Role of solutal free convection on interdiffusion in a horizontal microfluidic channel

We theoretically investigate the role of solutal free convection on the diffusion of a buoyant solute at the microfluidic scales, $\simeq 5$–$500\mu \mathrm{m}$. We first consider a horizontal microfluidic slit, one half of which is initially filled with a binary solution (solute and solvent) and the other half with pure solvent. The buoyant forces generate a gravity current that couples to the diffusion of the solute. We perform numerical resolutions of the 2D model describing the transport of the solute in the slit. This study allows us to highlight different regimes as a function of a single parameter, the Rayleigh number Ra which compares gravity-induced advection to solute diffusion. We then derive asymptotic analytical solutions to quantify the width of the mixing zone as a function of time in each regime and establish a diagram that makes it possible to identify the range of Ra and times for which buoyancy does not impact diffusion. In a second step, we present numerical resolutions of the same model but for a 3D microfluidic channel with a square cross section. We observe the same regimes as in the 2D case and focus on the dispersion regime at long timescales. We then derive the expression of the 1D dispersion coefficient for a channel with a rectangular section and analyze the role of the transverse flow in the particular case of a square section. Finally, we show that the impact of this transverse flow on the solute transport can be neglected for most of the microfluidic experimental configurations.

Figure 1

(a) Schematic diagram of the diffusion experiment in a microfluidic channel of height $H$ and width $L$. The red arrows represent the gravity current $U_{\mathrm{B}}$ induced by the difference in density. The color code indicates the concentration of the solute. (b) Width of the mixing zone $W$ versus time $T$. The dashed line is the case for which buoyancy does not affect the spreading of the solute.

Figure 2

For $\text{Ra}={10}^5$ and $\text{Sc}={10}^5$: (a) concentration fields $\phi (x,z,t)$ at the times indicated by the corresponding symbols in (b), the $z$ scale is different in each image. (b) Width of the mixing zone $w\left(t\right)$. The magenta dashed line is given by Eq. (27) and corresponds to early advection with $w\sim t$. The yellow dashed line is given by Eq. (29) and corresponds to late advection with $w\sim \sqrt{t}$. The green dashed line is computed from the numerical resolution of Eq. (31) and includes both the 1D dispersion regime $w\sim t^{1/4}$ and late diffusion $w\sim \sqrt{t}$. The blue dashed line is the diffusion law $w=\sqrt{t}$. See also movie M1 corresponding to these data in the SM.

Figure 3

(a) Height-averaged concentration profiles ${\phi }_0(z,t)$ versus $z$ for timescales ranging from $t={10}^{-7}$ to $t\le 1.6\times {10}^{-6}$ (seven curves, $\text{Ra}={10}^5$ and $\text{Sc}={10}^5$). (b) Same data plotted against the reduced variable $z/\sqrt{t}$, the dashed line is given by Eq. (19).

Figure 4

(a) Velocity vector field superimposed with the concentration field at $t=1.3\times {10}^{-6}$ ($\mathrm{Ra}={10}^5, \mathrm{Sc}={10}^5$). (b) Maximal value of the component $u_z$ in the plane $z=0$ versus $t$ for $\mathrm{Ra}={10}^5$ and $\mathrm{Sc}={10}^2$ ($★$), ${10}^3$ ($\square$), ${10}^4$ ($\circ$), ${10}^5$ ($▿$). The horizontal dashed line is $\simeq 1060$. (c) Maximal value of $u_z(x,z=0)$ rescaled by $\mathrm{Ra}$ versus $\mathrm{Sc}t$ for $\mathrm{Ra}={10}^5$ (blue), ${10}^4$ (magenta), and ${10}^3$ (cyan) and $\mathrm{Sc}={10}^2$ ($★$), ${10}^3$ ($\square$), ${10}^4$ ($\circ$), ${10}^5$ ($▿$). The horizontal dashed line is estimated using Eq. (22) and given by $\simeq 0.0106$, the vertical dashed line is $\mathrm{Sc}t=0.0571$.

Figure 5

(a) $u_x(x=1/2,z)$ versus $z$ and (b) $u_z(x,z=0)$ versus $x$ for timescales ranging from $t=1.6\times {10}^{-6}$ to ${10}^{-4}$ (10 curves, $\mathrm{Ra}={10}^5$ and $\mathrm{Sc}={10}^5$). The thin black lines are given by Eq. (21) in (a) and by Eq. (22) in (b).

Figure 6

Concentration fields $\phi$ at times (a) $t\simeq 6.3\times {10}^{-5}$ and (b) $t\simeq 6.3\times {10}^{-4}$ for $\mathrm{Ra}={10}^5$ and $\mathrm{Sc}={10}^5$. The magenta dashed lines are the deformations estimated by Eq. (25). (c) Average concentration profiles against $z/\left(\mathrm{Ra}t\right)$ for timescales ranging from $t\simeq 2\times {10}^{-5}$ to $1.3\times {10}^{-4}$ (9 curves), the dashed line is given by Eq. (26). Inset: same profiles against $z$.

