Flow through a circular tube with a permeable Navier slip boundary.
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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.
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 |
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Mentions: In Figure 2, we show the flow field resulting from a low value of ε = 0.01, which corresponds to an almost linear pressure gradient in a nanotubes of radius a = 2 nm and length L = 100 nm. Corresponding graphs are displayed in Figures 3 and 4 which show the flow fields for values of ε = 1 and 100, respectively. The leftmost graphs (ℓ = 0) show that as expected the inflow at the tube wall is perpendicular to the tube axis when there is no slip on the tube wall boundary. It also shows that as ε increases, in other words as the quadratic pressure term dominates, then the outflow originates exclusively from the tube wall and the inflow at the tube opening is negligible. The corresponding graphs on the right shown in Figures 2, 3 and 4 are for a slip length of ℓ = 3 nm. In these graphs we again see that as ε increases and the quadratic pressure term dominates, the flow at the tube opening becomes negligible. The salient difference between the graphs on the left and those on the right is that in the rightmost graphs the flow lines at the tube wall are not perpendicular to the axis, which is a feature of the Navier slip condition at that boundary. |
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Affiliation: Nanomechanics Group, School of Mathematical Sciences, University of Adelaide, SA 5005, Australia. barry.cox@adelaide.edu.au.
No MeSH data available.