Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 329 / 1

Use the Papkovich–Neuber representation in the formwhere $\mathbf{\Phi }$ and $\chi$ are harmonic functions. A torque is a axial vector, so rotational covariance and decay select the harmonic vector fieldHere $\mathbf{x}\cdot \mathbf{\Phi }=0$ and $\nabla \cdot \mathbf{\Phi }=0$, soThis rotlet equals the rotating sphere in Stokes flowwhen $\mathbf{\Omega }=\mathbf{G}/(8\pi \mu a^3)$. It satisfies the no-slip boundary condition on $r=a$, decays at infinity, and its Newtonian fluid stress tensor transmits the applied couple $\mathbf{G}$.

Put $S(\mathbf{x})=\mathbf{x}\cdot \mathbf{A}\cdot \mathbf{x}$. Since $\mathbf{u}=-S{r}^{-5}\mathbf{x}$ and $\nabla \times \mathbf{x}=0$, the product rule givesFor the symmetric stresslet tensor $\mathbf{A}$,The final term is parallel to $\mathbf{x}$ and drops out of the cross product, leaving

The Linearity of Stokes flow makes every velocity linear in ${\mathbf{G}}_1$. The only available isotropic polar vector built from the axial vector ${\mathbf{G}}_1$ and separation vector $\mathbf{R}$ is ${\mathbf{G}}_1\times \mathbf{R}$. Dimensional analysis then givesAn angular velocity is axial, so the two independent isotropic possibilities are ${\mathbf{G}}_1$ and $({\mathbf{G}}_1\cdot \mathbf{R})\mathbf{R}$. Hencefor dimensionless scalar functions ${\beta }_i,{\gamma }_i$.

Write $\mathbf{\Omega }={\mathbf{G}}_1/(8\pi \mu a^3)$. The incident rotlet of sphere 1 at sphere 2 isand it is harmonic away from sphere 1. Since sphere 2 is force-free, Faxén's first law therefore givesThe vorticity of the rotlet isBecause ${\mathbf{G}}_2=0$, Faxén's rotational law gives

The symmetric rate-of-strain tensor of the incident rotlet at sphere 2 isAfter translation and rotation have matched the uniform and antisymmetric parts of the incident flow, the leading perturbation from sphere 2 is the stresslet part of the supplied straining-sphere solution:Part b gives its vorticity asAt the centre of sphere 1, $\mathbf{x}=-\mathbf{R}$, soApplying Faxén's rotational law to sphere 1 produces half this ambient vorticity and proves

To hold sphere 2 fixed against the incident angular velocity from part ii, its applied couple must generate the bare rotationThe resulting rotlet advects the force-free sphere 1. Since the field is harmonic, Faxén's first law gives

The returned rotlet has velocity $O(\Omega a^6/R^5)$ and rate of strain $O(\Omega a^6/R^6)$ at sphere 1. That strain induces a stresslet of strength $O(\mu \Omega a^9/R^6)$, whose velocity at sphere 2 is $O(\Omega a^9/R^8)$. This method of reflections for Stokes flow gives the stated order of the next correction to ${\mathbf{U}}_2$.