= Solution
Use the <Papkovich–Neuber representation> in the convention
$$
\mathbf u=\boldsymbol\Phi-\tfrac12\nabla(\mathbf x\cdot\boldsymbol\Phi+\chi),
\qquad p=-\mu\nabla\cdot\boldsymbol\Phi,
\qquad \nabla^2\boldsymbol\Phi=\mathbf0,\quad\nabla^2\chi=0.
$$
It gives $\nabla\cdot\mathbf u=0$ and $\mu\nabla^2\mathbf u=-\mu\nabla(\nabla\cdot\boldsymbol\Phi)=\nabla p$. Additive pressure constants are immaterial.
Let $q=\mathbf x\cdot\mathbf E\mathbf x$, with $\mathbf E$ symmetric and traceless. The harmonic vector $\mathbf E\mathbf x/r^3$ is a derivative of $1/r$. The scalar $q/r^5$ is a decaying degree-two harmonic because $q$ is a harmonic homogeneous quadratic. These <Papkovich potentials for a strained sphere> have precisely the strain symmetry and decay required for the disturbance, so try
$$
\boldsymbol\Phi=A\frac{\mathbf E\mathbf x}{r^3},\qquad \chi=B\frac q{r^5}.
$$
Substitution gives
$$
\mathbf u'=\frac{3A}{2}\frac{q\mathbf x}{r^5}-B\frac{\mathbf E\mathbf x}{r^5}
+\frac{5B}{2}\frac{q\mathbf x}{r^7}.
$$
At $r=a$, require $\mathbf u'=-\mathbf E\mathbf x$. The independent vector structures give $B=a^5$ and $A=-5a^3/3$. \b[The disturbance for a <sphere in a uniform straining Stokes flow> is]
$$
\boxed{\mathbf u'=-\frac{a^5}{r^5}\mathbf E\mathbf x
-\frac{5a^3}{2r^5}\left(1-\frac{a^2}{r^2}\right)q\mathbf x,\qquad
p'=-5\mu a^3\frac q{r^5}.}
$$
It decays at infinity and satisfies no slip exactly at the sphere. Its leading far field is a <stresslet>, of velocity scale $Ea^3/r^2$.
A force-free sphere in a linear background translates with $\mathbf U_0$, by <Faxén's first law>, and a couple-free sphere rotates with the background rigid-body rotation $\boldsymbol\Omega$, by <Faxén's rotational law>. Uniform translation and solid-body rotation can then be matched without disturbance. The remaining relative boundary velocity is exactly $-\mathbf E\mathbf x$. The <Linearity of Stokes flow> and <Uniqueness of Stokes flow> therefore give the same disturbance. The pure strain produces no resultant force or couple, by its symmetric degree-two angular structure.
Back to article page
