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Unformatted text preview: exerts shear force (x), so fluid
must exert equal and opposite force
on walls, provided by external force U. Srinivasan © τw
τw Ux
W τ w = −η ∂U x
∂y =
y =h ∆Ph
2L Q
Wh
2
h ∇P 2
= 3 U max
U=
12ηL
U= 24 12 Poiseuille Flow
• Flow in channels of circular cross section Ux (r
= o 2 − r 2 )∆P
4ηL πro 4 ∆P
Q=
32ηL • Flow in channels of arbitrary cross section Dh = 4 × Area
Perimeter ∆P = f D (1 ρU 2 )
2 L
Dh f D Re D = dimensionless
constant EE C245 • Lumped element model for Poiseuille flow R pois = ∆P 12ηL
=
Q Wh 3 U. Srinivasan © 25 Velocity Profiles
• Velocity profiles for a
combination of
pressuredriven
(Poiseuille) and plate
motion (Couette) flow • Stokes, or creeping, flow
• If Re << 1 , inertial term may
be neglected compared to
viscous term • Development length
EE C245 • Distance before flow assume
steadystate profile U. Srinivasan © 26 13 Lecture Outline
• Today’s Lecture
Basic Fluidic Concepts
Conservation of Mass → Continuity Equation
Newton’s Second Law → NavierStokes Equation
Incompressible Laminar Flow in Two Cases
SqueezeFilm Damping in MEMS EE C245 •
•
•
•
• 27 U. Srinivasan © Squeezed Film Damping EE C245 • Squeezed film damping in
parallel plates
• Gap h depends on x, y, and t
• When upper plate moves
downward, Pair increases and
air is squeezed out
• When upper plate moves
upward, Pair decreases and
air is sucked back in
• Viscous drag of air during
flow opposes mechanical
motion U. Srinivasan © F
moveable h
fixed 28 14 Squeezed Film Damping
• Assumptions
•
•
•
•
•
• Gap h << width of plates
Motion slow enough so gas moves under Stokes flow
No ∆P in normal direction
Lateral flow has Poiseuille like velocity profile
Gas obeys Ideal Gas Law
No change in T
∂( Ph ) EE C245 • Reynold’s equation 12η ∂t = ∇ ⋅ [(1 + 6K n )h 3 P∇P ] • NavierStokes + continuity + Ideal Gas
Law
• Knudsen number Kn is ratio of mean
free path of gas molecules λ to gap h
• Kn < 0.01, continuum flow
• Kn > 0.1, slip flow becomes important
• 1 µm gap with room air, λ = h
29 U. Srinivasan © Squeezed Film Damping
• Approach EE C245 • Small amplitude rigid
motion of upper plate, h(t)
• Begin with nonlinear partial
differential equation
• Linearize about operating
point, average gap h0 and
average pressure P0
• Boundary conditions
• At t = 0, plate suddenly
displaced vertically amount
z0 (velocity impulse)
• At t > 0+, v = 0
• Pressure changes at edges
of plate are zero: dP/dh = 0
at y = 0, y = W U. Srinivasan © ∂( Ph ) h 3
[∇ 2 P 2 ]
=
24η
∂t
h = h0 + ∂h and p = P0 + ∂P h02 P0 ∂ 2 p 1 dh
∂p
−
=
∂t 12ηW 2 ∂ξ 2 h0 dt
∂P
y
p=
and ξ =
P0
W
30 15 General Solution
• Laplace transform of response to general timedependent source z(s) 96ηLW 3 1
sz ( s )
F(s ) = ∑4
43
S π h0 n odd n (1 + αn ) EE C245 plate velocity b
96ηLW 3
π 2h02 P0
F(s ) =
sz ( s ), b =
, ωc =
1 + ωSc
12ηW 2
π 4 h03
1st term for small
amplitude oscillation damping
constant cutoff
frequency
31 U. Srinivasan © Squeeze Number
• Squeeze number σd is a measure of relative
importance of viscous forces to spring forces
• ω < ωc : model reduces to linear resistive damping element
• ω > ωc : stiffness of gas increases since it does not have
time to squeeze out π 2ω 12ηW 2
=2
ω
h0 P0
ωc EE C245 squeeze number σ d = U. Srinivasan © 32 16 Examples EE C245 • Analytical and numerical
results of damping and spring
forces vs. squeeze number for
square plate
• Transient responses for two
squeeze numbers, 20 and 60 U. Srinivasan © Senturia group, MIT 33 17...
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This note was uploaded on 04/01/2014 for the course CHBE 6200 taught by Professor Staff during the Spring '08 term at Georgia Institute of Technology.
 Spring '08
 Staff

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