Anomalous diffusion and transport by a reciprocal convective flow

Under low-Reynolds-number conditions, dynamics of convection and diffusion are usually considered separately because their dominant spatial and temporal scales are different, but cooperative effects of convection and diffusion can cause diffusion enhancement [Koyano et al., Phys. Rev. E 102, 033109 (2020)]. In this paper, such cooperative effects are investigated in detail. Numerical simulations based on the convection-diffusion equation revealed that anisotropic diffusion and net shift as well as diffusion enhancement occur under a reciprocal flow. Such anomalous diffusion and transport are theoretically derived by the analyses of the Langevin dynamics.

Figure 1

Profile of the reciprocal flow expressed in Eqs. (2) and (3). The direction of the flow is inverted periodically. The gray curves show streamlines, while arrows show the direction of the flow. The modes of the reciprocal flow are (a) $n=2$, (b) $n=3$, (c) $n=4$.

Figure 2

Time evolutions of the concentration profile $c$ during one period $T$ of the reciprocal flow, which are numerically calculated based on Eq. (1). The white spots indicate the high-concentration areas, which are convected and deformed by the reciprocal flow. The modes of the reciprocal flow are (a) $n=2$, (b) $n=3$, and (c) $n=4$. The black region indicates the area occupied by fluid elements in a circular region with a radius of 1 at $t=0$ to show the configuration changes of fluid elements. The thermal diffusion coefficient $D$ and the amplitude of the flow $\varepsilon$ were set to be $D={10}^{-4}$ and $\varepsilon ={10}^{-1}$, respectively.

Figure 3

Snapshots of the concentration difference $c(\mathit{x},T)-c_{\mathrm{diff}}(\mathit{x},T)$. The modes of the reciprocal flow are (a), (b) $n=2$; (c), (d) $n=3$; and (e), (f) $n=4$. The red and blue regions show positive and negative values, respectively. The thermal diffusion coefficient was set as $D={10}^{-4}$ and the amplitude of the reciprocal flow was set as (a), (c), (e) $\varepsilon ={10}^{-3}$ and (b), (d), (f) $\varepsilon ={10}^{-1}$. The gray region indicates a circular region with a radius of 1.

Figure 4

Time series of (a), (c), (e) $\delta r\left(t\right)$ and (b), (d), (f) $\delta \phi \left(t\right)$. The initial conditions were set to be $r_{\mathrm{ini}}=0.5$ and ${\theta }_{\mathrm{ini}}=0$ (red solid curve), $\pi /\left(2n\right)$ (light green broken curve), $\pi /n$ (light blue solid curve), and $3\pi /\left(2n\right)$ (dark blue broken curve) for the $n\mathrm{th}$ mode flow [(a), (b) $n=2$; (c), (d) $n=3$; and (e), (f) $n=4$]. Other parameters were set to be $\varepsilon =0.1$ and $D={10}^{-4}$.

Figure 5

Plots of $\delta r\left(T\right)$ against (a), (c), (e) the flow amplitude $\varepsilon$ and (b), (d), (f) the diffusion coefficient $D$. The initial conditions were $r_{\mathrm{ini}}=0.5$ and ${\theta }_{\mathrm{ini}}=0$ for the $n\mathrm{th}$ mode [(a), (b) $n=2$; (c), (d) $n=3$; and (e), (f) $n=4$]. The parameters were fixed as $D={10}^{-4}$ for (a), (c), (e) and $\varepsilon ={10}^{-1}$ for (b), (d), (f). The initial size of the spot was fixed for $\sigma =0.02$. The black solid curves show the theoretical estimation, which is given in Eq. (36).

Figure 6

Profile of $q_3\left(\mathit{x}\right)$ in Eq. (29). The parameters were set to be $D={10}^{-4}, \varepsilon ={10}^{-3}$, and $\sigma =0.02$.

Figure 7

Plots of $X_r$ depending on (a), (c), (e) $\varepsilon$ and (b), (d), (f) $r_{\mathrm{ini}}$ for mode $n$ [(a), (b) $n=2$; (c), (d) 3; and (e), (f) 4]. The thermal diffusion coefficient $D$ and initial angle ${\theta }_{\mathrm{ini}}$ were set to be $D={10}^{-3}$ and ${\theta }_{\mathrm{ini}}=0$, respectively. The parameter was fixed as $r_{\mathrm{ini}}=0.5$ for the left panels and as $\varepsilon =0.1$ for the right panels. The black solid curves show the theoretical results, which are given in Eq. (30).

Figure 8

Plots of $S_{rr}$ and $S_{\theta \theta }$ depending on (a), (c), (e) $\varepsilon$ and (b), (d), (f) $r_{\mathrm{ini}}$ for the mode $n$ [(a), (b) $n=2$; (c), (d) 3; and (e), (f) 4]. The red and cyan points show $S_{rr}$ and $S_{\theta \theta }$, respectively. The thermal diffusion coefficient $D$ and initial angle ${\theta }_{\mathrm{ini}}$ were set to be $D={10}^{-3}$ and ${\theta }_{\mathrm{ini}}=0$, respectively. The parameter was fixed as $r_{\mathrm{ini}}=0.5$ for the left panels and as $\varepsilon =0.1$ for the right panels. The black solid and dashed curves show the theoretical results for $S_{rr}$ and $S_{\theta \theta }$, which are given in Eqs. (B6) and (B8), respectively.

Figure 9

Plots of $Y_{rrr}$ and $Y_{r\theta \theta }$ depending on (a), (c), (e) $\varepsilon$ and (b), (d), (f) $r_{\mathrm{ini}}$ for the mode $n$ [(a), (b) $n=2$; (c), (d) 3; and (e), (f) 4]. The red and cyan points show $Y_{rrr}$ and $Y_{r\theta \theta }$, respectively. The thermal diffusion coefficient $D$ and initial angle ${\theta }_{\mathrm{ini}}$ were set to be $D={10}^{-3}$ and ${\theta }_{\mathrm{ini}}=0$, respectively. The parameter was fixed as $r_{\mathrm{ini}}=0.5$ for left panels, and as $\varepsilon =0.1$ for right panels. The black solid and dashed curves show the theoretical results for $Y_{rrr}$ and $Y_{r\theta \theta }$, which are given in Eqs. (B13) and (B15).

Figure 10

Plots of (a), (c), (e) $Y_{rrr}$ and $Y_{r\theta \theta }$, and (b), (d), (f) $Y_{rr\theta }$ and $Y_{\theta \theta \theta }$ depending on $\theta$ for mode $n$ [(a), (b) $n=2$; (c), (d) 3; and (e), (f) 4]. The red points show (a), (c), (e) $Y_{rrr}$ and (b), (d), (f) $Y_{rr\theta }$, and the cyan points show (a), (c), (e) $Y_{r\theta \theta }$ and (b), (d), (f) $Y_{\theta \theta \theta }$. The thermal diffusion coefficient $D$, amplitude of the flow field $\varepsilon$, and initial radius $r_{\mathrm{ini}}$ were set to be $D={10}^{-3}, \varepsilon =0.1$, and $r_{\mathrm{ini}}=0.5$, respectively. The black solid curves show the theoretical results for (a), (c), (e) $Y_{rrr}$ and (b), (d), (f) $Y_{rr\theta }$, and the black dashed curves show the theoretical results for (a), (c), (e) $Y_{r\theta \theta }$ and (b), (d), (f) $Y_{\theta \theta \theta }$, which are given in Eqs. (B13, B14, B15, B16).