Download Lecture 17 Fluid Dynamics: handouts

Fluidic Dynamics
EE C245
Dr. Thara Srinivasan
Lecture 17
Picture credit: A. Stroock et al.,
Microfluidic
a chip Lab
Picture
credit:mixing
Sandiaon
National
Lecture Outline
• Reading
• From S. Senturia, Microsystem Design, Chapter 13, “Fluids,”
p.317-334.
• Today’s Lecture
Basic Fluidic Concepts
Conservation of Mass → Continuity Equation
Newton’s Second Law → Navier-Stokes Equation
Incompressible Laminar Flow in Two Cases
Squeeze-Film Damping in MEMS
EE C245
•
•
•
•
•
U. Srinivasan ©
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Viscosity
• Fluids deform continuously in presence of shear forces
• For a Newtonian fluid,
•
•
•
•
Shear stress = Viscosity × Shear strain
N/m2 [=] (N s/m2) × (m/s)(1/m)
Centipoise = dyne s/cm2
No-slip at boundaries
•
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h
U
τw
τw
Ux
ηair = 1.8 × 10-5 N s/m2
ηwater = 8.91 × 10-4
ηlager = 1.45 × 10-3
ηhoney = 11.5
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Density
• Density of fluid depends on pressure and
temperature…
• For water, bulk modulus =
• Thermal coefficient of expansion =
• …but we can treat liquids as incompressible
• Gases are compressible, as in Ideal Gas Law
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PV = nRT
U. Srinivasan ©
 R 
P = ρm 
T
 MW 
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Surface Tension
• Droplet on a surface
• Capillary wetting
2r
γ
γ
∆P
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h=
2γ cosθ
ρgr
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Lecture Outline
• Today’s Lecture
Basic Fluidic Concepts
Conservation of Mass → Continuity Equation
Newton’s Second Law → Navier-Stokes Equation
Incompressible Laminar Flow in Two Cases
Squeeze-Film Damping in MEMS
EE C245
•
•
•
•
•
U. Srinivasan ©
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Conservation of Mass
• Control volume is region fixed in space through which fluid moves
rate of accumulation = rate of inflow − rate of outflow
rate of mass efflux
• Rate of accumulation
dS
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• Rate of mass efflux
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Conservation of Mass
∂ρ m
dV + ∫∫S ρ m U ⋅ n dS = 0
∫∫∫V
∂t
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dS
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Operators
• Gradient and divergence
∂U y ∂U z
∂U x
i+
j+
k
∂z
∂x
∂y
∂
∂
∂
∇ = e x + e y + ez
∂x
∂y
∂z
∂U x ∂U y ∂U z
∇ ⋅ U = div U =
+
+
∂x
∂y
∂z
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∇U =
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Continuity Equation
• Convert surface integral to volume integral using Divergence
Theorem
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• For differential control volume
 ∂ρ

∫∫∫V  + ∇ ⋅ ( ρ U )  dV = 0
 ∂t

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∂ρ
+ ∇ ⋅ (ρ U) = 0
∂t
Continuity Equation
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Continuity Equation
• Material derivative measures time rate of change of a
property for observer moving with fluid
∂ρ
+ (U ⋅ ∇ )ρ + ρ∇ ⋅ U = 0
∂t
Dρ ∂ρ
=
+ ( U ⋅ ∇ )ρ
Dt ∂t
∂
∂
∂
∂
D ∂
+ U y + Uz
= + U ⋅ ∇ = + Ux
∂z
∂t
∂x
∂y
Dt ∂t
For incompressible fluid
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Dρ
+ ρ∇ ⋅ U = 0
Dt
Dρ
+ ρ∇ ⋅ U = 0
Dt
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Lecture Outline
• Today’s Lecture
Basic Fluidic Concepts
Conservation of Mass → Continuity Equation
Newton’s Second Law → Navier-Stokes Equation
Incompressible Laminar Flow in Two Cases
Squeeze-Film Damping in MEMS
EE C245
•
•
•
•
•
U. Srinivasan ©
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Newton’s Second Law for Fluidics
• Newton’s 2nd Law:
• Time rate of change of momentum of a system equal to net
force acting on system
Sum of forces
acting on control
volume
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∑F =
Rate of
momentum efflux
=
from control
volume
dp
d
= ∫∫∫V Uρ dV
dt
dt
Net momentum
accumulation rate
Rate of
accumulation of
momentum in
control volume
+
+ ∫∫S Uρ (U ⋅ n ) dS
Net momentum
efflux rate
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Momentum Conservation
• Sum of forces acting on fluid
∑F =
pressure and shear
stress forces
gravity force
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• Momentum conservation, integral form
∫∫S (− Pn + τ )dS + ∫∫∫V ρgdV =
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d
∫∫∫ ρUdV + ∫∫S ρU(U ⋅ n ) dS
dt V
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Navier-Stokes Equation
• Convert surface integrals to volume integrals
η∇ 2 U + η ∇(∇ ⋅ U )  dV
=
τ
dS
∫∫
∫∫∫ 

S
V 
3

∫∫ − Pn dS = ∫∫∫ − ∇P dV
S
V
∫∫ ρU( U ⋅ n ) dS = ∫∫∫ ρU( ∇ ⋅ U ) dV
V
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S
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Navier-Stokes Differential Form
η


