Poiseuille Channel Flow

Poiseuille periodic flow can be described as a flow between two stationary plates, caused by a pressure gradient \(\mathrm{d}p/\mathrm{d}x\) parallel to the plates. This is illustrated in the figure below.

We’re mainly interested in the velocity profile from \(y=0`\) to \(y=1\) (top plate) in the flow direction. This profile is analytically defined by equation (1)

(1)\[\frac{u(y)}{u_{\mathrm{avg}}}=6\left(y - y^2\right)\]

And \(u_{\mathrm{avg}}\) is the average velocity, which can be obtained numerically, calculating its value using the simulation data, or using the analytical equation:

(2)\[u_{\mathrm{avg}}=\left(-\frac{\mathrm{d}p}{\mathrm{d}x}\right)\frac{h^{2}}{12\rho\nu}\]

where the pressure gradient can be set through pressure inlet/outlet boundary conditions, or adding a external force to the bulk \(F_{x}=-\mathrm{d}p/\mathrm{d}x\).