Density and confinement effects on fluid velocity slip

Understanding the velocity slip of fluids in channels of molecular length scales is essential for accurately manipulating their flow in engineering applications, such as in water desalination or, by reversing the process, osmotic power generation. This study addresses two major open questions, namely (a) the range of validity of the Navier-Stokes equations with slip boundary conditions in describing fluid flows within channels of molecular confinement, and (b) the effect of fluid density and confinement as well as surface properties, such as microscopic roughness and curvature, on fluid slippage along walls. Using a hard-sphere fluid model and describing the fluid-wall interactions by the Maxwell scattering kernel, an event-driven molecular dynamics simulation study was performed in planar and cylindrical channels of different sizes and accommodation coefficients with varying fluid densities. The results show that the well-known criterion for the slip solution, which is valid in rarefied conditions, also holds for a dense gas, i.e., the predictive accuracy improves with decreasing Knudsen numbers, as long as the channel size is larger than about ten molecular diameters. We also find that the key quantity influencing the friction coefficient of a fluid is the peak density at the walls, rather than the nominal density, the fluid confinement, or the channel curvature, as found in previous studies. In particular, higher nominal densities lead to higher density peaks, and therefore a decrease in slip, while tighter confinements lead to lower density peaks, and therefore an increase in slip, regardless of channel geometry. Furthermore, the velocity slip depends on the microscopic roughness via the Smoluchowski factor. These findings are a stepping stone towards a deeper understanding of the molecular mechanisms underlying fluid velocity slip in molecular-scale channels.

Figure 1

The schematic depicts a planar channel with nominal height $H$ and characteristic length $L=H-\sigma$, with typical density (left) and velocity (right) profiles. The density profile shows the structuring of the fluid close to the walls, with the first dip marking the boundary of the $\mathrm{\Delta }$ layer, which is approximately one molecular diameter away from the walls. As highlighted in the inset, three different velocity slips can be defined, namely as the mean velocity in the $\mathrm{\Delta }$ layer, assuming that the density there is constant $u_{s,\mathrm{\Delta }}$, the apparent velocity at the boundary which gives the Navier-Stokes velocity profile in the fluid bulk $u_{s,\mathrm{app}}$, and the real fluid velocity at the boundary $u_{s,\mathrm{real}}$.

Figure 2

Interfacial friction coefficients ${\xi }_{0,\mathrm{\Delta }}$ versus the nominal reduced densities for (a) planar and (b) cylindrical channels with fully diffuse walls ($\alpha =1$). Results are presented for two confinement ratios, $R=50$ (orange color) and $R=10$ (red color), produced through nonequilibrium (solid symbols) and equilibrium (empty symbols) approaches. The solid line represents the kinetic theory prediction for rarefied gases, Eq. (11). See Appendix pp2 for results over a wider range of confinement ratios and in tabular form.

Figure 3

Accuracy of the slip solution for (a) planar and (b) cylindrical channels with fully diffuse walls ($\alpha =1$). The color scale bar shows the root mean square error between the numerical velocity profile and the slip analytical prediction. The error increases from green (lowest) to red (highest). The dashed lines represent the fluid conditions corresponding to $\text{Kn}=0.1$, above which the slip solution gives errors below 0.02 for $R⪆10$.

Figure 4

(a) Friction coefficient versus peak density for planar (blue squares) and cylindrical (red circles) channels with fully diffuse walls ($\alpha =1$). (b) Residuals and $Q-Q$ plots. Of the numerical results presented in Fig. 3, only those corresponding to conditions where the slip solution accuracy is met (i.e., root mean square error lower than 0.02) are shown here. The dashed black line in panel (a) represents the best linear fit of the results, while the solid black line represents the kinetic theory prediction for rarefied gases, Eq. (11). The inset zooms into this regime in the lower left.

Figure 5

Comparison of the peak density next to the wall and the nominal reduced density within (a) planar and (b) cylindrical channels for different confinement ratios with fully diffuse walls ($\alpha =1$). The insets show the density depletion in the middle of the channel.

Figure 6

Friction coefficient versus peak density for different wall accommodation coefficients. As in the fully diffuse case, results do not depend on the channel geometry for partial TMAC (planar geometries are represented with empty symbols as a guide to the eye). Colored dashed lines are obtained by scaling the dashed black line, from Fig. 4 corresponding to the full accommodation case, with the Smoluchowski prefactor.

Figure 7

Summary of the main results from this work, showing schematically how specific fluid and wall properties ultimately influence the friction coefficient. Relationships can be seen through some of the figures in the paper, as indicated in the connections between entities.

Figure 8

Friction coefficient versus peak density for MD, in the same range of confinement ratios and nominal densities to those of EDMD simulations. The solid line corresponds to the best linear fit of the MD values, whereas the dashed lines represents the EDMD fit from Fig. 4 with the TMAC value from MD simulations ($\alpha =0.49$).