Fluid Dynamics | Part IB, 2003

Show that the velocity field

u=U+Γ×r2πr2,\mathbf{u}=\mathbf{U}+\frac{\boldsymbol{\Gamma} \times \mathbf{r}}{2 \pi r^{2}},

where U=(U,0,0),Γ=(0,0,Γ)\mathbf{U}=(U, 0,0), \mathbf{\Gamma}=(0,0, \Gamma) and r=(x,y,0)\mathbf{r}=(x, y, 0) in Cartesian coordinates (x,y,z)(x, y, z), represents the combination of a uniform flow and the flow due to a line vortex. Define and evaluate the circulation of the vortex.

Show that

CR(un)udl12Γ×U as R\oint_{C_{R}}(\mathbf{u} \cdot \mathbf{n}) \mathbf{u} d l \rightarrow \frac{1}{2} \boldsymbol{\Gamma} \times \mathbf{U} \quad \text { as } \quad R \rightarrow \infty

where CRC_{R} is a circle x2+y2=R2,z=x^{2}+y^{2}=R^{2}, z= const. Explain how this result is related to the lift force on a two-dimensional aerofoil or other obstacle.

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