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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 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 ( $\partial (...)/\partial t=0$ ).
2. The radial and swirl components of the fluid velocity are zero ( $u_{r}=u_{\theta }=0$ ).
3. The flow is axisymmetric ( $\partial (...)/\partial \theta =0$ ) 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:

${\frac {1}{r}}{\frac {\partial }{\partial r}}\left(r{\frac {\partial u_{z}}{\partial r}}\right)={\frac {1}{\mu }}{\frac {\partial p}{\partial z}}$ where $\mu$ is the dynamic viscosity of the fluid.

The solution is

$u_{z}={\frac {1}{4\mu }}{\frac {\partial p}{\partial z}}r^{2}+c_{1}\ln r+c_{2}$

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

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

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:

${u_{z}}_{\mathrm {avg} }={\frac {1}{\pi R^{2}}}\int _{0}^{R}u_{z}\cdot 2\pi rdr=0.5{u_{z}}_{\mathrm {max} }.$

The Hagen–Poiseuille equation relates the pressure drop $\Delta p$ across a circular pipe of length $L$ to the average 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:

$\Delta P={\frac {32\mu L~{u_{z}}_{\mathrm {avg} }}{D^{2}}}.$

References

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See also

• Poiseuille's Law
• Couette flow
• Pipe flow