Fluids A. Joel Voldman* Massachusetts Institute of Technology *(with thanks to SDS)

Transcription

1 Cite as: Joel Voldan, course aterials for 6.777J / 2.372J Design and Fabrication of Microelectroechanical Devices, Spring MIT JV: 6.777J/2.372J Spring 2007, Lecture 14-1 Fluids A Joel Voldan* Massachusetts Institute of Technology *(with thanks to SDS)

2 Cite as: Joel Voldan, course aterials for 6.777J / 2.372J Design and Fabrication of Microelectroechanical Devices, Spring MIT JV: 6.777J/2.372J Spring 2007, Lecture 14-2 Outline > Intro to icrofluidics > Basic concepts: viscosity and surface tension > Governing equations > Incopressible lainar flow

3 Cite as: Joel Voldan, course aterials for 6.777J / 2.372J Design and Fabrication of Microelectroechanical Devices, Spring MIT JV: 6.777J/2.372J Spring 2007, Lecture 14-3 Microfluidics > The anipulation and use of fluids at the icroscale > Most fluid doains are in use at the icroscale Explosive therofluidic flows» Inkjet printheads» We will not cover this regie High-speed gas flows» Micro-turboachinery» We will not cover this regie Low-speed gas flows» Squeeze-fil daping» We ll do a bit of this to get b for SMD Liquid-based slow flow» This will be the focus Iage reoved due to copyright restrictions. Iage reoved due to copyright restrictions. 3-D cutaway of a icroachined icroengine. Photograph by Jonathan Protz.

4 Microfluidics Cite as: Joel Voldan, course aterials for 6.777J / 2.372J Design and Fabrication of Microelectroechanical Devices, Spring MIT JV: 6.777J/2.372J Spring 2007, Lecture 14-4 > This has been one of the ost iportant doains of MEMS Even though ost icrofluidics is not MEMS And there are few coercial products > The overall driver has been the life sciences Though the only ajor coercial success is inkjets > The initial driver was analytical cheistry Separation of organic olecules > More recently, this has shifted to biology Manipulation of DNA, proteins, cells, tissues, etc.

5 Microfluidics exaples > H-filter Developed by Yager and colleagues at UWash in id-90 s Being coercialized by Micronics > An intrinsically icroscale device Uses diffusion in lainar flow to separate olecules Yager et al., Nature 2006 Courtesy of Paul Yager, Thayne Edwards, Elain Fu, Kristen Helton, KjellNelson, Milton R. Ta, and Bernhard H. Weigl. Used with perission. Courtesy of Paul Yager, Thayne Edwards, Elain Fu, Kristen Helton, KjellNelson, Milton R. Ta, and Bernhard H. Weigl. Used with perission. Yager, P., T. Edwards, E. Fu, K. Helton, K. Nelson, M. R. Ta, and B. H. Weigl. "Microfluidic Diagnostic Technologies for Global Public Health." Nature 442 (July 27, 2006): Cite as: Joel Voldan, course aterials for 6.777J / 2.372J Design and Fabrication of Microelectroechanical Devices, Spring MIT JV: 6.777J/2.372J Spring 2007, Lecture 14-5

6 Microfluidics > Multi-layer elastoeric icrofluidics (Quake, etc.) Use low odulus of silicone elastoers to create hydraulic valves Move liquids around Use diffusivity of gas in elastoer to enable dead-end filling Iage reoved due to copyright restrictions. Iage reoved due to copyright restrictions. Please see: Figure 3 on p. 582 in: Thorsen, T., S. J. Maerkl, and S. R. Quake. "Microfluidic Large-Scale Integration." Science, New Series 298, no (October 18, 2002): Iages reoved due to copyright restrictions. Fluidig Cite as: Joel Voldan, course aterials for 6.777J / 2.372J Design and Fabrication of Microelectroechanical Devices, Spring MIT JV: 6.777J/2.372J Spring 2007, Lecture 14-6

