Local <mass conservation> of each particle species gives the <system of conservation laws>
$$
\partial_t\phi_i+\partial_z(\phi_iW_i)=0,
\qquad i=1,2.
$$
For $\boldsymbol\phi=(\phi_1,\phi_2)^T$ and flux $F_i=\phi_iW_i$, the <flux Jacobian> is
$$
A(\boldsymbol\phi)=
\begin{pmatrix}
(1-2\phi_1)\widehat W_1-\phi_2\widehat W_2&-\phi_1\widehat W_2\\
-\phi_2\widehat W_1&(1-2\phi_2)\widehat W_2-\phi_1\widehat W_1
\end{pmatrix}.
$$
Writing its entries as $A_{ij}$, the two <characteristic speeds> are its <eigenvalues>
$$
\lambda_\pm
=\frac{A_{11}+A_{22}}2
\pm\frac12\sqrt{(A_{11}-A_{22})^2+4A_{12}A_{21}}.
$$
The discriminant is nonnegative because $A_{12}A_{21}=\phi_1\phi_2\widehat W_1\widehat W_2\geq0$, so positive concentrations of two settling species give real characteristics.

Solution

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

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2024/iii/paper-345/1/b/i/solution