Blood flow of an Oldroyd-B fluid in a blood vessel incorporating a Brownian stress

The mechanical behavior of non-Newtonian fluids can be modeled by several constitutive differential equations. The Oldroyd model is viewed as one of the successful models for describing the response of a subclass of polymeric liquids, in particular the non-Newtonian behavior exhibited by these fluids. In this paper, we are concerned with the study of the unsteady flows of an incompressible viscoelastic fluid of an Oldroyd-B type in a blood vessel acting on a Brownian force. First we derive the orientation stress tensor considering Hookean dumbbells on Brownian configuration fields. Then we reformulate the three-dimensional Oldroyd-B model with the total stress tensor which consists of the isotropic pressure stress tensor, the shear stress tensor, and the orientation stress tensor. Finally we present the numerical simulations of the model and analyze the effect of the orientation stress tensor in the vessel.