 Solution

For a slender viscous thread, leading axial velocity is uniform over a section. Incompressibility gives $u_r=-r{w}_z/2$. The radial and azimuthal strain rates are $-w_z/2$, and the axial one is $w_z$. The Newtonian fluid stress tensor then has ${\sigma }_{rr}=-p-\mu w_z$. Matching the inviscid exterior normal stress requires $p=p_a-\mu w_z$, so ${\sigma }_{zz}=-p_a+3\mu w_z$. The coefficient three is the Trouton ratio.

A short section has end-stress resultant ${\partial }_z(A{\sigma }_{zz})dz$. The side's hydrostatic traction adds $p_a{A}_{z}dz$, and gravity adds $\rho gAdz$. Neglecting inertia and capillarity, their sum is zero:Since $(p_a{)}_z={\rho }_{a}g$, the balances areThe second is cross-sectional volume conservation, equivalently $D_{t}A=-A{w}_z$. Small interface slope justifies the sectionwise extensional approximation; the slow-flow assumption is additionally needed to neglect inertia.