Flow through a circular tube with a permeable Navier slip boundary
© Cox and Hill; licensee Springer. 2011
Received: 29 January 2011
Accepted: 17 May 2011
Published: 17 May 2011
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.
A body of evidence currently exists in the literature [1, 2] that suggests that fluid flow rates in carbon nanotubes are considerably in excess of that predicted by the conventional Poiseuille flow field, even taking into account a slip boundary condition. Some of this evidence has been re-appraised and certain errors in experimental measurements have been strongly suspected . Despite such findings, there is also a body of independent evidence to suggest that individual molecules (say of water) may achieve flow velocities in carbon nanotubes in vacuo as high as 1,000 m/s . As noted in  these high velocities are greatly reduced if allowance if made for non-vacuum effects. On balance there is sufficient evidence to suggest that fluid flow through carbon nanotubes may be quite different to conventional Poiseuille flow. In this article we ask, under what conditions might other flow fields become available for Newtonian viscous flow in a tube subject to a linear Navier slip boundary condition? At the nanoscale the continuum hypothesis does not apply and the particular point of view adopted in this article is that the Navier-Stokes equations are the best available approximation to nanoscale fluid flow, which together with the linear Navier slip boundary condition might reflect certain nanoscale effects. It is widely believed that confined flows at the nanoscale exhibit both density and viscosity variations closer to the boundaries  which arise in part from molecular van der Waals interactions with the boundaries. In this situation the basic governing flow equations would need to be modified to incorporate the density and viscosity variations. However, even with such modifications, one would still expect comparable formal mathematical solutions to those arising from the case of constant density and viscosity as outlined below.
Conventional Poiseuille flow [6, 7] arises from a pressure which is linear in the dependence along the length of the tube. We find that an exact flow arises satisfying a linear Navier slip boundary condition and arising from a pressure which is quadratic in the dependence along the length of the tube. However, quite remarkably, this second new flow field only exists for a prescribed permeability on the boundary. That is, assuming that the pressure is determined from experimental data, the radial flow velocity at the boundary must be prescribed quite precisely to achieve the quadratic pressure flow field. In the context of carbon nanotubes embedded in a matrix, the boundary may well be permeable at the molecular level, either naturally or arising from defects. Alternatively, if the radial boundary flow is prescribed, then the new flow field exists only for a given quadratic pressure. In other words, both the radial boundary flow and the quadratic pressure cannot be prescribed simultaneously. At present there is insufficient evidence in the literature to conclude whether or not carbon nanotubes have permeable boundaries. Our purpose here is to report that an exact solution of the Navier boundary layer equations with a linear Navier slip boundary can be determined which is different from conventional Poiseuille flow and corresponds to a quadratic pressure and a prescribed permeability on the boundary.
First we comment that there exists in the literature a number of solutions of the Navier-Stokes equations relating to laminar flow in both porous rectangular channels (see for example Berman  and Yuan ) and porous cylindrical pipes (see for example White , Terrill and Thomas  and Terrill [12, 13]), and there are additional references cited in these articles. Second, assuming axially symmetric flow, it transpires that since the radial fluid velocity u(r,z) is assumed to be a function of the radial coordinate r only, namely u(r), then the full Navier-Stokes equations happen to reduce to boundary layer flow in a cylindrical pipe. Accordingly, in the problem studied here, we do not need to assume boundary layer flow but, as described in the following section, the final governing equations are identical to those arising from the boundary layer approximation. This observation is consistent with the observation of Burde , where it is noted that certain solutions of the boundary layer equations for axially symmetric pipe flow are also exact solutions of the full axially symmetric Navier-Stokes equations. Finally, we also comment that for fluid flow at the nanoscale, it is generally believed that there are insufficient numbers of molecules for the Newtonian fluid flow equations to apply. However, any as yet unformulated alternative theory would necessarily be probabilistic in nature and no doubt far more complicated. Again we emphasise that the approach adopted here is to recognise that at present the Newtonian fluid flow equations offer the best approximation, and that the use of the Navier slip boundary condition might well be sufficient to reflect certain aspects of nanoscale fluid flow behaviour.
In the following section we present the Navier-Stokes governing equations for axially symmetric Newtonian flow in a tube, and we determine two coupled non-linear ordinary differential equations arising from incompressible flow with radial velocity u as a function of the radius r only. In the section thereafter we detail results arising from a full numerical solution and compare them with two analytical approximate solutions. Details of the approximate analytical solutions are presented in Appendix A and B. Some brief conclusions are presented in the final section of the article.
