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

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

N. V. Maksimova^I; R. G. Akhmetov^II

^IM.Sc: Bashkortostan State Pedagogical University, Oktyabr'skoy Revolutsii str.3a, Ufa, 450000, Russia

^IIProfessor: Bashkortostan State Pedagogical University, Oktyabr'skoy Revolutsii str.3a, Ufa, 450000, Russia

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.

Keywords: convective diffusion equation, the method of matched asymptotic expansions, the diffusion boundary layer, the saddle point, the stream function, quasilinear parabolic degenerate equation, the stability condition for difference scheme.

1 INTRODUCTION

The stationary convective diffusion equation in the presence of a bulk chemical reaction is given by (e.g., see [1, 2])

where

is the stream function [3], r and are polar coordinates, is the Laplace operator, Pe is the Peclet number, and 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 , ; and , the solution to problem (1.1) and (1.2) is simplified.

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 is assumed to have a constant value.

It is assummed that F(C) is continuous and and the asymptotic is

and the asymptotic is

2 THE DIFFUSION BOUNDARY LAYER

In this report the quantity 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 is introduced for convenience, and Eq. (1) is rewritten as

When the Eq. (1) has the suddle points and and the equation is equivalent the dynamical system.

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 (up to the line ). The natural variables in the diffusion boundary layer are . The AE of the solution is sought as

From (2.1), (2.2) and (1.1) - (1,4), in variables , determining in the domain , we obtain the problem

The asymptotics of the solution to the problem (2.3), (2.4) function as is [11]

where

.

3 THE ASYMPTOTICS AS .

The asymptotics of the function as is sought in the view

where the function is constructed [12] for small as the solution for the problem

Theorem 1. Let F(u) satisfies conditions (1.5), (1.6) and , then at the solution of the equation (3.2) asymptotics holds

where

, , , ,

, , .... .

The idea of the proof is similar to works [13, 14]. Let us search the function in the form of the sum

where

.

Substituting sum (3.5) into equation (3.2), we obtain the problem

where , -sufficiently small.

Let's consider the problem

where problem (3.6), (3.7) is equivalent to a problem (3.8), (3.9) and

For construction the solution of the problem (3.8), (3.9) we obtain integral equation

where are linearly independent solutions to the linear homogeneous equation:

is the Wronskian.

We have asymptotics for , , using the results of the works [15, 16]

where - is small. Such solutions of the equation (3.12) exists.

For we find estimate

We proceed by applying the method of successive approximations.

We choose

From (3.6), (3.13) - (3.17) it is find estimate

then by formulas (3.10), (3.13) - (3.18) we have

From (3.19) we obtain , and .

There exist that for solution of the equation (3.11) inequality is hold .

4 NUMERICAL SOLUTION AND FINDING THE CONSTANT e .

We rewrite Eq. (3.2) in the form of the system

Consider system (4.1) on the interval [0, ], Following [17, 18], we first discuss the stability conditions for the explicit Euler scheme

Replacing by the sum and assuming that and are known and , we find a solution to difference scheme (4.2).

The stability condition for difference scheme (4.2) is fulfilled [17-19] if .

This implies that one should take and integrate backwards (i. e., with increments ) in the interval [0,]. The initial conditions at the point the form

where are found from (3.4)

,

For example, according to (1.6) let

The results of the numerical analysis of the problem (4.1), (4.3) for are (for , )

= 0.5, = 1.7216, z(0) = -0.1173; = 1, = 1.3943, z(0) = -1.3943;

= 1.5, = 1.0452, z(0) = -0.2994; = 2, = 0.6737, z(0) = -0.3747.

This research was supported by the Presidents Grant for Leading Scientific Schools of the Russian Federation, project no. NSh - 6249.2010.1, and by the Federal Task Program "Research and educational professional community of innovation Russia" (contract 02.740.11.0612).

5 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

at where

A similar structure 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.

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].

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Publication Dates

• Publication in this collection
  25 Feb 2013
• Date of issue
  Jan 2013