^{1,a)}and Arvind Raman

^{2,b)}

### Abstract

We analyze the hydrodynamic coupling between long, slender micromechanical beams (microbeams) deployed in an array and oscillating in a viscous, incompressible fluid. The unsteady Stokes equations are solved using a boundary integral technique in a two-dimensional plane containing the microbeam cross sections. The oscillations of nearest neighbor and the next neighbor microbeams couple hydrodynamically in unanticipated ways depending on the gap, frequency, and the relative phase and amplitude of their oscillation. A rational basis is provided for choosing the gap between neighboring microbeams in an array in order to either decouple their hydrodynamics or to couple them strongly. The results clearly suggest that the dynamics of microbeams in an array can be tuned in a cooperative manner so as to minimize or maximize the hydrodynamic resistance on individual microbeams.

Financial support for this research provided by the Sandia National Laboratories under the Contract No. 623235, “Fluid-structure interactions in microsystems” is gratefully acknowledged.

I. INTRODUCTION

II. BOUNDARY INTEGRALMODEL

A. Formulation of the model

III. HYDRODYNAMIC COUPLING OF TWO INFINITESIMALLY THIN MICROBEAMS

A. Numerical solution for two microbeams vibrating transversely

B. Physical interpretation of complex pressure and unsteady Reynolds numbers

C. Effects of gap, relative phase, and amplitude ratio between the microbeams

1. Effect of the gap

2. Effect of the relative phase

3. Effect of the amplitude ratio

D. Hydrodynamic decoupling

E. Hydrodynamicfunctions for coupled microbeams

IV. ARRAY OF INFINITESIMALLY THIN MICROBEAMS

V. CONCLUSIONS

### Key Topics

- Hydrodynamics
- 119.0
- Viscosity
- 19.0
- Reynolds stress modeling
- 14.0
- Oscillators
- 11.0
- Real functions
- 7.0

## Figures

(a) Array of microbeams of arbitrary cross sections in fluid. (b) Sketch of the boundary value problem for oscillating cylinders of arbitrary cross sections. Axes of the cylinders lie along the axis. Each cross-hashed region is a microbeam cross section and is the contour of integration. (c) Sketch of the boundary value problem for two oscillating rectangular cross-sectional microbeams. is the vector basis corresponding to the , , and coordinate system.

(a) Array of microbeams of arbitrary cross sections in fluid. (b) Sketch of the boundary value problem for oscillating cylinders of arbitrary cross sections. Axes of the cylinders lie along the axis. Each cross-hashed region is a microbeam cross section and is the contour of integration. (c) Sketch of the boundary value problem for two oscillating rectangular cross-sectional microbeams. is the vector basis corresponding to the , , and coordinate system.

Variation of (a) the real part and (b) the imaginary part of nondimensional pressure jump across the microbeam 1 with gap ratio . The axis in this plot was chosen to be from to 0 as the microbeam’s absolute position varied as the gap was changed. Plots for (dotted line), (dashed line), (dash-dotted line), (line joining circles), (line joining squares), and (solid line) are shown. (c) Variation of the nondimensional transverse hydrodynamic force and (d) the nondimensional hydrodynamic torque on microbeam 1 with varying . Parameters used are , , and . The transverse force and torque per unit length are nondimensionalized by .

Variation of (a) the real part and (b) the imaginary part of nondimensional pressure jump across the microbeam 1 with gap ratio . The axis in this plot was chosen to be from to 0 as the microbeam’s absolute position varied as the gap was changed. Plots for (dotted line), (dashed line), (dash-dotted line), (line joining circles), (line joining squares), and (solid line) are shown. (c) Variation of the nondimensional transverse hydrodynamic force and (d) the nondimensional hydrodynamic torque on microbeam 1 with varying . Parameters used are , , and . The transverse force and torque per unit length are nondimensionalized by .

(a) Real part of the component of fluid velocity for an infinite wall vibrating along the direction at . (b) Real part of the component of fluid velocity for a microbeam . (c) Variation of phase of the fluid velocity along . The length scales for both plots are nondimensionalized by . is used for the microbeam velocity profile.

