Hi everyone and thank you in advance for any help. I am struggling to find an analytical solution to either a 2D or 3D Poiseuille flow in a rectangular duct. All I can find is 1D example. Can someone please point me out to the right direction. I have already checked out the Hagen Poiseuille for pipes but I need for rectangular channel.

  • $\begingroup$ This source describes the 2D solution. The short version: Fourier transform in one dimension to get a $sin$ series solution and then solve for the other dimension with $cosh$. Not sure about a 3D solution. Do you want entrance effects? $\endgroup$ – user3823992 May 7 '15 at 4:01
  • $\begingroup$ Yes my entrance will have a uniform flow, my exit will have fully developed flow. $\endgroup$ – Amani Lama May 12 '15 at 16:49

Hele-Shaw flow - the flow between two closely spaced parallel plates can be regarded as a spaciel case of 3D Poiseuille flow. It's governing equations are:

$\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}=0$

$\frac{1}{\mu}\frac{\partial p}{\partial x}=\frac{\partial^2 u}{\partial z^2}$

$\frac{1}{\mu}\frac{\partial p}{\partial y}=\frac{\partial^2 v}{\partial z^2}$

$\frac{\partial p}{\partial z}=0$

where $u,v$ are the velocities in the x and y directions respectively. As can be seen, the momentum equations are non-coupled and thus can be solved separately to yield a "Poiseuille flow" in each direction.

$\bar{u}=\frac{h^2}{12\mu}\frac{\partial p}{\partial x}$

$\bar{v}=\frac{h^2}{12\mu}\frac{\partial p}{\partial y}$.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.