*Article* **Exact Solution for Viscoelastic Flow in Pipe and Experimental Validation**

**Ekaterina Vachagina, Nikolay Dushin, Elvira Kutuzova and Aidar Kadyirov \***

Institute of Power Engineering and Advanced Technologies, FRC Kazan Scientific Center, Russian Academy of Sciences, 420111 Kazan, Russia; vachaginae@mail.ru (E.V.); ndushin@bk.ru (N.D.); elvira.kutuzova@list.ru (E.K.) **\*** Correspondence: aidarik@rambler.ru; Tel.: +7-843-292-7597

**Abstract:** The development of analytical methods for viscoelastic fluid flows is challenging. Currently, this problem has been solved for particular cases of multimode differential rheological equations of media state (Giesekus, the exponential form of Phan-Tien-Tanner, eXtended Pom-Pom). We propose a parametric method that yields solutions without additional assumptions. The method is based on the parametric representation of the unknown velocity functions and the stress tensor components as a function of coordinate. Experimental flow visualization based on the SIV (smoke image velocimetry) method was carried out to confirm the obtained results. Compared to the Giesekus model, the experimental data are best predicted by the eXtended Pom-Pom model.

**Keywords:** polymer solution; Giesekus; eXtended Pom-Pom; visualization; analytical solution
