Define the terms irrotational flow and incompressible flow. The two-dimensional flow of an incompressible fluid is given in terms of a streamfunction $\psi(x, y)$ as

$$\mathbf{u}=(u, v)=\left(\frac{\partial \psi}{\partial y},-\frac{\partial \psi}{\partial x}\right)$$

in Cartesian coordinates $(x, y)$. Show that the line integral

$$\int_{\mathbf{x}_{1}}^{\mathbf{x}_{\mathbf{2}}} \mathbf{u} \cdot \mathbf{n} d l=\psi\left(\mathbf{x}_{\mathbf{2}}\right)-\psi\left(\mathbf{x}_{\mathbf{1}}\right)$$

along any path joining the points $\mathbf{x}_{\mathbf{1}}$ and $\mathbf{x}_{\mathbf{2}}$, where $\mathbf{n}$ is the unit normal to the path. Describe how this result is related to the concept of mass conservation.

Inviscid, incompressible fluid is contained in the semi-infinite channel $x>0$, $0<y<1$, which has rigid walls at $x=0$ and at $y=0,1$, apart from a small opening at the origin through which the fluid is withdrawn with volume flux $m$ per unit distance in the third dimension. Show that the streamfunction for irrotational flow in the channel can be chosen (up to an additive constant) to satisfy the equation

$$\frac{\partial^{2} \psi}{\partial x^{2}}+\frac{\partial^{2} \psi}{\partial y^{2}}=0$$

and boundary conditions

if it is assumed that the flow at infinity is uniform. Solve the boundary-value problem above using separation of variables to obtain

$$\psi=-m y+\frac{2 m}{\pi} \sum_{n=1}^{\infty} \frac{1}{n} \sin n \pi y e^{-n \pi x}$$

$$\begin{aligned} & \psi=0 \quad \text { on } y=0, x>0, \\ & \psi=-m \quad \text { on } x=0,0<y<1, \\ & \psi=-m \quad \text { on } y=1, x>0, \\ & \psi \rightarrow-m y \quad \text { as } x \rightarrow \infty, \end{aligned}$$