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

Table of contents
- Solution ii

Solution

s = ϕ_{1} + ϕ_{2}, c = \frac{ϕ _{1}}{s},

where

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:

λ_{s} = W (1 - 2 s) [1 - ϵ (2 c - 1)] + O (ϵ^{2}),

λ_{c} = W [1 - s + ϵ (2 c - 1) (1 + s)] + O (ϵ^{2}) .

At

ϵ = 0

, both species move with the common hindered velocity

W (1 - s)

. Summing their conservation laws gives

s_{t} + W \partial_{z} [s (1 - s)] = 0,

so the total concentration is a nonlinear kinematic wave:

\frac{d z}{d t} = W (1 - 2 s), \frac{d s}{d t} = 0.

Taking the ratio

c = ϕ_{1} / s

instead gives the composition wave in a bidisperse suspension

c_{t} + W (1 - s) c_{z} = 0,

and hence

\frac{d z}{d t} = W (1 - s), \frac{d c}{d t} = 0.

These are the requested leading-order ordinary differential equations along the two characteristic families.

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