The polymer solution for polymer flooding is a viscoelastic fluid. There exist both shear flow and elongational flow when the polymer solution flows in a porous medium, where an additional dissipation is involved. The additional dissipation caused by elongational deformation is often ignored while studying the flow of the fluid in a porous medium. For a complex polymer solution, the generated elongational pressure drop cannot be ignored. In a capillary of fixed diameter, the polymer solution is only impacted by the shear force, and its rheological property is pseudoplastic. Therefore the variable diameter capillary and the converging-diverging flow model with different cross sections are required to describe the flow characteristics of the polymer solution in porous media more accurately. When the polymer solution flows through the port, we have the elongational flow and the polymer molecules undergo elongational deformation elastically. By using the mechanical energy balance principle and the minimum energy principle, a mathematical model of non-Newtonian fluid inlet flow was established by Binding. On the basis of the Binding theory, with the application of the theory of viscoelastic fluid flow in the circular capillary and the contraction - expansion tube, the relations between the viscoelastic fluid flow rate and the pressure drop are obtained.