Inertia dominates viscosity when
$$
w=\frac{\tau}{t_0}\gg1,
$$
equivalently when the imposed root-mean-square velocity gradient satisfies $\tau^{-1}\ll t_0^{-1}$. In the inertial-capillary regime,
$$
\Lambda\sim\left(\frac{\sigma\tau^2}{\rho}\right)^{1/3},
\qquad
U\sim\frac{\Lambda}{\tau}.
$$
The corresponding <Reynolds number> is
$$
\operatorname{Re}\sim\frac{\rho U\Lambda}{\eta}
=\frac{\rho\Lambda^2}{\eta\tau}
\sim\left(\frac{\tau}{t_0}\right)^{1/3}.
$$
The geometry is therefore not fixed as the stirring rate changes. Slower stirring permits much larger domains, and the growth of $\Lambda^2$ more than offsets the reduction of $U/\Lambda$. This is why the inertial regime occurs at low imposed velocity gradient in this self-adjusting coarsening problem.

Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 344 / 2 / f / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2024/iii/paper-344/2/f/solution