 Hagen–Poiseuille Flow From The Navier–Stokes Equations

# Hagen–Poiseuille flow from the Navier–Stokes equations

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Description:
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:

1. The flow is steady ( [itex] partial(...)/partial t = 0 [/itex] ).
2. The radial and swirl components of the fluid velocity are zero ( [itex] u_r = u_theta = 0 [/itex] ).
3. The flow is axisymmetric ( [itex] partial(...)/partial theta = 0 [/itex] ) and fully developed ([itex] partial u_z/partial z = 0 [/itex] ).

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

[itex] fracfracleft(r fracright)= frac frac[/itex] where [itex]mu[/itex] is the dynamic viscocity of the fluid.
The solution is
[itex] u_z = frac fracr^2 + c_1 ln r + c_2......

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