A1.4 - Semi-Numerical Simulation of a Miniaturized Vibrating Membrane - Rheometer
- SENSOR+TEST Conferences 2009
2009-05-26 - 2009-05-28
Congress Center Nürnberg
- Proceedings SENSOR 2009, Volume I
- A1 - Microacoustic Sensors
- T. Voglhuber-Brunnmaier, E. Reichel, B. Jakoby - Johannes Kepler University, Linz, Austria
- 41 - 46
For online process monitoring purposes micro rheometers based upon quartz resonators are commonly used but suffer from weak comparability of the results compared to those obtained from traditional laboratory viscometers. This discrepancy is partly explained by the low agitation of the measured liquid within these microrheometers - this issue is addressed by a larger deformation at lower frequencies in cur present measurement setup. In our contribution we investigate a miniaturized clamped membrane device. This system consists of a sample cell of 5.6 X 12 X 1 mm³ size filled with the liquid under test. Its bottom and top side both are sealed by 52 µm thick PMMA-membranes which are excited to vibrations. For the analysis of this problem finite element (FE) solvers fail due to the large amount of elements required for adequate discretization. The viscosity coefficient of the liquid under test is linearly related to the power dissipation due to shear velocity gradients in the sample fluid which causes a finite Q-factor cf the resonance peak. This paper shows that moreover considerable gradients do mainly occur within a thin - tens of microns - layer close to the membrane-fluid interface. Therefore a FE discretization Of the outermost liquid regions would have to be sufficiently fine as well.
In the presented contribution the problem of high numerical complexity is addressed using an alternatrve approach, namely a semi-numerical representation in the spectral domain. The membranes are described by using equations of motion and the constitutive equations of linear visco-elastic bulk material. The fluid sample volume is modeled employing Navier-Stokes equations, which can be linearized in case of the present problem (as will be discussed in the paper). The description of the fluid includes first and second viscosity coefficients and pressure-density coupling via compressibility coefficient as well. Our approach is based upon the conversion of the thus obtained partial differential equations (PDEs) to ordinary differential equations (ODEs) by expressing the field variables (displacements and stresses) by a harmonic representation in the spectral domain. This yields two Eigenvalue problems - one for the membranes and one for the fluid. The computational effort is reduced drastically this way. The principle solution of the obtained Eigenvalue problem describing the fluid is coupled to the membrane equations by means of a propagator matrix, which is derived from the solution of the Eigenvalue problem of the membrane. Taking into account boundary conditions and rearranging the solution system finally leads to the frequency response of the sensor. By analyzing the frequency response in the vicinity of the resonance peak, a behaviour similar to a damped second order system is revealed. The numerically found frequency response is compared to experimental results obtained with the micro-rheometer. For time-harmonic excitation of the membranes via Lorentz forces conductive paths are patterned. Read-aut is also achieved by a further pair of conductive paths. Imposed forces result in time-harmonic displacements which induce voltages in the read-out path. The frequency response showing dependency to liquid and membrane properties is measured by a lock-in amplifier setup. In the following sections the model is outlined in detail and numerical sample results are presented.