= Capillary tensile force of a hollow slender thread
{title2=$T/\pi=3\mu Aw_z+\gamma(R_e+R_i)$}
For a hollow <slender viscous thread>, $A=R_e^2-R_i^2$ and both surfaces contribute axial <surface tension>. The liquid cut <force> relative to exterior <pressure>, including these surface terms, is $\pi[3\mu Aw_z+(p_i-p_e)R_i^2+\gamma(R_e+R_i)]$. The axial traction on the sloping inner gas surface contributes $-\pi(p_i-p_e)(R_i^2)_z$ to <force> balance. For constant gas <pressures> the extra derivative cancels, giving $3\mu(Aw_z)_z+\rho gA+\gamma(R_e+R_i)_z=0$. The displayed $T$ is this effective tension after accounting for gas-pressure work, not the unadjusted liquid cut <force>.
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