THE VORTEX FORMATION IN A HORIZONTAL FINITE CYLINDER BY ALTERNATING ELECTRIC CURRENT 1

. We have investigated and calculated the distribution of electromagnetic ﬁelds and forces induced by a three-phase axially-symmetric system of electric current with six electrodes in a cylinder of a ﬁnite length [1, 3]. In this paper the alternating current is fed to each of nine discrete circular conductors-electrodes, which are placed on the internal wall of the cylinder. This new mathematical model describes a real device [7], which transforms electric energy into heat. The viscous incompressible ﬂow of weakly electroconductive liquid-electrolyte is obtained by the ﬁnite-difference method, using the monotonous vector ﬁnite difference schemes. The average axially-symmetric motion of electrolyte and vortex distribution in a cylinder has been considered. The dependence of electromagnetic forces and velocity distribution at the inlet of the cylinder is investigated in the case of: 1) the vortex formation from the Lorentz


Introduction
In many technological applications it is important to mix an electroconductive liquid using various magnetic fields [4].In this work we consider a finite cylinder Ω = (r, z) : 0 < r < a, 0 < z < Z with nine metal coils-electrodes L i = (r, z) : r = a, z = z i , 0 < z i < Z, i = 1, . . ., 9 , positioned on it's inner surface with a fixed distance one from the other.Each of nine discrete circular conductors are fed with alternating current with density j i = j 0 cos(ωt + (i − 1)θ), here j 0 is the amplitude, ω = 2πf and f are the angular frequency and frequency of the alternating current, θ = const is the phase and t is the time.Usually we use θ = 120 0 , f = 50Hz.
In the weakly conductive liquid-electrolyte the current creates the radial B r and axial B z components of the magnetic field as well as azimuthal component of the induced electric field E φ , which, in it's turn, creates axial F z and radial F r components of the electromagnetic force (due to the Lorentz force).
For calculation of the electromagnetic fields, the averaging method over the time interval 2π/ω = 1/f is used [1].The averaged values of force < F r >, < F z > give rise to a motion of liquid (electrolyte), which can be described by the stationary Navier-Stokes equation.At the inlet of the cylinder we have a uniform velocity U 0 , but the swirl velocity is taken as the induced by the rigid body rotation with angular velocity Ω 0 .
The main aim of the work is to analyze the influence of different connection schemes of nine electrodes and swirl velocity at the inlet for vortex formation in the cylinder.In our previous works [1,3] we have developed the mathematical model for calculation of the electromagnetic field with six electrodes and force by using elliptical integrals.This approach makes it possible the consideration of alternating current connections of various type, with phase shifts θ and various arrangements of the conductors.

The Navier-Stokes equations
The complete system of viscous incompressible flow consists of the continuity equation and the Navier-Stokes equations.The axially-symmetric stationary Navier-Stokes equations for vorticity function ω, hydrodynamic-stream function ψ and circulation W are given in the cylindrical coordinates (r, φ, z) by the following system of non-dimensional equations [2,3]: is the averaged azimuthal component of the force curl vectors, is the conjugate expression for the Laplace operator of functions g with g = ψ or is the Taylor number, µ = 4π 10 −7 m kg s 2 A 2 is the magnetic permeability in vacuum.At the inlet of the cylinder we have the velocity U 0 ≈ 0.1 m s and the angular velocity The liquid has the following parameters: kinematic viscosity ν ≈ 10 −5 m 2 s , density ρ ≈ 1000 kg m 3 , η = ρν is the dynamic viscosity, and the electric conductivity The amplitude of the current density is j 0 = 10 6 A m 2 .The radius a of the cylinder is 0.05m, the length Z of the cylinder is 0.35m.
We write equations (2.1) in the dimensionless form by scaling all the lengths to r 0 = a (the inlet radius of the tube), the axial velocity v z to U 0 (the uniform inlet axial velocity), swirl velocity to W 0 = r 0 V 0 , the azimuthal velocity v φ to V 0 = r 0 Ω 0 , the vorticity ω to ω 0 = U 0 /r 2 0 and stream function ψ to ψ 0 = U 0 r 2 0 .We have the following dimensionless form of the boundary conditions: 1) in the part of the inlet (z = 0, 0 ≤ r < r 1 ) the axial streams are assumed to have a uniform velocity U 0 in the other part of the inlet (z = 0, r 1 ≤ r ≤ 1) the swirl velocity profile is induced by the rigid body rotation with the angular velocity Ω 0 2) the symmetry conditions along the axis (r = 0) , ω = ω w , where ω w is the dimensionless wall-vorticity obtained within the frame of the finitedifference method from no-slip conditions [3], r, r 1 are the dimensionless coordinates, β ≈ 0.1 is the velocity ratio of the coaxial free stream velocity to axial jet velocity U 0 .

