= Solution
For a <slender viscous bubble>, approximate the interface normal by the radial direction and its curvature by $1/a$. Axial curvature and axial derivatives of the leading external motion are smaller by the slenderness factors. The exterior leading <Stokes flow> is locally radial. Incompressibility gives $\partial_r(ru_r)=0$, while the <kinematic boundary condition> gives $u_r(a)=a_t$ to leading order, hence
$$
u_r(r,z,t)=\frac{a(z,t)a_t(z,t)}r.
$$
The radial viscous Laplacian of this $1/r$ <velocity> vanishes, so exterior <pressure> is constant to leading order; set its reference value to zero. Its radial <normal stress> at the interface is $2\mu\partial_ru_r|_a=-2\mu a_t/a$. Inner viscous <normal stress> is smaller by the viscosity ratio $\lambda$, and the bubble's leading <normal stress> is $-P$. The <interfacial stress balance with variable surface tension>, here with constant $\gamma$, consequently gives
$$
P-\frac{2\mu a_t}a=\frac\gamma a,\qquad\boxed{P=\frac{2\mu a_t}a+\frac\gamma a.}
$$
These are local leading-order relations: the outer axial flow and tangential traction enter at higher slenderness order. The sign also has a useful check: a uniform cylinder with no imposed excess <pressure> contracts, $a_t=-\gamma/(2\mu)$, under <surface tension>.
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