(a) Real part of the component of fluid velocity for an infinite wall vibrating along the direction at . (b) Real part of the component of fluid velocity for a microbeam . (c) Variation of phase of the fluid velocity along . The length scales for both plots are nondimensionalized by . is used for the microbeam velocity profile.

The real (a) and imaginary (b) parts of the component of fluid velocity due to a microbeam vibrating in unbounded fluid. Plots for are shown.

The real (a) and imaginary (b) parts of the component of fluid velocity due to a microbeam vibrating in unbounded fluid. Plots for are shown.

Variation of the nondimensional transverse hydrodynamic force per unit length on microbeam 1 with the relative phase between the two microbeams. Parameters used are , , and . The transverse force per unit length is nondimensionalized by .

Variation of the nondimensional transverse hydrodynamic force per unit length on microbeam 1 with the relative phase between the two microbeams. Parameters used are , , and . The transverse force per unit length is nondimensionalized by .

Variation of the nondimensional transverse hydrodynamic force per unit length with amplitude ratio on microbeam 1 for in-phase and out-of-phase oscillation of the two microbeams. Other parameters used are and . The transverse force per unit length is nondimensionalized by .

Variation of the nondimensional transverse hydrodynamic force per unit length with amplitude ratio on microbeam 1 for in-phase and out-of-phase oscillation of the two microbeams. Other parameters used are and . The transverse force per unit length is nondimensionalized by .

Variation of (a) the real part and (b) the imaginary part of nondimensional transverse hydrodynamic force across microbeam 1 with gap ratio for in-phase oscillations of the two microbeams. These forces are normalized by their corresponding values at the same in unbounded fluid.

Variation of (a) the real part and (b) the imaginary part of nondimensional transverse hydrodynamic force across microbeam 1 with gap ratio for in-phase oscillations of the two microbeams. These forces are normalized by their corresponding values at the same in unbounded fluid.

Plot of the nondimensional gap required for the transverse hydrodynamic force [(Eq. (16)] to reach 99% of the unbounded fluid values versus the nondimensional frequency. Parameters used are and .

Plot of the nondimensional gap required for the transverse hydrodynamic force [(Eq. (16)] to reach 99% of the unbounded fluid values versus the nondimensional frequency. Parameters used are and .

Imaginary and real part of the hydrodynamic function as a function of of microbeam 1 for different nondimensional gaps (a), different amplitude ratios (b), and different relative phases (c) between the two microbeams. Parameters used in (a) are and ; parameters used in (b) are and ; parameters used in (c) are and .

Imaginary and real part of the hydrodynamic function as a function of of microbeam 1 for different nondimensional gaps (a), different amplitude ratios (b), and different relative phases (c) between the two microbeams. Parameters used in (a) are and ; parameters used in (b) are and ; parameters used in (c) are and .

An array of equally sized and equally spaced thin microbeams.

An array of equally sized and equally spaced thin microbeams.

Variation of (a) the imaginary part and (b) the real part of hydrodynamic function for the first three microbeams in the five-microbeam array of Fig. 10 for in-phase oscillations of the microbeams with .

Variation of (a) the imaginary part and (b) the real part of hydrodynamic function for the first three microbeams in the five-microbeam array of Fig. 10 for in-phase oscillations of the microbeams with .

## Tables

Convergence study for the nondimensional hydrodynamic transverse force and torque per unit length acting on microbeam 1 with , , , and . These calculations are performed for two infinitesimally thin, oscillating microbeams coupled hydrodynamically to each other. The transverse force and torque per unit length are nondimensionalized by .

Convergence study for the nondimensional hydrodynamic transverse force and torque per unit length acting on microbeam 1 with , , , and . These calculations are performed for two infinitesimally thin, oscillating microbeams coupled hydrodynamically to each other. The transverse force and torque per unit length are nondimensionalized by .

Range of parameter values used in this paper.

Range of parameter values used in this paper.

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