Past exam of the mathematics course of the University of Cambridge / 2017 / iii / Paper 344 / 2 / d / Solution

A stationary microporous network removes fluid momentum through linear drag rather than through a domain-scale viscous stress. With $[\bar{\eta }]=\mathrm{M}{\mathrm{L}}^{-3}{\mathrm{T}}^{-1}$, the single-scale force estimate becomesThis drag-limited hydrodynamic coarsening closure uses a fixed isotropic drag coefficient and domain sizes large compared with the pore scale. The only length and time made from $\rho ,\sigma ,\bar{\eta }$ areIndeed $L_1^3=\sigma t_1^2/\rho$, while the choice of $t_1$ equates inertial and drag prefactors. With $L=L_{1}g(u)$ and $u=t/t_1$, all force terms share $\rho L_1/t_1^2=\bar{\eta }{L}_1/t_1=\sigma /L_1^2$. ThereforeAs before, a parameter-free function of this one argument refers to the asymptotic advective scaling model with fixed morphology and negligible extra scales. Retaining the diffusive mobility or resolved network geometry can introduce additional dimensionless parameters. The original $\eta$ does not remain as an independent viscous coefficient after the stipulated replacement.