A slender viscous thread has a cross-sectional scale much smaller than its axial variation scale. For an axisymmetric Newtonian thread of area and nearly uniform axial velocity , mass conservation and the Trouton ratio givewhen inertia and surface tension are negligible and points downward.
In steady viscous-thread drawing the conserved volume flux couples thinning directly to acceleration. The axial tensile stress is , so solutions with larger positive thin more rapidly.
Viscous-sheet drawing stretches a thin ribbon between rollers. Mass conservation, axial force balance, lateral surface tension, and edge kinematics jointly determine its half-thickness, half-width, and axial speed.
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