^{1}, Donald L. Koch

^{1}and Rustom B. Bhiladvala

^{2}

### Abstract

Nanoelectromechanical oscillators are very attractive as sensing devices because of their low power requirements and high resolution, especially at low pressures. While many experimental studies of such systems are available in the literature, a fundamental theoretical understanding over the entire range of operating conditions is lacking. In this article, we use our newly developed Bhatnagar–Gross–Krook based low Mach number direct simulation Monte Carlo method to study the noncontinuum drag force acting on a cylinder oscillating normal to a wall. We explore quasisteady flows in which as well as unsteady flows for which . Here is the oscillation frequency and is the characteristic time for the development of the gas flow. The drag force per unit length acting on a long cylindrical wire is studied as a function of the Knudsen number, defined in terms of the mean free path and the radius of the cylinder as . For quasisteady flows, we also present theoretical calculations for the slip regime, , and the free molecular flow regime, . Simulations of unsteady gas flow around a sinusoidally oscillating cylinder near a wall indicate that the drag force per unit length nondimensionalized by approaches constant values for (quasisteady flow) and for . Here is the gas viscosity and is the maximum value of the nanowirevelocity. The simulation results are compared with experimental measurements in the quasisteady regime.

This work was supported by NSF Grant No. CBET-0730579.

I. INTRODUCTION

II. PROBLEM STATEMENT AND SIMULATION METHOD

III. THEORY FOR QUASISTEADY DRAG FORCE

A. Continuum regime

B. Slip regime

C. Free molecular flow regime

IV. DISCUSSION

A. Quasisteady regime

B. Unsteady regime

V. CONCLUSIONS

### Key Topics

- Nanowires
- 65.0
- Free molecular flows
- 21.0
- Flow instabilities
- 16.0
- Silicon
- 11.0
- Boundary value problems
- 10.0

## Figures

Simulation domain.

Simulation domain.

Convergence of drag force results with simulation box size for a system with .

Convergence of drag force results with simulation box size for a system with .

Problem geometry.

Problem geometry.

Dimensionless inverse drag force per unit length as a function of dimensionless distance between nanowire and bottom wall for continuum flow. The solid line is obtained by using Eq. (23), the dotted-dashed line represents lubrication theory results for small given in Eq. (24), and the dashed line represents the approximation for given in Eq. (25).

Dimensionless inverse drag force per unit length as a function of dimensionless distance between nanowire and bottom wall for continuum flow. The solid line is obtained by using Eq. (23), the dotted-dashed line represents lubrication theory results for small given in Eq. (24), and the dashed line represents the approximation for given in Eq. (25).

Nondimensionalized slip correction to the drag force, , plotted as a function of dimensionless distance between nanowire and bottom wall.

Nondimensionalized slip correction to the drag force, , plotted as a function of dimensionless distance between nanowire and bottom wall.

as a function of dimensionless distance between nanowire and bottom wall.

as a function of dimensionless distance between nanowire and bottom wall.

Drag force per unit length as a function of Knudsen number for corresponding to dimensions reported in Ref. 7. Legends are as shown in figure. Arrow near the -axis represents the Jeffrey–Onishi predictions in the continuum limit . The inset shows the same data on a linear-linear scale (free molecular flow results omitted for clarity). For simulations, error bars are smaller than symbol size.

Drag force per unit length as a function of Knudsen number for corresponding to dimensions reported in Ref. 7. Legends are as shown in figure. Arrow near the -axis represents the Jeffrey–Onishi predictions in the continuum limit . The inset shows the same data on a linear-linear scale (free molecular flow results omitted for clarity). For simulations, error bars are smaller than symbol size.

Normalized drag force per unit length as a function of Kn for different as given in the legend. Lines represent predictions obtained using the semiempirical expression given in Eq. (37) and curve fits for coefficients in that equation. Symbols represent simulation data at . Error bars in simulations are smaller than symbol size.

Normalized drag force per unit length as a function of Kn for different as given in the legend. Lines represent predictions obtained using the semiempirical expression given in Eq. (37) and curve fits for coefficients in that equation. Symbols represent simulation data at . Error bars in simulations are smaller than symbol size.

Simulation results for the normalized drag coefficient as a function of nondimensionalized oscillation frequency at different Kn where is the BGK relaxation time defined in Sec. II. The arrows near the -axis represent quasisteady simulation results. Error bars in simulations are smaller than symbol size.

Simulation results for the normalized drag coefficient as a function of nondimensionalized oscillation frequency at different Kn where is the BGK relaxation time defined in Sec. II. The arrows near the -axis represent quasisteady simulation results. Error bars in simulations are smaller than symbol size.

Low and high frequency asymptotes for the normalized drag coefficient plotted as a function of Kn. Symbols represent simulation data and the dashed line represents the drag coefficient for an isolated nanowire in the free molecular flow regime. This free molecular flow result is given by Eq. (36) with (Ref. 34). Error bars in simulations are smaller than symbol size.

Low and high frequency asymptotes for the normalized drag coefficient plotted as a function of Kn. Symbols represent simulation data and the dashed line represents the drag coefficient for an isolated nanowire in the free molecular flow regime. This free molecular flow result is given by Eq. (36) with (Ref. 34). Error bars in simulations are smaller than symbol size.

## Tables

Coefficients , , and in Eq. (37) as a function of . The coefficients can be well-fitted by the following equations: , , and .

Coefficients , , and in Eq. (37) as a function of . The coefficients can be well-fitted by the following equations: , , and .

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