LANGEVIN MODELS FOR SHEAR-STRESS FLUCTUATIONS IN FLOWS OF VISCOELASTIC LIQUIDS

From the principle of virtual work it is shown that, if equilibrium stress fluctuations are described by a Langevin equation, there is only one possible extension of this equation to describe stress fluctuations in a shear flow. It is shown that the resulting equation is consistent with linear-response theory. The formalism developed may be looked upon as a method for extending differential constitutive relations to incorporate thermal fluctuations. A few simple models are discussed as illustrations. These include a model where the stress fluctuates freely below a certain limit, and a model constructed to mimic the Tobolsky-Eyring phenomenological theory [J. Chem. Phys. 11, 125 (1942)] of viscoelastic liquids. It is concluded that these and similar models, however, do not realistically describe real polymeric liquids. To reach this goal, models involving several stress degrees of freedom will have to be considered.