Hagen-Poiseuille flow from the Navier-Stokes equations

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:

# The flow is steady ( $partial(...)/partial\; t\; =\; 0$ ).
# The radial and swirl components of the fluid velocity are zero ( $u\_r\; =\; u\_\; heta\; =\; 0$ ).
# The flow is axisymmetric ( $partial(...)/partial\; heta\; =\; 0$ ) and fully developed ($partial\; u\_z/partial\; z\; =\; 0$ ).

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:

:$frac\{1\}\{r\}frac\{partial\}\{partial\; r\}left(r\; frac\{partial\; u\_z\}\{partial\; r\}\; ight)=\; frac\{1\}\{mu\}\; frac\{partial\; p\}\{partial\; z\}$:The solution is :$u\_z\; =\; frac\{1\}\{4mu\}\; frac\{partial\; p\}\{partial\; z\}r^2\; +\; c\_1\; ln\; r\; +\; c\_2$Since $u\_z$ needs to be finite at $r\; =\; 0$, $c\_1\; =\; 0$. The no slip boundary condition at the pipe wall requires that $u\_z\; =\; 0$ at $r\; =\; R$ (radius of the pipe), which yields

:$c\_2\; =\; -frac\{1\}\{4mu\}\; frac\{partial\; p\}\{partial\; z\}R^2.$

Thus we have finally the following parabolic velocity profile:

:$u\_z\; =\; -frac\{1\}\{4mu\}\; frac\{partial\; p\}\{partial\; z\}\; (R^2\; -\; r^2).$

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

:$\{u\_z\}\_\{max\}=frac\{R^2\}\{4mu\}\; left(-frac\{partial\; p\}\{partial\; z\}\; ight).$

The average velocity can be obtained by integrating over the pipe cross-section:: $\{u\_z\}\_mathrm\{avg\}=frac\{1\}\{pi\; R^2\}\; int\_0^R\; u\_z\; cdot\; 2pi\; r\; dr\; =\; 0.5\; \{u\_z\}\_mathrm\{max\}.$

The Hagen-Poiseuille equation relates the pressure drop $Delta\; p$ across a circular pipe of length $L$ to theaverage flow velocity in the pipe $\{u\_z\}\_mathrm\{avg\}$ and other parameters. Assuming that the pressure decreases linearly across the length of the pipe, we have $-\; frac\{partial\; p\}\{partial\; z\}\; =\; frac\{Delta\; p\}\{L\}$ (constant). Substituting this and the expression for $\{u\_z\}\_mathrm\{max\}$ into the expression for $\{u\_z\}\_mathrm\{avg\}$, and noting that the pipe diameter $D\; =\; 2R$, we get: : $\{u\_z\}\_\{avg\}\; =\; frac\{D^2\}\{32\; mu\}\; frac\{Delta\; P\}\{L\}.$Rearrangement of this gives the Hagen-Poiseuille equation:: $Delta\; P\; =\; frac\{32\; mu\; L\; \{u\_z\}\_mathrm\{avg\{D^2\}.$

ee also

*Poiseuille's Law
*Couette flow
*Pipe flow