= Solution
For a <slender viscous thread>, leading axial velocity is uniform over a section. Incompressibility gives $u_r=-rw_z/2$. 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 $\rho gA\,dz$. Neglecting inertia and capillarity, their sum is zero:
$$
3\mu(Aw_z)_z-A(p_a)_z+\rho gA=0.
$$
Since $(p_a)_z=\rho_a g$, \b[the balances are]
$$
\boxed{\frac{3\mu}{A}(Aw_z)_z+(\rho-\rho_a)g=0,\qquad
A_t+(Aw)_z=0.}
$$
The second is cross-sectional volume conservation, equivalently $D_tA=-Aw_z$. Small interface slope justifies the sectionwise extensional approximation; the slow-flow assumption is additionally needed to neglect inertia.
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