Annular flow of viscoelastic fluids: Analytical and numerical solutions

Abstract

This work provides analytical and numerical solutions for the linear, quadratic and exponential Phan–Thien–Tanner (PTT) viscoelastic models, for axial and helical annular fully-developed flows under no slip and slip boundary conditions, the latter given by the linear and nonlinear Navier slip laws. The rheology of the three PTT model functions is discussed together with the influence of the slip velocity upon the flow velocity and stress fields. For the linear PTT model, full analytical solutions for the inverse problem (unknown velocity) are devised for the linear Navier slip law and two different slip exponents. For the linear PTT model with other values of the slip exponent and for the quadratic PTT model, the polynomial equation for the radial location (β) of the null shear stress must be solved numerically. For both models, the solution of the direct problem is given by an iterative procedure involving three nonlinear equations, one for β, other for the pressure gradient and another for the torque per unit length. For the exponential PTT model we devise a numerical procedure that can easily compute the numerical solution of the pure axial flow problemCOMPETE, FEDER and Fundação para a Ciência e a Tecnologia (FCT) through projects PEst-C/CTM/LA0025/2013 (Strategic Project – LA 25 – 2013-2014), PTDC/EQU-FTT/113811/2009 and PTDC/EME-MFE/113988/2009. LLF and AMA would also like to thank FCT for financial support through the scholarships SFRH/BD/37586/2007 and SFRH/BPD/75436/2010, respectively