Buoyancy-driven dispersion in confined drying of liquid binary mixtures

We investigate the impact of buoyancy on the solute mass transport in an evaporating liquid mixture (nonvolatile solute $+$ solvent) confined in a slit perpendicular to the gravity. Solvent evaporation at one end of the slit induces a solute concentration gradient which in turn drives free convection due to the difference between the densities of the solutes and the solvent. From the complete model coupling mass transport and hydrodynamics, we first use a standard Taylor-like approach to derive a one-dimensional nonlinear advection-dispersion equation describing the solute concentration process for a dilute mixture. We then perform a complete analysis of the expected regimes using both scaling analysis and asymptotic solutions of this equation. The validity of this approach is confirmed using a thorough comparison with the numerical resolution of both the complete model and the 1D advection-dispersion equation. Our results show that buoyancy-driven free convection always impacts solute mass transport at long timescales, dispersing solutes in a steadily increasing length scale along the slit. Beyond this confined drying configuration, our work also provides an easy way for evaluating the relevance of buoyancy on mass transport in any other microfluidic configuration involving concentration gradients.

Figure 1

(a) Schematic view of the confined drying experiment. The slit is connected at $X\to -\infty$ to a tank containing the dilute mixture at concentration ${\mathrm{\Phi }}_i$. (b) Solvent evaporation at a rate $\dot{E}>0$ drives a flow concentrating continuously the nonvolatile solute at the tip of the slit, see also the colored gradient in (a). The density gradient in turn generates a buoyancy-driven flow, superimposed on the evaporation-induced Poiseuille flow. (c) Without buoyancy, $\delta$ initially grows as $\sqrt{DT}$ before reaching a steady value $\delta \sim D/\dot{E}$, whereas the diffusive layer invades the channel as $\delta \propto T^{2/5}$ when buoyancy-driven free convection dominates at long timescales. These two curves are slightly shifted for the sake of clarity.

Figure 2

(a) Snapshots of $U_X$ (colors) and $\mathrm{\Phi }(X,T)$ (contours) at $T=10$, ${10}^2$, ${10}^3$, ${10}^4$, and ${10}^5$ s, from top to bottom. The range of the color map is scaled by the maximal velocity in each plot (b) Height-averaged concentration profiles ${\mathrm{\Phi }}_0(X,T)$ and (c) velocity profiles $U_X(Z,T)$ at $X=-300\mu \mathrm{m}$, at the same times $T$ as in (a). The thin dark line is the evaporation-driven Poiseuille profile $U_P=6\dot{E}Z(h-Z)/h^2$.

Figure 3

(a) Numerical solution $\delta$ vs. $t$ of the 1D advection-dispersion model Eqs. (23, 24, 25, 26, 27) for different $\text{Pe}\text{Ra}$: ${10}^{-2}$, ${10}^0$, ${10}^2$, and ${10}^4$ (from dark to light blue). The magenta dashed line is the theoretical prediction Eq. (44) in the diffusive regimes D1 and D2. The black lines are the approximations Eq. (45) for the dispersive regime C2. (b)–(d) $\phi (x,t)$ vs. $x$ computed from Eqs. (23, 24, 25, 26, 27) at $\text{Pe}\text{Ra}=1$. (b) Regime D1: $t<0.5$; (c) D2: $1<t<100$; and (d) C2: $t>1000$. In panels (b) and (c), the analytical solutions given by Eq. (B1) (black lines) are superimposed with the numerical resolution. In panel (d), black lines correspond to the approximate solution Eq. (C5).

Figure 4

Diagram highlighting the different regimes of solute mass transport predicted by the 1D advection-dispersion model, Eqs. (23, 24, 25, 26, 27). The thick line corresponds to $t_{\text{D2}\to \text{C2}}$ estimated from Eqs. (49)–(B2). The shaded area corresponds to the diffusive regimes D1 and D2. The dotted line corresponds to the transition between the dispersive regimes C1 $\to$ C2, estimated from Eq. (50). The red bullets represent the transition from a diffusive to a dispersive regime, estimated with the 2D model, Eqs. (A1, A2, A3, A4, A5, A6, A7, A8), for $\mathrm{Pe}\simeq 0.252$ and $\mathrm{Ra}$ varying from 4 to 2000 [critical time estimated from Eq. (47)].

Figure 5

(a) Average dimensionless concentration profiles ${\phi }_0(x,t)$ for both the 2D model (same parameters as in Sec. 2b leading to $\text{Pe}\simeq 0.344$ and $\text{Ra}\simeq 910$), and the solution of the advection-dispersion Eq. (23) for $\text{Pe}\text{Ra}\simeq 313$ (black dotted lines), at times $t\simeq 2.29\times {10}^{-3}$, $2.29\times {10}^{-2}$, 0.229, 2.29, and 22.9. The inset shows a zoom close to the drying interface $x=0$. (b) Extent of the diffusive layer $\delta$ defined by Eq. (29) for the 2D model and the 1D model (black dotted line) with the same parameters as in (a).