(Solved): 3. One places a small oil droplet on a substrate. The droplet forms a spherical cap with a radius o ...

3. One places a small oil droplet on a substrate. The droplet forms a spherical cap with a radius of $R(t)$ on the bottom and a volume of $V$ and spreads on the surface. The droplet totally wets the surface with a precursor film with a dynamic contact angle of ${?}_{\mathrm{d}}(t)?1$. The surface tension and the viscosity of the oil are $?$ and $?$, respectively, and the spreading parameter is $S$. 1) By neglecting the precursor film, estimate the droplet radius $R(t)$ in terms of other given parameters. 2) With the consideration of the precursor film, estimate the droplet radius $R(t)$ in terms of other given parameters. (Hint: For a small dynamic contact angle, the speed $u$ of the contact line scales as $u?$ $\left({?}_d/?\right)F$, where $F$ is the net traction force per unit length of the contact line.)