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

Abstract

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.

Implementation and Testing of a New Openfoam Solver for Pressure-Driven Liquid Flows on the Nanoscale

“…The idea is based on transforming the velocity vector field within the classical equations in a way that depends on the driving mechanism of the flow (for liquid flow in CNTs, this is the pressure gradient). The mass flow rate derived from this model as compared with experimental data showed reasonable agreement [16].…”

“…In Stamatiou et al [16], a novel modelling approach was introduced in which the flow enhancement is caused by a diffusion mechanism that only becomes apparent at the nanoscale. A unifying recasting methodology was proposed by which a new class of continuum models termed Recast Navier-Stokes equations (RNS), can be directly derived from the Navier-Stokes equations [17], [18].…”

“…The objective of the current work is to present and validate a numerical implementation of the Recast Navier-Stokes equations in a new OpenFOAM solver (rnsLiquidFoam). For creeping flows in cylindrical tubes of small aspect ratio, a perturbation analysis yielding analytical expressions for the pressure and velocity fields are obtained [16]. These solutions are reviewed and used to validate the solver.…”

“…The low Reynolds' number is typical of creeping flows so that inertial terms can be neglected in the momentum equation. A rigorous perturbation analysis on the non-dimensionalised equations with Re p = O(ε 2 ) and α * = O(1) reveals that the first two pressure terms in eqn (11) are independent of the radial coordinate r (see [16] for details). In other words, p0 = p0 (z) and p1 = p1 (z).…”

Reinterpreting shock wave structure predictions using the Navier–Stokes equations

“…The difference is attributable to the fact that hydrodynamic field variables from the recast equations no longer operate as in the original equations (also as boundary conditions are set based on redefined hydrodynamic variables rather than those in the original equations, see Ref. [20]). The recast Navier-Stokes equations with a Mach number-dependent mass diffusion coefficient, κ m 0 (see Table 1 for its values), and a viscosity-temperature exponent, s = 0.75, show better agreements with Alsmeyer's [10] experimentally measured density profiles in argon gas.…”