7 Goals Cite as: Joel Voldan, course aterials for 6.777J / 2.372J Design and Fabrication of Microelectroechanical Devices, Spring MIT JV: 6.777J/2.372J Spring 2007, Lecture 14-7 > To design icrofluidics we need to understand What pressures are needed for given flows» How do I size y channels? What can fluids do at these scales» What are the relevant physics? What things get better as we scale down» Mixing ties, reagent volues What things get worse, and how can we anage the» Surface tension

8 Cite as: Joel Voldan, course aterials for 6.777J / 2.372J Design and Fabrication of Microelectroechanical Devices, Spring MIT JV: 6.777J/2.372J Spring 2007, Lecture 14-8 Outline > Intro to icrofluidics > Basic concepts: viscosity and surface tension > Governing equations > Incopressible lainar flow

9 Cite as: Joel Voldan, course aterials for 6.777J / 2.372J Design and Fabrication of Microelectroechanical Devices, Spring MIT JV: 6.777J/2.372J Spring 2007, Lecture 14-9 Viscosity > When a solid experiences shear stress, it defors (e.g., strains) Shear Modulus relates the two > When a fluid experiences shear stress, it defors continuously Viscosity relates the two > Constitutive property describing relationship between shear stress [Pa] and shear rate [s -1 ] > Units: Pa-s Water: Pa-s Air: 10-5 Pa-s h τ τ y U U τ = η h and, in the differential liit U x τ = η y A related quantity : Kineatic Viscosity η η* = ρ ( ρ τ = η * y U x ) Iage by MIT OpenCourseWare. Adapted fro Figure 13.1 in Senturia, Stephen D. Microsyste Design. Boston, MA: Kluwer Acadeic Publishers, 2001, p ISBN: x This is a diffusivity for oentu [ 2 /s]

10 Surface Tension Cite as: Joel Voldan, course aterials for 6.777J / 2.372J Design and Fabrication of Microelectroechanical Devices, Spring MIT JV: 6.777J/2.372J Spring 2007, Lecture > A liquid drop iniizes its free energy by iniizing its surface area. The effective force responsible for this is called surface tension (Γ) [J/ 2 = N/] > The surface tension creates a differential pressure on the two sides of a curved liquid surface Γ ( ) ( 2 2πr Γ = ΔP πr ) 2r P solving for ΔP Γ ΔP = 2Γ r Iage by MIT OpenCourseWare. Adapted fro Figure 13.2 in Senturia, Stephen D. Microsyste Design. Boston, MA: Kluwer Acadeic Publishers, 2001, p ISBN:

11 Capillary Effects > Surface forces can actually transport liquids 2r θ > Contact angles deterine what happens, and these depend on the wetting properties of the liquid and the solid surface. ( 2 ρ gh πr ) h Iage by MIT OpenCourseWare. Adapted fro Figure 13.3 in Senturia, Stephen D. Microsyste Design. Boston, MA: Kluwer Acadeic Publishers, 2001, p ISBN: = 2πrΓcosθ 2Γcosθ ρ gr Cite as: Joel Voldan, course aterials for 6.777J / 2.372J Design and Fabrication of Microelectroechanical Devices, Spring MIT JV: 6.777J/2.372J Spring 2007, Lecture h =

12 Cite as: Joel Voldan, course aterials for 6.777J / 2.372J Design and Fabrication of Microelectroechanical Devices, Spring MIT JV: 6.777J/2.372J Spring 2007, Lecture > A hydrophobic valve Capillary Effects Hydrophobic barrier Fill Stop Iage reoved due to copyright restrictions. Burst

13 Cite as: Joel Voldan, course aterials for 6.777J / 2.372J Design and Fabrication of Microelectroechanical Devices, Spring MIT JV: 6.777J/2.372J Spring 2007, Lecture Surface tension > The scaling as 1/r is what akes dealing with surface tension HARD at the icroscale > Solutions Prie with low-surface tension liquids» Methanol (Γ=22.6 N/) vs. water (Γ=72.8 N/)» Or use surfactants Use CO 2 instead of air» Dissolves ore readily in water» Zengerle et al., IEEE MEMS 1995, p340 Use diffusivity of gas in PDMS Iage reoved due to copyright restrictions.

