Hagen–Poiseuille Flow From The Navier–Stokes Equations

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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.

## Derivation

The flow of fluid 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 equation in cylindrical coordinates by making the following set of assumptions:

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

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- The flow is steady ( <math> partial(...)/partial t = 0 </math> ).
- The radial and swirl components of the fluid velocity are zero ( <math> u_r = u_theta = 0 </math> ).
- The flow is axisymmetric ( <math> partial(...)/partial theta = 0 </math> ) and fully developed (<math> partial u_z/partial z = 0 </math> ).

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

- <math> fracfracleft(r fracright)= frac frac</math> where <math>mu</math> is the dynamic viscocity of the fluid.
- The solution is
- <math> u_z = frac fracr^2 + c_1 ln r + c_2......

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