Past exam of the flows, fluctuations and complexity course of the University of Oxford / 2016 / 1 / c / Solution

Choose the positive $z$ direction along the sphere's laboratory velocity $v\hat{\mathbf{z}}$. In its rest frame the far-field fluid velocity is $-v\hat{\mathbf{z}}$. Therefore the far-field potential is $-vr\cos \theta$. The impermeable sphere imposes ${\partial }_r\varphi =0$ at $r=R$. The monopole coefficient vanishes because the sphere is not a source or sink, and the dipole coefficient follows from the wall condition:ThusThe regular exterior boundary value problem has no forced higher multipoles; uniqueness fixes their coefficients to zero. Differentiating givesHere $u_{\theta }$ is the polar velocity component. An equivalent Cartesian form isOnly no-penetration, not no-slip, is appropriate to this ideal inviscid solution. The sphere density ${\rho }_0$ does not enter the velocity field for an externally prescribed constant speed.^[ref]