2.1 Symmetries of SC and BCC sphere lattices

The simple cubic (SC) and body-centred cubic (BCC) sphere arrays consist of lattices of uniform, close-packed unit radius spheres, as illustrated by their cubic unit cells in figure 2(a,b). These lattices are defined in the 3-D space $\unicode[STIX]{x1D6FA}:[X,Y,Z]=(-\infty ,\infty )\times (-\infty ,\infty )\times (-\infty ,\infty )$ in terms of the base vectors $\boldsymbol{s}^{(i)}$ , where the sphere centres are positioned at

(2.1) $$\begin{eqnarray}\displaystyle \boldsymbol{r}=\boldsymbol{r}_{0}+n_{1}\boldsymbol{s}^{(1)}+n_{2}\boldsymbol{s}^{(2)}+n_{3}\boldsymbol{s}^{(3)}, & & \displaystyle\end{eqnarray}$$

with integers $n_{i}=0$ , $\pm 1$ , $\pm 2$ , $\ldots \,$ . Here $\boldsymbol{r}_{0}=(-\ell /2,-\ell /2,-\ell /2)$ is such that the unit cell corresponding to $(n_{1},n_{2},n_{3})=(0,0,0)$ is centred at the origin, and $\ell$ is the length of the lattice unit cell. The base vectors for the SC and BCC lattices are respectively

(2.2a-c ) $$\begin{eqnarray}\displaystyle \boldsymbol{s}^{(1)}=(2,0,0),\quad \boldsymbol{s}^{(2)}=(0,2,0),\quad \boldsymbol{s}^{(3)}=(0,0,2), & & \displaystyle\end{eqnarray}$$

and

(2.3a-c ) $$\begin{eqnarray}\displaystyle \boldsymbol{s}^{(1)}=\frac{2}{\sqrt{3}}(1,1,-1)\quad \boldsymbol{s}^{(2)}=\frac{2}{\sqrt{3}}(-1,1,1)\quad \boldsymbol{s}^{(3)}=\frac{2}{\sqrt{3}}(1,-1,1), & & \displaystyle\end{eqnarray}$$

and so the respective lengths for the SC and BCC unit cells are $\ell =2$ and $4/\sqrt{3}$ . As shown in figure 2(a,b), the SC unit cell may be represented as a cube with a sphere at each vertex, and the BCC unit cell can be represented as the same with an additional sphere in the centre of the cube. Alternately the BCC unit cell can also be represented as an octahedron, where the central sphere now lies at one of the vertices of the octahedron. The cube and the octahedron are the dual of each other (i.e. each of their vertices lies in the centre of a face of the dual polyhedron), which has important implications for their basic symmetries (Ashcroft & Mermin Reference Ashcroft and Mermin1976).

The underlying symmetries of these sphere lattices are fundamental to their mixing characteristics. As the SC and BCC lattices are the dual of each other, they share the same symmetry point group, i.e. symmetries which consist only of reflections and rotations. This point group (termed m3m in Hermann–Mauguin notation) is comprised of two sets of reflection symmetries: $S_{1}^{m}=\{[100],[010],[001]\}$ and $S_{2}^{m}=\{[110],[101],[011],[1-10],[10-1],[01-1]\}$ , where the symmetry plane $[h_{1}h_{2}h_{3}]$ expressed in Miller indices denotes a plane orthogonal to the vector $h_{1}\boldsymbol{s}^{(1)}+h_{2}\boldsymbol{s}^{(2)}+h_{3}\boldsymbol{s}^{(3)}$ . Further to this, there also exist three axes of twofold rotational symmetry: $S^{r}=\{[100],[010],[001]\}$ . The normal vectors of the 9 reflection symmetry planes associated with $S_{1}^{m}$ , $S_{2}^{m}$ are expressed as $(\unicode[STIX]{x1D703},\unicode[STIX]{x1D719})\in (n\unicode[STIX]{x03C0}/4,m\unicode[STIX]{x03C0}/4)$ in spherical coordinates for all 9 permutations $n,m\in \{-1,0,1\}$ . Here $(\unicode[STIX]{x1D703},\unicode[STIX]{x1D719})=(0,0)$ , aligns with $\hat{\boldsymbol{e}}_{X}$ , as shown in figure 2(c).

For both the SC and BCC lattices, we study the Lagrangian kinematics of the autonomous advection equation

(2.4) $$\begin{eqnarray}\displaystyle \frac{\text{d}\boldsymbol{x}}{\text{d}t}=\boldsymbol{v}(\boldsymbol{x}), & & \displaystyle\end{eqnarray}$$

where $\boldsymbol{x}(t)$ is the position at time $t$ of a fluid particle which is advected by the steady 3-D Stokes flow $\boldsymbol{v}(\boldsymbol{x})$ . The local flow velocity $\boldsymbol{v}(\boldsymbol{x})$ in the fluid domain is associated with a mean flow orientation relative to the crystalline axes

(2.5) $$\begin{eqnarray}\displaystyle \boldsymbol{q}\equiv \frac{1}{|\langle \boldsymbol{v}\rangle |}\langle \boldsymbol{v}\rangle , & & \displaystyle\end{eqnarray}$$

where the angle brackets denote an average over the fluid volume between spheres. We consider the Lagrangian kinematics of the flow for both SC and BCC packings as function of the mean flow orientation $\boldsymbol{q}$ . As shown in figures 2(c) and 15(a), the point-group symmetries $S_{1}^{m}$ , $S_{2}^{m}$ of the SC and BCC lattices then reduce the set of unique mean flow orientations (denoted $\boldsymbol{q}=(\unicode[STIX]{x1D703}_{f},\unicode[STIX]{x1D719}_{f})$ in spherical coordinates, where $\unicode[STIX]{x1D703}$ , $\unicode[STIX]{x1D719}$ respectively are the azimuthal and elevation angles) to the parameter space ${\mathcal{H}}:(\unicode[STIX]{x1D703}_{f},\unicode[STIX]{x1D719}_{f})\in [\unicode[STIX]{x1D719}_{f},\unicode[STIX]{x03C0}/4]\times [0,\unicode[STIX]{x03C0}/4]$ . These results can then be mapped to the entire space of flow orientations ${\mathcal{H}}^{\ast }:(\unicode[STIX]{x1D703}_{f},\unicode[STIX]{x1D719}_{f})\in [-\unicode[STIX]{x03C0},\unicode[STIX]{x03C0}]\times [-\unicode[STIX]{x03C0}/2,\unicode[STIX]{x03C0}/2]$ as shown in figure 15(a).

To study the Lagrangian kinematics associated with the flow through both sphere lattices, we compute the steady 3-D Stokes velocity field $\boldsymbol{v}(\boldsymbol{x})$ through each lattice over select points within the mean flow orientation space ${\mathcal{H}}$ . The finite-volume computational fluid dynamics (CFD) package CFX-14 is used to compute these flows to high precision (root mean square of the finite-volume equation residuals is $10^{-14}$ ) over the interior fluid domain ${\mathcal{D}}$ of the primitive cell of these lattices. Periodic boundary conditions are imposed on the primitive cell faces and no-slip conditions are imposed on the sphere surfaces $\unicode[STIX]{x2202}{\mathcal{D}}$ . The flow is driven by an imposed pressure gradient, the orientation of which can be arbitrarily set to achieve any given flow orientation $(\unicode[STIX]{x1D703}_{f},\unicode[STIX]{x1D719}_{f})$ . We subsequently compute the advection equation (2.4) and Lyapunov exponents (detailed in appendix A) for these cases. Soulvaiotis, Jana & Ottino (Reference Soulvaiotis, Jana and Ottino1995) have extensively studied the impact of numerical errors in numerical calculation of both the velocity field and particle advection upon the Lyapunov exponent. Soulvaiotis et al. (Reference Soulvaiotis, Jana and Ottino1995) find that errors in the length of material lines grow with the square of advection time so long as the basic flow is sufficiently resolved. As these errors only grow algebraically in time, these errors do not impact estimation of the Lyapunov exponent.

