= Thermocapillary migration of an insulating bubble
{title2=$\mathbf U=-a\gamma_T\mathbf H/(2\mu)$}
For a clean spherical gas bubble with negligible internal <dynamic viscosity> and <thermal conductivity>, a weak temperature gradient produces the displayed migration <velocity>, where $\gamma_T=d\gamma/dT$. The exterior <Papkovich–Neuber representation> can use only the scalar potential $\chi=-a^3\mathbf U\cdot\mathbf r/(2r^3)$. Its surface normal velocity equals $\mathbf U\cdot\mathbf n$, and its tangential <traction> balances minus the surface <surface tension> gradient from the <temperature dipole around an insulating sphere>. For the usual $\gamma_T<0$ the bubble moves toward warmer liquid. Small <Reynolds number>, <Péclet number>, <capillary number>, and fractional surface-tension variation justify the approximation; <surfactants> can oppose the temperature-driven <Marangoni stress>.
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