The Finite-Difference Approximations
The presence of large parameters at the first order derivatives (Γ, Re) in the system of differential equations (2.1) causes additional numerical difficulties for the application of the general finite-difference schemes (a slow convergence rate, low precision).Thus special monotonous approximations are constructed in [2,5], using functions of matrix and the exponential functions The Patankar approximations [6] are given in the following form: Let us consider the 1D differential equation the finite-difference equations associated with grid point y k are given in the form where For the system of two differential equations (3.2) u is the vector (ω, W ), y = z and we get the vector finite-difference equations with matrix-functions A k , B k , s [5].We consider a uniform grid: Subscripts (i, j) refer to r, z indices, the mesh steps in the i, j directions are h 1 and h 2 , respectively.The solution of the finite-difference scheme is calculated by using the under relaxation method.

Numerical Results
The calculations are done for nine circular conductors L i , which are arranged in the axial direction at the points zj = z 1 , z 2 , z 3 , z 4 , z 5 , z 6 , z 7 , z 8 , z 9 , z i = 0.2i, i = 1, . . ., 9 , where z i are the dimensionless coordinates.
The results of numerical experiments for < F r >, < F z >, < f φ > and ψ, ω, W, in the dimensionless form were obtained with the help of the computer program MAPLE in the case of The numerical results show that the force fields induced by alternating current are concentrated on the cylinder's surface in the vicinity of the circular electrodes.The results depend on the arrangement of electrodes.
If the parameter Γ is increased, a vortex appears inside the cylinder, which, starting from Γ > 4, is developing into a vortex system.
The dependence of values of averaged forces < F z >, < F r >, and curl of forces < f φ > on the arrangement of nine conductors nj = [123456789] follows from results presented in Table 1.
Table 1.The extreme values of averaged forces and curl of forces.
In Fig. 1-16 we can see the vortex and vortex-breakdown formation in the cylinder depending on the arrangement of electrodes and the parameter Γ .The labels on the coordinates axis are given by 10r and 10z, where r ∈ (0, 1), z ∈ (0, l) are the dimensionless coordinates.

Conclusion
Squirt motion of weakly electrically conducting fluid influenced by alternating electromagnetic field in a finite cylinder is investigated.A real variable approach is used to describe time-averaged electromagnetic forces and fields, induced by alternating current, which is fed to every of 9 discrete circular conductors.An original monotonous difference scheme for approximation of this mathematical model is developed.The reported results of the numerical experiments with 9 circular conductors can give some new physical conclusions about the flow behavior in the cylinder.The averaged values of the electric field, electromagnetic forces and the azimuthal component of the curl of forces vector are calculated for different arrangement of the electrodes.Using monotone finite-difference schemes for calculations the vortex formation from the Lorentz force inside the cylinder and the vortex-breakdown at the inlet of the cylinder are obtained.The distribution of the temperature is depending of the vortex formation in the cylinder [1,3].
of the functions ψ and b, where b = ω or b

4 )
) the outflow boundary conditions at the outlet z = l = Zthe walls boundary conditions (r = 1)