= Planar viscous-sheet stretching equations
{title2=$h_t+(hu)_x=0,\quad(4\mu h u_x)_x+\gamma h h_{xxx}/2=0$}
A symmetric thin <viscous sheet> with shear-free faces has leading longitudinal <velocity> independent of thickness coordinate $z$, and <incompressibility> gives $w=-zu_x$. <Young–Laplace equation> and the <Newtonian fluid stress tensor> give $\sigma_{zz}=-p_a+\gamma h_{xx}/2$ and $\sigma_{xx}=-p_a+\gamma h_{xx}/2+4\mu u_x$. The effective longitudinal tension per unit transverse width is $\mathcal T=4\mu h u_x+\gamma hh_{xx}/2-\gamma h_x^2/4+2\gamma$. The last two terms include the horizontal projections of the two <surface tension> pulls. <Force balance> gives $\mathcal T_x=0$, yielding the displayed longitudinal equation; the face kinematic conditions give the displayed <conservation of mass> equation.
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