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

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

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Figure 6: Normalised pressure (P -PL)/(P0 -PL) as a function of normalised axial distance z/L. Five specific values of ε are shown.
Mentions: In Figure 6, we graph the normalised pressure (P - PL)/(P0 - PL) as a function of the normalised distance along the axis z/L for various values of the parameter ε. We comment that for ε ≪ 1 we obtain the expected linear relationship between pressure and distance. For ε ≫ 1 we obtain a pressure with a quadratic dependence on distance such that the derivative of the pressure dP/dz approaches zero at z = 0 and at z = L the derivative of the pressure dP/dz for large ε approaches exactly twice the value of the linear relationship obtained for ε ≪ 1.

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.

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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.

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