2
∫∫∫  − ∇P + ρg + η∇ U + ∇(∇ ⋅ U )  dV =
V 
3

 ∂U  dV + ρU(∇ ⋅ U ) dV
∫∫∫  ρ
∫∫∫

V 
V
∂t 
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 DU  dV
∫∫∫  ρ

V 
Dt 
ρ
DU
η
= −∇P + ρg + η∇ 2 U + ∇(∇ ⋅ U )
Dt
3
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Incompressible Laminar Flow
• Incompressible fluid
ρ
∇⋅U = 0
DU
η
= −∇P + ρg + η∇ 2 U + ∇(∇ ⋅ U )
Dt
3
DU
= −∇P + ρg + η∇ 2 U
Dt
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ρ
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Cartesian Coordinates
ρ
DU
= −∇P + ρg + η∇ 2 U
Dt
x direction
 ∂U x
∂U x
∂U x
∂U x 
+ Ux
+U y
+ Uz
=
∂x
∂y
∂z 
 ∂t
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ρ
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 ∂ 2U x ∂ 2U x ∂ 2U x 
∂Px
−
+ ρg x + η  2 + 2 +

∂x
∂y
∂z 2 
 ∂x
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Dimensional Analysis
DU
∇P
η∇ 2 U
=−
+g+
ρ
ρ
Dt
• Each term has dimension L/t2 so ratio of any two gives
dimensionless group
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• inertia/viscous
• pressure /inertia
• flow v/sound v
=
=
=
P/ρU2
=
=
=
Reynolds number
Euler number
Mach number
• In geometrically similar systems, if dimensionless numbers are
equal, systems are dynamically similar
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Two Laminar Flow Cases
U
h
τw
τw
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y
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τw
τw
High h
P
Low
P
y
Ux
U
Ux
Umax
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Couette Flow
• Couette flow is steady viscous flow
between parallel plates, where top plate is
moving parallel to bottom plate
• No-slip boundary conditions at plates
∇ ⋅ U = 0 U = Uxix
U = U x ( y )i x
DU x
η  ∂ 2U x ∂ 2U x 
1 ∂P
=−
+ gx + 
+

∂y 2 
Dt
ρ ∂x
ρ  ∂x 2
U x = 0 at y = 0
∂ 2U x
=
and
0
,
U x = U at y = h
∂y 2
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U
h
τw
τw
y
Ux =
Ux
U
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Couette Flow
• Shear stress acting on plate due to motion, $, is dissipative
• Couette flow is analogous to resistor with power dissipation
corresponding to Joule heating
τ w = −η
U
τw
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h
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τw
RCouette =
∂U x
∂y
=
y =h
τ w A ηA
U
=
h
PCouette = RU 2
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Poiseuille Flow
• Poiseuille flow is a pressure-driven flow between stationary
parallel plates
• No-slip boundary conditions at plates
 ∂ 2U x ∂ 2U x 
DU x
1 ∂P
=−
+ gx +η
+

2
2
∂
∂
Dt
x
y
ρ ∂x


∂P
∆P
=−
L
∂x
∂ 2U x
∆P
=−
,
2
∂y
Lη
U x = 0 at y = 0, h
Ux =
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y
τw
τw
Highh
P
Low
P
U max =
Ux
Umax
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Poiseuille Flow
• Volumetric flow rate Q ~ [h3]
Q = W ∫0 U x dy =
h
• Shear stress on plates, $, is
dissipative
• Force balance
• Net force on fluid and plates is zero
since they are not accelerating
• Fluid pressure force ∆PWh (+x)
balanced by shear force at the walls
of 2$WL (-x)
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• Wall exerts shear force (-x), so fluid
must exert equal and opposite force
on walls, provided by external force
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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=
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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 (12 ρU 2 )
L
Dh
f D Re D = dimensionless
constant
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• Lumped element model for Poiseuille flow
R pois =
∆P 12ηL
=
Q Wh 3
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Velocity Profiles
• Velocity profiles for a
combination of
pressure-driven
(Poiseuille) and plate
motion (Couette) flow
• Stokes, or creeping, flow
• If Re << 1 , inertial term may
be neglected compared to
viscous term
• Development length
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• Distance before flow assume
steady-state profile
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Lecture Outline
• Today’s Lecture
Basic Fluidic Concepts
Conservation of Mass → Continuity Equation
Newton’s Second Law → Navier-Stokes Equation
Incompressible Laminar Flow in Two Cases
Squeeze-Film Damping in MEMS
EE C245
•
•
•
•
•
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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
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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 )
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• Reynold’s equation
12η
∂t
= ∇ ⋅ [(1 + 6K n )h 3 P∇P ]
• Navier-Stokes + 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
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Squeezed Film Damping
• Approach
EE C245
• Small amplitude rigid
motion of upper plate, h(t)
• Begin with non-linear 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
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∂( 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
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General Solution
• Laplace transform of response to general time-dependent source z(s)
96ηLW 3

1
sz ( s )
F(s ) = 
∑
4 3
4
S 
 π h0 n odd n (1 + αn )
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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
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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
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squeeze number σ d =
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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
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