A plain <material derivative> of the <conformation tensor> is not an <objective time derivative>: an observer undergoing a time-dependent rigid rotation would infer a different constitutive response, and even rigid-body rotation could appear to change polymer deformation. The <upper-convected derivative> subtracts deformation and rotation carried by the velocity gradient and is frame indifferent.

The <Oldroyd-B model> is often inadequate because its Hookean dumbbells are infinitely extensible. It predicts constant shear viscosity rather than shear thinning, zero second <normal-stress difference>, and an unbounded extensional viscosity at a finite extension rate. Real polymer chains have finite extensibility and commonly exhibit shear thinning, bounded extensional stress, multiple relaxation times, and nonlinear solvent or concentration effects.

Solution

Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 352 / 2 / a / Solution

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The conformation tensor describes the average stretch and orientation of polymer molecules. Its equilibrium value is the identity tensor under a common normalization.


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