10/20/2014 Hagen–Poiseuille flow from the Navier–Stokes equations - Wikipedia, the free encyclopediahttp://en.wikipedia.org/wiki/Hagen%E2%80%93Poiseuille_flow_from_the_Navier%E2%80%93Stokes_equations 1/2 Hagen–Poiseuille flow from the Navier–Stokes equations From Wikipedia, the free encyclopedia In fluid dynamics, the derivation of the Hagen–Poiseuille flow from the Navier–Stokes equations shows how this flow is an exact solution to the Navier–Stokes equations. [1][2] Derivation The laminar flow through a pipe of uniform (circular) cross-section is known as Hagen–Poiseuille flow. The equations governing the Hagen–Poiseuille flow can be derived directly from the Navier–Stokes equations in cylindrical coordinates by making the following set of assumptions: 1. The flow is steady ( ). 2. The radial and swirl components of the fluid velocity are zero ( ). 3. The flow is axisymmetric ( ) and fully developed ( ). Then the second of the three Navier–Stokes momentum equations and the continuity equation are identically satisfied. The first momentum equation reduces to , i.e., the pressure is a function of the axial coordinate only. The third momentum equation reduces to: where is the dynamic viscosity of the fluid. The solution is Since needs to be finite at , . The no slip boundary condition at the pipe wall requires that at (radius of the pipe), which yields Thus we have finally the following parabolic velocity profile: The maximum velocity occurs at the pipe centerline ( ): The average velocity can be obtained by integrating over the pipe cross section: 10/20/2014 Hagen–Poiseuille flow from the Navier–Stokes equations - Wikipedia, the free encyclopedia http://en.wikipedia.org/wiki/Hagen%E2%80%93Poiseuille_flow_from_the_Navier%E2%80%93Stokes_equations 2/2 The Hagen–Poiseuille equation relates the pressure drop across a circular pipe of length to the average flow velocity in the pipe and other parameters. Assuming that the pressure decreases linearly across the length of the pipe, we have (constant). Substituting this and the expression for into the expression for , and noting that the pipe diameter , we get: Rearrangement of this gives the Hagen–Poiseuille equation: References 1. ^ White, Frank M. (2003). "6". Fluid Mechanics (5 ed.). McGraw-Hill. 2. ^ Bird, Stewart, Lightfoot (1960). Transport Phenomena. See also Poiseuille's Law Couette flow Pipe flow Retrieved from "http://en.wikipedia.org/w/index.php?title=Hagen–Poiseuille_flow_from_the_Navier– Stokes_equations&oldid=551319879" Categories: Fluid dynamics Fluid mechanics This page was last modified on 20 April 2013 at 17:30. Text is available under the Creative Commons Attribution-ShareAlike License; additional terms may apply. By using this site, you agree to the Terms of Use and Privacy Policy. Wikipedia® is a registered trademark of the Wikimedia Foundation, Inc., a non-profit organization.
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