Laminar Flow in the Entrance Region of Elliptical Ducts

A closed-form analytical solution is developed to hitherto unsolved problem of steady laminar flow of a Newtonian fluid in the entrance region of elliptical ducts. The analysis is based on the Karman-Pohlhausen integral method and entails solution of the integrated forms of the mass and the momentum balance equations. According to this analysis, the hydrodynamic entrance length based on 99 percent approach to the fully developed flow is equal to 0.5132λ/(1+λ²) where λ is the aspect ratio. Also, the fully developed incremental pressure defect is found to be 7/6 which is independent of the aspect ratio. In the limit when the flow becomes fully developed, the solution converges to the known exact asymptotic solution. Available, wide-ranging velocity measurements for a circular tube agree with the analytical predictions within 7 percent. Also, available pressure drop measurements near the inlet of a circular tube agree with the analytical predictions within 2 percent.