Fluid–Solid Reaction in Porous Media as a Chaotic Restart Process

Chemical and biological reactions at fluid-solid interfaces are central to a broad range of porous material applications and research. Pore-scale solute transport limitations can reduce reaction rates, with marked consequences for a wide spectrum of natural and engineered processes. Recent advances show that chaotic mixing occurs spontaneously in porous media, but its impact on surface reactions is unknown. We show that pore-scale chaotic mixing significantly increases reaction efficiency compared to nonchaotic flows. We find that reaction rates are well described in terms of diffusive first-passage times of reactants to the solid interface subjected to a stochastic restart process resulting from Lagrangian chaos. Under chaotic mixing, the shear layer at no-slip interfaces sets the restart rate and leads to a characteristic scaling of reaction efficiency with Péclet number, in excellent agreement with numerical simulations. Reaction rates are insensitive to the flow topology as long as flow is chaotic, suggesting the relevance of this process to a broad range of porous materials.

Figure 1

Simulations of flow (line a) and fluid-solid reactive transport (lines b, c) in a BCC, for two flow orientations: $(\theta ,\varphi )=(0,0)$ and $(5,1)\pi /40$, leading to nonchaotic (column 1) and chaotic (column 2) advection. Streamlines colored with velocity magnitudes $v$ normalized by the maximum $v_{\max }$ are shown in (a). Reactive transport of a solute instantaneously consumed at the solid surfaces (infinite Damköhler number) is shown for Péclet numbers ${10}^5$ (b) and ${10}^2$ (c). Initially-uniform reactant concentrations are depleted by surface reaction. Concentrations $c$ normalized by the initial value $c_0$ are shown after 20 (b) and 5 (c) advection times ${\tau }_A$. Chaotic flow leads to exponential fluid stretching and accelerated consumption compared to the nonchaotic case.

Figure 2

Asymptotic reaction efficiency as a function of Péclet number for Stokes flow through a BCC, for Damköhler number ${10}^2$ and four flow orientations, three yielding fully-chaotic (squares: ${\lambda }_{\infty }\approx 7.53\times {10}^{-3}$; triangles: $\approx 1.40\times {10}^{-1}$; asterisks: $\approx 2.52\times {10}^{-1}$) and one nonchaotic flow (circles).

Figure 3

Near-interface dynamics controlling the restart time in fully-chaotic, no-slip flow. Under fully-chaotic advection in the bulk, dynamics near no-slip interfaces represent the limiting factor for advection-induced exploration of the domain and set the restart time. This leads to restart times controlled by interface shear $\gamma$, which result from balancing advective and diffusive escape times near the interface.

Figure 4

Moments of first-passage times (a) and asymptotic reaction efficiency (b) as a function of Péclet number in a BCC. Symbols (superimposed) are based on numerical computation of first-passage and return times for three chaotic flow orientations. Solid lines show theoretical predictions using the restart model [Eqs. (1), (2), and (7)]. The dashed line in (a) is the large-Pe prediction using Eqs. (4) and (7). Dashed lines in (b) are small-Pe [Eq. (1) for pure diffusion] and large-Pe [Eqs. (5) and (7)] predictions, for $w_h⩽w_h^D$ and ${\tau }_r>w_h^D$, respectively.