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Hagen-Poiseuille flow

The flow of fluid through a pipe of uniform (circular) cross-section is known as Hagen-Poiseuille flow. The Hagen-Poiseuille flow is an exact solution of the Navier-Stokes equations in fluid mechanics. The equations governing the Hagen-Poiseuille flow can be derived from the Navier-Stokes equation in cylindrical coordinates by making the following set of assumptions:

  1. The flow is steady ( \partial(...)/\partial t = 0 ).
  2. The radial and swirl components of the fluid velocity are zero ( ur = uθ = 0 ).
  3. The flow is axisymmetric ( \partial(...)/\partial \theta = 0 ) and fully developed (\partial u_z/\partial z = 0 ).

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Then the second of the three Navier-Stokes momentum equations and the continuity equation are identically satisfied. The first momentum equation reduces to \partial p/\partial r = 0, i.e., the pressure p is a function of the axial coordinate z only. The third momentum equation reduces to:

\frac{1}{r}\frac{\partial}{\partial r}\left(r \frac{\partial u_z}{\partial r}\right)= \frac{1}{\mu} \frac{\partial p}{\partial z}
The solution is
u_z = \frac{1}{4\mu} \frac{\partial p}{\partial z}r^2 + c_1 \ln r + c_2

Since uz needs to be finite at r = 0, c1 = 0. The no slip boundary condition at the pipe wall requires that uz = 0 at r = R (radius of the pipe), which yields

c_2 =  -\frac{1}{4\mu} \frac{\partial p}{\partial z}R^2.

Thus we have finally the following parabolic velocity profile:

u_z = -\frac{1}{4\mu} \frac{\partial p}{\partial z} (R^2 - r^2).

The maximum velocity occurs at the pipe centerline (r=0):

{u_z}_{max}=\frac{R^2}{4\mu} \left(-\frac{\partial p}{\partial z}\right).

The average velocity can be obtained by integrating over the pipe cross-section:

{u_z}_{avg}=\frac{1}{\pi R^2} \int_0^R u_z \cdot 2\pi r dr = 0.5 {u_z}_{max}.

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Poiseuille's Law Couette flow

This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article "Hagen-Poiseuille_flow". A list of authors is available in Wikipedia.
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