= Temperature dipole around an insulating sphere
{title2=$T=T_0+(\mathbf H\cdot\mathbf r)(1+a^3/(2r^3))$}
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 $\partial_rT=0$ 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 $3/(2+K)$.
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