Slender viscous thread
= Slender viscous thread
A slender viscous thread has a cross-sectional scale much smaller than its axial variation scale. For an axisymmetric Newtonian thread of area $A$ and nearly uniform axial velocity $w$, <mass conservation> and the <Trouton ratio> give
$$
A_t+(Aw)_z=0,
\qquad
\rho gA+\partial_z(3\mu A w_z)=0
$$
when inertia and <surface tension> are negligible and $z$ points downward.