Methods | Part IB, 2003

The velocity potential ϕ(r,θ)\phi(r, \theta) for inviscid flow in two dimensions satisfies the Laplace equation

Δϕ=[1rr(rr)+1r22θ2]ϕ(r,θ)=0\Delta \phi=\left[\frac{1}{r} \frac{\partial}{\partial r}\left(r \frac{\partial}{\partial r}\right)+\frac{1}{r^{2}} \frac{\partial^{2}}{\partial \theta^{2}}\right] \phi(r, \theta)=0

(a) Using separation of variables, derive the general solution to the equation above that is single-valued and finite in each of the domains (i) 0ra0 \leqslant r \leqslant a; (ii) ar<a \leqslant r<\infty.

(b) Assuming ϕ\phi is single-valued, solve the Laplace equation subject to the boundary conditions ϕr=0\frac{\partial \phi}{\partial r}=0 at r=ar=a, and ϕrUcosθ\frac{\partial \phi}{\partial r} \rightarrow U \cos \theta as rr \rightarrow \infty. Sketch the lines of constant potential.

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