Nonuniversal Self-Similarity for Jump-to-Contact Dynamics between Viscous Drops under van der Waals Attraction

When two small viscous drops are sufficiently close, van der Waals force overcomes surface tension and deforms the surfaces into contact, initiating coalescence. The dynamics of surface deformation across an inviscid gap are self-similar as contact is approached, with both radial and gap scales varying as $t^{\prime 1/3}$ for time until contact $t^{\prime }$. Van der Waals and viscous forces are dominant. The self-similar profiles are both nonuniversal and of the second kind: the observed $t^{\prime 1/3}$ behavior is selected only by the subdominant surface tension.

Figure 1

Two viscous drops with surfaces $\tilde{z}=\pm \tilde{h}(\tilde{r},\tilde{t})$ attracted by van der Waals force across an inviscid gap.

Figure 2

Surface profile $d(r,t)$ for minimum heights $d_0(t)=0.9$, 0.7, 0.5, 0.3, 0.1, and ${10}^{-8}$.

Figure 3

$\ln |{\dot{d}}_0|$ (solid) and $-\ln r^{*}$ (dashed) plotted against $-\ln d_0(t)$. After an initial transient, the solid line has gradient 2 and the dashed line has gradient 1.

Figure 4

Rescaled surface profile $d(r,t)/d_0(t)$ plotted against rescaled radial variable $r/d_0(t)$ for minimum heights $d_0(t)=0.3$, 0.1, 0.05, 0.01, and ${10}^{-8}$ (solid). The similarity solution [Eq. (13)] is also plotted (dots).

Figure 5

Scaled curvature $d_0{d}_0^{\prime \prime }(t)$ plotted against $-\ln d_0(t)$ for different initial conditions: $d=d_e(r)-0.1e^{-r^2/100}$ with $d_e(0)=1$ (solid) and $d=d_e(r)-0.02e^{-r^2/100}$ with $d_e(0)=0.6$ (dashed). The two simulations asymptote to $d_0{d}_0^{\prime \prime }=2.39$ and $d_0{d}_0^{\prime \prime }=2.98$, respectively.