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Evaporation-induced saline Rayleigh convection inside a colloidal droplet
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Image of FIG. 1.

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FIG. 1.

Flow inside evaporating droplets for different NaCl concentrations: (a) 0.01 wt.% (evaporation time (ET) ∼ 39 min); (b) 0.1 wt.% (ET ∼ 45.7 min); (c) 1 wt.% (ET ∼ 50.5 min); and (d) 10 wt.% (ET ∼ 63 min). Exposure time was 20 s for (a), (b), and (c) and 2 s for (d). The substrate was placed on a glass substrate coated with the amorphous fluoropolymer Teflon. The temperature and relative humidity inside the chamber were kept at 25 °C ± 0.5 °C and 45% ± 5%, respectively.

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FIG. 2.

Experimental setup to measure the flow field inside a droplet.

Image of FIG. 3.

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FIG. 3.

Speculative diagram of the movement on a droplet surface.

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FIG. 4.

Images of tracer particles near the droplet surface. The time interval between the two images is 5 s. The droplet volume is 3 μL and the NaCl concentration is 1 wt.%.

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FIG. 5.

Schematic diagram of the axisymmetric coordinate system.

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FIG. 6.

Time evolution of the effective concentration for = 1 and = 0.

Image of FIG. 7.

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FIG. 7.

Peclet-number dependence of the effective concentration for = 0 at τ = 0.2. The solid line, dashed line, and dashed-dotted line correspond to the case of Pe = 0.1, 1, and 4, respectively.

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FIG. 8.

Change in surface area and volume versus evaporation time for a 3 μL NaCl droplet at a concentration of 1 wt.%. The relative uncertainties shown by the error bar are within 5%.

Image of FIG. 9.

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FIG. 9.

Evaporation flux ( ) versus evaporation time for a 3 μL NaCl droplet at a concentration of 1 wt.%. The solid line in the figure shows the best linear-fit calculated lines for both experimental data (symbols). The relative uncertainties shown by error bar are within 5%.

Image of FIG. 10.

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FIG. 10.

Flow field (right) and concentration distribution (left) for initial NaCl concentration of 1 wt.% and volume of 3 μL: (a) slip condition; (b) no-slip condition; (c) experimental results. In the right-hand side of figures (a) and (b), the lines are for the streamline and the color contour represents the velocity magnitude. (Drawings are in gray.) The relative uncertainties shown by error bar are within 5%.

Image of FIG. 11.

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FIG. 11.

Comparison of the velocity magnitude along the symmetric axis for an initial NaCl concentration of 1 wt.% and volume of 3 μL. The symbols show the velocity magnitude along the line of symmetry, and the lines are the numerical results under slip and no-slip conditions. The relative uncertainties shown by the error bars are within 3%.

Image of FIG. 12.

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FIG. 12.

Temporal evolution of flow field and concentration distribution for = 0.197, = 5.85 × 10: (a) τ = 0.02; (b) τ = 0.03; (c) τ = 0.06; (d) τ = 0.09; (e) τ = 0.12; (f) τ = 0.15. The left half of the plane is for the concentration distribution and the right half of the plane is for the velocity field.

Image of FIG. 13.

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FIG. 13.

Distributions of concentration (a) within droplet and (b) effective body force (λ( )). The distributions are taken at the non-dimensional time of τ = 0.15 (half of the hemispherical cap on the left) and τ = 0.30 (on the right) at Pe = 0.197 and = 5.85 × 10. Each scale is shown on the contour line.

Image of FIG. 14.

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FIG. 14.

Computed and measured velocity versus evaporation time at point P(0,0.5) for a 3 μL NaCl droplet at a concentration of 1 wt.%. The snapshots of evaporating droplets are also shown at each salient time. The relative uncertainties shown by the error bar are within 5%.

Image of FIG. 15.

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FIG. 15.

Temporal evolution of flow velocity calculated numerically at point P for various Rayleigh numbers of 5 × 10, 10, 5 × 10, and 10 at the number of 0.197. The velocity is almost saturated before τ = 0.25. The dot symbols on the line denote the flow velocity within the deviation of ± 1% of the equilibrium speed.

Image of FIG. 16.

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FIG. 16.

Effect of Rayleigh number on the concentration distribution and flow field at τ = 0.3 for = 0.197: (a) = 5 × 10; (b) = 10; (c) = 5 × 10; and (d) = 10. The left half of the plane is for the concentration distribution and the right half of the plane is for the velocity field.

Image of FIG. 17.

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FIG. 17.

Computed and measured velocity versus evaporation time at point P for various Rayleigh numbers of 1.8 × 10, 5.8 × 10, 1.8 × 10, 5.8 × 10, 1.7 × 10, and 5.7 × 10. The relative uncertainties shown by the error bar are within 5%.


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Table I.

Characteristic time scales and dimensionless groups.

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Table II.

Numerical and experimental results for vortex-core position and velocity at point P(0,0.5). The vortex-core position is represented by a dimensionless value.


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Inside evaporating two-component sessile droplets, a family of the Rayleigh convection exists, driven by salinity gradient formed by evaporation of solvent and solute. In this work, the characteristic of the flow inside an axisymmetric droplet is investigated. A stretched coordinate system is employed to account for the effect of boundary movement. A scaling analysis shows that the flow velocity is proportional to the (salinity) Rayleigh number ( ) at the small-Rayleigh-number limit. A numerical analysis for a hemispherical droplet exhibits the flow velocity is proportional to the non-dimensional number , at high Rayleigh numbers. A self-similar condition is established for the concentration field irrespective of the Rayleigh numbers after a moderate time, and the flow field is invariant with time at this stage. The scaling relation for the high Rayleigh numbers is verified experimentally by using aqueous NaCl droplets.


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Scitation: Evaporation-induced saline Rayleigh convection inside a colloidal droplet