Past exam of the MPhil in micro- and nanotechnology enterprise course of the University of Cambridge / 2019 / Paper 2 / Section A(iii) / 19 / iii / Solution

For the hydraulic ladder network, using the frontier star just defined, start at a single groove with $(x_1,y_1,z_1)=(0,0,R_g)$. Put $B=y_n+R_c$, $C=z_n+R_c$ and $S=B+C+R_g$. The exact update isAfter all grooves have been included, the left frontier is a blind end, so no external flow passes through $y_N$. The inlet-to-outlet resistance is $R_N=x_N+z_N$. Iterate 49 times to obtain $R_{50}$.

An equivalent closed form, for $R_c,R_g>0$, isThus substitute $N=50$; no single number follows without $R_c/R_g$. Checks are $R_1=R_g$, $R_2=(R_g+R_c)/2$, and $R_N\to R_g/N$ as $R_c\to 0$. The formula counts only the $N-1$ inter-groove segments in each rail; extra inlet/outlet lead resistances, if present, must be added separately. All elements obey linear laminar-flow pressure-drop laws.