# Viscosity, No Slip, Permeable Couette plate

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1. May 18, 2015

### Tom79Tom

I am trying to understand the no slip condition (parallel velocity =0) in the presence of a permeable surface (in this case a Reverse Osmosis filter mesh ).

Does the presence of permeability affect the no slip condition ?

Intuitively

For a gas Where viscous effects are dominated by momentum gradients and molecular surface roughness . I can see permeability (above the mean free path nm scale) affect the no slip condition as the molecules will not interact with the surface (pass thru the permeable sections ) maintaining there average momentum.

For a liquid I see the no slip condition holding at much greater scales (perhaps mm) as the forces of cohesion and adhesion dominate liquid viscosity and will affect average momentum (reducing to 0 ) over a much greater scale.

To me this suggests a scale limit to permeability for both Gas and Liquid scenarios before the no slip assumption is affected

How can I quantify or test this . I cannot find anything that deals with this in my searching?

2. May 20, 2015

### Andy Resnick

Permeable surfaces violate the no-slip condition because the fluid velocity is not zero at the interface- there is mass flow 'through' the surface. IF you do a search for something like 'pipe flow with permeable walls' you will find infinite derivations, for example:

3. May 22, 2015

### Tom79Tom

Perhaps if I rephrase my question I can simplify what I am ultimately trying to understand - In the presence of a surface, impermeable to any component of ux, (Represented in my diagram by the angled surface of my membrane )
• will the static fluid area beyond this impermeable surface be available to carry this momentum according to Bernoulli's
• will flow through this surface be only dependent on an overall Total pressure gradient ?

I have read what I can of this article and while i understand there is a 'through' radial velocity -I am still confused about how the no slip condition, zero parallel velocity =0 ( axial =0 ) is broken, from the article -

'At the tube surface inside the test section, we require zero axial velocity, ux=0, and a radial velocity given by Starling’s equation uσ =Lp (p− pa)

This makes sense to me as there will only be a flow in the presence of a pressure gradient * - without a pressure gradient the flow could be treated as an impermeable wall with additional surface roughness /momentum penetration

What you have stated about mass transfer also makes sense as any fluid mass transiting the surface will carry with it a component of ux momentum but wasn't that was going to happen anyway ? (without permeability ) the difference being was this momentum was only going to be expended to the surface as friction. (Friction coefficient) what doesn't occur (or is diminished in the presence of mass transfer) is the aspect of the friction co-efficient resulting from the return of static pressure from the boundary degrading the flow.

Intuitively mass transfer (arising from permeability) alone does seem to be the mechanism increasing momentum loss

I have found another article of Interest (albeit on turbulent flows) http://onlinelibrary.wiley.com/doi/10.1029/2012GL052369/pdf which outlines the effect of vorticity and momentum penetration on increasing the friction coefficient and depth where ux=0 becomes zero but it still suggests that ux still applies within the boundary. From this article I see a similar Starling term ud that would be driven from a total pressure difference

* I am aware that this pressure gradient will always result from the application of an additional dynamic pressure term to only one component of fluids with equal static pressures.

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Last edited: May 22, 2015
4. May 22, 2015

### Andy Resnick

I'm having a harder time understanding your question now. The no-slip condition means that fluid in contact with a solid *must* have the same velocity as the solid, in order to avoid a singularity in the stress tensor. However, for a permeable surface, the fluid velocity is not that of the wall- there is a component of velocity normal to the permeable surface, as fluid flows through the membrane, and that velocity is typically given as Starling's equation which involves the permeability of the wall and pressure drop across the wall.