Investigating enhanced mass flow rates in pressure-driven liquid flows in nanotubes

Over the past two decades, several researchers have presented experimental data from pressure-driven liquid flows through nanotubes. They quote flow velocities which are four to five orders of magnitude higher than those predicted by the classical theory. Thus far, attempts to explain these enhanced mass flow rates at the nanoscale have focused mainly on introducing wall-slip boundary conditions on the fluid mass velocity. In this paper, we present a different theory. A change of variable on the velocity field within the classical Navier–Stokes equations is adopted to transform the equations into physically different equations. The resulting equations, termed re-casted Navier–Stokes equations, contain additional diffusion terms whose expressions depend upon the driving mechanism. The new equations are then solved for the pressure driven flow in a long nano-channel. Analogous to previous studies of gas flows in micro- and nano-channels, a perturbation expansion in the aspect ratio allows for the construction of a 2D analytical solution. In contrast to slip-flow models, this solution is specified by a no-slip boundary condition at the channel walls. The mass flow rate can be calculated explicitly and compared to available data. We conclude that the new re-casting methodology may provide an alternative theoretical physical explanation of the enhanced mass flow phenomena.


Introduction
Advances in technology have highlighted the importance of understanding and modelling fluids in micro-and nano-structures found in many mechanical and bio-mechanical devices. An important example of these structures are micro-and nano-electro-mechanical systems (MEMS/NEMS) devices such as micromotors, comb microdrives, blood flow analysers and particle separators [1,2]. Other areas of interest include transport in aquaporin water channels in biological cell membranes [3] and gas flow through shale rock formations [4]. A key feature of fluid flow in these confined spaces is a significant departure from predictions of the classical Navier-Stokes equations. Experimental evidence on both gases and liquids points towards enhanced mass flow rates in micro-and nano-sized ducts when compared to the classical no-slip Hagen-Poiseuille flow law. This was first observed by Knudsen [5] while investigating the flow of rarefied gases in narrow tubes. His findings were later confirmed and extended for gases in various states of rarefaction [6,7]. The extent to which a gas is rarefied is conveniently expressed by the dimensionless Knudsen number Kn=λ/h, where λ is the mean free path of a gas molecule and h the height/diameter of the confining duct. An extensive experimental data-set covering the widest range of Knudsen number (0.01<Kn<100) to date is given by Ewart et al [8] for the flow of helium in long rectangular micro-channels of 9.38 μm height and the reported data clearly capture the famous Knudsen minimum at Kn≈0. 8. In stark contrast to Navier-Stokes based predictions, the mass flow rate increases further as the gas becomes more rarefied.
While enhanced flow rates of rarefied gases have been known since Knudsen's original findings, the phenomenon of enhanced liquid flows in micro-and nano-channels is a more recent discovery. Early indications of this were given by Pfahler et al, who found n-propanol flow rates to be three times higher than expected in rectangular channels of 800 nm long [9]. At much smaller scales, both Majumder et al [10,11] and 2. Re-casted Navier-Stokes equations for isothermal pressure-driven flow For flow under isothermal conditions, we can restrict attention to the transport of mass and momentum only. In the classical Navier-Stokes formalism, these are governed by the following balance laws: Here, ρ=ρ(x t , ) is the fluid density, p= ( ) x p t , is the pressure and Π (NS) is the Newtonian stress tensor which is given in terms of the fluid's dynamic viscosity (μ) as: The unknown velocity field U m = ( ) U x t , m is that of the conventional mean mass velocity. In our new theory, an externally applied pressure gradient is the principal driving mechanism of the flow; hence we assume that the classical mass velocity can be written in terms of a new pressure diffusion velocity U p as: The second term on the right hand side represents a mass diffusion mechanism driven by the pressure gradient. This distinction between U m and U p is analogous to the idea of a volume velocity in gas [22,25,28]. The precise functional form of the molecular pressure diffusivity coefficient κ p is not yet understood. For simplicity, it is assumed to be constant in the following derivations. Substituting equation (4) into the Navier-Stokes equations (1)- (2) and later re-arranging terms leads to the re-casted Navier-Stokes (RNS) equations for mass and momentum: The last term on the left-hand side of equation (6) contains the new stress tensor: where Π p denotes the transformed Newtonian stress tensor, We can also write the new momentum equation (6) in the following form: with the tensor T (RNS) on the right-hand side of this equation given by, Here, ( ) U D p denotes the symmetric part of the velocity gradient. A parallel can be drawn between the structure of tensor T (RNS) and Korteweg's stress tensor T [33]. Korteweg augmented the Newtonian stress tensor with the dyadic product ∇ρ⊗∇ρ to represent forces experienced by fluids during phase transitions. His complete tensor may be written as [34]: where the material coefficients α 0 , α 1 , β, μ, λ may depend on ρ as well. On comparing equation (11) with (10), we note that all terms involved in the structure of the Korteweg stress tensor are found in the re-casted Navier-Stokes tensor, but written with p rather than ρ.

