Approximate analytical solution for the flow of a Phan-Thien–Tanner fluid through an axisymmetric hyperbolic contraction with slip boundary condition

Abstract

An analytic approximation for the flow of a linear Phan-Thien–Tanner model fluid through an axisymmetric semi-hyperbolic contraction is presented. Such an approximation allows us to compute velocity and pressure response for the flow through axisymmetric contraction geometries; in particular, we have considered here the semi-hyperbolic contraction, which is a geometry where an almost constant extension-rate is reached at different radial positions. In addition, we present a semi-analytic solution capable of representing the exponential version of the selected viscoelastic model; this solution was compared to the results of commercial software, demonstrating the excellent approximation level of the semi-analytic model proposed. Alternatively, for both approaches (linear and exponential Phan-Thien–Tanner), the flow model equations are solved by considering the Navier boundary condition, which allows these models to represent flows with some degree of slip at the geometry wall.