Fully Developed Noncircular Pipe Flow
2.2.2 Ducts
2.2.2.2 Fully developed noncircular pipe flow
K. Gersten
A. Laminar flow
In a duct of arbitrary cross-section, the axial velocity distribution u (y, z) is a solution of the following linear partial differential equation
\[\label{eq1}\frac{\partial}{\partial y}\left(\eta\frac{\partial u}{\partial y}\right) +\frac{\partial}{\partial z}\left(\eta\frac{\partial u}{\partial z}\right)=\frac{dp}{dx}=\mbox{const}\tag{1}\]
where x is the main flow direction parallel to the duct axis. On the periphery of the cross section, the no-slip condition u = 0 has to be satisfied For isothermal flows this equation reduces to Poisson's equation, where the constant on the right-hand side is proportional to the pressure gradient and hence to the friction factor.
The reference length for noncircular cross section is the equivalent diameter or hydraulic diameter
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