Effect of gravity on the shape of a droplet on a fiber: Nearly axisymmetric profiles with experimental validation

We study the effect of gravity on barrel-shaped droplets on fibers using theoretical analyses, numerical simulations, and experiments. By performing a perturbation analysis in the limit of small Bond numbers and small dimensionless droplet volumes, we formulate a nearly axisymmetric solution that describes the shape of the droplet. The leading-order solution yields the axisymmetric profile and the first-order correction incorporates the effect of gravity. Thus, we report the droplet shape as a function of dimensionless droplet volume, contact angle, and Bond number. We find that, due to gravity, the contact line position varies sinusoidally with the azimuthal angle around the fiber. We validate our solution by comparing droplet shape and contact line predictions with experiments and numerical simulations.

Figure 1

The barrel-like shape of a drop on a fiber is described as $r=f(z,\theta )$, where $z$ is the axial direction, $r$ is the radial direction, and $\theta$ is the azimuthal direction. (a) The front view of the droplet. $z=0$ plane divides the drop into two symmetric halves. We denote the axial wetted half-length as $L\left(\theta \right)$. (b) The end-on view of the droplet. The azimuthal angle $-\pi \le \theta \le \pi$, such that $-\pi \le \theta \le 0$ and $0\le \theta \le \pi$ denote the bottom portion and top portion of the drop, respectively.

Figure 2

The nearly axisymmetric solution as described in Table 1. Front ($x\text{−}y$ plane) and end-on views ($y\text{−}z$ plane) of the droplet for $V=125$, $\alpha =\frac{\pi }{12}$, and (a) $\beta =0$, (b) $\beta =5\times {10}^{-3}$, and (c) $\beta ={10}^{-2}$. The contact line as viewed from the top looking in the direction of gravity is also provided. (d) The axial wetted half length, i.e., $L=L_0+\beta L_{1}sin\theta$ for $V=125$ and $\alpha =\frac{\pi }{12}$.

Figure 3

Comparison of nearly axisymmetric solution with surface evolver simulations for $V=27$, $\alpha =\frac{\pi }{12}$, and [(a)–(c)] $\beta ={10}^{-3}$ and [(d)–(f)] $\beta ={10}^{-2}$. The black lines represent the nearly axisymmetric solution and the red lines represent the surface evolver simulations. Comparisons of the front view ($y\text{−}z$ plane), end-on view ($x\text{−}y$ plane), and contact line position $\left[L\left(\theta \right)-L\left(0\right)\right]$ are provided.

Figure 4

Validation of nearly axisymmetric solution with experiments and surface evolver simulations. (a) Front view ($y\text{−}z$ plane), (b) end-on view ($x\text{−}y$ plane), and (c) the half-wetted length, i.e., $L\left(\theta \right)-L\left(0\right)$ for $V=962$, $\alpha =0$, and $\beta =3.5\times {10}^{-3}$. (d) Front view, (e) end-on view, and (f) $L\left(\theta \right)-L\left(0\right)$ for $V=34$, $\alpha =0$, and $\beta =2.49\times {10}^{-2}$. Black circles are experimental data, the black lines are nearly axisymmetric solutions, and the red lines are surface evolver simulations.

Figure 5

Comparison of contact line from experiments (circular data points) and nearly axisymmetric solution (solid line) for (a) $V=2274$ and (b) $V=3695$. $\beta =1.16\times {10}^{-3}$, and $\alpha =0$.

Figure 6

Image of our experimental setup.