Depinning of water droplets from a horizontal solid surface by wall-bounded shear flows

The onset of droplet motion along a horizontal anodized aluminum surface arising from aerodynamic loading imposed by an accelerating wall-bounded shear flow was investigated for water droplet volumes ranging from 75 to $120\mu \mathrm{L}$. Two accelerating shear flows were considered, a flat plate boundary layer and an impinging jet with orientation angles spanning ${30}^{\circ }$ to ${90}^{\circ }$. The flows were linearly accelerated at three different rates, 1.2, 2.2, and $4.4\mathrm{m}/{\mathrm{s}}^2$, to a maximum flow speed of $20\mathrm{m}/\mathrm{s}$. Droplets in the flat plate boundary layer were observed to have a constant depinning threshold Weber number of ${\mathrm{We}}_{h,\mathrm{crit}}=7.5\pm 0.5$. By contrast, droplets in impinging jets exhibited lower thresholds in the range of $2⪅{\mathrm{We}}_{h,\mathrm{crit}}⪅4$. Droplet volume and flow acceleration had marginal influence on depinning threshold for the considered parameter range. Rigorous dimensional analysis, supplemented by statistical examination of the present data and results from previous studies, is employed to identify dominant dimensional groups governing ${\mathrm{We}}_{h,\mathrm{crit}}$. A new dimensionless group that encapsulates interrelated parameters connected to droplet shape and substrate wettability is proposed, producing good collapse of available data and allowing for a simplified empirical estimation of critical depinning velocity.

Figure 1

Experimental setup for droplet depinning under the impact of wall-bounded shear flows formed by (a) flat plate boundary layer and (b) impinging jet.

Figure 2

Side-view geometry of a $120\mu \mathrm{L}$ droplet (a) in the sessile state, and (b) prior to depinning, with geometric parameters annotated in the images. Note that the droplet image is reflected about the ground plane.

Figure 3

Sliding average of velocity sampled in the freestream or at the jet exit (blue lines) and three near-wall locations as the flow speed ramps up at $1.2\mathrm{m}/{\mathrm{s}}^2$ (column 1), $2.2\mathrm{m}/{\mathrm{s}}^2$ (column 2), and $4.4\mathrm{m}/{\mathrm{s}}^2$ (column 3), overlaid with instantaneous velocities measured over five runs (gray lines). Steady near-wall velocity profile (column 4) measured at a free stream velocity or jet exit velocity of $U=10\mathrm{m}/\mathrm{s}$, with dots showing the wall-normal locations where the near-wall velocity ramp-up time histories are sampled. Near-wall velocity profiles and ramp-up time histories measured at $x_i=550\mathrm{mm}$ downstream of the flat plate leading edge (row 1), and at $x^{*}=20\mathrm{mm}$ downstream of the stagnation point of impinging jets oriented at $\alpha ={45}^{\circ }$ (row 2) are shown as examples.

Figure 4

Example of a $120\mu \mathrm{L}$ droplet subjected to a shear flow generated by an accelerating impinging jet with $\alpha ={45}^{\circ }$ and $\mathrm{d}{U}_j/\mathrm{d}t=4.4\mathrm{m}/{\mathrm{s}}^2$. The droplet is placed at $x^{*}=70\mathrm{mm}$ downstream of the jet stagnation point. (a) Velocity ramp-up profile at jet exit. (b-1)–(b-6) Typical droplet deformation and runback at various time points indicated in panel (a).

Figure 5

Typical (a) displacement and (b) velocity of droplet contact points (blue and orange lines: upstream and downstream contact points, respectively; inset: zoom-in view of contact point velocity in the jet exit velocity range of $U_j\le 12\mathrm{m}/\mathrm{s}$), (c) droplet contact length (red line) and height (green line), and (d) contact angle hysteresis (purple line) with increasing jet exit velocity of a $120\mu \mathrm{L}$ droplet under the impact of shear flow formed at $x^{*}=70\mathrm{mm}$ downstream of the stagnation point of a $\alpha ={45}^{\circ }$ accelerating impinging jet at $\mathrm{d}{U}_j/\mathrm{d}t=4.4\mathrm{m}/{\mathrm{s}}^2$ is shown as exemplar. Trend lines (gray dashed lines) are acquired by computing the moving average with a window size corresponding to $\mathrm{\Delta }t=0.25\mathrm{s}$.

Figure 6

Critical droplet depinning velocities (a) measured in the freestream, $U_{\infty }$, or at the jet exit, $U_j$, and (b) measured at droplet height $U_h$ as a function of droplet volume under background flow configurations listed in Table 1.

Figure 7

Critical droplet depinning velocities, $U_{h,\mathrm{crit}}$, averaged across all droplet volumes, as a function of (a) background flow orientation angle $\alpha$, and (b) relative submergence ${\delta }^{*}/h$ at depinning.

Figure 8

Mean side-view geometry of droplets prior to depinning under shear flows formed (a) by a flat plate boundary layer, (b) for $\alpha ={45}^{\circ }$ impinging jet with $x^{*}=20\mathrm{mm}$, and (c) for $\alpha ={90}^{\circ }$ impinging jet with $x^{*}=70\mathrm{mm}$. Lengths in horizontal and vertical directions are normalized by initial droplet contact length and height, respectively.

Figure 9

Droplet (a) contact length, (b) height, and (c) contact angle hysteresis at depinning, averaged across all droplet volumes as a function of background flow orientation angle $\alpha$.

Figure 10

Critical Weber number ${\mathrm{We}}_{h,\mathrm{crit}}$ as a function of (a) $\sqrt{\mathrm{La}}$, (b) $\mathrm{AR}$, (c) ${\delta }^{*}/h$, (d) ${\overline{\theta }}_c$, and (e) $\mathrm{CAH}$.

Figure 11

Critical Weber number based on droplet height ${\mathrm{We}}_{h,\mathrm{crit}}$ as a function of volumetric shape factor $\mathcal{K}$ at depinning; the latter compares the droplet volume to that of an ellipsoid with base diameter $L_b$ and height $h$.