Lecture 17 Fluid Dynamics: Handouts

Fluidic Dynamics

Dr. Thara Srinivasan Lecture 17

EE C245 Picture credit: A. Stroock et al., PictureMicrofluidic credit: mixingSandia on National a chip Lab

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 2 U. Srinivasan ©

1 Viscosity • Fluids deform continuously in presence of shear forces • For a Newtonian fluid, • Shear stress = Viscosity × Shear strain 2 2 • N/m [=] (N • s/m ) × (m/s)(1/m) • Centipoise = dyne s/cm2 U • No-slip at boundaries τw h U τw x

-5 2 ηair = 1.8 × 10 N s/m -4 ηwater = 8.91 × 10 -3 ηlager = 1.45 × 10

ηhoney = 11.5 EE C245 3 U. Srinivasan ©

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

 R  PV = nRT P = ρm  T  MW  EE C245 4 U. Srinivasan ©

2 Surface Tension

• Droplet on a surface • Capillary wetting 2r

γ γ ∆P

2γ cosθ h = ρgr EE C245 5 U. Srinivasan ©

Lecture Outline

3 Conservation of Mass

• Control volume is region fixed in space through which fluid moves rate of accumulation = rateof inflow − rate of outflow

rate of mass efflux • Rate of accumulation

• Rate of mass efflux EE C245 7 U. Srinivasan ©

Conservation of Mass

∂ρ ∫∫∫ m dV + ∫∫ ρ U ⋅ndS = 0 V ∂t S m

dS EE C245 8 U. Srinivasan ©

4 Operators

• Gradient and divergence

∂U ∂U ∂U ∇U = x i + y j + z k ∂x ∂y ∂z ∂ ∂ ∂ ∇ = e + e + e ∂x x ∂y y ∂z z ∂U ∂U ∂U ∇ ⋅ U = div U = x + y + z ∂x ∂y ∂z EE C245 9 U. Srinivasan ©

Continuity Equation

• Convert surface integral to volume integral using Divergence Theorem

• For differential control volume  ∂ρ  ∂ρ ∫∫∫  + ∇ ⋅(ρU)dV = 0 + ∇ ⋅(ρU) = 0 V  ∂t  ∂t EE C245 Continuity Equation 10 U. Srinivasan ©

5 Continuity Equation

• Material derivative measures time rate of change of a property for observer moving with fluid

∂ρ Dρ ∂ρ + (U ⋅∇)ρ + ρ∇ ⋅ U = 0 = + (U ⋅∇)ρ ∂t Dt ∂t

D ∂ ∂ ∂ ∂ ∂ = + U ⋅∇ = +U +U +U Dt ∂t ∂t x ∂x y ∂y z ∂z

For incompressible fluid Dρ Dρ + ρ∇ ⋅ U = 0 + ρ∇ ⋅ U = 0 EE C245 Dt Dt 11 U. Srinivasan ©

Lecture Outline

6 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

Rate of Sum of forces Rate of momentum efflux accumulation of acting on control + = from control momentum in volume volume control volume

dp d ∑F = = ∫∫∫ Uρ dV + ∫∫ Uρ()U ⋅n dS dt dt V S Net momentum Net momentum accumulation rate efflux rate EE C245 13 U. Srinivasan ©

Momentum Conservation

• Sum of forces acting on fluid

∑ F =

pressure and shear gravity force stress forces

• Momentum conservation, integral form d ∫∫ ()− Pn + τ dS + ∫∫∫ ρgdV = ∫∫∫ ρUdV + ∫∫ ρU()U ⋅n dS S V dt V S EE C245 14 U. Srinivasan ©

7 Navier-Stokes Equation

• Convert surface integrals to volume integrals

 2 η  ∫∫τdS = ∫∫∫η∇ U + ∇()∇ ⋅ U dV S V  3  ∫∫− PndS = ∫∫∫− ∇P dV S V ∫∫ ρU(U ⋅n)dS = ∫∫∫ρU(∇ ⋅ U)dV S V EE C245 15 U. Srinivasan ©

Navier-Stokes Differential Form

 2 η  ∫∫∫− ∇P + ρg +η∇ U + ∇()∇ ⋅ U dV = V  3   ∂U  ∫∫∫ ρ dV + ∫∫∫ρU()∇ ⋅ U dV V  ∂t  V

 DU  ∫∫∫ ρ dV V  Dt  DU η ρ = −∇P + ρg +η∇2U + ∇()∇ ⋅ U

EE C245 Dt 3 16 U. Srinivasan ©

8 Incompressible Laminar Flow

• Incompressible fluid ∇ ⋅ U = 0

DU η ρ = −∇P + ρg +η∇2U + ∇()∇ ⋅ U Dt 3

DU ρ = −∇P + ρg +η∇2U Dt EE C245 17 U. Srinivasan ©

Cartesian Coordinates

DU ρ = −∇P + ρg +η∇2U Dt

x direction

 ∂U x ∂Ux ∂Ux ∂Ux  ρ +Ux +U y +Uz  =  ∂t ∂x ∂y ∂z  2 2 2 ∂Px  ∂ Ux ∂ Ux ∂ Ux  − + ρg x +η + +  ∂x  ∂x 2 ∂y 2 ∂z 2  EE C245 18 U. Srinivasan ©