The underlying symmetries of the velocity field $\boldsymbol{v}(\boldsymbol{x})$ play a fundamental role in controlling the Lagrangian kinematics (Franjione et al. Reference Franjione, Leong and Ottino1989; Mezić & Wiggins Reference Mezić and Wiggins1994; Haller & Mezic Reference Haller and Mezic1998; Yannacopoulos et al. Reference Yannacopoulos, Mezić, Rowlands and King1998). We denote the symmetry planes on the faces and mid-planes, respectively, of the SC and BCC cubic unit cells as $L_{i}^{\pm }$ and $L_{i}^{\prime }$ (with $i\in \{X,Y,Z\}$ ). As shown in figure 2(d), these correspond to the planes oriented normal to the $Z$ -coordinate as

(2.6) $$\begin{eqnarray}\displaystyle & \displaystyle L_{Z}:Z=\pm \frac{\ell }{2}, & \displaystyle\end{eqnarray}$$

(2.7) $$\begin{eqnarray}\displaystyle & \displaystyle L_{Z}^{\prime }:Z=0, & \displaystyle\end{eqnarray}$$

with similar symmetry planes orientated normal to the $X$ , $Y$ coordinates. The associated reflection symmetries in the $Z$ -direction are then

(2.8) $$\begin{eqnarray}\displaystyle & \displaystyle {\mathcal{L}}_{Z}:(X,Y,Z)\mapsto (X,Y,-Z\pm \ell ), & \displaystyle\end{eqnarray}$$

(2.9) $$\begin{eqnarray}\displaystyle & \displaystyle {\mathcal{L}}_{Z}^{\prime }:(X,Y,Z)\mapsto (X,Y,-Z), & \displaystyle\end{eqnarray}$$

with similar reflections in the $X,$ $Y$ directions. Note that the symmetry planes defined in (2.8), (2.9) constitute a subset of the total reflection symmetries of the SC and BCC lattices described at the start of § 2.1, and all of these symmetry planes repeat throughout the lattice due to periodicity of the unit cell.

The reversibility of Stokes flow means that if the mean flow direction $\boldsymbol{q}=(\unicode[STIX]{x1D703}_{f},\unicode[STIX]{x1D719}_{f})$ is oriented normal to any of the reflection symmetries of the granular lattice, then the associated velocity field $\boldsymbol{v}(\boldsymbol{x})$ inherits this reflection symmetry, which ultimately constrains the Lagrangian kinematics to be completely regular and integrable. This is shown in figure 1(b), which depicts periodic, integrable streamlines (with zero transverse hydrodynamic dispersion) when the flow direction $(\unicode[STIX]{x1D703}_{f},\unicode[STIX]{x1D719}_{f})=(0,0)$ is normal to the ${\mathcal{L}}_{X}^{\pm }$ and ${\mathcal{L}}_{X}^{\prime }$ reflection symmetries in the BCC lattice. When this symmetry is broken however (e.g. for $(\unicode[STIX]{x1D703}_{f},\unicode[STIX]{x1D719}_{f})=(\unicode[STIX]{x03C0}/40,7\unicode[STIX]{x03C0}/90)$ ), there is a sharp transition in the Lagrangian kinematics, leading to strong chaotic mixing with significant fluid stretching and folding (and the onset of transverse hydrodynamic dispersion), as shown in figure 1(b). Such breaking of reflection symmetries is also a necessary condition for the onset of chaotic mixing in the SC lattice, although as shown in figure 1(a), significantly weaker mixing results, despite the SC and BCC lattices sharing the same symmetry point group.

These differences in mixing dynamics between the two sphere lattices are further highlighted in figure 3, which shows the distribution of the infinite-time Lyapunov exponent $\unicode[STIX]{x1D706}$ over ${\mathcal{H}}$ . Here $\unicode[STIX]{x1D706}$ is defined in terms of the mean fluid deformation over the entire fluid domain ${\mathcal{D}}$ per unit cell length distance $\ell$ advected downstream, see appendix A for details. From the distribution of the Lyapunov exponent $\unicode[STIX]{x1D706}$ over the parameter space ${\mathcal{H}}$ it is then possible to estimate the mean $\langle \unicode[STIX]{x1D706}\rangle$ and standard deviation $\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D706}}$ over the ensemble of all flow orientations (see appendix B for details). From figure 3, it is clear that chaotic mixing is significantly stronger and more variable within the BCC lattice ( $\langle \unicode[STIX]{x1D706}\rangle =0.1280$ , $\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D706}}=0.0856$ ) than in the SC lattice ( $\langle \unicode[STIX]{x1D706}\rangle =0.0240$ , $\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D706}}=0.0133$ ). For each flow orientation in ${\mathcal{H}}$ , the Lagrangian topology of the flow is illustrated by its corresponding Poincaré section, constructed by recording the intersection of streamlines with the planes shown in figure 2(a,b). A Poincaré section provides an illustration of the Lagrangian kinematics of the flow; it is constructed by recording the intersection of a number of particle trajectories with a 2-D surface (the Poincaré section) embedded within the fluid domain ${\mathcal{D}}$ of the unit cell over many flow periods. To construct the Poincaré section, periodic boundary conditions are applied to the particle trajectories within the unit cell such that they intersect multiple times with the Poincaré section. Coherent regions of the Poincaré section correspond to non-mixing regions (i.e. Kolmogorov-Arnol’d-Moser (KAM) islands (Ottino Reference Ottino1989)), whereas regions of apparently random particle locations indicate ergodicity and chaotic dynamics (chaos).

As shown in figure 5(b,d,f), the Poincaré sections evidence a rich array of Lagrangian topologies throughout the BCC flow orientation space, ranging from completely regular (figure 5 b) to mixed regular and chaotic (figure 5 d) to seemingly globally chaotic dynamics (figure 5 f). Conversely, the Lagrangian topologies for the SC lattice are predominantly regular (figure 5 a,c), and the regions which are chaotic (figure 5 e) are only weakly so.

The role of reflection symmetries is highlighted by the completely regular dynamics (with $\unicode[STIX]{x1D706}=0$ ) observed at the three corners of the parameter space ${\mathcal{H}}$ of both the SC and BCC lattices (see, respectively, figures 5(a,c,e) and 5(b,d,f)), which correspond to mean flows oriented normal to their reflection symmetries.

Conversely, there exist very significant differences along the $\unicode[STIX]{x1D719}_{f}=0$ transect (where the mean flow is parallel to the $L_{Z}$ , $L_{Z}^{\prime }$ planes); in the BCC lattice this results in strong chaotic mixing (almost the strongest throughout ${\mathcal{H}}$ ), whereas for the SC lattice the mixing dynamics is completely regular ( $\unicode[STIX]{x1D706}=0$ ) along $\unicode[STIX]{x1D719}_{f}=0$ . These differences in Lagrangian kinematics along $\unicode[STIX]{x1D719}_{f}=0$ are then responsible for the much stronger mixing in the BCC lattice throughout the entire parameter space ${\mathcal{H}}$ .

In Turuban et al. (Reference Turuban, Lester, Borgne and Méheust2018), we discovered that the origin of these significant differences in the Lagrangian kinematics are due to the fact that whilst the SC and BCC lattices share the same symmetry point group, their symmetry space groups (which include translations as well as reflections and rotations) differ due to the presence of the central sphere in the BCC unit cell. This additional sphere then admits a set of glide symmetries in the BCC lattice, which are a type of symmetry that consists of a combination of a translation and a reflection. One common example of a glide symmetry is a set of footprints shown in figure 4(a); here the left and right footprints map into each other by translation along the direction of travel and reflection across the travel direction. In the BCC lattice, one glide symmetry arises from translation of an exterior sphere across a unit cell face as shown in figure 4(b), followed by reflection through a plane parallel to this face (denoted by the light grey outline). This combination of translation and reflection is represented by the mapping

(2.10) $$\begin{eqnarray}\displaystyle {\mathcal{G}}_{Z}^{\pm }:(X,Y,Z)\mapsto (X+\ell /2,Y+\ell /2,\pm \ell /2-Z), & & \displaystyle\end{eqnarray}$$

and there also exist other similar glide symmetries corresponding to $X$ , $Y$ reflections. As shown in figure 2(d), these glide symmetries are oriented about the planes $G_{Z}^{\pm }:Z=n\ell \pm \ell /4$ , with similar $X$ , $Y$ glide planes associated with the $X$ , $Y$ glide symmetries respectively. In the Stokes limit both the sphere lattice and the relative mean flow fully dictate the entire fluid velocity field $\boldsymbol{v}(\boldsymbol{x})$ . As such, when the mean flow direction $\boldsymbol{q}$ is oriented tangentially to the glide planes $G_{Z}^{\pm }$ (i.e. when $\unicode[STIX]{x1D719}_{f}=0$ ), the local flow field $\boldsymbol{v}(\boldsymbol{x})=(v_{X},v_{Y},v_{Z})$ then inherits the glide symmetry (2.10) as

