Simple shear flow invariants

The Invariants of the Rate of Strain Tensor in Simple Shear and Simple Elogational Flows Calculate the invariants of a simple shear flow and elonga-tional flow. 

The apparent viscosity tj in the above equation is a function of the first, second and third invariants of the rate of deformation tensor. For incompressible fluids, the first invariant 7 becomes identically equal to zero. The third invariant III vanishes for simple shear flows and is normally neglected in non-viscometric flows as well. The apparent viscosity then is a function of the second invariant II alone. Hence equation (2.5a) is written in the simplified form as... 

In simple shear flow, = 0, and is a function of the second invariant only, and the y tensor has the form ... 

This equation is most often applied to steady simple shear flows in which the absolute value of the second invariant becomes (eq. 2.2.23)... 

The strain-memory function is derived from the first and second invariants of the Finger strain tensor. For simple shear flow, the strain-memory function is given as... 

The most important flow process in polymer liquids is shear flow. Polymer liquids differ from simple liquids, first in that the shear viscosity is invariably extremely large, and second in that Newton s empirical equation giving a linear relationship between shear stress r and shear strain rate y with constant shear viscosity ft... 

The map of invariants of the rate of deformation tensor lie in the shaded region bounded by simple shear and uniaxi extension for all flows of an incompressible material ly) — 0. 

Here h(/i, 12), the damping function , is a function of the invariants of the Finger strain tensor given in equations (31) and (32) the damping function is determined by requiring the constitutive equation to describe shear and elongational flow data. Extensive comparisons with experimental data show that this rather simple empiricism is extremely useful. Equation (47) gives a value of zero for the second normal stress coefficient. 

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