Review of Lagrangian and Eulerian descriptions: Jacobian, Euler’s identity and Reynold’s transport theorem. The continuity equation and incompressibility condition.

这是一份Bath巴斯大学MA40255作业代写的成功案

to $\sigma^{2}$ as $d_{i j}$ tends to zero. In some cases, there will further baseline variability and the

Solve the Jeffery-Hamel wedge-flow relation, Eq. (3-195), for creeping flow, $R e=0$ but $\alpha \neq 0$. Show that the proper solution is
$$
f(\eta)=1+\frac{1}{2} \csc ^{2} \alpha\left[\sin \left(\frac{\pi}{2}-2 \alpha \eta\right)-1\right]
$$
Show also that the constant $C=4 \alpha^{2} \cot ^{2} \alpha$ and sketch a few profiles. Show that backflow always occurs for $\alpha>90^{\circ}$.
In spherical polar coordinates, when the variations $\partial / \partial \phi$ vanish, an incompressible stream function $\psi(r, \theta)$ can be defined such that
$$
u_{r}=\frac{\partial \psi / \partial \theta}{r^{2} \sin \theta} \quad u_{\theta}=-\frac{\partial \psi / \partial r}{r \sin \theta}
$$
The particular stream function
$$
\psi(r, \theta)=\frac{2 v r \sin ^{2} \theta}{1+a-\cos \theta} \quad a=\text { const }
$$

MA40255 COURSE NOTES ：

So let us prove that the entries increase until the middle (then they begin to decrease by the symmetry of the table). We want to compare two consecutive entries:
$$
\left(\begin{array}{l}
n \
k
\end{array}\right) ?\left(\begin{array}{c}
n \
k+1
\end{array}\right) .
$$
If we use the formula in , we can write this as
$$
\frac{n(n-1) \cdots(n-k+1)}{k(k-1) \cdots 1} ? \frac{n(n-1) \cdots(n-k)}{(k+1) k \cdots 1} .
$$
There are many common factors on both sides that are positive, and so we can simplify. We get the really simple comparison
$$
1 ? \frac{n-k}{k+1} \text {. }
$$
Rearranging, we get
$$
k ? \frac{n-1}{2} .
$$