We review the FENE-P dumbbell model of diluted polymer solutions, based on molecular dynamics. Although simplified, this model has proven to be a useful tool for qualitative and quantitative studies of non-Newtonian fluids.

It is demonstrated that under reasonable assumptions, the equations of fluid dynamics with the FENE-P constitutive equations can be solved analytically for some simple flows. In this report, we obtain and investigate the analytical solutions for laminar flows of FENE-P fluids in straight circular tubes and slits of constant width. This includes the expressions for the velocity and shear rate profiles, volume flow rates, pressure gradients, stress tensor components, and viscometric functions. The results are formulated in a manner allowing their direct practical use. A connection to several relations utilized by petroleum engineers is established.

The solutions are generalized to describe the flow through capillary bundles and grids consisting of multiple slits --- these can be thought as a simplified model of porous media. We explain why the behaviour of polymeric fluids in such media is essentially different from that in a single tube or slit, and demonstrate how this difference can be accounted for, if the pore size distribution is known.


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