydrodynamic problems. Basically, finite difference method, finite element method, boundary integral element method and others are used for the numerical analysis of such problems. These methods are easy to implement, but do not have the necessary property of universality: when moving to a new area (especially non-classical geometry), it is necessary to generate a new grid, and often to replace complex sections of the boundary with simple ones, composed of, for example, straight line segments. The use of the R-functions structuralvariational method [7,8] by V. L. Rvachev, the Academician of Ukrainian National Academy of Sciences, is an alternative to the existing methods of numerical computation of hydrodynamic problems.
The R-functions method in computational hydrodynamics was applied in [9][10][11][12][13][14]. The task of viscous fluid external flow around bodies of revolution in a spherical coordinate system, which is complicated by mass transfer, was solved in [15,16] with using the R-functions method.
In this study, we propose to apply the R-functions and Galerkin method for mathematical modeling the problem of mass transfer of a cylindrical body with a uniform translational flow.

MATERIALS AND METHODS
The method for solving the task for the stream function, based on the application of R-functions, successive approximations and Galerkin method, is described in [17,18]. Substituting the stream function ) , ( ϕ ψ r so obtained into equation (2), let us solve the problems (1), (5), (6) and (2), (5), (6) by the R-functions method. To do this, using the constructive means of the R-functions theory [7,8] let us construct the structure of the boundary value task solution, i.e. the functions bundle that exactly satisfies the boundary condition and the condition at infinity.
Let us consider a sufficiently smooth function [19] ⎪ ⎩ The function (10) ) , and satisfies the conditions: In the problems (3), (5), (6), and (4)-(6) we make the replacement The choice of such a substitution is due to the fact that the function ) satisfies the boundary condition (5) and the condition at infinity (6).
Then for l u , 2 , 1 = l , we obtain the tasks Let us find the generalized solution u of tasks (11)-(13) as the limit, when ∞ → n , of solutions n u of equations (11), which are considered in a sequence of domains } { n Ω , that is a monotonic exhaustion of an infinite area Ω.
In domains n Ω we will consider the boundary value tasks l l n l l n where functions l n u , are continued by zero outside of n Ω .
An approximate solution of tasks (14) -(15) for each ,... 2 , 1 = n according to the Bubnov-Galerkin method, will be sought in the form: The numbers where To construct the coordinate sequence, the complete system of particular solutions of the Laplace equation [20] and the R-functions method [7,8] will be used.
We have proved that for any choice of sufficiently smooth functions 1 Φ , 2 Φ and at the requirement that exactly satisfies the boundary conditions (12) and (13), that is, it is the structure of the solution of the boundary value problem (11)- (13).
Approximations of the functions 1 Φ и 2 Φ in the domain n Ω will be seek in the form Then the sequence of functions ,which is complete with relatively to the whole plane, has the form: The values of the coefficients k α ( 1 ,..., 2 , 1 m k = ) and j β ( 2 ,..., 2 , 1 m j = ) in accordance with the Bubnov-Galerkin method will be found from the condition of residual orthogonality to the first N ( 2 1 m m N + = ) elements of the sequence (18), which leads to a system of linear algebraic equations in the form (17).
We have proved the convergence of the Galerkin approximations

RESULTS
The concentration lines for a circular cylinder are shown in Fig. 1-3. Fig. 4-6 show the concentration lines for elliptical cylinder.

DISCUSSION
At small Reynolds and Peclet numbers the substance is transported uniformly, dissolving into the liquid. As the Reynolds and Peclet numbers increase, the particles of substance begin to move with the flow. These results are consistent with the physics of the process. The efficiency of the proposed method for a spherical coordinate system was verified on the problem of flow past a sphere [13], for which an exact solution is known [20].

CONCLUSIONS
The numerical method for calculating the mass transfer of a cylindrical body with a uniform translational flow, which based on the joint application of the R-functions method and Galerkin method, is proposed for the first time in this study. By using the R-functions method the structure of the solution of the problems of flow past bodies with allowance for mass transfer, which precisely satisfying the boundary condition on the boundary and the condition at infinity, was constructed, and this made it possible to lead tasks in the infinite domain to tasks in the finite domain. To approximate the uncertain components of the solution structure, the Galerkin method was applied. The stationary problem of flow past a cylindrical body in a cylindrical coordinate system for a circular and elliptical cylinders has been solved numerically for various Reynolds and Peclet numbers.

ACKNOWLEDGMENTS
The work was carried out at the department of Applied Mathematics at Kharkiv National University of Radio Electronics and department of Advanced Mathematics at O. M. Beketov National University of Urban Economy in Kharkiv within the framework of collaborative scientific research conducted by the departments.