= Extensional equations for a slender Newtonian column
{title2=$(a^2)_t+(a^2w)_z=0$}
A slender axisymmetric column of a <Newtonian fluid>, with radius $a(z,t)$, axial <velocity> $w$, external <pressure> $p_e$ and axial body-force density $f$, obeys $(a^2)_t+(a^2w)_z=0$ and $3\mu(a^2w_z)_z/a^2=p_{e,z}-f$ when inertia, <surface tension> and leading external shear are negligible. The factor three is the <Trouton ratio>. Radial <mass conservation> gives $u_r=-rw_z/2$; the <Newtonian fluid stress tensor> gives $p=p_e-\mu w_z$ and axial excess <stress> $3\mu w_z$. This reduces a three-dimensional <Stokes flow> to one-dimensional evolving geometry.
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