(2.11) $$\begin{eqnarray}\displaystyle \left.\begin{array}{@{}c@{}}\displaystyle v_{X}(X,Y,Z)=v_{X}(X+\ell /2,Y+\ell /2,\pm \ell /2-Z),\\ \displaystyle v_{Y}(X,Y,Z)=v_{Y}(X+\ell /2,Y+\ell /2,\pm \ell /2-Z),\\ \displaystyle v_{Z}(X,Y,Z)=-v_{Z}(X+\ell /2,Y+\ell /2,\pm \ell /2-Z).\end{array}\right\} & & \displaystyle\end{eqnarray}$$

The resultant glide symmetry of the velocity field (2.11) itself then generates a series of ‘eggtray’-shaped material surfaces ${\mathcal{M}}_{Z}^{\pm }$ in the BCC lattice (material surfaces are defined as being invariant with the flow, and hence consisting of streamlines); ${\mathcal{M}}_{Z}^{+}$ and ${\mathcal{M}}_{Z}^{-}$ are centred about the planes $G_{Z}^{\pm }$ (as shown in figure 2 d) and are invariants of both the flow and the glide symmetries ${\mathcal{G}}_{Z}^{\pm }$ .

Figure 4.

(a) Glide symmetry associated with a set of footprints, consisting of a translation (T) followed by a reflection (R). (b) Glide symmetry of the BCC unit cell, consisting of a translation (T) of $+\ell /2$ in the $X$ - and $Y$ -direction, followed by a reflection (R) through the plane $Z=\ell /4$ , indicated by the light grey outline.

Figure 5.

Poincaré sections for the SC (a,c,e) and BCC (b,d,f) lattice for different flow orientations $(\unicode[STIX]{x1D703}_{f},\unicode[STIX]{x1D719}_{f})$ . The orientation and location of the planes used to compute the Poincaré section is illustrated in figure 2 for the SC (a) and BCC (b) unit cells. (a,b) $(\unicode[STIX]{x1D703}_{f},\unicode[STIX]{x1D719}_{f})=(0,0)$ , (c,d) $(\unicode[STIX]{x1D703}_{f},\unicode[STIX]{x1D719}_{f})=(\unicode[STIX]{x03C0}/10,0)$ , (e,f) $(\unicode[STIX]{x1D703}_{f},\unicode[STIX]{x1D719}_{f})=(\unicode[STIX]{x03C0}/8,\unicode[STIX]{x03C0}/40)$ . Each particle in the Poincaré sections is shaded according to the finite-time Lyapunov exponent $\hat{\unicode[STIX]{x1D706}}$ (A 4) computed along the streamline from its initial location. Note that the same number of initial points were used for all Poincaré sections, but the completely integrable flow in (a) leads to periodic trajectories.

Whilst the shape of the material surface formed by these streamlines can deform with the orientation angle $\unicode[STIX]{x1D703}_{f}$ , the gross eggtray shape persists as it is constrained by the glide symmetry ${\mathcal{G}}_{Z}^{\pm }$ , where contact points between spheres in the BCC lattice must all belong to the invariant surfaces ${\mathcal{M}}_{Z}^{\pm }$ . Turuban et al. (Reference Turuban, Lester, Borgne and Méheust2018) find that the glide symmetries ${\mathcal{G}}_{Z}^{\pm }$ and the associated material surfaces are directly responsible for the generation of Lagrangian chaos for mean flows with $\unicode[STIX]{x1D719}_{f}=0$ in the BCC lattice. Conversely, the glide symmetries and associated material surfaces do not occur in the SC lattice as there is no central sphere in the unit cell. In § 4 we briefly review the specific mechanisms which generate these chaotic dynamics.

Therefore in these cases it appears that both the reflection and glide symmetries control the mixing dynamics. It is well known that breaking of the reflection symmetry is a necessary condition for the attainment of chaotic dynamics in all flows. The novel observation in Turuban et al. (Reference Turuban, Lester, Borgne and Méheust2018) is that for Stokes flows in the SC and BCC lattices with the mean flow direction oriented along the $\unicode[STIX]{x1D719}_{f}=0$ transect, the presence of a glide symmetry is a further necessary condition for chaotic dynamics. The implication of this observation is then that chaotic dynamics is possible for Stokes flows over the BCC and SC lattices which are not constrained to $\unicode[STIX]{x1D719}_{f}=0$ , regardless of the presence of a glide symmetry. However, for $\unicode[STIX]{x1D719}_{f}\neq 0$ , the strength of chaos in this region of orientation space is influenced by the dynamics along $\unicode[STIX]{x1D719}_{f}=0$ , hence weaker mixing overall is observed in the SC lattice than the BCC. The glide symmetry is responsible for this observation.

Thus, in contrast to classical understanding (Szabó & Tél Reference Szabó and Tél1989) that symmetry breaking is connected with the onset of chaotic dynamics, in Turuban et al. (Reference Turuban, Lester, Borgne and Méheust2018) we find that for certain applications glide symmetries can also play a governing role with respect to the generation of Lagrangian chaos. This surprising result may be attributed to the fact that space-group symmetries have received significantly less attention in the study of chaotic dynamical systems than their point-group counterparts. The extent to which this observation applies to other systems which possess space group symmetries is an open question. This observation points to a possible extension of field-invariant symmetry analyses (Mezić & Wiggins Reference Mezić and Wiggins1994; Haller & Poje Reference Haller and Poje1998) based on point-group symmetries to include the impacts of space-group symmetries upon dynamical systems in general and fluid mixing in particular.

It is also interesting to note that the strongest mixing case for the BCC lattice for the points computed in ${\mathcal{H}}$ lies on the $\unicode[STIX]{x1D719}_{f}=0$ transect. As shall be shown in § 4, along this transect, streamlines (and hence mixing) are confined to material regions that are bound between the $L_{Z}$ , $L_{Z}^{\prime }$ symmetry planes and the ${\mathcal{M}}_{Z}^{\pm }$ material surface (as shown in figure 12 c). This means that for $\unicode[STIX]{x1D719}_{f}=0$ there are regions of strong mixing in the flow, however these regions are bound and so the net transverse dispersion is zero. These bounding symmetries (including the glide symmetry ${\mathcal{G}}_{Z}^{\pm }$ ) are broken by a small perturbation in $\unicode[STIX]{x1D719}_{f}$ , resulting in global chaos (defined as chaotic dynamics throughout the entire flow domain) and transverse dispersion. Hence in the BCC lattice the strongest mixing case over ${\mathcal{H}}$ arises close to the maximum computed value of $\unicode[STIX]{x1D706}$ along $\unicode[STIX]{x1D719}_{f}=0$ . In contrast, the SC lattice displays weak mixing throughout the parameter space ${\mathcal{H}}$ .

3.1 Mechanisms of chaotic mixing in discrete porous media

In addition to the role of symmetries, it is also instructive to consider the underlying mechanisms for generation of chaotic mixing in discrete (i.e. granular) porous media. Whilst these mechanisms are well established for continuous porous media (i.e. that consisting of a continuous solid phase) (Lester et al. Reference Lester, Metcalfe and Trefry2013), it is unclear how these mechanisms apply to discrete porous media such as granular matter. In discrete porous media, the point-like nature of the grain contacts renders the fluid–solid boundary non-smooth, invalidating the underlying theorems (e.g. Poincaré–Hopf) central to understanding these mechanisms in continuous media (Lester et al. Reference Lester, Metcalfe and Trefry2013). As shall be shown, the point-like nature of the contacts between spheres (or grains in the general case) has a significant impact upon the generation of chaotic mixing in discrete porous media. In the following Sections we shall explore the detailed role of these singular grain contacts, and use these insights to develop a predictive model of the Lyapunov exponent distribution over ${\mathcal{H}}$ . To begin, we shall first briefly review the mechanisms which lead to chaotic mixing in steady pore-scale flows.

