COMSOL 6.4 - Viscous Damping of a Microperforated Plate in the Slip-Flow Regime

Viscous Damping of a Microperforated
Plate in the Slip-Flow Regime

The advances in the fabrication of micromechanical devices and transducers has reduced the dimensions of thermoviscous acoustic (microacoustic) systems to require the handling of high Knudsen’s numbers. The Knudsen number is high when the characteristic dimensions of a system becomes comparable to the mean free path of the gas molecules. This either occurs at low pressure where the mean free path becomes large or in systems with small geometric dimensions. In the slip-flow regime 0.001 < Kn < 0.1 the effects can be included by using the so-called slip-velocity boundary condition instead of the traditional no-slip boundary condition. At even higher Knudsen numbers it is necessary to use the Transitional Flow or Free Molecular Flow interface available in the Molecular Flow Module. However, these are not formulated for acoustic problems.

This tutorial model is of a perforated membrane/plate (also known as a micro perforated plate or MPP) backed by a vibrating structure. This is a typical configuration in, for example, a MEMS microphone. The vibrating structure is not modeled explicitly, but just assigned a vibration velocity. The vibrating structure creates a pressure field that squeezes air through the perforated membrane. In the system there are two main sources for viscous losses: viscous losses in the squeezing flow between the gap between the vibrating structure and perforated membrane, and the viscous losses from the flow through the holes in the perforates. In this model the optimal hole size, to achieve minimal resistive losses, is analyzed for a given gap height and a fixed porosity (area ratio) for the perforates. The results are compared to the analytical results by Homentcovschi and Miles in Ref. 1.

The model also investigates the importance of using a slip-velocity boundary condition instead of a no-slip boundary condition. A boundary condition with a slip velocity will lead to smaller gradients in the velocity field near boundaries and thus less damping. The optimal geometry is roughly the same for the two boundary conditions, however the magnitude of the resistive loss changes significantly.

Model Definition

The model represents a hexagonal unit cell using symmetry planes, to only model one twelfth of the unit cell (see Figure 1). The model consists solely of an air domain described by the Thermoviscous Acoustics, Frequency Domain interface. The model consists of a vibrating surface placed d = 1.5 μm from the perforated membrane/plate, and on the other side the air domain ends in a Port using a Plane Wave. The port is placed at a distance equal to twice the thermal boundary layer thickness away from the perforated membrane. This is to ensure that generated entropy waves vanish and do not interact with the Port. The area of the hole is set to be 20% of the unit cell area. The geometry is defined by the distance between the vibrating surface and the perforated membrane d = 1.5 μm, the thickness of the perforated membrane h = 1.0 μm, the area factor AR = 0.2, and the radius of the holes R. See Figure 1 for the geometry of the hexagonal unit cell. A parametric sweep is run over the radius R to find the optimal radius. The optimal radius is chosen to be the radius with lowest resistive loss (real part of the input impedance). The optimal radius is compared to the analytical results of Ref. 1. Here the optimal amount of holes per area for an incompressible gas is given as

(1)

The optimal radius is given by the hole density Nopt and the area factor AR as

(2)

For the parameters h = 1.0 μm, d = 1.5 μm, and AR = 0.2 the optimal radius is analytically given as Ropt = 2.3 μm. The model is run with both the Slip Wall boundary condition and the classical No Slip boundary condition (the No slip option of the Wall condition) to see the importance of including the slip velocity formulation at high Knudsen number.

Figure 1: Geometry of unit cell. There are three vertical symmetry planes to create a hexagonal unit cell. At acoustics is actuated by a vibration of the boundary at z = 0, and a port i placed at the top boundary. The perforated plate is assumed to be fixed.

Results and Discussion

The model is run from radius R = 1.0 μm to R = 4.0 μm. The acoustic velocity with the Slip Wall boundary condition is shown in Figure 2 for three radii R. Since a fixed area factor AR is used the size of the unit cell increases with the radius R. At the large radius of R = 4.0 μm the largest velocity is between the membrane and perforated plate in the xy-plane, while for the smallest radius of R = 1 μm the highest velocity is located in the perforated holes and is in the z direction.

Figure 2: Acoustic velocity for R = 1, 2.3, 4 μm the velocity profile consists of a horizontal squeezing flow between the two plates and a vertical flow through the holes in the perforated plate.

The input (specific) impedance is evaluated on the vibrating surface as

(3)

Where the integration area is the vibrating membrane at z = 0, A is the area of the integration surface, and w is the actuation velocity set to 1 nm/s. The impedance as a function of R can be seen in Figure 3 for both the Slip Wall boundary condition and the No Slip boundary condition. The location of the minimum impedance is in good agreement with the analytical prediction of Ropt = 2.3 μm.

Figure 3: The real part of the impedance as a function of the hole radius R. In blue the model with Slip Wall boundary conditions and in green the model with No Slip boundary conditions.

The dissipation is lowest around R = 2.3 μm. At smaller radii the dissipation is dominated by the viscous losses from the vertical flow through the holes of the perforated membrane and for larger radii the viscous losses is dominated by the horizontal squeezing flow between the two plates. In Figure 4 the total dissipation of the acoustic field is shown for the three different radii, this shows how the dissipation occurs at different locations in the model at large and small radii. The dissipation for R = 1.0 μm is located near the vertical wall in the perforated holes, while the dissipation for R = 4.0 μm is largest between the two plates at the edge of the perforated holes. For R = 2.3 μm the losses is shared between the horizontal squeezing flow and the vertical flow through the holes, but also including losses beneath the hole where the flow changes from a horizontal to vertical flow.

Figure 4: Acoustic dissipation for R = 1, 2.3, 4 μm. The dissipation is either dominated by the horizontal squeezing flow or the vertical flow through the holes.

Notes About the COMSOL Implementation

The implementation uses the boundary condition , available in the thermoviscous acoustics interfaces, to model the slip velocity for high Knudsen numbers. The theoretical background for the boundary condition can be found in the documentation of the Slip Wall boundary condition found in the Acoustics Module User’s Guide.

Reference

1. D. Homentcovschi and R.N. Miles, “Viscous damping of perforated planar micromechanical structures,” Sensors and Actuators A: Physical, vol. 119, no. 2, pp. 544–552, 2005.

Acoustics_Module/Tutorials,_Thermoviscous_Acoustics/viscous_damping_mpp

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