 # Flow through a cylinder with pores I'm building a model to study the flow of fluid through a cylind Iyana Macdonald 2022-05-17 Answered
Flow through a cylinder with pores
I'm building a model to study the flow of fluid through a cylinder with pores on its surface.
For flow through a cylinder, the velocity of the fluid flow is given by the Hagen-Poiseuille equation.
I would like to ask for suggestions on references from which I can look at derivations for a cylinder with porous walls. By porous, I mean openings on the surface of the cylinder. Example, the pores present in the fenestrated capillary.
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Let Q(z) be the volumetric flow rate along the tube. Then, from a mass balance on the flow,
$\begin{array}{}\text{(1)}& \frac{dQ}{dz}=-\pi Dq\end{array}$
where q(z) is the superficial flow velocity through (i.e., normal to) the porous wall at location z. From Darcy's law, q is related to the pressure difference between inside and outside of the tube P(z) by
$\begin{array}{}\text{(2)}& q=\frac{k}{\mu }\frac{P}{w}\end{array}$
where w is the wall thickness, k is the permeability of the porous medium, and $\mu$ is the fluid viscosity. Combining Eqns. 1 and 2 gives:
$\begin{array}{}\text{(3)}& \frac{dQ}{dz}=-\frac{\pi Dk}{\mu w}P\end{array}$
From the Hagen-Poiseuille equation,
$\begin{array}{}\text{(4)}& \frac{dP}{dz}=-\frac{128\mu }{\pi {D}^{4}}Q\end{array}$
Eqns. 3 and 4 provide two coupled linear ordinary differential equations for the variations in pressure and volumetric flow rate along the tube. You can eliminate P between these equations to solve for Q(z), and then use that to determine P.