Influence of outer medium viscosity on the motion of rolling droplets down an incline

The role of medium viscosity on a non-deformable rolling body has long been understood while its deformable counterpart still awaits clarification. In this article, we present a scaling analysis with experimental evidences to demonstrate that in creeping flow, medium viscosity significantly alters how the descent speed of a drop responds to an increase in drop size. While descending down an incline in a viscous medium, a rolling drop may travel with either increasing or decreasing velocities as its size increases. The boundary between these two motion behaviors can be maneuvered by altering the viscosity of the surrounding medium.

Figure 1

(a) Schematic of the experimental setup for observing the droplet motion along an inclined plane (not to scale). Snapshots of the moving droplet along its trajectory parallel to the bottom plane of the cuvette are provided, which were used to perform our analysis. The effect of the difference in the liquid reservoir level, denoted as $\mathrm{\Delta }h$, is ignored in this analysis. The inset shows the instantaneous contact angle ($\theta$) of the drop during its motion. The observed motion is assumed to be a rolling motion for very low contact angle hysteresis $(\mathrm{\Delta }\theta \le 7^{\circ })$ and high contact angle of the non-wetting drop [4]. (b) The lowering of the drop's center of mass due to its weight has been denoted by $\delta$ as shown in the schematic [7]. As a drop moves forward, we tracked a small marker on the drop which exhibits cycloidal trajectory. The schematic presented below the experimental data further depicts the motion of the marker as the drop rolls; the linear motion of the center of mass of the drop is denoted by U, the straight arrow indicates the direction of this velocity (U) and the curved arrow shows the direction of the droplet's rotation.

Figure 2

(a) Droplet trajectory at different inclination angles show weak dependency of the rate of displacement (velocity) on the angle of inclination. Top-left inset shows the displacement for varying drop size $\kappa R$, where $1/\kappa$ is the capillary length and $R$ is drop radius (typically less than 1 mm). Bottom-right inset depicts the role of medium viscosity at the same inclination angle of $5^{\circ }$. Both the insets are presented for displacement versus time with same axes limits as figure (a). (b) Theoretical prediction for droplet velocity is plotted over nondimensional drop size $\kappa R$. Mahadevan and Pomeau model [7] (solid line) ignores the medium viscosity, whereas, the present model (dashed lines) accounts for resistance from the medium. Three different cases with ${\mu }_R$ of the order of $10,\sim {10}^3$, and ${10}^4$ demonstrate the role of medium viscosity. A viscous medium results in a regime (shaded region) where increase in size results in increase in the velocity. In the inset, progression of a critical or crossover length scale $\left(\kappa R_{cr}\right)$ towards higher drop size is observed as viscosity ratio decreases—this crossover size demarcates between the two velocity behaviors of the drop for a specific drop-to-medium viscosity ratio as shown by the dashed lines.

Figure 3

Droplet motion for Regime I under different operating parameters: (a) Silicon-oil drop rolling in water medium, ${\mu }_R\sim 110$, at different inclination angles (open diamond—$\alpha =5^{\circ }$, open rectangle—${10}^{\circ }$, and open circle—${15}^{\circ }$) (b) water-glycerol drop rolling in canola oil medium, ${\mu }_R\sim 0.10$, (open diamond—$\alpha ={10}^{\circ }$, open rectangle—${15}^{\circ }$, and open circle—${20}^{\circ }$). Maximum error obtained for experiment is $\pm 8\%$. (c) Non-dimensional descent velocity of liquid droplets increases with $\kappa R$ in Regime I for different drop-medium combinations. Inset (reconstructed from experimental observations of Bico et al. (2009): [14]) shows power law dependency between the two parameters for a non-deformable body.

Figure 4

Occurrence of both regimes for drop descent with double-emulsion drop on a $3^{\circ }$ incline. The increment in velocity with increase in size [regime I of Fig. 2] is obtained by increasing the overall size of the drop. Increment in the droplet size is presented on the top x axis. As presented in the hatched areas of Cases (a)–(e), Regime II behavior (decrements in the velocity) is observed with an increment in the dissipation by inserting an inner drop for a fixed outer drop size. Symbols are experimental results while the dashed and solid lines denote the velocity trend observed from the experimental data.