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Flow through a circular tube with a permeable Navier slip boundary.

Cox BJ, Hill JM - Nanoscale Res Lett (2011)

Bottom Line: Alternatively, if the radial boundary flow is prescribed, then the new flow field exists only for a given quadratic pressure.Our primary purpose here is to demonstrate the existence of a new pipe flow field for a permeable Navier slip boundary and to present a numerical solution and two approximate analytical solutions.The maximum flow rate possible for the new solution is precisely twice that for the conventional Poiseuille flow, which occurs for constant inward directed flow across the boundary.

View Article: PubMed Central - HTML - PubMed

Affiliation: Nanomechanics Group, School of Mathematical Sciences, University of Adelaide, SA 5005, Australia. barry.cox@adelaide.edu.au.

ABSTRACT
For Newtonian fluid flow in a right circular tube, with a linear Navier slip boundary, we show that a second flow field arises which is different to conventional Poiseuille flow in the sense that the corresponding pressure is quadratic in its dependence on the length along the tube, rather than a linear dependence which applies for conventional Poiseuille flow. However, assuming that the quadratic pressure is determined, say from known experimental data, then the new solution only exists for a precisely prescribed permeability along the boundary. While this cannot occur for conventional pipe flow, for fluid flow through carbon nanotubes embedded in a porous matrix, it may well be an entirely realistic possibility, and could well explain some of the high flow rates which have been reported in the literature. Alternatively, if the radial boundary flow is prescribed, then the new flow field exists only for a given quadratic pressure. Our primary purpose here is to demonstrate the existence of a new pipe flow field for a permeable Navier slip boundary and to present a numerical solution and two approximate analytical solutions. The maximum flow rate possible for the new solution is precisely twice that for the conventional Poiseuille flow, which occurs for constant inward directed flow across the boundary.

No MeSH data available.


Related in: MedlinePlus

Flow rates Q for tubes of radius a = 2 nm, length L= 100 nm and a slip length of ℓ ∈{0,3,6,12} nm. Note that the units are 10-18Ls-1 = a Ls-1.
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Figure 5: Flow rates Q for tubes of radius a = 2 nm, length L= 100 nm and a slip length of ℓ ∈{0,3,6,12} nm. Note that the units are 10-18Ls-1 = a Ls-1.

Mentions: In Figure 5, we graph the flow rate Q for a nanotube of radius a = 2 nm, length L = 100 nm and various slip lengths against the parameter ε. We note from this graph that for that most values of the slip length ℓ the ratio of flow rates for ε ≪ 1 and ε ≫ 1 is precisely 1.2. However, we note that for a slip length of l = 3 μm (not graphed here), we find that this ratio begins to degrade and is approximately 1:1.72. This indicates that for larger slip lengths the inflow from the permeable nanotube wall cannot completely replace all the inflow from the open tube end at L = 0. We would expect that this ratio reduces even further for larger values of the tube radius a and the slip length ℓ.


Flow through a circular tube with a permeable Navier slip boundary.

Cox BJ, Hill JM - Nanoscale Res Lett (2011)

Flow rates Q for tubes of radius a = 2 nm, length L= 100 nm and a slip length of ℓ ∈{0,3,6,12} nm. Note that the units are 10-18Ls-1 = a Ls-1.
© Copyright Policy - open-access
Related In: Results  -  Collection

License
Show All Figures
getmorefigures.php?uid=PMC3211482&req=5

Figure 5: Flow rates Q for tubes of radius a = 2 nm, length L= 100 nm and a slip length of ℓ ∈{0,3,6,12} nm. Note that the units are 10-18Ls-1 = a Ls-1.
Mentions: In Figure 5, we graph the flow rate Q for a nanotube of radius a = 2 nm, length L = 100 nm and various slip lengths against the parameter ε. We note from this graph that for that most values of the slip length ℓ the ratio of flow rates for ε ≪ 1 and ε ≫ 1 is precisely 1.2. However, we note that for a slip length of l = 3 μm (not graphed here), we find that this ratio begins to degrade and is approximately 1:1.72. This indicates that for larger slip lengths the inflow from the permeable nanotube wall cannot completely replace all the inflow from the open tube end at L = 0. We would expect that this ratio reduces even further for larger values of the tube radius a and the slip length ℓ.

Bottom Line: Alternatively, if the radial boundary flow is prescribed, then the new flow field exists only for a given quadratic pressure.Our primary purpose here is to demonstrate the existence of a new pipe flow field for a permeable Navier slip boundary and to present a numerical solution and two approximate analytical solutions.The maximum flow rate possible for the new solution is precisely twice that for the conventional Poiseuille flow, which occurs for constant inward directed flow across the boundary.

View Article: PubMed Central - HTML - PubMed

Affiliation: Nanomechanics Group, School of Mathematical Sciences, University of Adelaide, SA 5005, Australia. barry.cox@adelaide.edu.au.

ABSTRACT
For Newtonian fluid flow in a right circular tube, with a linear Navier slip boundary, we show that a second flow field arises which is different to conventional Poiseuille flow in the sense that the corresponding pressure is quadratic in its dependence on the length along the tube, rather than a linear dependence which applies for conventional Poiseuille flow. However, assuming that the quadratic pressure is determined, say from known experimental data, then the new solution only exists for a precisely prescribed permeability along the boundary. While this cannot occur for conventional pipe flow, for fluid flow through carbon nanotubes embedded in a porous matrix, it may well be an entirely realistic possibility, and could well explain some of the high flow rates which have been reported in the literature. Alternatively, if the radial boundary flow is prescribed, then the new flow field exists only for a given quadratic pressure. Our primary purpose here is to demonstrate the existence of a new pipe flow field for a permeable Navier slip boundary and to present a numerical solution and two approximate analytical solutions. The maximum flow rate possible for the new solution is precisely twice that for the conventional Poiseuille flow, which occurs for constant inward directed flow across the boundary.

No MeSH data available.


Related in: MedlinePlus