The uniaxial extensional viscosity is the tensile normal-stress difference divided by the imposed extension rate.
A slender axisymmetric column of a Newtonian fluid, with radius , axial velocity , external pressure and axial body-force density , obeys and when inertia, surface tension and leading external shear are negligible. The factor three is the Trouton ratio. Radial mass conservation gives ; the Newtonian fluid stress tensor gives and axial excess stress . This reduces a three-dimensional Stokes flow to one-dimensional evolving geometry.
In a closed-base cylindrical container, mass conservation makes the annular volume flux opposite to the core flux. With and annular viscosity , Couette-Poiseuille flow in a thin gap gives , where removes the annular hydrostatic pressure. Hence . The resulting column equation is . Curvature and the Couette flux change only the neglected relative order .
For a uniform initial column, annular return flow around a slumping column competes with axial extensional stress over . With , choose velocity and time . Equivalently , where and in the leading thin-annulus model. This length describes the vertical range of the basal constraint: a much taller column has a localized basal boundary layer, while a much shorter column responds predominantly through extension.
Using the extensional screening length of a slumping column, an initially uniform column of length obeys , and . Thus and its initial area-fraction growth is . For , the velocity is outside exponentially small end corrections; for , it is . The first limit balances excess weight with annular hydraulic drag, the second with extensional viscous stress.
A column of viscosity in an annulus of viscosity and gap drives return flow of speed in a closed container. Annular shear scales as , inducing relative core-velocity variation . Thus the extensional equations for a slender Newtonian column admit a leading plug-like axial velocity when . The small viscosity ratio alone is insufficient when the gap is very narrow.
The axial strain rate is the axial component of the rate-of-strain tensor. For axial velocity it is . In a uniform uniaxial extensional flow, and incompressible flow requires ; positive stretches the axial direction and contracts the transverse directions.
The Trouton ratio is extensional viscosity divided by shear viscosity. It approaches three for an incompressible Newtonian fluid in uniaxial extension.
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Extensional viscosity is a measure of a fluid's resistance to deformation under extensional (stretching) flow conditions. Unlike shear viscosity, which describes a fluid's resistance to flow when layers slide past one another, extensional viscosity relates to how the fluid behaves when it is being stretched or elongated. In extensional flow, a fluid element experiences forces that cause it to be pulled apart, creating a change in shape without necessarily changing volume.