Symmetry Reduction and Numerical Solution of Von K á rm á n Swirling Viscous Flow

In this paper, the numerical solutions of von Kármán swirling viscous flow are obtained based on the effective combination of the symmetry method and the Runge-Kutta method. Firstly, the multi-parameter symmetry of von Kármán swirling viscous flow is determined based on the differential characteristic set algorithm. Secondly, we used the symmetry to reduce von Kármán swirling viscous flow to an initial value problem of the original differential equations. Finally, we numerically solve the initial value problem of the original differential equations by using the Runge-Kutta method.

As it is well known, the similarity transformation is used frequently in solving nonlinear PDEs problems.The Lie transformation group of PDEs can yield a more general form of similarity transformation, and these transformations have more significance for mathematics and physics.So the symmetry method has a higher superiority than the similarity transformation in BVP of nonlinear PDEs.At present, combining the symmetry method with other methods to solve BVP of the nonlinear PDEs are the new research subjects.Recently, we have studied this topic based on the differential characteristic set algorithm [26][27][28].
It is an new research to the application of the Lie symmetry method in the BVP for nonlinear PDEs in fluid mechanics.We will study the symmetry reduction and the numerical solutions of von K árm án swirling viscous flow based on the effective combination of the Lie symmetry method and the Runge-Kutta method.This investigation will widen the application of Lie symmetry.The rest of the paper is organized as follows.In Section 2, we give the formulation of invariance for a BVP of PDEs.In Section 3, we obtain the symmetry of von K árm án swirling viscous flow based on the differential characteristic set algorithm and then we reduce it.In Section 4, we give numerical solutions of von

Formulation of Invariance for a BVP of PDEs
Consider a BVP for kth order scalar PDEs (k ≥ 2) (where ) are m dependent variables and defined on a domain Ω x in x-space with boundary conditions prescribed on boundary surfaces Assume that BVP (1)-( 3) has a unique solution.Consider an infinitesimal generator of the form which defines a one-parameter Lie group of transformations in x-space as well as in (x, u)-space.

The Symmetry and Symmetry Reduction of von K árm án sWirling Viscous Flow
Let us consider von K árm án swirling viscous flow which is a famous classical problem in fluid mechanics.The governing equations are as follows: where V r , V θ , V z , p are all functions of r, θ, z. ρ is the fluid density, ν is the kinematic viscosity, p is the pressure.The boundary conditions of Equations ( 11)-( 14) are where ) are the functions of (r, θ) to be determined later.

First Symmetry Reduction
The symmetry group of Equations ( 11)-( 14) will be generated by the vector field of the form where are the infinitesimal functions of the symmetry.We obtain the determining equations of symmetry (19) by using the Lie algorithm, but it is too difficult to get their solutions.However, we use the differential characteristic set algorithm to obtain the following equivalent system of the determining equations [29].
By solving the above PDEs, we get where a 1 , a 2 , a 3 are arbitrary symmetry parameters, and f (θ) is an arbitrary function.Then the corresponding infinitesimal vector has the following form The characteristic equations for the symmetry X 1 are as follows By solving dr a 1 r = dθ a 2 and dθ a 2 = dz a 1 z+a 3 , we obtain two invariants as follows By using the invariant form method, we get the solutions of Equations ( 23) By substituting (25) into Equations ( 11)-( 14), we obtain PDEs as follows According to invariance for a BVP of the PDEs, the symmetry X 1 leaves the boundary conditions ( 15)-( 18) invariant, namely

Second Symmetry Reduction
By the same manner, we get the following infinitesimal vector of symmetry for the system ( 26)- (29), where b 1 , b 2 are arbitrary symmetry parameters.
In the following, the BVP for PDEs ( 26)-(49) will be reduced to the initial value problem of the ordinary differential equations (ODEs) by using the invariant form method [2].

Conclusions
In this paper, the application of the symmetry method on BVP for nonlinear PDEs is studied.Firstly, we have got the multi-parameter symmetry of von K árm án swirling viscous flow based on the differential characteristic set algorithm.Via twice symmetry reducions, BVP (11)-( 18) became an initial value problem of ODEs.Secondly, we solved numerically the initial value problem of ODEs by using the Runge-Kutta method.The differential characteristic set algorithm is a key factor which influences the calculation of the symmetry of PDEs.
We considered that the boundary conditions are the arbitrary functions B i (r, θ).However, B i (r, θ) are determined by using the invariance of the boundary conditions under a multi-parameter Lie group of transformations.This approach is different from other research.For example, the boundary conditions are given by the following forms V r (r, θ, 0) = 0, V z (r, θ, 0) = 0 in [30].The Lie symmetry and Runge-Kutta methods are effective methods which are applied to solving PDEs.Hence, their combination will advance the availability of solutions.At present, it is very valuable to solve nonlinear PDEs by combining the symmetry method, the differential characteristic set algorithm and other methods.