Chaotic advection occurs in steady 3-D flows when 2-D stable ${\mathcal{W}}_{2D}^{S}$ and unstable ${\mathcal{W}}_{2D}^{U}$ manifolds (shown in figure 7) in the fluid domain ${\mathcal{D}}$ intersect transversely (MacKay Reference MacKay2001) to form a homoclinic or heteroclinic tangle, the hallmark of chaos in continuous systems. In steady flows, 2-D stable and unstable manifolds are material surfaces in the flow that are more commonly known in fluid mechanics as flow reattachment and separation surfaces. They respectively undergo exponential contraction and stretching, and are hence termed hyperbolic. Due to volume conservation, these exponentially contracting stable and exponentially expanding unstable manifolds must respectively repel and attract neighbouring fluid streamlines at an exponential rate. Heteroclinic and homoclinic tangles refer to a transverse (non-tangential) intersection of stable and unstable manifolds, where homoclinic refers to manifolds that originate from a common location, and heteroclinic refers to manifolds from different locations. As the steady Stokes flow $\boldsymbol{v}(\boldsymbol{x})$ typically does not exhibit flow reversal (aside from e.g. localised Moffatt vortices (Moffatt Reference Moffatt1964)), only heteroclinic intersections (involving un/stable manifolds that originate from different locations) arise as unstable manifolds propagate downstream, whereas stable manifolds propagate upstream. Given the central role of 2-D un/stable manifolds, we shall consider the mechanisms that lead to their generation in discrete porous media and the subsequent formation of transverse heteroclinic intersections, leading to chaotic mixing as illustrated in supplementary movie 2.

As shown by Surana, Grunberg & Haller (Reference Surana, Grunberg and Haller2006), flow separation and reattachment (and hence 2-D un/stable manifolds) are generated by hyperbolic (exponential) stretching of the skin friction field

(3.1) $$\begin{eqnarray}\displaystyle \boldsymbol{u}(x_{1},x_{2})\equiv \left.\frac{\unicode[STIX]{x2202}\boldsymbol{v}}{\unicode[STIX]{x2202}x_{3}}\right|_{x_{3}=0}, & & \displaystyle\end{eqnarray}$$

over the fluid boundary $\unicode[STIX]{x2202}{\mathcal{D}}$ (figure 6) on the sphere surfaces, where $(x_{1},x_{2})$ and $x_{3}$ are tangent and normal respectively to $\unicode[STIX]{x2202}{\mathcal{D}}$ . Typical skin friction fields and associated vector field lines for the SC and BCC lattices are shown in figures 6 and 7(a). Vector field lines that act as strongly hyperbolic (as defined by Surana et al. (Reference Surana, Grunberg and Haller2006)) attractors and repellers in $\boldsymbol{u}$ are termed as critical lines $\unicode[STIX]{x1D701}$ . From volume conservation, 2-D unstable and stable manifolds are generated respectively by separation $\unicode[STIX]{x1D701}_{S}$ ( $\unicode[STIX]{x1D735}_{\bot }\boldsymbol{\cdot }\boldsymbol{u}<0$ ) or reattachment $\unicode[STIX]{x1D701}_{R}$ ( $\unicode[STIX]{x1D735}_{\bot }\boldsymbol{\cdot }\boldsymbol{u}>0$ ) critical lines (as shown in figure 7 a), where $\unicode[STIX]{x1D735}_{\bot }\equiv (\unicode[STIX]{x2202}_{1},\unicode[STIX]{x2202}_{2},0)$ . Attracting and repelling vector field lines which are not strongly hyperbolic are termed pseudo-critical lines (denoted $\hat{\unicode[STIX]{x1D701}}$ ); we find these typically arise from the termination of 2-D un/stable manifolds emanating from neighbouring spheres in the primitive cell (as shown in figure 7 b). We compute the 2-D stable (respectively, unstable) manifolds of the flow by placing tracer particles a small distance $\unicode[STIX]{x1D716}\sim 2.5\times 10^{-4}$ (in unit cell sizes) away from the separation (respectively, reattachment) critical lines in the direction normal to the sphere surface, and advect these particles backwards (respectively, forwards) in time. Due to exponential contraction (respectively, expansion) of the stable (respectively, unstable) manifold itself, particles are attracted to these manifolds in backwards (respectively, forwards) time, so any small error in the initial position of the tracer with respect to the true manifold location decays exponentially.

Figure 6.

Forward (light grey (red online)) and backward (dark grey (blue online)) field lines (thin) of the skin friction field $\boldsymbol{u}$ over the sphere surface ${\mathcal{D}}$ (in terms of the spherical coordinates $(\unicode[STIX]{x1D703},\unicode[STIX]{x1D719})$ ) corresponding to the cases (a–f) described in figure 5. Due to periodicity all spheres share the same skin friction field. Also shown are the mean flow orientation (light grey, (yellow online) dots), node $\boldsymbol{x}_{p}^{N}$ (light grey (green online) dots), saddle $\boldsymbol{x}_{p}^{S}$ (light grey (green online) crosses) and contact $\boldsymbol{x}_{c}$ points (black discs). The separation (light grey (red online)) and reattachment (dark grey (blue online)) critical $\unicode[STIX]{x1D701}$ (thick, black dashed) and pseudo-critical $\tilde{\unicode[STIX]{x1D701}}$ lines (thick) are coloured according to $(\unicode[STIX]{x1D735}_{\bot }\boldsymbol{\cdot }\boldsymbol{u})/\Vert \boldsymbol{u}\Vert$ .

Figure 7.

(a) Three-dimensional view of the skin friction field shown in figure 6(f), showing fluid streamlines (black), stable ${\mathcal{W}}_{2D}^{S}$ (dark grey (blue online)) and unstable ${\mathcal{W}}_{2D}^{U}$ (light grey (red online)) 2-D manifolds emanating from pseudo-critical $\tilde{\unicode[STIX]{x1D701}}$ and critical lines $\unicode[STIX]{x1D701}$ . (b) Same as for (a) but with primitive cell (black lines) and neighbouring spheres and associated (transparent) periodic translations of the separation ( ${\mathcal{W}}_{2D}^{U}$ ) and reattachment ( ${\mathcal{W}}_{2D}^{S}$ ) surfaces, indicating the pseudo-critical lines $\tilde{\unicode[STIX]{x1D701}}$ arise as non-hyperbolic intersections of these translated manifolds with the reference spheres. See supplementary movies 3 and 4 for 3-D animations of this figure. Adapted from Turuban et al. (Reference Turuban, Lester, Borgne and Méheust2018).

All critical lines in the skin friction field $\boldsymbol{u}$ must terminate at either stable spirals, limit cycles or critical points $\boldsymbol{x}_{p}$ . Whilst stable spirals and limit cycles can occur in discrete porous media due to the presence of Moffatt vortices (Moffatt Reference Moffatt1964) near contact points, in practice Moffatt vortices are rare. Across all simulations we only observe Moffatt vortices for the flow orientation $(\unicode[STIX]{x1D703}_{f},\unicode[STIX]{x1D719}_{f})=(0,0)$ (reflecting observations (Davis et al. Reference Davis, O’Neill, Dorrepaal and Ranger1976) of such vortices for two isolated spheres aligned with the mean flow direction), but none are apparent for any other flow orientation.

Hence we focus our attention upon the formation of critical lines from critical points. Critical points, defined as $\boldsymbol{u}(\boldsymbol{x}_{p})=\mathbf{0}$ , are characterised in terms of the linearised skin friction tensor

(3.2) $$\begin{eqnarray}\displaystyle \left.\unicode[STIX]{x1D63C}\equiv \frac{\unicode[STIX]{x2202}\boldsymbol{u}}{\unicode[STIX]{x2202}\boldsymbol{x}}\right|_{\boldsymbol{x}=\boldsymbol{x}_{p}}, & & \displaystyle\end{eqnarray}$$

which is the boundary counterpart of the strain-rate tensor $\unicode[STIX]{x1D641}=\unicode[STIX]{x2202}\boldsymbol{v}/\unicode[STIX]{x2202}\boldsymbol{x}|_{\boldsymbol{x}=\boldsymbol{x}_{0}}$ that quantifies fluid deformation local to interior stagnation points $\boldsymbol{x}_{0}$ . Critical points are termed non-degenerate if $\det (\unicode[STIX]{x1D63C})\neq 0$ , i.e. fluid deformation local to $\boldsymbol{x}_{p}$ can be quantified to leading order in terms of $\unicode[STIX]{x1D63C}$ . For incompressible flow the eigenvalues $\unicode[STIX]{x1D708}_{i}$ of $\unicode[STIX]{x1D641}$ satisfy $\sum _{i=1}^{3}\unicode[STIX]{x1D708}_{i}=\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{v}=0$ , whereas the eigenvalues $\unicode[STIX]{x1D702}_{i}$ of $\unicode[STIX]{x1D63C}$ satisfy $\sum _{i=1}^{3}\unicode[STIX]{x1D702}_{i}=\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{u}=-\unicode[STIX]{x1D702}_{3}$ , where $\unicode[STIX]{x1D702}_{1}$ and $\unicode[STIX]{x1D702}_{2}$ have corresponding eigenvectors tangent to the skin friction boundary $\unicode[STIX]{x2202}{\mathcal{D}}$ , while $\unicode[STIX]{x1D702}_{3}$ has a corresponding eigenvector normal to $\unicode[STIX]{x2202}{\mathcal{D}}$ .