However, we emphasise that this equivalence hinges on the assumption u(r,z) = u(r), but as previously noted, Burde  gives other examples of solutions of the boundary layer equations which are also exact solutions of the full Navier-Stokes equations.
must remain finite along the axis r = 0, we have the additional requirements that both A'(1) and B' (1) remain finite. We comment that Terrill and Thomas  provide a comprehensive account of asymptotic and numerical solutions of the single equation (20) for the case of no slip. White  also provides a power series solution of the same problem but again for the case of no slip. As far as the authors are aware, to date an analysis for both A(x) and B(x) non-zero and for the case of slip has not appeared in the literature.
which is the standard equation for fully developed laminar flow with a slip length ℓ.
In this section we illustrate the general features of the flow using both numerical and approximate analytical solutions. We find that for general values of the parameters α, β and γ, the solution only exists for a particular value of the permeability A 0. This is also a feature of the approximate solutions which are given in the appendices. Also we show that the usual boundary value problem with a specific value of A 0 will generally not converge and only a precise value of A 0 enables the general numerical boundary value solution to be obtained. For these reasons we adopt the following procedure to find our numerical solution.
We first begin with the Bessel function approximate solution as outlined in Appendix A. Since the analytical solution has a logarithmic singularity at x = 1 one of the arbitrary constants is immediately forced to zero and therefore only two boundary conditions are necessary to fully determine the solution of the third-order equation. However, the full numerical solution of the non-linear third-order ordinary differential equation (20) requires three boundary conditions and therefore we provide the boundary condition A' (1) = A 1 as determined from the approximate solution and then supply the approximate solution to the numerical solver which then converges to the required solution. A similar issue exists to determine the solution for B(x) and again the approximate analytical solution is used to provide the boundary condition B' (1) = B 1 for the numerical solution. We comment that while these boundary conditions can be thought of as artificial, the approach is justifiable since the linearisation of the governing differential equations is performed using values for A and its derivative at x = 1, and therefore one should expect that the approximate solution will be most accurate at this point.
Table of constant parameters for the analysis of water flow through a carbon nanotube
2 × 10-9
100 × 10-9
where ΔP = P 0-P L . We comment that when ε = 0 then the pressure term is entirely linear, C 1 = 0 and C 2=ΔP/ρL and the solution is standard Poiseuille flow. As ε approaches infinity then C 1→2ΔP/ρL 2 and C 2→0 and the pressure profile is entirely quadratic in the sense that dP/dz = 0 at z = 0.
The problem of fluid flow through carbon nanotubes is believed to involve flow rates which are well in excess of conventional Poiseuille pipe flow. Here we have determined a new exact pipe flow from the Navier-Stokes equations which only becomes available for a certain prescribed flow through an assumed permeable boundary. While such a flow field is non-physical for conventional pipe flow, for a carbon nanotube embedded in a porous matrix, the higher than normal flow rates may well be due to additional permeable boundary flow. Simply not enough is known regarding flow in nanotubes and this possibility may be physically realistic. However, the new flow field gives rise to enhanced flow rates which are at most twice the conventional Poiseuille flow rates and occur for an injected radial flow velocity across the permeable boundary, but certainly does not explain some of the extraordinarily high flow rates that have been reported in the literature for carbon nanotubes.
A Bessel function approximate solution
which is a transcendental equation that may be solved numerically to determine the value of A 1. With the constant A 1 so determined, the problem is over determined in the sense that the tube permeability A 0, must be prescribed and also determined from (38) by substituting x = 0. The interpretation of this result is that the new flow field found here is only physically meaningful for a precise combination of the three variable, the change in pressure, the slip length and the tube permeability, namely α,γ and A 0. This result is also supported by the numerical analysis where we find that a convergent numerical solution is only available for a specific value of A 0, which lies within a limited range.
B Log sine function approximate solution
We note that again we have an equation for A 0 in terms of α and γ and therefore the system may be considered to be over determined such that the solution only exists for a precise combination of these parameters.
The authors acknowledge the support of the Australian Research Council (ARC) through the Discovery Projects scheme and providing an Australian Professorial Fellowship for JMH and an Australian Postdoctoral Fellowship for BJC. The authors are also grateful to Dr Duangkamon Baowan for assistance and useful comments.
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