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

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- Solution i

Solution

Local mass conservation of each particle species gives the system of conservation laws

\partial_{t} ϕ_{i} + \partial_{z} (ϕ_{i} W_{i}) = 0, i = 1, 2.

For

ϕ = (ϕ_{1}, ϕ_{2})^{T}

and flux

F_{i} = ϕ_{i} W_{i}

, the flux Jacobian is

A (ϕ) = ((1 - 2 ϕ_{1}) W_{1} - ϕ_{2} W_{2} - ϕ_{2} W_{1} - ϕ_{1} W_{2} (1 - 2 ϕ_{2}) W_{2} - ϕ_{1} W_{1}) .

Writing its entries as

A_{ij}

, the two characteristic speeds are its eigenvalues

λ_{\pm} = \frac{A _{11} + A _{22}}{2} \pm \frac{1}{2} (A_{11} - A_{22})^{2} + 4 A_{12} A_{21} .

The discriminant is nonnegative because

A_{12} A_{21} = ϕ_{1} ϕ_{2} W_{1} W_{2} \geq 0

, so positive concentrations of two settling species give real characteristics.

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