Hence the attraction/repulsion of vector lines to critical points and lines in the skin friction field are directly related to flow separation/reattachment, where $\unicode[STIX]{x1D702}_{3}$ characterises the rate of stretching (and thus separation/attachment) rate of the associated 2-D un/stable manifolds. Non-degenerate critical points $\boldsymbol{x}_{p}$ may then be classified respectively as node $\boldsymbol{x}_{p}^{N}$ , saddle $\boldsymbol{x}_{p}^{S}$ or centre $\boldsymbol{x}_{p}^{C}$ points based on whether the real parts of $\unicode[STIX]{x1D702}_{1}$ , $\unicode[STIX]{x1D702}_{2}$ have the same sign, different sign or are both zero. Figure 6 illustrates the diversity of skin friction field topologies observed in the SC and BCC lattices; here the impact of lattice symmetries can be observed, along with the formation of non-degenerate critical points, critical and pseudo-critical lines. In general, the critical points and their type (saddle, node, centre) play a key role in organising the topology of the skin friction field and hence flow separation and reattachment. An exception arises for degenerate critical points, in which case higher order (i.e. higher than first order in (3.2)) expansion of the skin friction field $\boldsymbol{u}$ is required to characterise local deformation.

Surana et al. (Reference Surana, Grunberg and Haller2006) use the Poincaré–Bendixson theorem and invariant manifold theory to show that only four types of critical lines are structurally stable (i.e. have stable topology with respect to second-order perturbations of the flow), and only one of these does not involve spirals or limit cycles. The remaining critical line type is formed by the connection between a saddle point and node point, as shown in figure 6. As such, the presence of non-degenerate saddle points in $\boldsymbol{u}$ are necessary for the formation of 2-D un/stable manifolds in the fluid interior. For any differentiable (smooth) manifold, the number of saddle points is related to the Poincaré–Hopf theorem, which connects the indices $\unicode[STIX]{x1D6FE}$ of the critical points $\boldsymbol{x}_{p}$ (where $\unicode[STIX]{x1D6FE}=-1,0,1,$ for saddles, centres and nodes respectively) to the topological genus $g$ (which quantifies the number of handles or holes an object has; i.e. a sphere has genus zero, a torus genus one, a pretzel genus two or more) of the manifold according to

(3.3) $$\begin{eqnarray}\displaystyle \mathop{\sum }_{i}\unicode[STIX]{x1D6FE}(\boldsymbol{x}_{p}^{i})=2(1-g). & & \displaystyle\end{eqnarray}$$

This theorem was used by Lester et al. (Reference Lester, Metcalfe and Trefry2013) to form a lower bound for the number density of saddle points in continuous porous media (such as open porous networks) based upon topological complexity of the porous medium, as characterised by the number density of the topological genus $g$ , which is typically large for such porous materials. In turn, this lower bound suggests a significant number density of 2-D un/stable manifolds emanate from the skin friction field at the fluid–solid boundary $\unicode[STIX]{x2202}{\mathcal{D}}$ , which can lead to chaotic mixing in continuous porous media. In principle, this mechanism also applies to discrete porous media, as the large number density of particle contacts means the number density of the topological genus must also be large. However the cusp-shaped contacts between particles renders the fluid–solid boundary $\unicode[STIX]{x2202}{\mathcal{D}}$ non-smooth, and so the Poincaré–Hopf theorem does not apply to such non-differentiable manifolds, as shown in figure 8. In the following we explore the implications of this result upon chaotic mixing in discrete porous media.

3.2 Topology of discrete porous media and implications for mixing

Figure 8.

(a) The simplest possible skin friction skeleton for a periodic array of laterally connected spheres with smooth contacts. Negative curvature of the solid surface local to the contact region promotes the formation of saddle points, whilst node points form on the positively curved sphere surfaces. (b) Simplest possible skin friction skeleton for the same system with cusp-shaped contacts. Here the pairs of non-degenerate saddle points shown in (a) coalesce into a single topological saddle point which is degenerate.

Whilst The Poincaré–Hopf theorem (3.3) does not apply to non-smooth manifolds, it is useful to consider a similar smooth manifold such as that depicted in figure 8(a), where the cusp-shaped sphere contacts have been smoothed to form finite-sized contacts. Figure 8(a) shows the simplest possible skin friction skeleton (comprising of critical points and critical lines) for a periodic array of laterally connected spheres, whereas figure 8(b) shows the corresponding system for cusp-shaped, non-smooth contacts. As saddle points tend to form on surface regions of negative curvature, whereas nodes tend to form on surface regions of positive curvature, the nodes in figure 8(a) arise on the sphere surfaces, whereas the saddle points are localised to the contact regions. For each sphere in the lateral array (which has a total topological genus of one when accounting for periodicity), the Poincaré–Hopf theorem predicts an equal number of nodes and saddles (accounting for the sharing of saddle points at contacts), resulting in the distribution shown in figure 8(a). If we then consider shrinking the smooth, finite contacts in figure 8(a) to form the point-wise, non-smooth contacts in figure 8(b), the negative curvature of these contacts diverges to form a cusp, and the pair of non-degenerate saddle points coalesce into a single degenerate topological saddle. As these saddle points are degenerate, the simplest possible skin friction skeleton (figure 8 b) is the same as that for an isolated sphere.

The role of grain contacts is also illustrated by the Gauss–Bonnett theorem, which connects the mean Gaussian curvature $\langle {\mathcal{C}}_{G}\rangle =(1/V)\int _{S\in \unicode[STIX]{x2202}{\mathcal{D}}}{\mathcal{C}}_{1}{\mathcal{C}}_{2}\,\text{d}S$ (with ${\mathcal{C}}_{1}$ , ${\mathcal{C}}_{2}$ the local principle curvatures, $V$ the sphere volume) to the topological genus $g$ as

(3.4) $$\begin{eqnarray}\displaystyle \langle {\mathcal{C}}_{G}\rangle =4\unicode[STIX]{x03C0}(1-g/V). & & \displaystyle\end{eqnarray}$$

A continuous porous medium is characterised by a large topological genus $g$ per unit volume, hence $\langle {\mathcal{C}}_{G}\rangle$ is strongly negative and the pore geometry is predominantly hyperbolic, promoting formation of non-degenerate saddle points (Lester et al. Reference Lester, Metcalfe and Trefry2013). For discrete media $g$ is also large and the mean curvature $\langle {\mathcal{C}}_{G}\rangle$ is also strongly negative, but now the negative curvature (and hence saddle points) is localised to the cusp-shaped contact points $\boldsymbol{x}_{c}$ (whilst the spheroidal grains have predominantly positive curvature), rendering these saddle points degenerate.

For a lattice of spheres with finite-sized connections the topological genus per sphere is $g=n_{c}/2$ , where $n_{c}$ is the number of contacts $\boldsymbol{x}_{c}$ per sphere, as each pair of periodic contact points on the spheres forms a topological ‘handle’ once periodicity of the lattice is taken into account. Hence $g=3$ , $4$ for the SC and BCC lattices respectively (note that only 4 contact points are apparent for the SC lattices shown in figure 6 as the contact points at $\unicode[STIX]{x1D719}=\pm \unicode[STIX]{x03C0}/2$ are not shown). If we denote $n_{n}$ as the number of node points per sphere, then from (3.3), the number $n_{s}$ of non-degenerate saddle points per sphere is

(3.5) $$\begin{eqnarray}\displaystyle n_{s}=n_{n}+n_{c}-2, & & \displaystyle\end{eqnarray}$$

i.e. from figure 8(a), there are two node points per spheres, two contact points and two non-degenerate saddle points (due to periodicity).

