Analytical and numerical investigation of Poiseuille flow through semi-elliptic annulus

Abstract

A fully developed laminar flow through semi-elliptic annulus formed between two confocal elliptical ducts, driven by a constant pressure-gradient, has been analyzed. The elliptic cylindrical coordinate system has been used to determine the exact solutions for “wide” and “narrow” semi-elliptic annuli with cross sections being symmetric about the minor and major axes of the confocal elliptic boundaries, respectively. For both configurations, exact analytical expressions have been obtained for velocity distribution, volume flow rate, shear stress, and Poiseuille number. The results are expressed in terms of two non-dimensional physical parameters: the ratio of the length of the semi-minor axis to the semi-major axis of the outer boundary, ro, 0 ≤ro<1, and the ratio of the length of semi-major axes of inner and outer elliptic boundaries, rma, c ≤rma<1, with c being the non-dimensional focal distance of the elliptic boundaries. Based on the analytical expressions, the graphical and tabulated results of the flow fields are presented for representative values of ro and rma to illustrate the characteristic features of the flow. Numerical evaluation of the analytical expressions shows that the flow field and the corresponding distributions of velocity and shear stresses are characteristically different for wide and narrow semi-elliptic annuli. In addition to the analytical results, a bivariate Chebyshev pseudospectral method is formulated in the elliptic-cylindrical coordinate system for obtaining the numerical solution of the problem. The numerical results show that the proposed method yields “exponential convergence” or “infinite order of accuracy,” as expected from a spectral method; exact agreement has been observed between the analytical and numerical results.