Numerical investigations of Lagrangian unsteady extensional flows of viscoelastic fluids in 3-D rectangular ducts with sudden contractions

Abstract

A numerical study focusing on the macroscopic flow patterns and the Lagrangian unsteady extensional flow behaviour of a viscoelastic fluid in a rectangular duct with a sudden contraction is carried out using a three-dimensional (3-D) finite volume method (FVM). The flow geometry consists of an upstream duct with a square cross-section and a smaller downstream duct with a square or a rectangular cross-section, which corresponds to the so-called „square/square” (Sq/Sq) contraction and the planar contraction, respectively. The Phan-Thien Tanner (PTT) model, including the upper convected Maxwell (UCM) model as a special case, is adopted for the viscoelastic fluid description. Different vortex features observed experimentally in the two different types of contraction flows for the same fluid are predicted. It is found numerically that the macroscopic flow features are well correlated to the fluid‘s Lagrangian unsteady extensional behaviour in the Sq/Sq (or similarly in the axi-symmetric) contraction flow. In the planar contraction flow, however, the predicted Lagrangian unsteady extensional behaviour is different from its steady-state case, and no direct correlation can be made. The large transient extensional stress and the bounded transient extensional viscosity in the planar contraction flow may be responsible for the differences in the vortex patterns in the two contraction flows of viscoelastic fluid with a constant viscosity.