In the limit of vanishing contact area, these non-degenerate saddle points coalesce into a topological saddle (Brøns & Hartnack Reference Brøns and Hartnack1999), as shown in figure 8(b). This degenerate saddle point has index $\unicode[STIX]{x1D6FE}=-2$  (Brøns & Hartnack Reference Brøns and Hartnack1999), and so we may augment (3.5) to include the number $n_{d}$ of degenerate topological saddles for point-wise contacts as

(3.6) $$\begin{eqnarray}\displaystyle n_{s}+2n_{d}=n_{n}+n_{c}-2. & & \displaystyle\end{eqnarray}$$

As such, the Poincaré–Hopf theorem can be extended to non-smooth manifolds via suitable treatment of local critical points in the neighbourhood of the contact points. If each pair of contact points $\boldsymbol{x}_{c}$ yields a degenerate saddle ( $n_{c}=2n_{d}$ ), then the number of non-degenerate saddles is then related to the number of nodes as

(3.7) $$\begin{eqnarray}\displaystyle n_{s}=n_{n}-2. & & \displaystyle\end{eqnarray}$$

Compared to (3.5), (3.7) is the same relationship as for an isolated spheroid, i.e. $n_{c}=g=0$ . Therefore, for discrete porous media the simplest possible skin friction field over each particle involves two non-degenerate node points on the sphere surfaces and $n_{c}$ degenerate topological saddle points at the grain contact points $\boldsymbol{x}_{c}$ , as shown in figure 8(b). The simplest possible non-degenerate skin friction field then has the same topology as that of an isolated sphere with genus $g=0$ , i.e. two non-degenerate node points (as reflected by (3.7)).

As such, the basic topological complexity of discrete (granular) porous media alone is not sufficient to generate 2-D un/stable manifolds in the fluid interior due to the degenerate nature of the topological saddle points. This is in contrast to continuous porous media, which must admit a large number density of non-degenerate saddle points and hence 2-D un/stable manifolds (Lester et al. Reference Lester, Metcalfe and Trefry2013). This may appear surprising as the basic mechanism for mixing is the same in both systems: impingement of fluid streams on solid surfaces, implying flow separation and exponential stretching. Both discrete and continuous porous media contain regions of the solid surface which have negative curvature and so saddle points tend to arise in these regions. Discrete porous media also have regions of positive curvature (i.e. the spheres or grains themselves) at which node points tend to accumulate, whereas continuous porous media have a predominantly negative curvature throughout (although there may be local exceptions) and so node points are less likely.

The major difference between these systems however is that discrete porous media involve regions of divergent negative curvature (cusps) which render the saddle points degenerate, as discussed above. These point-wise contacts also render the effective topological genus of each grain as zero; a contact point is neither a node nor a saddle of the skin friction field, but is rather a degenerate point, akin to a centre (with index zero). As such, the physical interactions of the flow are quite different in the two systems; continuous porous media have predominantly non-degenerate saddle points throughout with few nodes, whereas discrete porous media have an even mix of non-degenerate nodes and degenerate saddles. As non-degenerate saddles generate chaos, chaotic advection is inherent to continuous porous media but not to discrete porous media. By inherent, we mean that chaotic mixing in continuous media does not require a bifurcation away from the simplest possible skin friction field, whereas chaotic mixing in discrete media does.

Despite this restriction, discrete porous media can still exhibit 2-D flow separation and chaotic mixing if the skin fiction field bifurcates away from the simplest possible topological state (as determined by (3.6)). The flow topology can lead to (i) bifurcation of the degenerate topological saddles to non-degenerate saddle points (e.g. figure 6 e,d,f) or (ii) bifurcation of node points in the skin friction field into an isolated saddle point and a pair of nodes (e.g. figure 9). As the skin friction field topology on the grains is controlled by the mean flow orientation with respect to the lattice, the presence or absence of chaotic advection depends upon this orientation, in contrast to continuous porous media for which chaotic advection is expected for all but perfectly symmetric media (Lester et al. Reference Lester, Metcalfe and Trefry2013).

Figure 9.

Skin friction field over a sphere in the BCC lattice for flow orientation $(\unicode[STIX]{x1D703}_{f},\unicode[STIX]{x1D719}_{f})=(\unicode[STIX]{x03C0}/20,\unicode[STIX]{x03C0}/20)$ , illustrating the bifurcation of simplest possible skin friction topology (as discussed in § 3.2) to admit a pair of node points $\boldsymbol{x}_{p}^{N}$ and an isolated non-degenerate saddle point $\boldsymbol{x}_{p}^{S}$ . This in contrast to the skin friction fields shown in figure 6(c–f) which only contain saddle points which are degenerate.

More generally, the non-inherent nature of the non-degenerate saddle points (and hence 2-D flow separation) in discrete porous media means fluid mixing in such media differs markedly from that of continuous media. This fundamental difference may explain the weaker mixing observed for the sphere lattices herein (with mean Lyapunov exponent $\langle \unicode[STIX]{x1D706}\rangle \approx 0.0760$ averaged over both the SC and BCC lattices), compared to the open porous networks considered in Lester et al. (Reference Lester, Metcalfe and Trefry2013) (where $\langle \unicode[STIX]{x1D706}\rangle \approx 0.1178$ ). Note that discrete porous media considered herein are ordered and the mean Lyapunov exponents for the SC and BCC lattices ( $\langle \unicode[STIX]{x1D706}\rangle \approx 0.0240$ , $\langle \unicode[STIX]{x1D706}\rangle \approx 0.1280$ respectively) span the measured value ( $\langle \unicode[STIX]{x1D706}\rangle \approx 0.1178$ ) for the continuous porous media studied in Lester et al. (Reference Lester, Metcalfe and Trefry2013). Whilst this value was reported for a random open porous network, it was shown (Lester et al. Reference Lester, Metcalfe and Trefry2013) that this value also corresponds to the ensemble average of the Lyapunov exponent over the ensemble of all possible ordered configurations of the open porous network. Whether this behaviour extends to discrete porous media is an open question, and further work on chaotic mixing in random discrete systems is required to answer this question and to firmly establish whether random discrete or continuous porous media exhibits stronger chaotic mixing.

5.1 Scaling of Lyapunov exponent with inter-sphere distance

As hyperbolic stretching of the 2-D un/stable manifolds is highly localised around the critical lines $\unicode[STIX]{x1D701}$ on the sphere surfaces $\unicode[STIX]{x2202}{\mathcal{D}}$ , the net fluid deformation arising from a pair of un/stable manifolds which emanate from spheres $S_{0}$ and $S_{1}$ (figure 12 b) is largely independent of the sphere separation distance $d$ . To develop a model for the Lyapunov exponent distribution based on this observation, we assume that within the boundary $\unicode[STIX]{x1D6E4}_{0}$ ; (i) the fluid deformation associated with separation over each sphere is constant, and (ii) the net deformation that persists once manifolds intersect is also constant. Whilst this is clearly not true in general (such as when smooth connections between manifolds occur, leading to zero net stretching) this assumption within $\unicode[STIX]{x1D6E4}_{0}$ serves as a reasonable basis for the model. Under this assumption the fluid deformation rate is then expected to scale inversely with the minimum separation distance $d$ between interacting spheres, which is inversely proportional to the spatial frequency of manifold intersections, i.e.

(5.1) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D706}(\unicode[STIX]{x1D703}_{f})\propto \frac{1}{d(\unicode[STIX]{x1D703}_{f})}. & & \displaystyle\end{eqnarray}$$

This scaling is tested by comparing the values of the Lyapunov exponent from the numerical simulations (using the methods described in appendix A) to estimates given by (5.1) based upon values of $d$ determined by identifying from the numerical simulations which sphere form a transverse manifold connection for a given value of $\unicode[STIX]{x1D703}_{f}$ . The good agreement between these values shown in figure 12(d) forms the basis for the theoretical estimation of $\unicode[STIX]{x1D706}(\unicode[STIX]{x1D703}_{f})$ based upon (5.1).

