On the flow past a needle in a cylindrical tube

Abstract

In this paper, the uniform flow past a needle placed at the centre-line of a tube is analysed for a class of constitutive equations of the Maxwell type using a finite element implementation of the explicitly elliptic momentum equation (EEME) formulation. In the coupled finite element approach, the Galerkin method is applied to the modified momentum equations and the continuity equation, while the treamline upwind Petrov-Galerkin method is applied to the constitutive equations of the viscoelastic fluids. For the power-law and the Phan-Thien-Tanner (PTT) fluids, an asymptotic analysis valid for slender needles is also given. The flow is of practical importance since it forms the basis of an existing commercial viscometer. The viscometer would provide the true shear stress-shear rate relationship if the nominal shear rate is given by KU/R, where U is the falling speed of the needle, R is the radius of the tube, and K is a constant. For the PTT model and the geometry used in this paper, this constant is approximately 4, which is also the value for power-law fluids with power-law index n = 1 3. © 1993.