Differential-flow-induced transition of traveling wave patterns and wave splitting

We have analyzed the differential flow-induced instability in the presence of diffusive transport in a reaction-diffusion system following activator-inhibitor kinetics. The conspicuous interaction of differential flow and differential diffusivity that leads to pattern selection during transition of the traveling waves from stripes to rotating spots propagating in hexagonal arrays subsequent to wave splitting has been explored on the basis of a few-mode Galerkin scheme.

Figure 1

Phase diagram corresponding to linear stability analysis of Eqs. (2.1) and (2.2) in the absence of the flow term; $V=0.0$ (arb. units). Hopf (solid blue) and Turing (dashed red) lines are shown to divide the parametric space into a steady homogeneous region, an oscillatory homogeneous region, and a stationary Turing pattern region.

Figure 2

Dispersion relation for flow-induced instability in the presence of diffusion; plot of the real part of the eigenvalue $\text{Re}\left({\lambda }_{+}\right)$ vs $ln\left(K\right)$ at different flow velocities for $\sigma =0.2$ for the chosen homogeneous steady state $P$ of Fig. 1 with $\mu =1.0$, and $D=0.1$ (arb. units).

Figure 3

Numerical simulation of Eqs. (2.1) and (2.2): (a)–(d) Snapshots of traveling stripes of $u$ resulting from instability of state $P$ ($\mu =1.0, D=0.1$) for $\sigma =0.2$ and $V=0.75$ (flow-induced instability threshold as shown in Fig. 2) at time (a) $t=3500$ t.u., (b) $t=4000$ t.u., (c) $t=4500$ t.u., and (d) $t=5000$ t.u. (arb. units). The last member (e) of the first column represents the variation of $u$ with $x$ for a fixed $y$ at 15.0 for each snapshot. (f)–(i) Same as in (a)–(d) but for $V=2.0$ (above the instability threshold). The last member (j) of the second column shows that stripes become uniform and move fast. (k)–(o) Same as in (f)–(j) but for $V=4.0$. Stripes become uniform.

Figure 4

Wave splitting: (a) and (b) Same as in Fig. 3 but for $V=7.6$ at time (a) $t=4500$ t.u. and (b) $t=5000$ t.u.; traveling stripes get fragmented. (c)–(f) Same as in Fig. 3 but for $V=8.0$; traveling spots move in hexagonal arrays. The movement of hexagons in time is shown by marking each spot by numbers.

Figure 5

Effect of domain size: (a)–(d) Same as in Fig. 3 but for $V=8.0$ and domain size $120\times 120$; traveling stripes get fragmented. (e)–(h) Same as in (a)–(d) but for domain size $300\times 300$ and $\mathrm{\Delta }x=2.0$. No fragmentation is observed as the amplitudes of the waves are small.

Figure 6

Variation of the critical wave number $k_{c\left(\text{spot}\right)}$ for spots as a function of the ratio of the diffusion coefficients, $D$, for the set of parameters mentioned in the text. A comparison of the theoretical scheme based on Galerkin analysis [Eq. (3.18)] with numerical findings is shown (arb. units).