*Article* **A Semi-Analytical Method for Channel and Pipe Flows for the Linear Phan-Thien-Tanner Fluid Model with a Solvent Contribution**

**Matheus Tozo de Araujo 1,†, Laison Furlan 1,†, Analice Brandi 2,† and Leandro Souza 1,\* ,†**

	- Presidente Prudente 19060-900, Brazil

**Abstract:** This work presents a semi-analytical method for laminar steady-state channel and pipe flows of viscoelastic fluids using the Linear Phan-Thien-Tanner (LPTT) constitutive equation, with solvent viscosity contribution. For the semi-analytical method validation, it compares its results and two analytical solutions: the Oldroyd-B model and the simplified LPTT model (without solvent viscosity contribution). The results adopted different values of the dimensionless parameters, showing their influence on the viscoelastic fluid flow. The results include the distribution of the streamwise velocity component and the extra-stress tensor components in the wall-normal direction. In order to investigate the proposed semi-analytical method, different solutions were obtained, both for channel and pipe flows, considering different values of Reynolds number, solvent viscosity contribution in the homogeneous mixture, elongational parameter, shear parameter, and Weissenberg number. The results show that the proposed semi-analytical method can find a laminar solution using the non-Newtonian LPTT model with solvent viscosity contribution and verify the effect of the parameters in the resulting flow field.

**Keywords:** Phan-Thien-Tanner constitutive equation; semi-analytical method; solvent viscosity contribution; pipe flow; channel flow