Hence we seek a model for the dependence of the distance $d$ between the sphere $S_{0}$ and the boundary $\unicode[STIX]{x1D6E4}_{0}$ as a function of $\unicode[STIX]{x1D703}_{f}$ when $\unicode[STIX]{x1D719}_{f}=0$ . Since ${\mathcal{W}}_{2D}^{S}$ , ${\mathcal{W}}_{2D}^{U}$ are material surfaces, a transverse connection occurs between manifolds which originate from spheres whose relative position vector is approximately parallel to the mean flow direction $\unicode[STIX]{x1D703}_{f}$ . Changes in the mean flow orientation can result in the destruction of existing connections between un/stable manifolds emanating from a pair of given spheres and the formation of new connections between other spheres pairs. This can lead to sudden fluctuations in $d$ and hence $\unicode[STIX]{x1D706}$ with changes in $\unicode[STIX]{x1D703}_{f}$ along $\unicode[STIX]{x1D719}_{f}=0$ , as shown in figure 12(c).

Figure 12.

(a) Heteroclinic intersections in the BCC lattice for four different mean flow orientations $\unicode[STIX]{x1D703}_{f}$ (black arrow) with $\unicode[STIX]{x1D719}_{f}=0$ . The boundary $\unicode[STIX]{x1D6E4}_{0}$ (dashed line) is defined by the closest set of spheres $S_{1}$ which form heteroclinic intersections with the reference sphere $S_{0}$ . (b) Array of spheres (light grey (green online)) in $L_{Z}$ plane of the BCC lattice, with backward $\unicode[STIX]{x1D6FC}^{S}$ (dark grey (blue online)) and forward $\unicode[STIX]{x1D6FC}^{U}$ (light grey (red online)) streamlines emanating from spheres $S_{0}$ and $S_{1}$ respectively for various values of the flow orientations $\unicode[STIX]{x1D703}_{f}$ . Inset: bi-periodic cell of the $L_{Z}$ plane with the $\unicode[STIX]{x1D6FC}^{S}$ and $\unicode[STIX]{x1D6FC}^{U}$ streamlines shown for $(\unicode[STIX]{x1D703}_{f},\unicode[STIX]{x1D719}_{f})=(\unicode[STIX]{x03C0}/10,0)$ . (c) Forward $\unicode[STIX]{x1D6FC}^{U}$ and backward $\unicode[STIX]{x1D6FC}^{S}$ streamlines, emerging from $S_{0}$ and $S_{1}$ , respectively, for the mean flow direction $(\unicode[STIX]{x1D703}_{f},\unicode[STIX]{x1D719}_{f})=(\unicode[STIX]{x1D703}_{o},0)$ . The light grey (red online) and dark grey (blue online) arrows represent the respective directions in which $\unicode[STIX]{x1D6FC}^{U}$ and $\unicode[STIX]{x1D6FC}^{S}$ move with increasing $\unicode[STIX]{x1D703}_{f}$ . (d) Comparison between the normalised Lyapunov exponent $\unicode[STIX]{x1D706}(\unicode[STIX]{x1D703}_{f})/\langle \unicode[STIX]{x1D706}\rangle$ for the case $\unicode[STIX]{x1D719}_{f}=0$ in the BCC lattice from numerical simulations (filled circles), scaling $\unicode[STIX]{x1D706}\sim 1/d$ (empty circles), and predicted from (5.12) (dashed lines).

Figure 13.

(a) Geometry of the BCC lattice for $\unicode[STIX]{x1D719}_{f}=0$ . Whilst the light grey (cyan online) line denotes the connection between the medium grey (red online) reference sphere $S_{0}$ and the dark grey (blue online) sphere on the lower branch of the boundary $\unicode[STIX]{x1D6E4}_{0}$ , the corresponding flow direction is indicated by the light grey (red online) straight line. In contrast, the light grey (green online) sphere on the upper branch of $\unicode[STIX]{x1D6E4}_{0}$ is connected with the reference sphere $S_{0}$ when the flow orientation is aligned with the purple line connecting the centres of the two spheres. (b) The approximate manifold distance $d(\unicode[STIX]{x1D703}_{f})$ (normalised by $\ell$ ) as predicted by the lower (dashed) and upper (dot-dashed) curves as a function of the mean flow orientation $\unicode[STIX]{x1D703}_{f}$ , and numerical data (open circles).

5.2 Geometric estimate of $d(\unicode[STIX]{x1D703}_{f})$ in the BCC lattice

To derive an estimate of $d(\unicode[STIX]{x1D703}_{f})$ , we consider geometric arguments based upon the boundary $\unicode[STIX]{x1D6E4}_{0}$ shown in figures 12(b), 13(a). We denote as $S_{1}$ the sphere nearest to the reference sphere $S_{0}$ for which a transverse heteroclinic intersection is possible for a given flow orientation $\unicode[STIX]{x1D703}_{f}$ (figure 12 a). As shown in figure 12(a), aside from a few exceptions (e.g. $\unicode[STIX]{x1D703}_{f}=3\unicode[STIX]{x03C0}/20$ ), the sphere $S_{1}$ predominantly lies on the boundary $\unicode[STIX]{x1D6E4}_{0}$ , hence $\unicode[STIX]{x1D6E4}_{0}$ serves as an appropriate basis for the construction of a simple predictive model for $d(\unicode[STIX]{x1D703}_{f})$ .

For the BCC lattice, the spheres in the $L_{Z}$ plane have centres which lie at $(X,Y)=(\ell j,\ell i)$ for integers $j$ , $i$ , as shown in figure 13(a). The upper and lower branches of the boundary $\unicode[STIX]{x1D6E4}_{0}$ then correspond to $i=j-1$ , $i=1$ , respectively. Consequently the distance $d$ from the reference sphere $S_{0}$ to the boundaries $\unicode[STIX]{x1D6E4}_{0}$ expressed in terms of $j$ is

(5.2) $$\begin{eqnarray}\displaystyle & \displaystyle d_{upper}=\ell \sqrt{j^{2}+(j-1)^{2}}, & \displaystyle\end{eqnarray}$$

(5.3) $$\begin{eqnarray}\displaystyle & \displaystyle d_{lower}=\ell \sqrt{1+j^{2}}, & \displaystyle\end{eqnarray}$$

which respectively correspond to the purple and cyan lines shown in figure 13(a). It then remains to couch these expressions in terms of the flow orientation angle $\unicode[STIX]{x1D703}_{f}$ .

The orientation angle $\unicode[STIX]{x1D703}_{f}$ that corresponds to the transition between manifold intersections occurring between spheres located on the upper and lower branches of $\unicode[STIX]{x1D6E4}_{0}$ is denoted $\unicode[STIX]{x1D703}_{o}$ , where this orientation angle corresponds to the shortest distance $d$ and hence largest Lyapunov exponent for $\unicode[STIX]{x1D719}_{f}=0$ . At this transition (5.2), (5.3) must also satisfy

(5.4) $$\begin{eqnarray}\displaystyle d_{upper}|_{\unicode[STIX]{x1D703}_{f}=\unicode[STIX]{x1D703}_{o}}=d_{lower}|_{\unicode[STIX]{x1D703}_{f}=\unicode[STIX]{x1D703}_{o}}. & & \displaystyle\end{eqnarray}$$

As the $L_{Z}$ plane in which these spheres reside is a material plane, the streamlines $\unicode[STIX]{x1D6FC}^{S}$ , $\unicode[STIX]{x1D6FC}^{U}$ associated with the 2-D un/stable manifolds must deform around the spheres in $L_{Z}$ , as shown in figure 12(b). Due to such bending, the flow angle $\unicode[STIX]{x1D703}_{f}$ associated with a manifold connection $S_{0}$ and a sphere on the lower branch of $\unicode[STIX]{x1D6E4}_{0}$ is more accurately represented by a line tangent to the spheres immediately to the right of $S_{0}$ and left of the associated sphere in $\unicode[STIX]{x1D6E4}_{0}$ , of respective coordinates $(1,0)$ and $(j-1,1)$ , as indicated by the light grey (red online) line in figure 13(a).

