Space-Group Symmetries Generate Chaotic Fluid Advection in Crystalline Granular Media

The classical connection between symmetry breaking and the onset of chaos in dynamical systems harks back to the seminal theory of Noether [Transp. Theory Statist. Phys. 1, 186 (1918)]. We study the Lagrangian kinematics of steady 3D Stokes flow through simple cubic and body-centered cubic (bcc) crystalline lattices of close-packed spheres, and uncover an important exception. While breaking of point-group symmetries is a necessary condition for chaotic mixing in both lattices, a further space-group (glide) symmetry of the bcc lattice generates a transition from globally regular to globally chaotic dynamics. This finding provides new insights into chaotic mixing in porous media and has significant implications for understanding the impact of symmetries upon generic dynamical systems.

Figure 1

(a) Primitive (green) and unit (red) cells for the SC (left) and bcc (right) packings (spheres are shrunk for visualization). (b) Selected $S_2^m$ reflection symmetry planes (yellow discs) of the SC and bcc unit cells $\{[1-10],[10-1],[01-1]\}$ which intersect along $(\theta ,\varphi )=(\pi /4,\pi /4)$. The purple region indicates the flow parameter space $\mathcal{H}:({\varphi }_f,{\theta }_f)=[0,\pi /4]\times [0,\pi /4]$. Adapted from [33].

Figure 2

(a) Particle trajectories for two flow orientations in the BCC lattice: 1 $({\theta }_f,{\varphi }_f)=(0,0)$, 2 $({\theta }_f,{\varphi }_f)=(\pi /40,7\pi /90)$, illustrating the sharp transition from regular flow to strong chaotic mixing under small changes in the mean flow orientation. (b) Intersection (black line) of a continuously injected material line with the window shown in (a) for flow orientation 2; the red circle indicates the upstream injection line (see video Anim1 [36]). (c) Distribution of the Lyapunov exponent $\lambda$ over $\mathcal{H}$ for BCC (main figure) and SC (upper inset) packings, where gray denotes regular dynamics ($\lambda =0$) due to alignment of the mean flow with the normal to a reflection symmetry plane. (d)–(f) Poincaré sections [normal to (1,1,1)] for the BCC lattice, for flow orientations $({\theta }_f,{\varphi }_f)$: (d) $(\pi /4,\pi /4)$, (e) $(\pi /5,\pi /5)$, (f) $(\pi /8,\pi /40)$, with corresponding $\lambda$ shown in (c).

Figure 3

(a) Vertically exploded view of sphere layers in the BCC lattice (with color convention identical to that in Fig. 5), $L_Z$, $L_Z^{\prime }$ reflection (gray) and $G_Z^{\pm }$ glide (blue) symmetry planes and material surfaces ${\mathcal{M}}_Z^{\pm }$ (pink) for ${\varphi }_{\mathrm{f}}=0$. (b) Forward (red) and backward (blue) skin friction lines (thin) in spherical coordinates ($\theta ,\varphi$) over the sphere surface $\partial \mathcal{D}$ in the BCC lattice for $({\theta }_{\mathrm{f}},{\varphi }_{\mathrm{f}})=(\pi /8,\pi /40)$, with mean flow orientation (yellow dots), node ${\mathbf{x}}_p^N$ (green dots), saddle ${\mathbf{x}}_p^S$ (green crosses), and contact ${\mathbf{x}}_c$ points (black discs) shown. The separation (red) and reattachment (blue) critical lines $\zeta$ (thick, black dashed lines) and degenerate critical lines $\tilde{\zeta }$ (thick lines) are colored according to $(\nabla ·\mathbf{u})/\parallel \mathbf{u}\parallel$ (left/right color bars). (c) 3D view of the same skin friction field, showing fluid streamlines (black), stable (blue), and unstable (red) 2D manifolds emanating from critical lines $\zeta$.

Figure 4

Smooth heteroclinic connections for (a) the SC lattice with $({\theta }_f,{\varphi }_f)=(3\pi /20,0)$ (see also 3D animation Anim3 in Ref. [36]) and (b) the bcc lattice with $({\theta }_f,{\varphi }_f)=(\pi /4,0)$. (c) Transverse heteroclinic intersection for the bcc lattice with $({\theta }_{\mathrm{f}},{\varphi }_{\mathrm{f}})=(\pi /20,0)$ (see also 3D animation Anim2 in Ref. [36]).

Figure 5

(a) Projection in the $Z$ direction of spheres in the $L_Z$ (green) and $L_Z^{\prime }$ (gray) symmetry planes of the bcc lattice for $({\theta }_{\mathrm{f}},{\varphi }_{\mathrm{f}})=(\pi /10,0)$. The $S_0$ (red) and $S_1$ (blue) spheres in $L_Z$ interact via their respective stable ${\mathcal{W}}_{2\mathrm{D}}^S$ (blue) and unstable ${\mathcal{W}}_{2\mathrm{D}}^U$ (red) manifolds and associated streamlines ${\alpha }^S$, ${\alpha }^U$ (thick lines) in $L_Z$. (b),(c) Projection of spheres and manifolds in $L_f^{\prime }$ plane in the bcc and SC lattices respectively. (b) Concavity of the (un)stable manifolds [as indicated by the red and blue arrows in (a)] results in a transverse heteroclinic connection away from $L_Z$, $L_Z^{\prime }$ (see 3D animation Anim2 [36]). The pink dotted lines denote the material surfaces ${\mathcal{M}}_Z^{\pm }$, and the manifolds wrapping around the spheres are forward or backward iterates [35] of the manifolds shown in the center. (c) For the SC lattice, convexity of the manifolds about $l_Z$ results in smooth connections for all ${\theta }_{\mathrm{f}}$ along ${\varphi }_{\mathrm{f}}=0$.