Temperature dipole around an insulating sphere ($T=T_0+(\mathbf{H}\cdot \mathbf{r})(1+a^3/(2r^3))$)

In the absence of heat advection, an insulating sphere of radius $a$ in a uniform remote temperature gradient $\mathbf{H}$ has the displayed exterior harmonic function as its temperature field. The reflected dipole term makes ${\partial }_{r}T=0$ at $r=a$. The surface temperature is $T_0+(3/2)a\mathbf{H}\cdot \mathbf{n}$, so the tangential gradient is amplified by $3/2$. If the sphere has thermal conductivity ratio $K$ relative to its surroundings, temperature and heat-flux continuity instead give surface gradient factor $3/(2+K)$.