If the angle of this line is given by the flow orientation $\unicode[STIX]{x1D703}_{f}$ , then the $(X,Y)$ coordinates of the tangent contact points at each end of the light grey (red online) line is then

(5.5) $$\begin{eqnarray}\displaystyle & \displaystyle (X_{1},Y_{1})=\left(\ell +\cos \left(\frac{\unicode[STIX]{x03C0}}{2}+\unicode[STIX]{x1D703}_{f}\right),\sin \left(\frac{\unicode[STIX]{x03C0}}{2}+\unicode[STIX]{x1D703}_{f}\right)\right), & \displaystyle\end{eqnarray}$$

(5.6) $$\begin{eqnarray}\displaystyle & \displaystyle (X_{2},Y_{2})=\left(\ell (j-1)+\cos \left(\frac{3\unicode[STIX]{x03C0}}{2}+\unicode[STIX]{x1D703}_{f}\right),\ell +\sin \left(\frac{3\unicode[STIX]{x03C0}}{2}+\unicode[STIX]{x1D703}_{f}\right)\right). & \displaystyle\end{eqnarray}$$

The slope of this line is then

(5.7) $$\begin{eqnarray}\displaystyle \tan \unicode[STIX]{x1D703}_{f}=\frac{\ell -2\cos \unicode[STIX]{x1D703}_{f}}{\ell (j-2)+2\sin \unicode[STIX]{x1D703}_{f}}, & & \displaystyle\end{eqnarray}$$

which yields the dependence of $j$ on $\unicode[STIX]{x1D703}_{f}$ on the lower branch of the boundary as

(5.8) $$\begin{eqnarray}\displaystyle j_{lower}(\unicode[STIX]{x1D703}_{f})=2+\cot \unicode[STIX]{x1D703}_{f}-\frac{2}{\ell \sin \unicode[STIX]{x1D703}_{f}}. & & \displaystyle\end{eqnarray}$$

Rather than constrain $j$ to integer values corresponding to sphere locations within the BCC lattice, we now consider $j$ as the set of real numbers $j\geqslant 1$ to yield a continuous set of flow angles $\unicode[STIX]{x1D703}_{f}$ . Substitution of (5.8) into (5.3) then yields a continuous estimate of $d(\unicode[STIX]{x1D703}_{f})$ for the lower boundary of $\unicode[STIX]{x1D6E4}_{0}$ , as

(5.9) $$\begin{eqnarray}\displaystyle d_{lower}(\unicode[STIX]{x1D703}_{f})=\ell \sqrt{1+\left(2+\cot \unicode[STIX]{x1D703}_{f}-\frac{2}{\ell \sin \unicode[STIX]{x1D703}_{f}}\right)^{2}}\quad \text{for }0<\unicode[STIX]{x1D703}_{f}\leqslant \unicode[STIX]{x1D703}_{o}. & & \displaystyle\end{eqnarray}$$

For manifold connections between the reference sphere $S_{0}$ and interaction spheres $S_{1}$ on the upper boundary of $\unicode[STIX]{x1D6E4}_{0}$ , the line which directly connects the centre of sphere $S_{0}$ to that of the corresponding sphere on the upper boundary of $\unicode[STIX]{x1D6E4}_{0}$ , shown by the purple line in figure 13(a), aligns closely with the corresponding flow orientation $\unicode[STIX]{x1D703}_{f}$ . From this there is a simple relationship between $j_{upper}$ and $\unicode[STIX]{x1D703}_{f}$ as

(5.10) $$\begin{eqnarray}\displaystyle j_{upper}(\unicode[STIX]{x1D703}_{f})=\frac{1}{1-\tan \unicode[STIX]{x1D703}_{f}}. & & \displaystyle\end{eqnarray}$$

Again if we consider $j$ as a continuous rather than discrete variable, then by substituting this into (5.2) we obtain an expression for the distance from $S_{0}$ to the corresponding sphere on the upper boundary of $\unicode[STIX]{x1D6E4}_{0}$ as a function of flow orientation $\unicode[STIX]{x1D703}_{f}$ , as

(5.11) $$\begin{eqnarray}\displaystyle d_{upper}(\unicode[STIX]{x1D703}_{f})=\frac{\ell }{\cos \unicode[STIX]{x1D703}_{f}-\sin \unicode[STIX]{x1D703}_{f}}\quad \text{for }\unicode[STIX]{x1D703}_{o}<\unicode[STIX]{x1D703}_{f}\leqslant \unicode[STIX]{x03C0}/4. & & \displaystyle\end{eqnarray}$$

Combining (5.9), (5.11) the distance $d$ from the sphere $S_{0}$ to the boundary $\unicode[STIX]{x1D6E4}_{0}$ is

(5.12) $$\begin{eqnarray}\displaystyle d(\unicode[STIX]{x1D703}_{f})=\left\{\begin{array}{@{}l@{}}\displaystyle \ell \sqrt{1+\left(2+\cot \unicode[STIX]{x1D703}_{f}-\frac{2}{\ell }\csc \unicode[STIX]{x1D703}_{f}\right)^{2}}\quad \text{for }0\leqslant \unicode[STIX]{x1D703}_{f}<\unicode[STIX]{x1D703}_{o},\\ \displaystyle \frac{\ell }{\cos \unicode[STIX]{x1D703}_{f}-\sin \unicode[STIX]{x1D703}_{f}}\quad \text{for }\unicode[STIX]{x1D703}_{o}\leqslant \unicode[STIX]{x1D703}_{f}\leqslant \frac{\unicode[STIX]{x03C0}}{4},\end{array}\right. & & \displaystyle\end{eqnarray}$$

where the transition angle $\unicode[STIX]{x1D703}_{o}\approx 0.46$ satisfies (5.4). This angle yields the minimum distance $d$ (and therefore the strongest chaotic mixing) and corresponds to the intersection of $\unicode[STIX]{x1D6FC}^{S}$ , $\unicode[STIX]{x1D6FC}^{U}$ shown in figure 12(c). As shown in figure 13(b), equation (5.12) corresponds very well with the distance as determined directly from the manifold intersections in the numerical simulations.

The scaling of the Lyapunov exponent with $\unicode[STIX]{x1D703}_{f}$ along the $\unicode[STIX]{x1D719}_{f}=0$ transect of the parameter space ${\mathcal{H}}$ shown in figure 3(b) is then obtained by combining (5.12) with the scaling of (5.1), which yields

(5.13) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D706}(\unicode[STIX]{x1D703}_{f})\propto \left\{\begin{array}{@{}l@{}}\displaystyle \frac{1}{\ell \sqrt{1+\left(2+\cot \unicode[STIX]{x1D703}_{f}-\frac{2}{\ell }\csc \unicode[STIX]{x1D703}_{f}\right)^{2}}}\quad \text{for }0\leqslant \unicode[STIX]{x1D703}_{f}<\unicode[STIX]{x1D703}_{o},\\ \displaystyle \frac{\cos \unicode[STIX]{x1D703}_{f}-\sin \unicode[STIX]{x1D703}_{f}}{\ell }\quad \text{for }\unicode[STIX]{x1D703}_{o}\leqslant \unicode[STIX]{x1D703}_{f}\leqslant \frac{\unicode[STIX]{x03C0}}{4}.\end{array}\right. & & \displaystyle\end{eqnarray}$$

Normalising $\unicode[STIX]{x1D706}$ by $\langle \unicode[STIX]{x1D706}\rangle \equiv \int _{0}^{\unicode[STIX]{x03C0}/4}\unicode[STIX]{x1D706}(\unicode[STIX]{x1D703}_{f})\,\text{d}\unicode[STIX]{x1D703}_{f}$ , this simple predictive model (5.13) agrees quite well with the numerical data, as shown in figure 12(d). As expected, some discrepancies are observed due to (i) fluctuations in distance $d$ of manifold connections, and (ii) fluctuations in the fluid deformation across various manifold intersections. However, equation (5.13) does capture the gross trend of the Lyapunov exponent with $\unicode[STIX]{x1D703}_{f}$ , and in principle this geometric model could be extended to the BCC parameter space ${\mathcal{H}}$ , or more generally to the parameter space of any given lattice geometry, to predict the phase diagram of figure 3(b) more broadly. This could be achieved by identifying the governing symmetries that control mixing in a given periodic porous architecture, using these to construct an envelope similar to $\unicode[STIX]{x1D6E4}_{0}$ in figure 13, estimating the distance of that envelope to the reference sphere as a function of the flow orientation and then predicting the Lyapunov exponent distribution from the scaling (5.1). Conversely, the distribution of the Lyapunov exponent across the SC parameter space (figure 3 a) only exhibits weak deviations ( $\unicode[STIX]{x1D706}\in [0,0.027]$ ) from the completely regular dynamics $\unicode[STIX]{x1D706}=0$ at the boundaries of ${\mathcal{H}}$ due constraints imposed by reflection symmetries.