Exact solution to a 2D/3D Poiseuille flow in a channel

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.

• 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? – user3823992 May 7 '15 at 4:01
• Yes my entrance will have a uniform flow, my exit will have fully developed flow. – 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.

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

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