Contact angles of a drop pinned on an incline

For a drop on an incline with small tilt angle $\alpha$, when the contact line is a circle of radius $r$, we derive the relation $mgsin\alpha =\gamma r\frac{\pi }{2}(cos{\theta }^{\min }-cos{\theta }^{\max })$ at first order in $\alpha$, where ${\theta }^{\min }$ and ${\theta }^{\max }$ are the contact angles at the back and at the front, $m$ is the mass of the drop and $\gamma$ the surface tension of the liquid. We revisit in this way the Furmidge model for a large range of contact angles. We also derive the same relation at first order in the Bond number $B=\rho g{R}^2/\gamma$, where $R$ is the radius of the spherical cap at zero gravity. The drop profile is computed exactly in the same approximation. Results are compared with surface evolver simulations, showing a surprisingly large range of contact angles for applicability of first-order approximations.

Figure 1

Water drop on hydrophobic incline at angle $\alpha ={30}^{\circ }$. Volume $V\simeq 42\mu \mathrm{L}$. Pinned base radius $r_0\simeq 2.0\mathrm{mm}$. Simulated with surface evolver.

Figure 2

The ratio Eq. (13). As (a) function of $\alpha$: $100\mu \mathrm{l}$ droplet, with fit $1+c{sin}^2\alpha$. As (b) function of drop volume: $+(\alpha ={30}^{\circ })$ and $\square$($\alpha ={60}^{\circ }$), each with fit $1+aV+b{V}^2$. Simulated with surface evolver.

Figure 3

Drop on ${30}^{\circ }$ incline: spherical cap $z_{00}$, first-order $z_{00}+B{z}_1$, and surface evolver profiles. (a) Volume $20\mu \mathrm{l}$, base radius $6\times 0.2^{1/3}\mathrm{mm}$, Bond number $B=5.89$. (b) Volume $100\mu \mathrm{l}$, base radius $r_0=6\mathrm{mm}$, Bond number $B=17.2$. Abscissa $x$ along incline, downwards, and ordinate $z$ perpendicular to incline, both in meters.

Figure 4

Spherical coordinates.

Figure 5

Drop on ${30}^{\circ }$ hydrophobic incline: spherical cap $z_{00}$ with contact angle $2\pi /3$, first-order $z_{00}+B{z}_1$ and surface evolver profiles. (a) Bond number $B$ = 0.5, corresponding to a volume $V\simeq 25\mu \mathrm{l}$ and base radius $r_0\simeq 1.7\mathrm{mm}$. (b) $B$ = 0.8, corresponding to $V\simeq 51\mu \mathrm{l}$ and $r_0\simeq 2.1\mathrm{mm}$. The profiles are scaled by a factor $r_0^{-1}$ for comparison. The dimensionless radius of the contact circle is thus set to one in both figures.