The asymptotic solutions for boundary value problem to a convec- tive diffusion equation with chemical reaction near a cylinder

Latin American Journal of Solids and Structures 10(2013) 123 – 131 Abstract The work deals with a boundary value problem for a quasilinear partial elliptical equation. The equation describes a stationary process of convective diffusion near a cylinder and takes into account the value of a chemical reaction for large Peclet numbers and for large constant of chemical reaction. The quantity the rate constant of the chemical reaction and Peclet number is assumed to have a constant value. The leading term of the asymptotics of the solution is constructed in the boundary layer as the solution for the quasilinear ordinary differential equation. In this paper, we construct asymptotic expansion of solutions for a quasilinear partial elliptical equation in the boundary layer near the cylinder.


The asymptotic solutions for boundary value problem to a convective diffusion equation with chemical reaction near a cylinder 1 INTRODUCTION
The stationary convective diffusion equation in the presence of a bulk chemical reaction is given by (e.g., see [1,2]) where b Professor: Bashkortostan State Pedagogical University, Oktyabr'skoy Revolutsii str.3a,Ufa, 450000, Russia.

( , )
sin rr r (1.4) is the stream function [3], r and are polar coordinates, is the Laplace operator, Pe is the Peclet number, and k is parameter depending on the chemical reaction rate.The angle is measured relative to the free-stream direction.
Problems analogous to (1.1) and (1.2), and a broader class of problems, were considered in [1,2], [4-6, 9, 10, 21, 22, 24 ].In the absence of chemical reaction, problem (1.1) and (1.2) was analyzed in [4,5] by the method of matched asymptotic expansions [7,8,23].It is well known (see, for example, [2, Chapter 5, (6.1)-(6.3)])that, in the limit cases In the case when the volume chemical reaction of the first order (F(u) = u) the asymptotics of solution in all space outside the drop was constructed in [9].In study, the number k Pe is assumed to have a constant value.
It is assummed that F(C) is continuous and and the asymptotic is

THE DIFFUSION BOUNDARY LAYER
In this report the quantity k Pe is assumed to have a constant value.In this case, all terms in Eq. ( 1) are similar in order of magnitude in the neighborhoods of saddle points.The small parameter 1/2 () Pe is introduced for convenience, and Eq. ( 1) is rewritten as The asymptotic expansions (AE) of the solution in the diffusion boundary layer was considered in a earlier study [11].This solution was continued up to the front stagnation point The asymptotics of the solution to the problem (2.3), (2.4) function 0 ( , ) ut as is [11] 22 0,0 ( ) where 2 0,0 ( ) 3 THE ASYMPTOTICS 0 ( , ) ut AS 0 .
The asymptotics of the function where the function 0 () Vt is constructed [12] for small as the solution for the problem where . The idea of the proof is similar to works [13,14].Let us search the function 0 () Vt in the form of the sum where ) For construction the solution () wx of the problem (3.8), (3.9) we obtain integral equation  We proceed by applying the method of successive approximations.then by formulas (3.10), (3.13) -(3.18) we have

NUMERICAL SOLUTION AND FINDING THE CONSTANT c .
We rewrite Eq. (3.2) in the form of the system Consider system (4.1) on the interval [0, 0 X ], Following [17,18], we first discuss the stability conditions for the explicit Euler scheme Replacing () n Fv by the sum v and assuming that 00 ,, tv and 0 z are known and , we find a solution to difference scheme (4.2).The stability condition for difference scheme (4.2) is fulfilled [17][18][19] if This implies that one should take 0 X and integrate backwards (i.e., with increments The initial conditions at the point 0 X the form where 00 , VZ are found from (3.4)

CONCLUSION
The problem considered in this work describes the phenomenon convective diffusion in the neighborhood the cylinder taking into account chemical reaction, a streamlined cross-flow of ideal fluid.In the vicinity of the cylinder there are few boundary layers.In this work the problem was investigated in the diffusion boundary layer.The leading term the asymptotic solution in the vicinity of the saddle point corresponding to the point of the liquid dripping with a cylinder has the form Were previously considered the problem of convective diffusion in the neighborhood a spherical particle and a cylinder in the flow of a viscous incompressible fluid.In these cases, there are also a few boundary layers.If we compare the results obtained in the diffusion boundary layer, the structure of the asymptotics in the vicinity of the saddle point about spherical particle is simpler than those considered in this work and have the form [17].A similar structure the asymptotics takes place in the flow around a cylinder cross-flow of a viscous incompressible fluid [26].
This work can be extended taking into account thermal conductivity of the same article [22].Mathematical problems arising in this work and require further study in the internal boundary layers, containing the singular points of saddle type, there are also problems of hydrodynamics.Similar problems have been studied in [20], [25].
Maksimova and Akhmetov / The asymptotic solutions for boundary value problem to a convective diffusion equation with chemical reaction… of the asymptotics (up to a linear change of variable t) takes place in the diffusion boundary layer of convection-diffusion problem about the drop, streamlined flow of a viscous incompressible fluid.