Microscopic determination of macroscopic boundary conditions in Newtonian liquids

We study boundary conditions applied to the macroscopic dynamics of Newtonian liquids from the view of microscopic particle systems. We assume the existence of microscopic boundary conditions that are uniquely determined from a microscopic description of the fluid and the wall. By using molecular dynamical simulations, we examine a possible form of the microscopic boundary conditions. In the macroscopic limit, we may introduce a scaled velocity field by ignoring the higher-order terms in the velocity field that is calculated from the microscopic boundary condition and standard fluid mechanics. We define macroscopic boundary conditions as the boundary conditions that are imposed on the scaled velocity field. The macroscopic boundary conditions contain a few phenomenological parameters for an amount of slip, which are related to a functional form of the given microscopic boundary condition. By considering two macroscopic limits of the nonequilibrium steady state, we propose two different frameworks for determining macroscopic boundary conditions.

Figure 1

Schematic illustration of our model.

Figure 2

Velocity profiles for the applied force $f=2.0$. The wall parameters are chosen as $(a,c_{\mathrm{BW}})=(0.5,0.6)$ (blue), (0.6,1.0) (orange), and (0.7,1.0) (green). Inset: linear fits of the velocity profile away from the walls.

Figure 3

Stress tensor as a function of shear rate. The applied force $f$ is varied from 0.2 to 4.8. The wall parameters are chosen as $(a,c_{\mathrm{BW}})=(0.5,0.6)$ (blue), (0.6,1.0) (orange), and (0.7,1.0) (green).

Figure 4

Plot of $g\left(u_w\right)$. The wall parameters and the applied force are chosen as $a=0.5,c_{\mathrm{BW}}=0.6$, and $0.2\le f\le 4.8$, which is the same as Fig. 3.

Figure 5

Slip length as a function of local velocity at the $z=0$ wall, $u_w$ (left-hand side), and local shear rate at the $z=0$ wall, ${\dot{\gamma }}_w$ (right-hand side). The parameter settings are the same as those in Fig. 4.

Figure 6

Schematic graph of $y=g\left(u_w\right)$ and $y=\eta (U-u_w)/L$. The intersection of these graphs corresponds to the solution of (19): (a) behavior of the solution in the quasiequilibrium limit and (b) behavior of the solution in the hydrodynamic limit.

Figure 7

Schematic graph of $y=g\left(u_w\right)$ and $y=\eta (U-u_w)/L$. The intersection of these graphs corresponds to the solution in (19): (a) behavior of the solution for $\eta U/L$ satisfying (44) in the hydrodynamic limit and (b) behavior of the solution for $\eta U/L$ satisfying (45) in the hydrodynamic limit

Figure 8

Schematic image of $g\left(u_w\right)$ for the wall $a=0.5$, $c_{\mathrm{FS}}=0.6$.

Figure 9

Schematic image of $D({\mathit{r}}_i,{\mathit{r}}_j;{\mathcal{R}}_z)$. $D({\mathit{r}}_i,{\mathit{r}}_j;{\mathcal{R}}_z)$ is given as the ratio of black line and red dotted line.