Model equations for flows in micro-and nanotubes
In the experiment described by Majumder et al [11], different liquids were forced to flow through carbon nanotube (CNT) membranes by means of a constant externally applied pressure gradient of 1 bar. The crosssectional area of CNT membrane exposed to the flow was 0.785 cm 2 , with a length L of only 126 μm. The millions of fluid-carrying pores were envisaged as aligned cylindrical tubes of diameter d and length L, where d was determined to be about 7 nm. The volume of fluid exiting the membrane was collected over a period of time. Taking into account the membrane's porosity, it was then possible to establish the volumetric flow rate through a single nanotube, which was compared with the theoretical flow rates predicted by the classical (no-slip) Hagen-Poiseuille flow law (see table 1 in section 5). Figure 1 shows a schematic of a CNT membrane (not to scale) with an individual nanopore enlarged to highlight its cylindrical geometry. A similar experimental set-up was adopted by other researchers [12][13][14][15].

Governing equations
For an analytical treatment of the flow configuration described in figure 1 using the re-casted Navier-Stokes equations (5)-(8), we can make several simplifications based on geometrical considerations. Firstly, we assume that the liquid density ρ and viscosity μ are constant. Furthermore, the maintenance of a constant pressure difference and the length of the tube imply that the flow can be considered steady, with minimal entrance/exit losses. Thus, the unknown fields U p and p are functions of position ( ) = x x y z , , only. With these assumptions in place, equations (5)- (8) can be simplified and re-written as follows: For the nanotube geometry shown in figure 1, it is most convenient to transform these equations into cylindrical polar coordinates (r, θ, z). The radial symmetry implies that the flow fields are independent of the swirl angle θ. Moreover, we can assume that the θ-component of the velocity is zero, so: Then, using the transformed expression for the gradient operator in cylindrical polar coordinates, equations (12)-(13) can be written in component form as follows:   -( ) In order to solve equations (15)- (17), we must stipulate appropriate boundary conditions on the velocity field ( ) U r z , p and the pressure distribution ( ) p r z , . The radial component of the pressure diffusion velocity, U p r , is chosen in such a way that no mass passes through the confining walls. This amounts to setting: Furthermore, we will adopt a no-slip condition on the tangential component of the pressure diffusion velocity: Finally, we set inlet/outlet values on the pressure field as usual: in out Note that imposing the no-slip condition (19) leads to the following expression for the tangential component of the mass velocity at the wall: Equation (21) highlights that, by imposing a no-slip condition on the pressure diffusion velocity, the mass transfer along the wall becomes a purely diffusive process driven by the pressure gradient. We believe this boundary condition to be physically more sound than that used in [28]. There, the authors imposed the conventional Maxwell slip expression on the volume velocity. On the other hand, using the condition expressed in equation (19), a form of wall-slip arises naturally from the new equations while retaining the traditional walladherence in connection with the advective transport of mass. No additional constitutive relation is imposed on the form of the velocity slip.
To proceed with our analysis we introduce the following dimensionless variables:˜˜˜˜( Here, U p is a typical value of the pressure diffusion velocity and p has been normalized based on reasoning provided in [24]. In addition to these variables, we define the Reynold's number by, Finally, we turn to the pressure diffusion coefficient, κ p . As stated earlier the precise nature of this coefficient is not yet known. However, in previous investigations on rarefied gas flows [30], it was assumed that where α * is a dimensionless number. This form of κ p is reminiscent of the kinematic viscosity ν=μ/ρ and it is useful for the dimensional analysis carried out in this section. Indeed, making use of (22)- (24), equations (15)

