Marangoni vortex rings in miscible spreading

We present the observation of an expanding vortex ring below a fast-spreading ethanol-water film over a deep substrate of water. The radius of the vortex ring ($r_v$) is shown to scale with time as $t^{1/2}$. The outer radius of the disturbed region ($r_o$), which comprises the expanding film of length $r_f$ and the plume of length $l_p$ at its periphery, scales in the same way as $r_v$. Based on this observation, contrary to the earlier proposal of the origin of such vortex rings to be due to evaporative cooling, we hypothesize that such vortex rings occur due to the separation of the shear layer below the plumes. Based on this hypothesis, we obtain scaling laws for the radius of the vortex ring ($r_v$) and its radial expansion velocity ($U_v$). These scaling laws are shown to match our measurements, thereby validating the phenomenology of such vortex rings.

Figure 1

(a) Schematic of the spreading film, the plumes developing on the film edge, and the vortex ring in the water layer. (b) Schematic of the side view of the film, the plume, and the vortex ring formed due to the separation and roll up of the shear layer.

Figure 2

Top view of the spreading ethanol-water film at the surface of a deep water layer, from a drop of radius $r_d=0.97$ mm having a concentration of ethanol of $C_e=100\%$. The image is time averaged over ten frames using instantaneous frames acquired at 800 fps. The schematic shows the circulating flow pattern observed within the plumes.

Figure 3

Instantaneous PIV vector fields, overlaid over the corresponding instantaneous vorticity fields, at different instants during the spreading of the ethanol-water film on the surface of a 50-mm-deep water layer, from an ethanol-water drop of $r_d=1\mathrm{mm}$ having an ethanol concentration $C_e=$ 80% ethanol (refer to Movie M3 in the Supplemental Material S10 [1] depicting the temporal variation of the vector field).

Figure 4

(a) Contours of ${\lambda }_2$ values obtained from the velocity field at $t=180\mathrm{ms}$. (b) Variation of the $\theta$ component of vorticity (${\omega }_{\theta }$) along a horizontal line passing through the vortex centers for $C_e=80\%$ and $r_d=1$ mm at $t=180\mathrm{ms}$ (blue dashed line) and $t=220\mathrm{ms}$ (solid red line). The solid line (black) in (a) represents the free surface.

Figure 5

Variation of (a) the experimentally measured vortex ring radius ($r_v$) and (b) the mean radial expansion velocity ($U_v$) of the vortex ring with time ($t$) for ethanol drops of $r_d=1$ mm deposited on a 50-mm-deep water layer having the following concentrations of ethanol in the drop: $△, C_e=40\%$; $\diamond , C_e=60\%$; $\square , C_e=80\%$; $▽, C_e=100\%$. The dashed line in (a) shows the scaling $r_v\sim t^{1/2}$, while the solid symbols show the variation of the experimentally measured plume tip radius ($r_o$) with time for a drop of $C_e=80\%$ and $r_d=0.97$ mm, deposited on a 5-mm-deep water layer. The solid line in (a) shows the scaling (10) for a drop of $C_e=100\%$ and $r_d=0.97$ mm. The inset in (a) shows $r_v$ vs $t$ for $C_e=100\%$ and $r_d=1$ mm, along with the curve fit $r_v=22.5t^{0.46}$ used for estimating $U_v$. In (b), the dashed line shows the scaling $U_v\sim t^{-1/2}$.

Figure 6

Scaling of (a) the dimensionless vortex ring radius ($r_v^{*}$) (10) normalized by ${\mathrm{\Gamma }}_1$(14) and (b) the dimensionless mean radial expansion velocity of the vortex ring ($U_v^{*}$) (17) normalized by ${\mathrm{\Gamma }}_2$(15) with the dimensionless time ($t^{*}=t/t_{\xi }$). The symbols denote $r_v^{*}/{\mathrm{\Gamma }}_1$ and $U_v^{*}/{\mathrm{\Gamma }}_2$ at the following concentrations of ethanol in the drop: $△, C_e=40\%$; $\diamond , C_e=60\%$; $\square , C_e=80\%$; $▽, C_e=100\%$ for a drop of radius $r_d=1$ mm. The dashed line in (a) shows the scaling $r_v^{*}/{\mathrm{\Gamma }}_1=0.7t^{{*}^{3/4}}$, while the dashed line in (b) shows the scaling $U_v^{*}/{\mathrm{\Gamma }}_2=0.72t^{{*}^{-3/4}}$.