Droplet depinning in a wake

Rapid Communication

Droplet depinning in a wake

Alireza Hooshanginejad and Sungyon Lee

Phys. Rev. Fluids 2, 031601(R) – Published 8 March 2017

Abstract

Pinning and depinning of a windswept droplet on a surface is familiar yet deceptively complex for it depends on the interaction of the contact line with the microscopic features of the solid substrate. This physical picture is further compounded when wind of the Reynolds number greater than 100 blows over pinned drops, leading to the boundary layer separation and wake generation. In this Rapid Communication, we incorporate the well-developed ideas of the classical boundary layer to study partially wetting droplets in a wake created by a leader object. Depending on its distance from the leader, the droplet is observed to exhibit drafting, upstream motion, and splitting, due to the wake-induced hydrodynamic coupling that is analogous to drafting of moving bodies. We successfully rationalize the onset of the upstream motion regime using a reduced model that computes the droplet shape governed by the pressure field inside the wake.

Received 16 May 2016

DOI:https://doi.org/10.1103/PhysRevFluids.2.031601

Physics Subject Headings (PhySH)

Boundary layer receptivity, stability & separation Drop & bubble phenomena

Fluid Dynamics

Authors & Affiliations

Alireza Hooshanginejad and Sungyon Lee^*

Department of Mechanical Engineering, Texas A&M University, College Station, Texas 77843, USA

^*sungyon.lee@tamu.edu

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Issue

Vol. 2, Iss. 3 — March 2017

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Images

Figure 1
(a) A water droplet is placed behind a steel hemispherical protuberance on a textured aluminum surface inside a miniature wind tunnel. The images from top-view and side-view cameras are used to analyze the evolving droplet shapes and to determine the onset of motion. (b) A typical side-view image of a droplet ( $V = 50 μ L$ ) placed at a distance, $d$ , downstream from the solid: With no external forcing, the droplet of width, $w$ , forms $θ_{i} = 49 . 1^{\circ}$ with the textured surface. (c) When the wind is turned on, the droplet visibly deforms asymmetrically in the direction of the wind. (d) The droplet at the onset of depinning at the critical wind velocity, $U_{c}$ : The drop forms the maximum contact angle, $θ_{a} = 63 . 5^{\circ}$ , on the advancing side and the minimum angle, $θ_{r} = 8 . 2^{\circ}$ , on the trailing edge.
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Figure 2
(a) $U_{c} / U_{c 0}$ as a function of $d / w_{s}$ for $V$ ranging from $50 μ L$ to $150 μ L$ : $U_{c}$ and $U_{c 0}$ correspond to the critical wind velocity at which a droplet depins downstream behind and without the solid, respectively. (b) The drop's initial width, $w$ , as a function of $V$ : Solid squares refer to experimental values of $w$ extracted from the top-down images (inset), while the solid line is the analytic solution based on the spherical cap geometry, $w = \frac{2 V^{1 / 3}}{tan (θ_{i} / 2)} {[\frac{6 {sin}^{2} (θ_{i} / 2)}{π (2 + cos θ_{i})}]}^{1 / 3}$ , with $θ_{i} = 49 . 1^{\circ}$ . The shaded region indicates the transitional droplets ( $V = 130 - 140 μ L$ ) whose $w \approx l_{r}$ , where $l_{r}$ is the reattachment length. (c) The parabolic function (solid line) empirically describes the pressure inside the reattachment zone of length, $l_{r}$ .
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Figure 3
(a) A $V - d / w_{s}$ phase diagram summarizes distinct droplet behaviors: “downstream” (filled triangle), “upstream” (open triangle), and “splitting” (square); the shaded region corresponds to the upstream regime. (b)i–(b)iv A droplet of $50 μ L$ depins upstream when it is located inside the reattachment zone behind the solid protrusion. (c)i–(c)iv A droplet of $150 μ L$ splits when its downstream portion intersects the reattaching flow, while the upstream portion lies inside the separated zone.
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Figure 4
(a) A 2D schematic of a droplet at the critical onset of depinning upstream when its trailing edge lies at the reattaching point. (b) A schematic of forces acting on a differential element of length, $d x$ , inside the drop. (c) The area of the largest droplet that moves upstream at a given initial distance $d / w_{s}$ is calculated based on a 2D model in qualitative agreement with the experiments. Here, the droplet is assumed to move upstream when it is completely immersed inside the reattachment zone for both case I and case II. Two droplet shapes (initial: dashed line; final: solid line) for case I ( $d / w_{s} = 0.45$ ) and case II ( $d / w_{s} = 0.18$ ) are included. Here the values of the maximum solid-drop distance, $d_{m} \equiv d (area \to 0)$ , and $d_{t}$ , the distance that marks the transition between cases I and II, correspond to $d_{m} / w_{s} \approx 0.55$ and $d_{t} / w_{s} \approx 0.35$ .
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Figure 5
(a)i Two drops of $V = 50 μ L$ separated by $d < 1.4 w$ : Drops deform asymmetrically with imposing wind [(a)ii], and the leader drop is observed to depin before the follower drop and merge with it [(a)iii]. (b)i–(b)iii Droplets of $50 μ L$ at $d > 1.4 w$ are observed to deform and depin approximately at the same time.
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Figure 6
The area of the largest droplet that moves upstream at a given initial distance $d / w_{s}$ is calculated based on the 2D model; the reference case from Fig. 4 is reproduced here in solid lines, with $l_{r} / w_{s} = 1, \hat{a} = 1$ , and ${\hat{p}}_{ext}^{*} = - 0.3$ . (a) Changing $l_{r} / w_{s}$ to 1.2 and 0.8 scales up (down) the overall size of the upstream regime, while the qualitative trend remains unchanged. (b) Varying the pressure gradient inside the reattachment zone by increasing $\hat{a}$ directly decreases $d_{t}$ and the slope of case II, while the value $d_{m}$ is unaffected; the inset of (b) shows that further decreasing ${\hat{p}}_{ext}^{*}$ slightly reduces the droplet sizes that fit case II.
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