9 Dimensional Analysis

DU ∇P η∇ 2 U = − + g + Dt ρ ρ

• Each term has dimension L/t2 so ratio of any two gives dimensionless group • inertia/viscous = = Reynolds number • pressure /inertia = P/ρU2 = Euler number • flow v/sound v = = Mach number

• In geometrically similar systems, if dimensionless numbers are EE C245 equal, systems are dynamically similar 19 U. Srinivasan ©

Two Laminar Flow Cases

τw τw Low h High h τw P τw P

y y

Ux Ux

EE C245 U Umax 20 U. Srinivasan ©

10 Couette Flow

• Couette flow is steady viscous flow ∇ ⋅ U = 0 U = U i between parallel plates, where top plate is x x

moving parallel to bottom plate U = Ux ( y)i x • No-slip boundary conditions at plates

2 2 DUx 1 ∂P η  ∂ Ux ∂ Ux  = − + gx +  +  Dt ρ ∂x ρ  ∂x 2 ∂y 2 

2 U = 0 at y = 0 ∂ U x x 2 = 0, and ∂y U x = U at y = h U y

τw h U U x = τ x

EE C245 w U 21 U. Srinivasan ©

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

∂U x τ w = −η = ∂y y=h U

τw τ A ηA h τ R = w = w Couette U h

2 PCouette = RU EE C245 22 U. Srinivasan ©

11 Poiseuille Flow

• Poiseuille flow is a pressure-driven flow between stationary parallel plates • No-slip boundary conditions at plates

2 2 DUx 1 ∂P  ∂ Ux ∂ Ux  = − + gx +η +  Dt ρ ∂x  ∂x 2 ∂y 2  ∂P ∆P ∂2U ∆P = − x = − , U = 0 at y = 0,h ∂x L ∂y 2 Lη x

U = y x

Poiseuille Flow • Volumetric flow rate Q ~ [h3] h • Shear stress on plates, $,is Q = W ∫ U dy = dissipative 0 x • Force balance • Net force on fluid and plates is zero ∂Ux ∆Ph since they are not accelerating τ w = −η = • Fluid pressure force ∆PWh (+x) ∂y y=h 2L balanced by shear force at the walls of 2$WL (-x) • Wall exerts shear force (-x), so fluid Q must exert equal and opposite force U = on walls, provided by external force Wh 2 h ∇P 2 τw U = = 3 Umax Ux 12ηL EE C245 τw W 24 U. Srinivasan ©

12 Poiseuille Flow

• Flow in channels of circular cross section 4 2 2 (r − r )∆P πro ∆P U = o Q = x 4ηL 32ηL

• Flow in channels of arbitrary cross section

× 2 L 4 Area ∆ = 1 ρ f Re = dimensionless Dh = P f D ()2 U D D Perimeter Dh constant

Velocity Profiles

• Velocity profiles for a • Stokes, or creeping, flow combination of • If Re << 1 , inertial term may pressure-driven be neglected compared to viscous term (Poiseuille) and plate motion (Couette) flow

13 Lecture Outline

Squeezed Film Damping

• Squeezed film damping in parallel plates • Gap h depends on x, y, and t F • When upper plate moves moveable

downward, Pair increases and air is squeezed out • When upper plate moves h

upward, Pair decreases and air is sucked back in fixed • Viscous drag of air during flow opposes mechanical motion EE C245 28 U. Srinivasan ©

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) 12η = ∇ ⋅[](1+ 6K )h 3P∇P • Reynold’s equation ∂t n • 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 h

Squeezed Film Damping

• Approach ∂(Ph) h 3 • Small amplitude rigid = []∇2P 2 motion of upper plate, h(t) ∂t 24η • Begin with non-linear partial differential equation • Linearize about operating h = h0 + ∂h and p = P0 + ∂P point, average gap h0 and average pressure P 0 ∂p h 2P ∂2 p 1 dh • Boundary conditions = 0 0 − • At t = 0, plate suddenly ∂t 12ηW 2 ∂ξ 2 h dt displaced vertically amount 0 z (velocity impulse) 0 ∂P y • At t > 0+, v = 0 p = and ξ = • Pressure changes at edges P0 W of plate are zero: dP/dh = 0

15 General Solution

• Laplace transform of response to general time-dependent source z(s)

96ηLW 3 1  = F(s )  4 3 ∑ 4 sz(s ) n odd S  π h0 n ()1+ αn  plate velocity

b 96ηLW 3 π 2h 2P F(s ) = sz(s ), b = , ω = 0 0 S 4 3 c 2 1+ ωc π h0 12ηW damping cutoff 1st term for small constant frequency EE C245 amplitude oscillation 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

16 Examples • Analytical and numerical results of damping and spring forces vs. squeeze number for square plate • Transient responses for two squeeze numbers, 20 and 60 EE C245 33 U. Srinivasan © Senturia group, MIT