= Capillary tensile force of a slender thread
{title2=$T=3\pi\mu a^2w_z+\pi\gamma a$}
For an axisymmetric <slender viscous thread>, the <Young–Laplace equation> and radial extensional <viscous stress> give $p=p_e+\gamma/a-\mu w_z$. The axial liquid stress relative to ambient <pressure> is $3\mu w_z-\gamma/a$. Its cross-sectional <integral> plus the circumferential <surface tension> <force> $2\pi\gamma a$ gives the displayed tensile <force>. Thus the net capillary contribution is $\pi\gamma a$, and axial <force> balance with downward gravity is $T_z+\pi\rho ga^2=0$.
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