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A simple numerical methodology for thermal-fluid-structural interactions of air damping over heated micro-cantilevers

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Abstract

This paper proposes a simple methodology to analyse numerically the effects of air damping on the oscillatory motion of heated micro-cantilevers. The proposed methodology is fully solved by the fluid solver, because the solid domain is embedded into a subroutine executed at the end of each time step. The methodology is first validated against numerical and experimental data yielding acceptable results. Next, it is applied to predict the quality factor of a micro-cantilever heated by an electric resistance inserted inside it. The transferred heat flow and the evolution in time of temperature are analysed and the quality factors of both heated and isothermal micro-cantilevers are compared.

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Abbreviations

A :

Area (m2)

b :

Width of the micro-cantilever (m)

B :

Constant to evaluate Young’s Modulus of silicon in equation 9 (15.8 MPa/K)

C :

Damping constant (kg/s)

c cr :

Critical damping (kg/s), \( C_{\text{cr}} = 2m_{\text{eq}} \omega_{n} \)

c eq :

Equivalent damping (kg/s)

E :

Young’s Modulus (GPa)

E 0 :

Young’s Modulus of silicon at T 0 (167.5 GPa)

f :

Frequency (Hz)

F Fl :

Fluid force (N)

F i :

Fluid force transferred to element i (N)

F ni :

Fluid force transferred to node i (N)

h :

Thickness of the micro-cantilever (m)

i :

Enthalpy (J)

k :

Thermal conductivity of air (W/m K)

K :

Stiffness constant (kg/s2)

Kn :

Knudsen number (dimensionless)

l :

Characteristic length (m)

L :

Length of the cantilever (m)

L 0 :

Cell size attached to the micro–cantilever walls (m)

L Out :

Cell size attached to the outer limits (m)

m eq :

Equivalent mass (kg), \( m_{\text{eq}} = 0.24\rho_{C} hbL \)

M :

Mass constant (kg)

n :

Oscillation (dimensionless), n = f × t

Nu :

Nusselt number (dimensionless), \( Nu = \frac{q \cdot l}{{A\left( {\bar{T}_{C} - T_{\infty } } \right)k}} \)

N y,Fluid :

Number of fluid cells between the micro-cantilever and the upper or bottom sides (dimensionless)

N y,Solid :

Number of fluid cells in the micro-cantilever thickness (dimensionless)

P :

Pressure (Pa)

P Op :

Operating pressure (101,325 Pa)

q :

Heat flow transferred through the micro-cantilever walls (W)

q gen :

Heat flow generated inside the micro-cantilever (W)

Q :

Quality factor (dimensionless)

R :

Ideal gas constant (8.314 J/mol K)

t :

Time (s)

T :

Temperature (K)

\( \bar{T}_{i} \) :

Volume weighted average of the temperature of the element i in which the micro-cantilever is divided (K)

\( \bar{T}_{C} \) :

Volume weighted average of the temperature of the micro-cantilever (K)

T* :

Dimensionless temperature defined in equation 10

T 0 :

Reference temperature to calculate Young’s Modulus of silicon (317 K)

T ∞ :

Fluid temperature far from the micro-cantilever (300 K)

U, V, W :

Components of the velocity vector (m/s)

x, y, z :

Spatial directions (m)

\( Y \) :

Micro-cantilever displacement (m)

\( \dot{Y} \) :

Micro-cantilever velocity (m/s)

\( \ddot{Y} \) :

Micro-cantilever acceleration (m/s2)

\( \alpha \) :

Thermal diffusivity (m2/s)

\( \lambda \) :

Mean free path of air molecules (m)

\( \nu \) :

Kinematic viscosity (m2/s)

\( \rho \) :

Fluid density (kg/m3)

\( \rho_{C} \) :

Micro-cantilever density (kg/m3)

\( \omega_{\text{n}} \) :

Undamped natural frequency (Hz), \( \omega_{\text{n}} = 3.52\frac{h}{{L^{2} }}\sqrt {\frac{E}{{12\rho_{C} }}} \)

\( \zeta \) :

Damping ratio (dimensionless), \( \zeta = \frac{{C_{\text{eq}} }}{{C_{\text{cr}} }} \)

∇·V :

Divergence of the velocity vector (s−1)

CFD:

Computational fluid dynamics

CSD:

Computational structural dynamics

FEM:

Finite element method

FFT:

Fast fourier transform

FSI:

Fluid-structure interaction

MEMS:

Micro-electro–mechanical systems

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Acknowledgments

This work was partially supported by the Research Project PI 110/08 “Diseño y desarrollo de arrays de sensores tipo micro-cantilevers para la detección de explosivos entre mezclas de gases” funded by the Government of Aragon. Moreover, GICSERV actions (87, 138) with CNM-CSIC are gratefully acknowledged.

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Authors and Affiliations

  1. Desarrollo y Servicios Tecnológicos, Área de Investigación, Instituto Tecnológico de Aragón, María de Luna 7, 50018, Zaragoza, Spain

    M. Miana, E. Bernal, J. Paniagua, J. R. Valdés & S. Izquierdo

  2. Instituto de Nanociencia de Aragón, Universidad de Zaragoza, C/Mariano Esquillor, Edificio I + D, 50018, Zaragoza, Spain

    I. Pellejero

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  1. M. Miana
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  2. E. Bernal
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  3. J. Paniagua
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  4. J. R. Valdés
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  5. S. Izquierdo
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  6. I. Pellejero
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Correspondence to M. Miana.

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Miana, M., Bernal, E., Paniagua, J. et al. A simple numerical methodology for thermal-fluid-structural interactions of air damping over heated micro-cantilevers. Microfluid Nanofluid 13, 131–140 (2012). https://doi.org/10.1007/s10404-012-0951-5

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