Influence of oscillatory centrifugal forces on the mechanism of Turing pattern formation

Constantly acting centrifugal forces on Turing pattern forming systems have been observed to induce orientation and wavelength changes on Turing structures. Here, we will consider a periodic modulation of such centrifugal forces and their effects on pattern formation. Depending on the oscillation period the system exhibits a wide variety of stationary (stripes, ${\mathrm{H}}_0$, etc.) or nonstationary patterns (black eyes, etc.), as well as transitions and instabilities such as Eckhaus, zigzag, etc. In this paper, a detailed description of the different patterns and patterning mechanisms will be described and understood within the previous context. The system considered is the Belousov-Zhabotinsky reaction encapsulated in AOT micelles modeled by the adapted version of the Oregonator model.

Figure 1

Effect of the oscillating forcing $\omega \left(t\right)$ on Turing pattern formation along a full oscillation for $T\sim \tau$. (a–h) correspond to $\omega \left(t\right)/{\omega }_0$: 0, 0.6, 0.7, 1.0, 0.8, 0.6, 0.3, and 0, respectively. The direction of the oscillatory force is fixed along the diagonal of the third quadrant [see arrow in (a)] with a maximum value fixed in ${\omega }_0=0.025\mathrm{rad}/\mathrm{t}.\mathrm{u}.$

Figure 2

Effect of the oscillating forcing $\omega \left(t\right)$ on Turing pattern formation along a full oscillation for $T\sim 10\tau$. (a–h) correspond to $\omega \left(t\right)/{\omega }_0$: 0, 0.5, 0.7, 0.8, 1, 0.7, 0.5, and 0.2, respectively. The direction of the oscillatory force is fixed along the diagonal of the third quadrant [see arrow in (a)] with a maximum value fixed in ${\omega }_0=0.025\mathrm{rad}/\mathrm{t}.\mathrm{u}.$

Figure 3

Effect of the oscillating forcing $\omega \left(t\right)$ on Turing pattern formation along a full oscillation for $T\sim 100\tau$. (a–h) correspond to $\omega \left(t\right)/{\omega }_0$: 0, 0.4, 0.6, 0.8, 0.9, 0.7, 0.5, and 0.1, respectively. The direction of the oscillatory force is fixed along the diagonal of the third quadrant [see arrow in (a)] with a maximum value fixed in ${\omega }_0=0.025\mathrm{rad}/\mathrm{t}.\mathrm{u}.$

Figure 4

Black eye pattern. Effect of the oscillating forcing $\omega \left(t\right)$ on Turing pattern formation along a full oscillation for $T\sim 0.1\tau$. (a–h) correspond to $\omega \left(t\right)/{\omega }_0$: 0, 0.3, 0.6, 1, 0.7, 0.5, 0.3, and 0, respectively. The direction of the oscillatory force is fixed along the diagonal of the third quadrant [see arrow in (a)] with a maximum value fixed in ${\omega }_0=0.025\mathrm{rad}/\mathrm{t}.\mathrm{u}.$

Figure 5

(a) Wavelength increments with respect to the standard pattern, $\Delta \lambda =\lambda /{\lambda }_0-1$, versus forcing. Blue squares, dotted green line, and red circles correspond to the following forcing periods taken from Figs. 1, 3, and 4: $T=\tau ,T=100\tau$, and $T=0.1\tau$, respectively. The shadowed region corresponds to theoretical increments (at $T=100\tau$) suitable for the Turing regime, with the dashed pink line as the most probable increment. (b) Morphological characterization of Turing patterns with Γ parameter versus forcing (for $T\sim \tau$). $\Gamma =-\alpha \mathrm{sign}\left(E\right)$, with E the Euler number and α the mean eccentricity of the patterns. All magnitudes in this figure were measured at ${\omega }_0=0.025\mathrm{rad}/\mathrm{t}.\mathrm{u}$ and $R=1000\mathrm{s}.\mathrm{u}.$

Figure 6

Turing instability coupled to an oscillatory centrifugal force close to the rotation center. The computational domain consists of a two-dimensional mesh of $300\times 300$ grid points, with a spatial step of 0.2 s.u. Three sequences of snapshots were taken for three different oscillation periods (a–c) at different forcings along a full oscillation, i.e., $\omega \left(t\right)/{\omega }_0=0.5,1.0,0.5,\mathrm{and}0,$ denoted as 1, 2, 3, and 4, respectively. The magnitude of the forcing remains at ${\omega }_0=0.40\mathrm{rad}/\mathrm{t}.\mathrm{u}.$ (a) Oscillation period $T=0.001\tau$. SP.A is the space-time plot of the mean value pattern showed in (a.1)–(a.4) for an interval of $(t=15\tau )$. (b) Oscillation period $T=\tau$. SP.B is the space-time plot of the circular patterns shown in (b.1)–(b.4) for an interval of $(t=15\tau )$. The zoom corresponds to an interval of $2\tau$. (c) Oscillation period $T=100\tau$. SP.C is the space-time plot of the radial pattern showed in (c.1)–(c.4) for an interval of $(t=150\tau )$.

Figure 7

Semilog plot of the drift velocity [$\ln (v)$, with ln as the natural log] for the Turing patterns subjected to the oscillating angular motion versus the position with respect to the rotation center $\left(x\right)$. The velocity was measured from the space-time plots (SP.A and SP.C) of Fig. 6. Circles (squares) correspond to the patterns subject to a period of forcing of $T=0.001\tau (T=100\tau )$. Paths 1 and 2 indicate the trajectories of any two patterns from the center to the disappearance region.