Bifurcation analysis in a vortex flow generated by an oscillatory magnetic obstacle

A numerical simulation and a theoretical model of the two-dimensional flow produced by the harmonic oscillation of a localized magnetic field (magnetic obstacle) in a quiescent viscous, electrically conducting fluid are presented. Nonuniform Lorentz forces produced by induced currents interacting with the oscillating magnetic field create periodic laminar flow patterns that can be characterized by three parameters: the oscillation Reynolds number, ${\text{Re}}_{\omega }$, the Hartmann number, Ha, and the dimensionless amplitude of the magnetic obstacle oscillation, $D$. The analysis is restricted to oscillations of small amplitude and $\text{Ha}=100$. The resulting flow patterns are described and interpreted in terms of position and evolution of the critical points of the instantaneous streamlines. It is found that in most of the cycle, the flow is dominated by a pair of counter rotating vortices that switch their direction of rotation twice per cycle. The transformation of the flow field present in the first part of the cycle into the pattern displayed in the second half occurs via the generation of hyperbolic and elliptic critical points. The numerical solution of the flow indicates that for low frequencies (v.e. ${\text{Re}}_{\omega }=1$), two elliptic and two hyperbolic points are generated, while for high frequencies (v.e. ${\text{Re}}_{\omega }=100$), a more complex topology involving four elliptic and two hyperbolic points appear. The bifurcation map for critical points of the instantaneous streamline is obtained numerically and a theoretical model based on a local analysis that predicts most of the qualitative properties calculated numerically is proposed.

Figure 1

Instantaneous streamlines for $D=0.01$, $\text{Ha}=100$, and ${\text{Re}}_{\omega }=1$: (a) $\varphi =-997\pi /1000$, (b) $\varphi =3\pi /1000$. The magnetic obstacle is in the region $-0.5<x<0.5$ and $-0.5<y<0.5$.

Figure 2

Instantaneous streamlines for $D=0.01$, $\text{Ha}=100$, and ${\text{Re}}_{\omega }=1$: (a) $\varphi =-927\pi /1000$, (b) $\varphi =-895\pi /1000$, (c) $\varphi =-831\pi /1000$, and (d) $\varphi =-752\pi /1000$.

Figure 3

(Color online) Position of critical points of the instantaneous streamlines for $D=0.01$, $\text{Ha}=100$, and ${\text{Re}}_{\omega }=1$: (a) $y$ position of elliptical points as a function of phase $\varphi$. Red (dark) and blue (light) lines refer to vortices rotating clockwise and counterclockwise, respectively. (b) $x$ position of hyperbolic points, as a function of the phase $\varphi$.

Figure 4

$u$-velocity component as a function of the axial coordinate $x$ for $y=0$ for ${\text{Re}}_{\omega }=1$. Labels near the lines indicate their phase in the cycle. The inset shows an amplification of the region close to the origin.

Figure 5

Instantaneous streamlines for $D=0.01$, $\text{Ha}=100$, and ${\text{Re}}_{\omega }=100$: (a) $\varphi =-553\pi /1000$, (b) $\varphi =-542\pi /1000$, (c) $\varphi =-534\pi /1000$, and (d) $\varphi =-489\pi /1000$.

Figure 6

$u$-velocity component as a function of the axial coordinate $x$ for $y=0$ for ${\text{Re}}_{\omega }=100$. Labels near the lines indicate their phase in the cycle. The inset shows an amplification of the region close to the origin.

Figure 7

Instantaneous streamlines for $D=0.01$, $\text{Ha}=100$, and ${\text{Re}}_{\omega }=100$ and $\varphi =469\pi /1000$.

Figure 8

(Color online) Bifurcation diagram for critical points of the instantaneous streamlines. Representative examples of flows are shown in Figs. 2, 5.

Figure 9

(Color online) Theoretical bifurcation diagram in the $c_2,c_0$ parameter plane for $c_1=0.2$. The dashed lines correspond to the sequences of topologies in Figs. 2, 5.