# 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.

• What's the typical size of pores? – Deep Sep 21 '18 at 15:08
• Have you tried formulating a model on your own? – Chet Miller Sep 21 '18 at 15:19
• @Deep The size of the pore is 6-12 nm. – Natasha Sep 21 '18 at 16:44
• @ChesterMiller I tried procceeding this way, considering the pores to be positioned at equal distance.I am not sure how the pressure drop can be accounted for, as the flow of fluid through these pores will decrease the pressure. In a cylinder, without pores, the average velocity of the fluid can be computed from the Hagen-Poiseuille equation. Likewise, I am trying to compute the average velocity of the fluid along the axial direction of the cylinder and also the average velocity through the pore. – Natasha Sep 21 '18 at 17:02

Let Q(z) be the volumetric flow rate along the tube. Then, from a mass balance on the flow, $$\frac{dQ}{dz}=-\pi D q\tag{1}$$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 $$q=\frac{k}{\mu}\frac{P}{w}\tag{2}$$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: $$\frac{dQ}{dz}=-\frac{\pi Dk}{\mu w}P\tag{3}$$ From the Hagen-Poiseuille equation, $$\frac{dP}{dz}=-\frac{128\mu}{\pi D^4}Q\tag{4}$$ 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.
• Thanks a lot for the clarification. Since the post mentioned porous medium,I got confused. Apologies – Natasha Sep 22 '18 at 3:07
• Yes, you are absolutely right. I got confused because I have learned these terminologies, porous media and superficial velocity, in the context of flow through packed beds, in the past.I failed to understand that porous medium, in this context, refers to the tiny openings in the wall of the cylinder. – Natasha Sep 22 '18 at 4:47