Slip-flow theory for thermo-osmosis based on a kinetic model with near-wall potential

TL;DR

This work develops a kinetic-slip framework to predict thermo-osmotic slip at micro- and nanoscale gas-fluid interfaces in the presence of near-wall potentials. By applying generalized slip-flow theory to a BGK model with diffuse-reflection boundaries, the authors decompose the flow into a Hilbert bulk part and Knudsen-layer corrections, obtaining slip and jump coefficients that depend on the near-wall potential through parameters $U$ and $oldsymbol{ extchi}$. They validate the theory against direct numerical BGK solutions for a thermo-osmosis setup between parallel plates, showing good agreement and revealing that repulsive near-wall interactions can reverse thermo-osmotic flow and strongly affect Knudsen-layer corrections. The framework links molecular-scale interfacial physics to macroscopic boundary conditions, enabling systematic prediction of the thermal-slip coefficient $K_{ ext{TS}}$ for gases and offering qualitative insights for liquids via a BGK toy model. The findings have implications for passive thermal-flow control in microfluidic devices and point to future work on more realistic potentials, curved geometries, and nonideal fluid effects.

Abstract

In this paper, thermal-slip coefficients in slip boundary conditions of the Stokes equation are derived using the generalized slip-flow theory, with special interest in the role of near-wall potential in micro- and nanoscale flows. As the model of fluids and fluid-solid interaction, we employ the model Boltzmann equation for dilute gases and the diffuse-reflection boundaries with near-wall potential, respectively. It is found that, when the mean free path of gas molecules and the effective range of potential are of the same order of magnitude, the thermal-slip boundary condition can be derived in the near-continuum limit. In the derived slip-flow theory, the thermal-slip coefficient and the boundary-layer corrections (i.e., Knudsen-layer corrections) are determined by solving the kinetic boundary-layer problems (i.e., Knudsen-layer problems) that include external-force terms and inhomogeneous terms both driven by the potential. As an application of the slip-flow theory, thermo-osmosis between two parallel plates with uniform temperature gradients is analyzed. The results of the slip-flow theory are validated by comparing them with those of the direct numerical analysis of the same problem. Furthermore, it is found that thermo-osmosis and thermal slip on the plates are significantly affected by the features of the near-wall potential; even the gas-flow direction can be reversed when the near-wall potential is repulsive. Such a flow reversal is qualitatively similar to thermo-osmosis in liquids reported in existing molecular dynamics simulation.

Figures (11)

• Figure 1: Schematic of the problem. (a) General setting of the slip-flow theory (Sec. \ref{['sec:slip-flow-theory']}). (b) Application of the slip-flow theory to the thermo-osmosis problem (Sec. \ref{['sec:thermo-osmosis']}). (c,d) Schematic examples of $\Psi$ for (c) purely-attractive and (d) purely-repulsive potential.
• Figure 2: Schematics of the slip-flow theory. As an illustrative example, a parallel flow near a stationary boundary is considered. The entire region is described by the fluid-dynamic equations with no-slip boundary conditions (red). Near the boundary, the Knudsen layer plays two roles: (i) the slip boundary condition modifies the fluid-dynamic part (blue) and (ii) the Knudsen-layer correction modifies the macroscopic quantities near the boundary. The Knudsen-layer correction rapidly decays as the distance from the boundary increases.
• Figure 3: (a) Dimensionless potential $U\psi_\delta(x_1)$ [Eq. \ref{['eq:psi-nd']}] for $\delta=0.01$ near the plate at $x_1=1/2$. The cases of $U=1$ and $U=-1$ are shown by red-solid and blue-dash curves, respectively. (b) Corresponding reference density profile $\alpha$ [Eq. \ref{['eq:alpha']}], where the inset shows the magnification near $\alpha=1$.
• Figure 4: Thermal-slip coefficients for attractive potential $U=1$. (a) The ratio $b_{2,\mathrm{num}}^{(1)}/b_{2}^{(1)}$ between the thermal-slip coefficient obtained by the numerical analysis $b_{2,\mathrm{num}}^{(1)}$ and that by the slip-flow theory $b_{2}^{(1)}$. (b) The thermal-slip coefficient $b_{2}^{(1)}$ as a function of the range of the potential $\chi$.
• Figure 5: Thermal-slip coefficients for repulsive potential. (a) The ratio $b_{2,\mathrm{num}}^{(1)}/b_{2}^{(1)}$ between the thermal-slip coefficient obtained by the numerical analysis $b_{2,\mathrm{num}}^{(1)}$ and that by the slip-flow theory $b_{2}^{(1)}$ for $U=-1$. (b)(c) The thermal-slip coefficient $b_{2}^{(1)}$ as a function of the range of the potential $\chi$ for (b) $U=-1$ and (c) $U<0$ with $|U|=\chi$.