= Solution
Let $\mathbf n$ point from fluid $-$ to fluid $+$, $P_s=I-\mathbf n\mathbf n$, $\nabla_s=P_s\nabla$, and $\mathcal K=\nabla_s\cdot\mathbf n$. A clean material interface without phase change has continuous <velocity>, with normal interface speed $V_n=\mathbf u_\pm\cdot\mathbf n$. Its surface stress is $\gamma P_s$. The <surface divergence> of that tensor is $\nabla_s\gamma-\gamma\mathcal K\mathbf n$. Balancing forces on a small interfacial patch therefore gives the <interfacial stress balance with variable surface tension>
$$
\boxed{(\boldsymbol\sigma_+-\boldsymbol\sigma_-)\mathbf n
=\gamma\mathcal K\mathbf n-\nabla_s\gamma,\qquad
\boldsymbol\sigma_\pm=-p_\pm I+\mu_\pm(\nabla\mathbf u_\pm+\nabla\mathbf u_\pm^{T}).}
$$
The normal condition is $\mathbf n\cdot(\boldsymbol\sigma_+-\boldsymbol\sigma_-)\mathbf n=\gamma\mathcal K$; the tangential condition is $P_s(\boldsymbol\sigma_+-\boldsymbol\sigma_-)\mathbf n=-\nabla_s\gamma$. On a stationary <sphere> with outward normal these reduce to $p_--p_+=2\gamma/a$ when viscous normal stresses vanish. This fixes the sign convention for the <Marangoni stress> rather than leaving the direction ambiguous.
For the bubble take negligible internal <dynamic viscosity>, no contaminants, and initially the standard effectively insulating gas-bubble approximation. With heat advection neglected, the exterior <temperature> satisfies <Laplace's equation>, approaches $T_0+\mathbf H\cdot\mathbf r$ remotely, and has zero normal heat flux at $r=a$. The <temperature dipole around an insulating sphere> gives
$$
T=T_0+(\mathbf H\cdot\mathbf r)\left(1+\frac{a^3}{2r^3}\right),\qquad
T_s=T_0+\frac32a\mathbf H\cdot\mathbf n.
$$
The zero-flux condition follows by differentiating the radial factor. Let $\gamma_T=(d\gamma/dT)_{T_0}$. To first order in the weak gradient,
$$
\gamma_s=\gamma_0+\frac32a\gamma_T\mathbf H\cdot\mathbf n,\qquad
\nabla_s\gamma_s=\frac32\gamma_TP_s\mathbf H.
$$
In the laboratory frame the force-free bubble can be described by the harmonic <Papkovich–Neuber representation> potentials
$$
\boldsymbol\Phi=0,\qquad \chi=-\frac{a^3\mathbf U\cdot\mathbf r}{2r^3}.
$$
They give constant exterior <pressure> and the decaying <source dipole> field
$$
\mathbf u=-\frac{a^3}{2r^3}[\mathbf U-3(\mathbf U\cdot\widehat{\mathbf r})\widehat{\mathbf r}].
$$
At the surface $\mathbf u\cdot\mathbf n=\mathbf U\cdot\mathbf n$, so the normal <kinematic boundary condition> holds for translation. In coordinates aligned with $\mathbf U$, $u_r=a^3U\cos\theta/r^3$ and $u_\theta=a^3U\sin\theta/(2r^3)$. Direct differentiation gives
$$
\sigma_{r\theta}=\mu\left(\partial_ru_\theta-u_\theta/r+r^{-1}\partial_\theta u_r\right)
=-3\mu a^3U\sin\theta/r^4.
$$
Equivalently, the surface tangential <traction> is $(3\mu/a)P_s\mathbf U$. The tangential stress condition therefore requires
$$
\frac{3\mu}{a}P_s\mathbf U=-\frac32\gamma_TP_s\mathbf H,
\qquad\boxed{\mathbf U=-\frac{a\gamma_T}{2\mu}\mathbf H.}
$$
For the usual negative $\gamma_T$, the bubble moves toward hotter liquid. The normal condition is consistent as well: its viscous dipole is $-6\mu\mathbf U\cdot\mathbf n/a=3\gamma_T\mathbf H\cdot\mathbf n$, exactly the dipolar part of $2\gamma_s/a$. The constant part fixes $p_{\rm gas}-p_\infty=2\gamma_0/a$. There is no <Stokeslet> term, hence no net applied <force> or drag imbalance.
The thermal assumption matters. The problem gives the exterior <thermal diffusivity> but no gas <thermal conductivity>. For an inviscid bubble with conductivity ratio $K=k_{\rm gas}/k_{\rm liquid}$, temperature and normal heat-flux continuity instead give
$$
T_{\rm out}=T_0+(\mathbf H\cdot\mathbf r)\left[1+\frac{1-K}{2+K}\frac{a^3}{r^3}\right],\qquad
T_s=T_0+\frac3{2+K}a\mathbf H\cdot\mathbf n,
$$
so the same stress calculation gives
$$
\boxed{\mathbf U=-\frac{a\gamma_T}{\mu(2+K)}\mathbf H.}
$$
The insulating-bubble result uses $K\ll1$. If instead an undisturbed surface temperature gradient is imposed, corresponding to $K=1$, the coefficient is $1/3$ rather than $1/2$. Exterior <thermal diffusivity> alone does not select between those thermal boundary conditions. This dependence is explicit rather than silently treating a missing gas property as a prescribed value.
The small parameters can now be stated precisely. With $U_*=a|\gamma_T||\mathbf H|/(2\mu)$ and $\nu=\mu/\rho$, require
$$
\boxed{\mathrm{Re}=U_*a/\nu\ll1,\qquad
\mathrm{Pe}=U_*a/\kappa\ll1,\qquad
\mathrm{Ca}=\mu U_*/\gamma_0\ll1,\qquad
\frac{a|\gamma_T||\mathbf H|}{\gamma_0}\ll1.}
$$
Small <Reynolds number> justifies <Stokes flow>; small <Péclet number> allows quasi-static <heat conduction> without advective distortion; small <capillary number> and fractional surface-tension variation support a nearly spherical interface and a weak perturbation of its base <surface tension>. These last two conditions are proportional for this velocity scale but describe different approximations. Linearizing $\gamma(T)$ also needs $a|\mathbf H||\gamma_{TT}/\gamma_T|\ll1$ if its derivative varies appreciably. Negligible internal viscosity and the insulating specialization require the corresponding gas-to-liquid viscosity and conductivity ratios to be small. Gravity has been excluded; if restored, its buoyancy-induced migration must be small compared with $U_*$, not merely small enough to avoid deformation.
Finally, <surfactants> change both the constitutive surface tension and the surface stress dynamics. The surface flow driven by the thermal <Marangoni effect> carries insoluble surfactant toward the high-tension side, lowering its tension and producing an opposing concentration-driven gradient. When surface diffusion or exchange is slow, this tends to immobilize the interface, suppressing the clean-bubble <thermocapillary migration of an insulating bubble>, sometimes nearly stopping it. If the surfactant stays uniformly equilibrated, it can instead change $\gamma_T$ without a large opposing gradient. Its concentration, surface diffusivity and adsorption kinetics must be supplied to compute a replacement speed; no universal numerical reduction factor follows merely from its presence.
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