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Research Papers: Fundamental Issues and Canonical Flows

# Experimental Observation of Inertia-Dominated Squeeze Film Damping in Liquid

[+] Author and Article Information
Antoine Fornari, Matthew Sullivan, Hua Chen

Schlumberger-Doll Research, 1 Hampshire Street, Cambridge, MA 02139

Christopher Harrison1

Schlumberger-Doll Research, 1 Hampshire Street, Cambridge, MA 02139

Kai Hsu

Schlumberger Sugar Land Product Center, 125 Industrial Boulevard, Sugar Land, TX 77478

Frederic Marty, Bruno Mercier

Ecole Supérieure d’Ingénieurs en Electronique et Electrotechnique, Noisy-Le-Grand 93162, France

1

Corresponding author.

J. Fluids Eng 132(12), 121201 (Dec 22, 2010) (10 pages) doi:10.1115/1.4003150 History: Received April 14, 2009; Revised November 22, 2010; Published December 22, 2010; Online December 22, 2010

## Abstract

We have studied the phenomenon of squeeze film damping in a liquid with a microfabricated vibrating plate oscillating in its fundamental mode with out-of-plane motion. It is paramount that this phenomenon be understood so that proper choices can be made in terms of sensor design and packaging. The influences of plate-wall distance $h$, effective plate radius $R$, and fluid viscosity and density on squeeze film damping have been studied. We experimentally observe that the drag force is inertia dominated and scales as $1/h3$ even when the plate is far away from the wall, a surprising but understandable result for a microfluidic device where the ratio of $h$ to the viscous penetration depth is large. We observe as well that the drag force scales as $R3$, which is inconsistent with squeeze film damping in the lubrication limit. These two cubic power laws arise due to the role of inertia in the high frequency limit.

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## Figures

Figure 1

Schematics showing sensors and measurement apparatus used in this paper. (a) Configuration of vibrating plate in plan view and orientation of magnetic field B. Oscillatory current I flowing through the plate produces an oscillatory force indicated by F. The plate has width w, thickness h0, and length l. The dimensions of the three chip sizes used in this paper can be found in Table 1. (b) Optical view (plan) of the microchip. The entire die has dimensions of 10×23 mm2 and the cantilever is substantially smaller. The orientation of the magnetic field is indicated by arrows. A thin glass plate of similar lateral dimensions to the chip is glued underneath. (c) The vibrating plate shown here (cantilever) is oriented normal to the page. The first glass wall is glued on the back of the silicon chip and the distance h (indicated by black double-headed arrow) between a second glass wall and the cantilever is controlled with the micrometer. The cantilever and the two glass walls are coplanar. The laser beam (indicated by arrow) passes through the transparent glass to measure the motion of the plate. The motion of the plate is increasingly damped as the distance between the glass wall and the plate is reduced.

Figure 2

Representative resonance spectrum. The in-phase data are denoted with squares, the out-of-phase by circles, and the amplitude by crosses. Fit to data using Eq. 1 are shown with solid lines. Resonance frequency is higher here than in subsequent figures as the density of the fluid is lower.

Figure 3

(a) Plate amplitude at tip versus applied current. The squares, circles, and triangles correspond to plates of lengths 400 μm, 600 μm, and 900 μm, respectively. Note the linearity of amplitude with applied current. (b) Same data but converted to the microstrain experienced at the hinge point.

Figure 4

(a) Resonance frequency fR versus excitation current for plate of length 600 μm. The resonance frequency is largely independent of the excitation current from 0.1 mA to 10 mA. For higher currents, the frequency begins to increase indicating the excitation of a second mode or a nonlinearity. (b) Quality factor Q demonstrates similar behavior.

Figure 10

Distance between wall and vibrating plate at which normalized resonant frequency (open squares) and quality factor (filled circles) decrease by 1 and 5%, respectively.

Figure 9

(a) The real DR (open symbols) and imaginary DI (closed symbols) portion of the complex drag versus Δ (the distance to the wall normalized by the effective plate radius). The blue diamonds, red squares, black circles, and green triangles correspond to toluene, decane, tetrachloroethylene, and 9.6 cP silicone oil, respectively. Note that DI⪢DR as the drag is dominated by inertia. (b) The angle of the complex drag for the data shown in (a). (c) DR and DI less their values when far away (DR∞ and DI∞) versus Δ. Note that both the real and the imaginary components of the data follow a power law close to −3, except for the smallest values of Δ, where our interpretation may break down. (d) The magnitude of the drag versus Δ again following a −3 power law

Figure 8

(a) The magnitude of the drag D for the vibrating plates shown in Fig. 7 is presented as a function of the distance to the wall after subtracting the drag measured very far away (D∞). The black squares, red circles, and green triangles correspond to plates of length 400 μm, 600 μm, and 900 μm, respectively. (b) Same as (a), but the data for the 600 μm and 900 μm long sensors are scaled by the effective radius cubed, exactly as was done in Fig. 7, resulting in a collapse onto a master curve.

Figure 7

Normalized shifts of both the (a) resonant frequency and the (b) quality factor as a function of distance h between the vibrating plate and a second wall when immersed in toluene. The black squares, red circles, and green triangles correspond to plates of length 400 μm, 600 μm, and 900 μm, respectively. f∞ and Q∞ correspond to the resonance frequency and quality factor measured when the plate is effectively infinitely far away from wall. Note that in each case the effect of the wall is minimal when h is large but grows rapidly as h decreases. In (c), the resonance frequency data shown in (a) are presented where the data for the plates of length 600 μm and 900 μm are multiplied by the cube of the ratio of their (effective) radii to that of the 400 μm plate. In (d), the same scaling is performed for the quality factor. Note that most data collapse onto a master curve, suggesting that scaling should be performed with the radius cubed. There is more scatter in the ((b) and (d)) quality factor measurements as compared with the ((a) and (c)) resonance frequency measurements as the frequency can be measured more precisely than the quality factor.

Figure 6

(a) Resonance spectra obtained at five different distances from a wall for a vibrating plate of length 600 μm. Crosses, 5 mm; squares, 0.4 mm; circles, 0.08 mm; triangles, 0.02 mm; diamonds, 0 mm. (b) Resonance frequency as a function of distance (small open squares). Note the sharp drop for distances less than 0.5 mm. The positions corresponding to the five spectra shown in (a) are indicated by the five large symbols.

Figure 5

(a) Amplitude of 600 μm long plate as a function of position. xl corresponds to the distance from the laser spot to the hinge point of the plate, centered along the plate width. The symbols ×, +, circle, and square correspond to currents of 4 mA, 8 mA, 16 mA, and 40 mA. (b) The resonance frequency fR does not vary significantly with position, although slightly with excitation current.

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