Explain why the irrotational flow of an incompressible fluid can be expressed in terms of a velocity potential $\phi$ that satisfies Laplace's equation.

The axis of a stationary cylinder of radius $a$ coincides with the $z$-axis of a Cartesian coordinate system $(x, y, z)$ with unit vectors $\left(\mathbf{e}_{x}, \mathbf{e}_{y}, \mathbf{e}_{z}\right)$. A fluid of density $\rho$ flows steadily past the cylinder such that the velocity field $\mathbf{u}$ is independent of $z$ and has no component in the $z$-direction. The flow is irrotational but there is a constant non-zero circulation

$$\oint \mathbf{u} \cdot d \mathbf{r}=\kappa$$

around every closed curve that encloses the cylinder once in a positive sense. Far from the cylinder, the velocity field tends towards the uniform flow $\mathbf{u}=U \mathbf{e}_{x}$, where $U$ is a constant.

State the boundary conditions on the velocity potential, in terms of polar coordinates $(r, \theta)$ in the $(x, y)$-plane. Explain why the velocity potential is not required to be a single-valued function of position. Hence obtain the appropriate solution $\phi(r, \theta)$, in terms of $a, U$ and $\kappa$.

Neglecting gravity, show that the net force on the cylinder, per unit length in the $z$-direction, is

$$-\rho \kappa U \mathbf{e}_{y}$$

Determine the number and location of stagnation points in the flow as a function of the dimensionless parameter

$$\lambda=\frac{\kappa}{4 \pi U a}$$