Lecture 17 Fluid Dynamics Handouts

Srinivasan 26 13 lecture outline todays lecture basic

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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 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 EE C245 • Distance before flow assume steady-state profile U. Srinivasan © 26 13 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 • • • • • 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 ] • 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 29 U. Srinivasan © 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 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 time-dependent 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.

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