14 Cite as: Joel Voldan, course aterials for 6.777J / 2.372J Design and Fabrication of Microelectroechanical Devices, Spring MIT JV: 6.777J/2.372J Spring 2007, Lecture Outline > Intro to icrofluidics > Basic concepts: viscosity and surface tension > Governing equations > Incopressible lainar flow

15 Fluid Mechanics Governing Equations Cite as: Joel Voldan, course aterials for 6.777J / 2.372J Design and Fabrication of Microelectroechanical Devices, Spring MIT JV: 6.777J/2.372J Spring 2007, Lecture > Mass is conserved Continuity equation > Moentu is conserved Navier-Stokes equation > Energy is conserved Euler equation > We consider only the first two in this lecture

16 Cite as: Joel Voldan, course aterials for 6.777J / 2.372J Design and Fabrication of Microelectroechanical Devices, Spring MIT JV: 6.777J/2.372J Spring 2007, Lecture > Conservation of ass Continuity equation > In this case, for a control or fixed fluid volue both S and V are constant in tie > Point conservation relation is valid for fixed or oving point F V b n d dt Iage by MIT OpenCourseWare. g S bdv F nds + = gdv b t = F + g d dt volue volue volue surface Apply the divergence theore: volue ρ t ρ t = ρ dv ρ dv ρ + ( ρu) dv 0 t = which iplies ρ + ( ρu) = 0 t = ρ Un surface ds dv + ρ Un ds = 0 + U ρ + ρ U = 0

17 Material Derivative Cite as: Joel Voldan, course aterials for 6.777J / 2.372J Design and Fabrication of Microelectroechanical Devices, Spring MIT JV: 6.777J/2.372J Spring 2007, Lecture > The density can change due to three effects: An explicit tie dependence (e.g. local heating) Flow carrying fluid through changing density regions Divergence of the fluid velocity > The first two of these are grouped into the aterial derivative Rate of change for an observer oving with the fluid ρ t ρ U ρ ( U) tie U U

18 Cite as: Joel Voldan, course aterials for 6.777J / 2.372J Design and Fabrication of Microelectroechanical Devices, Spring MIT JV: 6.777J/2.372J Spring 2007, Lecture > The first two of these are grouped into the aterial derivative Incopressible Material Derivative If ρ t + U ρ + ρ U If we define Dρ ρ = + U ρ Dt t or. ore generally, define the operator D = + U Dt t we can write the continuity equation as Dρ + ρ U = 0 Dt the density is unifor, then in steady state U = 0 = 0

19 Moentu Conservation Cite as: Joel Voldan, course aterials for 6.777J / 2.372J Design and Fabrication of Microelectroechanical Devices, Spring MIT JV: 6.777J/2.372J Spring 2007, Lecture > We want to write Newton s 2 nd Law for fluids > We start with a volue that travels with the fluid contains a constant aount of ass Material volue > Then go to an arbitrary control volue Must account for flux of oentu through surface dp d( U) F = = dt dt d F= ρudv dt V () t d F= dt ρ UdV + ρ U ( U U ) n ds s V() t S() t flux of ρ U thru surface velocity of surface

20 Cite as: Joel Voldan, course aterials for 6.777J / 2.372J Design and Fabrication of Microelectroechanical Devices, Spring MIT JV: 6.777J/2.372J Spring 2007, Lecture Moentu Conservation > Pull tie derivative into integral > Cancel ters > Apply divergence theore d F= ρ UdV + ρ U ( ) ds dt U U n s V() t S() t = ρ UdV + ρ U[ U n] ds + ρ U ( U U ) n ds s s t V() t S() t S() t Leibniz s rule = ρ UdV + ρ UU nds t V() t S() t an ds = adv S() t V() t

