Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 345 / 1 / b

Local mass conservation of each particle species gives the system of conservation lawsFor $\mathbf{\varphi }=({\varphi }_1,{\varphi }_2{)}^T$ and flux $F_i={\varphi }_i{W}_i$, the flux Jacobian isWriting its entries as $A_{ij}$, the two characteristic speeds are its eigenvaluesThe discriminant is nonnegative because $A_{12}{A}_{21}={\varphi }_1{\varphi }_2{\hat{W}}_1{\hat{W}}_2\ge 0$, so positive concentrations of two settling species give real characteristics.

Setwhere $s>0$. Applying first-order perturbation of a simple eigenvalue to the flux Jacobian, or expanding its quadratic formula directly, separates the characteristic that changes total concentration from the characteristic that changes composition:At $ϵ=0$, both species move with the common hindered velocity $\hat{W}(1-s)$. Summing their conservation laws givesso the total concentration is a nonlinear kinematic wave:Taking the ratio $c={\varphi }_1/s$ instead gives the composition wave in a bidisperse suspensionand henceThese are the requested leading-order ordinary differential equations along the two characteristic families.