The unit is the second part of a bespoke package that covers the applied mathematics aspect of the training, focussing on concrete contexts of industrial application of mathematics. The first unit (Mathematical Modelling for Industry), delivered in Semester 1, provides the relevant theoretical foundations.

这是一份Bath巴斯大学MA50284作业代写的成功案

$\quad E=\frac{1}{2} \rho v^{2}+\frac{1}{\gamma-1} P$,
where $\gamma$ is the ratio of the specific heats. For helium, $\gamma=c_{\mathrm{p}} / c_{\mathrm{v}}=5 / 3$.

When the pulse of gas is initiated, we assume that the shock tube is in thermal equilibrium with a room temperature of $T_{0}=300 \mathrm{~K}$ and that the gas is at rest, $v_{0}=0$. Across the valve is a jump of pressure:
$\quad P(x, 0)= \begin{cases}P_{\mathrm{L}}=4 \mathrm{~atm}, & xx_{0},\end{cases}$

MA50284 COURSE NOTES ：

$\frac{\mathrm{d} \rho}{\rho}+\frac{\mathrm{d} v}{v}+\frac{\mathrm{d} A}{A}=0$.
In the same steady one-dimensional limit, using , the conservation of momentum reduces to
$v \mathrm{~d} v+\frac{1}{\rho} \mathrm{d} P=v \mathrm{~d} v+\frac{c^{2}}{\rho} \mathrm{d} \rho=0$.
Eliminating $\mathrm{d} \rho / \rho$ from and rearranging give the expression
$$
\frac{\mathrm{d} A}{\mathrm{~d} v}=\frac{A}{v}\left(M^{2}-1\right),
$$
where the Mach Number $M=v / c$.