Regular perturbation solutions
As the aspect ratio ε of the nanotubes used in flow experiments cited earlier is typically very small, it is reasonable to seek regular perturbation expansions in ε for the pressure and velocity fields. To determine the coefficient functions in the above expansions, the derivatives involvingp ln in equations (25)- (27) need to be fully expanded. Regularized equations are then obtained by multiplying both sides with appropriate powers ofp. Substituting the expansions (28)-(30) into these regularized equations and collecting terms involving the same powers of ε leads to a set of equations from which the coefficients can be determined. Clearly, for this approach to work we need to know the magnitude of the dimensionless numbers Re p , α * in relation to the aspect ratio ε. Frequently (see e.g. [6,24]), for the analysis of gas flows in microchannels, the assumption is made that Re p =O(ε). Further, supposing that α * =O(1), the regularized mass and r-momentum balance equations yield the following two O(1/ε)-relations, rewritten in terms of p ln 0 : Hence, the general solution for the zeroth order term of the radial velocity is, p ,0 r where (˜) h z is an arbitrary function. As with the pressure before, the radial velocity should not become infinite at = r 0, so that we must have (˜) º h z 0 and therefore˜º U 0 p ,0 r . It follows that all terms in equations (35) and (36) vanish and the perturbation expansions take the form:

Analytical solution
Equations (43)-(45) may be solved analytically to obtain the stream-wise pressure and velocity variation along the nano-tube to zeroth order in ε.

Velocity and pressure profiles
Making use of the perturbation expansions (40)-(41), the O(ε) dimensionless no-penetration condition on the mass velocity (equation (18)) reads:  Figure 2 shows profiles of these fields with values based on the experiment described by Majumder et al [10]. The re-casted Navier-Stokes pressure profile is seen to deviate from the classical linear pressure drop predicted by the Navier-Stokes equations.

Mass flow rate
For calculation of the mass flow rate, we emphasize that the total mass flux ru m as defined through equation (4) consists of the advective mass flux ru p as well as the diffusive mass flux rk  p ln p . The (dimensional) tangential In the last step towards obtaining equation (63), we have made use of the pressure distribution (58). We recall further that the mass flow rate according to the classical no-slip Hagen-Poiseuille flow law is given by, This allows our new mass flow rate to be expressed as: where the flow enhancement factor γ is defined by,