21 Cite as: Joel Voldan, course aterials for 6.777J / 2.372J Design and Fabrication of Microelectroechanical Devices, Spring MIT JV: 6.777J/2.372J Spring 2007, Lecture Moentu Conservation > Final result does not depend on control volue = ρ UdV + ρ UUdV t V() t V() t ρ U U ρ U ρ U U dv = t t V() t F = ρ V DU Dt dv

22 Cite as: Joel Voldan, course aterials for 6.777J / 2.372J Design and Fabrication of Microelectroechanical Devices, Spring MIT JV: 6.777J/2.372J Spring 2007, Lecture > Add in force ters > Include body forces (e.g., gravity) > And surface forces (i.e., stresses) σ n Moentu Conservation s(n) ρ F F F F b s s = = s( n) s = = DU Dt V S ( t) s( n) ds = n σ S ( t) V ( t) ρ g dv n σds σdv = ρ g + σ

23 Cite as: Joel Voldan, course aterials for 6.777J / 2.372J Design and Fabrication of Microelectroechanical Devices, Spring MIT JV: 6.777J/2.372J Spring 2007, Lecture > Substitute in for stress tensor > Copressible Newtonian fluid constitutive relation > Copressible Navier-Stokes equations P = P ρ g r * Navier-Stokes Equations σ = Pn + τ σ = P + τ τ τ =η U x y 2 η η U + ( U) 3 DU 2 η ρ = P + η U + ( U) + ρg Dt 3 DU 2 η ρ = P* + η U + ( U) Dt 3

24 Cite as: Joel Voldan, course aterials for 6.777J / 2.372J Design and Fabrication of Microelectroechanical Devices, Spring MIT JV: 6.777J/2.372J Spring 2007, Lecture > Ters in copressible N-S equations Navier-Stokes Equations tie-dependence pressure copressibility U 2 η ρ + U U = P + η U + + t 3 ( U) ρ g inertial viscous stresses gravity

25 Diensionless Nubers Cite as: Joel Voldan, course aterials for 6.777J / 2.372J Design and Fabrication of Microelectroechanical Devices, Spring MIT JV: 6.777J/2.372J Spring 2007, Lecture > Fluid echanics is full of nondiensional nubers that help classify the types of flow > Reynolds nuber is ost iportant > Reynolds nuber: The ratio of inertial to viscous effects Ratio of convective to diffusive oentu transport Sall Reynolds nuber eans neglect of inertia Flow at low Reynolds nuber is lainar Re ~ ~ ~ ~ ~ ( ) 2 U U = ( A) P + U LU 0 0 LU 0 0 U0 Re = ρ η = η* = η * L ~ Non-diensionalized steady incopressible flow ~ 0 See Deen, Analysis of Transport Phenoena

26 Cite as: Joel Voldan, course aterials for 6.777J / 2.372J Design and Fabrication of Microelectroechanical Devices, Spring MIT JV: 6.777J/2.372J Spring 2007, Lecture Outline > Intro to icrofluidics > Basic concepts: viscosity and surface tension > Governing equations > Incopressible lainar flow

27 Incopressible Lainar Flow Cite as: Joel Voldan, course aterials for 6.777J / 2.372J Design and Fabrication of Microelectroechanical Devices, Spring MIT JV: 6.777J/2.372J Spring 2007, Lecture > The Navier-Stokes equation becoes very heat-flow-equation-like, although the presence of DU/Dt instead of U/ t akes the equation nonlinear, hence HARD Navier-Stokes becoes DU 2 ρ = P* + η U Dt to obtain a "diffusion-like" equation: DU 2 ρ = η U P* Dt

28 As aside on heat convection Cite as: Joel Voldan, course aterials for 6.777J / 2.372J Design and Fabrication of Microelectroechanical Devices, Spring MIT JV: 6.777J/2.372J Spring 2007, Lecture > Our heat-flow equation looked like > Copare to incopressible N-S eqn > If we allow fluid to ove to convect we can include convection in our heat conservation > At steady state, we get a relation that allows us to copare convective heat transport to conduction > This is the Peclet nuber > For icroscale water flows, L~100 μ, U~0.1 /s, D~150x /s ρ LU Pe = = D T 2 1 = D T + P% t C% DU 2 = η U P Dt * DT 2 1 = D T + P% Dt C% 2 1 U T = D T + P% C% sources ( 4 10 )( 10 4 / s) s sources sources ~0.1