Figure 7

(a) Concentration field $\phi$ and components (b) $u_z$, (c) $u_x$ at $t=1.6\times {10}^{-2}$ for $\mathrm{Ra}={10}^5$ and $\mathrm{Sc}={10}^5$. The magenta dashed line is given by the solution of Eq. (28) in the case of a slit. (c) Average concentration profiles ${\phi }_0(z,t)$ versus $\xi =z/\sqrt{\tilde{D}t}$ for timescales ranging from $t\simeq 0.003$ to 0.02 (nine curves), the dashed line is $\psi \left(\xi \right)$ solution of Eq. (28). Inset: Same profiles against $z$.

Figure 8

(a) Concentration field $\phi$, (b) component $u_z$, and (c) component $u_x$ at $t=2.5$ ($\mathrm{Ra}={10}^5$ and $\mathrm{Sc}={10}^5$). The black lines in (a) are isoconcentration lines. (d) Average concentration profiles ${\phi }_0(z,t)$ in the dispersion regime plotted against $\eta$ given by Eq. (34), $t$ ranges from $t=1$ to $2.5\times {10}^2$ (10 curves). The dashed line is Eq. (33). Inset: Same profiles against $z$. (e) Rescaled concentration profiles ${\phi }_0(z,t)$ in the late diffusion regime, $t={10}^4$ up to $1.6\times {10}^5$ (10 curves). The dashed line is Eq. (19). Inset: Same profiles against $z$.

Figure 9

(a) Diagram of the different regimes in the plane $\mathrm{Ra}$ versus $t$. The transition times $t$ versus $\mathrm{Ra}$ are given in Table 1. The vertical dotted lines correspond to the extent of the inertial regime $t\simeq 0.0571/\text{Sc}$ for $\mathrm{Sc}={10}^3$ and $\mathrm{Sc}={10}^5$, see Sec. 3b. (b) Width of the mixing zone $w\left(t\right)$ for $\mathrm{Sc}={10}^5$ and $\mathrm{Ra}={10}^5$ (red), ${10}^4$ (blue), ${10}^3$ (magenta). The black dashed line is the purely diffusive spreading $w=\sqrt{t}$. The symbols are the transition times given in Table 1 for $\mathrm{Ra}={10}^5$ and $\mathrm{Ra}={10}^4$.

Figure 10

Width of the mixing zone $w\left(t\right)$ obtained in the case of a channel with a square cross section for $\mathrm{Sc}={10}^3$ and $\mathrm{Ra}={10}^5$ (red), ${10}^4$ (blue), ${10}^3$ (magenta). Asymptotic models for $\mathrm{Ra}={10}^5$: the yellow dashed line given by Eq. (41) corresponds to the regime of late advection, the green dashed line is computed from the numerical resolution of Eqs. (31) and (32) with $\alpha \simeq 739872$. The black dashed line indicates the diffusion law $w=\sqrt{t}$.

Figure 11

3D case, $\mathrm{Ra}={10}^5, \mathrm{Sc}={10}^3$, and $t\simeq 4.7$. (a) Concentration field $\phi (x,y=0,z,t)$. (b) Velocity field $u_z(x,y=0,z,t)$ and (c) $u_x(x,y=0,z,t)$. (d) $\phi (x,y,z=0,t)$, (e) $u_z(x,y,z=0,t)$, (f) $u_x(x,y,z=0,t)$, and (g) $u_y(x,y,z=0,t)$. The black lines in (a) and (d) are isoconcentration lines.

Figure 12

(a) $u_z(x,y,z=0)$ and (b) ${\phi }_1(x,y,z=0)$ calculated using Eqs. (C2) and (C3) at $t\simeq 4.7$, see the corresponding numerical data shown in Figs. 11 and 11. The longitudinal density gradient is calculated by the numerical resolution of the dispersion equation Eq. (31). (c) and (d) show the comparisons along the line $y=z=0$ between the numerical data reported in Figs. 11 and 11(e) (squares) and the theoretical profiles given by Eqs. (C2) and (C3) (black lines).

Figure 13

Components $u_x$ (a) and $u_y$ (b) scaled by ${10}^{-8}{\left(\mathrm{Ra}\frac{\partial {\phi }_0}{\partial z}\right)}^2$ computed from the resolution of the biharmonic equation Eq. (D2). (c) Corresponding transverse flow field in the plane $x-y$. This solenoidal flow field results from the density gradients given by ${\phi }_1$, see Eq. (D2).

Figure 14

Maximal value of the components $u_z$ (red), $u_x$ (blue) and $u_y$ (magenta) in the plane $z=0$ computed from the numerical 3D model at (a) $\mathrm{Ra}={10}^3$, (b) ${10}^4$, and (c) ${10}^5$ ($\mathrm{Sc}={10}^3$ for these three cases). The black lines are the prediction given by Eq. (C2) and the dotted line the prediction given by Eq. (54). In these theoretical predictions, the longitudinal density gradient ${\partial }_z{\phi }_0$ at $z=0$ is calculated from the numerical solution of the 1D dispersion Eq. (31).

Figure 15

Prefactor $\alpha$ in Eq. (32) for a rectangular cross section of aspect ratio $\gamma$. The dashed dotted line is the asymptotic behavior Eq. (C5) valid for $\gamma \to 0$. The magenta line is the approximation Eq. (C6) valid for $\gamma \gg 1$. The dashed line is the value for a slit $\alpha =362880$. Inset: Zoom on the $\gamma$ range 1–4.