Discussion and comparison with experimental data
Considering analytical expressions for the mass flow rate as given in equations (65) and (66), several key features can be uncovered. First, we note that the flow enhancement factor γ is directly proportional to the viscosity coefficient, μ, and the diffusivity coefficient, κ p , while depending inversely on the square of the tube radius R. This ensures that the classical Hagen-Poiseuille flow law is recovered at large radii where surface effects become negligible. For a simple numerical illustration, let us take a diffusivity coefficient k = -10 p 8 m 2 s −1 , a dynamic . In a tube of radius R=1 nm, we then have a flow enhancement factor γ≈500. Increasing the tube radius to 100 nm, we find γ≈0.05, or about 5% enhancement over the Hagen-Poiseuille flow rate. This would explain why no enhancement was measured when investigating water flow through channels of 200 nm diameter and 800 nm diameter [9].
Second, equation (66) is explicitly dependent on the inlet and outlet pressures via: To investigate the influence of the inlet/outlet ratio  on the flow enhancement in more detail, let us fix the outlet pressure as P out =1·10 5 Pa and view equation (67) as a function of ΔP. Figure 3 shows the graph of this function (and hence γ) as well as the corresponding flow rates for pressure differences varying from ΔP=0 Pa to ΔP=1·10 6 Pa. The 1 bar pressure difference used in the experiments of Majumder et al [10] and Holt et al [13] is highlighted on these plots. It can be seen that, for fixed values of μ, κ p , R, the enhancement decreases as the applied pressure P in is increased further from this point. On the other hand, if P in is reduced, the enhancement increases. For ΔP=1 bar, equation (67) may be approximated as ( ) D »  P P ln 1 out . In fact, with D  P 0, we find g mk  R P 8 p 2 out . This limiting enhancement value is of course meaningless, as there is no flow in this case ( = =  ln ln1 0in equation (63)). Suppose now that we fix ΔP instead and consider the effect of the ambient pressure on γ. From equation (67), it is clear that a lower P out leads to a higher value of γ. Thus, the diffusive mechanism in our model becomes more important in a low pressure environment.
The dependence of the mass flow rate on 1/R 2 and the logarithmic term ( )  ln appears also in several models for rarefied gas flow. In fact, equation (65) and its derivation are synonymous to Dadzie and Brenner's equation (24) in [28] for gas flow in a rectangular micro-channel of width w and height h: Equation (69) was derived using a phenomenological approach based on an adsorption/desorption mechanism described by the Langmuir isotherm [37]. The authors subsequently add this term to the Hagen-Poiseuille flow law for a compressible gas, together with a term representing Knudsen diffusion. Solving the original incompressible Navier-Stokes equations written in terms of the mass velocity, equation (2), subject to the slip boundary condition on the mass velocity as given in equation (21), we obtain exactly the new mass flow equation (65). Equation (21), which originates from the no-slip condition on the pressure diffusion velocity, U p , appears as a slip condition on the mass velocity written in terms of a pressure gradient rather than in terms of a tangential velocity gradient as in a traditional Maxwell's slip expression [18]. We conclude therefore that, in this case, re-casting the Navier-Stokes equations is an alternative to introducing additional physics such as mass velocity slip or other surface diffusion effects at the wall.
To compare our new theoretical results with data available in the literature, the measured flow velocity, U exp , and the Hagen-Poiseulle predicted velocity, U HP , are used to define the flow enhancement factor, γ exp , as: To validate our new theory with the 1.66 nm to 44 nm diameter CNT experimental data in table 2 we collect existing measurement values and molecular dynamic simulation results for water. The quoted enhancement factors for CNT diameters between 1.66 nm and 44 nm are presented in figure 4. A significant discrepancy can be seen between the enhancement factors quoted by Du et al [12] (light green square) and Zhang et al [38] (red square) for the 10 nm diameter experiments. It has been suggested that the extremely large enhancement values reported by Majumder et al (dark green squares) and Du et al (light green squares) represent an over-estimation [39] and could be the result of an uncontrolled external variable such as an electric field [40]. In general, experiments involving flow through CNT membranes are prone to showing errors due to practical challenges in   spaces in table 2). This average α * -value corresponds to a pressure diffusivity coefficient κ p in the order of magnitude between 10 −7 and 10 −6 , which is comparable to the kinematic viscosity ν of water (≈1× 10 −6 m 2 s −1 ). The theoretical results for enhancement factor ( ) ( ) g a m r = D  R P 8 ln 2 2 * , with α * =0.47, are superimposed on figure 4 as blue filled circles joined by straight line segments. One can observe on this plot that our theoretical predictions using equation (66) are close to most of the various experimental and MD enhancement values except those of Majumder et al [11] and Du et al [12], for which the predictions are about two orders of magnitude less than the measured values. For the smaller diameter CNTs, it is noted that a reduced viscosity in a 'depletion layer' existing approximately within 0.7 nm of the tube wall may become significant and it has been modelled previously by Myers [35] in an attempt to explain the high flow rates without resorting to wall-slip conditions. If considered in our model, a reduced viscosity zone near the wall would lead to an overall decrease in the flow enhancement γ. This might explain why the theoretical prediction in figure 4 lies above the experimental data for the 1.66 nm and 2 nm channels.
As the experimental and molecular dynamics simulation data in table 2 involve a wide range of diameters and tube lengths, we also conducted another analysis based on the full dataset, this time including the highest enhancement values from the 7 nm and 10 nm diameter channels. We considered the possibility of adopting a geometry-dependent pressure diffusivity coefficient. This type of study was included as a scaling factor L/w in [28] for the analysis of gas flows in a rectangular channel of width w. Following the procedure described in [28], equation (24) may be written with that factor as,

Conclusion
We performed a theoretical analysis of isothermal pressure-driven liquid flows in nano-channels. The new equations used for the analysis termed re-casted Navier-Stokes equations are obtained by a change of variable on the mass velocity U m appearing in the classical conservation equations of mass and momentum. The specific form of this transformation is assumed to be determined by the main driving mechanism of the flow. For the problem of isothermal pressure-driven liquid flow discussed in the present paper, the mass velocity U m was written as the sum of a new velocity U p (the pressure diffusion velocity) and a diffusive term, κ p ∇p/p, involving the pressure gradient. A regular perturbation expansion in the channel aspect ratio was used to derive a simpler set of equations for the flow configuration. These were then solved analytically subject to a no-slip condition on U p , resulting in a theoretical expression for the mass flow rate. This expression was seen to be the classical Hagen-Poiseuille flow law with an additional extra mass transport term resulting from the mass diffusion at the wall. It was therefore noted that the re-casting methodology introduces new physics in the solutions. The theoretical mass flow rate was compared with existing literature data from experiments and molecular dynamic simulations reported on enhanced flow rates of liquid flows in nanotubes. The new equations and analytical solutions provide an alternative physical explanations of the data. The methodology may therefore be deployed to interpret other similar anomalous flow experiments.