Warner spring

Figure 3.18 Elastic spring force versus molecular extension for the Warner spring, for the freely jointed chain (which is described by the inverse Langevin function), and for the linear spring. (From Tanner, copyright 1985 by Oxford University Press, Inc. Used by permission of Oxford University Press, Inc.)...

Problem 3.6 Derive Eq. (3-60) for the dumbbell model with a Warner spring. 

Values of p22 — P33 = N2 appear to be negative and approximately 10-30% of Nj in magnitude (82). The conventional bead-spring models yield N2=0. Indeed, N2 in steady shear flow is identically zero for all free draining models, regardless of the force-distance law in the connectors (102a). Thus, finite extensibility and, by inference at least, internal viscosity do not in themselves provide non-zero N2 values. Bird and Warner (354) have recently analyzed the rigid dumbbell model with intramolecular hydrodynamic interaction, the latter represented by the Oseen approximation. In this case N2 turns out to be non-zero but positive. 

The unrealistic behavior of the UCM results from the fact that the Hookean spring allows extensions to go to infinity. A way of improvement is to use a more realistic force law in the model. Warner (1972) replaced the Hookean spring constant Hq by... 

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