29 Cite as: Joel Voldan, course aterials for 6.777J / 2.372J Design and Fabrication of Microelectroechanical Devices, Spring MIT JV: 6.777J/2.372J Spring 2007, Lecture > Pure shear flow with a linear velocity profile > No pressure gradient > Relative velocity goes to zero at the walls (the socalled no-slip boundary condition) h h y τ W τ W U Couette or Shear Flow Iage by MIT OpenCourseWare. Adapted fro Figure 13.4 in Senturia, Stephen D. Microsyste Design. Boston, MA: Kluwer Acadeic Publishers, 2001, p ISBN: U x U The flow is one-diensional du ρ dt U= U ( y)ˆ n 2 + U U = η U P* N-S Eqns collapse to the Laplace eqn x y x 2 U x 2 = 0 U = c y+ c 1 2 B.C.'s: U (0) = 0, U ( h) = U x U x y = U h x x

30 Poiseuille Flow Cite as: Joel Voldan, course aterials for 6.777J / 2.372J Design and Fabrication of Microelectroechanical Devices, Spring MIT JV: 6.777J/2.372J Spring 2007, Lecture > Pressure-driven flow through a pipe In our case, two parallel plates > Velocity profile is parabolic > This is the ost coon flow in icrofluidics Assues that h<<w y h high P τ W τ W low P U x U ax Iage by MIT OpenCourseWare. Adapted fro Figure 13.5 in Senturia, Stephen D. Microsyste Design. Boston, MA: Kluwer Acadeic Publishers, 2001, p ISBN:

31 Cite as: Joel Voldan, course aterials for 6.777J / 2.372J Design and Fabrication of Microelectroechanical Devices, Spring MIT JV: 6.777J/2.372J Spring 2007, Lecture Solution for Poiseuille Flow > Assue a unifor pressure gradient along the pipe > Assue x-velocity only depends on y > Enforce zero-flow boundary conditions at walls > Maxiu velocity is at center > Voluetric flowrate is dp = K dx 2 U x K = 2 y η 1 U ( ) x = y h y K 2η U ax h 0 = 2 h K 8η 3 Wh Q= W Uxdy = K 12η

32 Luped Model for Poiseuille Flow Cite as: Joel Voldan, course aterials for 6.777J / 2.372J Design and Fabrication of Microelectroechanical Devices, Spring MIT JV: 6.777J/2.372J Spring 2007, Lecture > Can get luped resistor using the fluidic convention > Note STRONG dependence on h > This relation is ore coplicated when the aspect ratio is not very high ΔP = effort = KL 12ηL ΔP = Q 3 Wh 12ηL RPois = 3 Wh

33 Developent Length Cite as: Joel Voldan, course aterials for 6.777J / 2.372J Design and Fabrication of Microelectroechanical Devices, Spring MIT JV: 6.777J/2.372J Spring 2007, Lecture > It takes a certain characteristic length, called the developent length, to establish the Poiseuille velocity profile > This developent length corresponds to a developent tie for viscous stresses to diffuse fro wall > Developent length is proportional to the characteristic length scale and to the Reynolds nuber, both of which tend to be sall in icrofluidic devices 2 L tie L * Re η U L ( tie) U Re L D

34 A note on vorticity Cite as: Joel Voldan, course aterials for 6.777J / 2.372J Design and Fabrication of Microelectroechanical Devices, Spring MIT JV: 6.777J/2.372J Spring 2007, Lecture > A coon stateent is to say that lainar flow has no vorticity > What is eant is that lainar flow has no turbulence > Vorticity and turbulence are different > Can the pinwheel spin? Then there is vorticity > Deonstrate for Poiseuille flow U x 1 = 2η [ y( h y) ]K ω = U U x U ω = ny n z z y K ω = n z ( h 2y) 2η x