1 Fluidic Dynamics Dr. Thara Srinivasan Lecture 17. EE C245. Picture credit: A. Stroock et al., Microfluidic Picture credit:mixing Sandiaon a chip Lab National Lecture Outline Reading From S. Senturia, Microsystem Design, Chapter 13, Fluids, . 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 N/m2 [=] (N s/m2) (m/s)(1/m).. Centipoise = dyne s/cm2 U.
2 No-slip at boundaries w h w Ux air = 10-5 N s/m2. water = 10-4. lager = 10-3. EE C245. honey = 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 = m T. MW . EE C245. 4. U. Srinivasan . 2. Surface Tension Droplet on a surface Capillary wetting 2r . P. 2 cos . h=. EE C245. gr 5. U. Srinivasan . 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.
3 EE C245. 6. U. Srinivasan . 3. 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. Rate of mass efflux EE C245. 7. U. Srinivasan . Conservation of Mass m V dV + S m U n dS = 0. t dS. EE C245. 8. U. Srinivasan . 4. Operators Gradient and divergence U x U y U z U = i+ j+ k x y z . = e x + e y + ez x y z U x U y U z U = div U = + +. x y z EE C245. 9. U. Srinivasan . Continuity Equation Convert surface integral to volume integral using Divergence Theorem For differential control volume . V + ( U ) dV = 0 + ( U) = 0.
4 EE C245. t t U. Srinivasan . Continuity Equation 10. 5. Continuity Equation Material derivative measures time rate of change of a property for observer moving with fluid D . + (U ) + U = 0 = + ( U ) . t Dt t D . = + U = + Ux + U y + Uz Dt t t x y z For incompressible fluid D D . EE C245. + U = 0 + U = 0. Dt Dt 11. U. Srinivasan . 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. 12. U. Srinivasan . 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 Rate of Sum of forces momentum efflux accumulation of acting on control = from control +.
5 Momentum in volume volume control volume dp d F = = V U dV + S U (U n ) dS. dt dt EE C245. Net momentum Net momentum accumulation rate efflux rate 13. U. Srinivasan . Momentum Conservation Sum of forces acting on fluid F =. pressure and shear gravity force stress forces Momentum conservation, integral form d S ( Pn + )dS + V gdV = UdV + S U(U n ) dS. EE C245. dt V. 14. U. Srinivasan . 7. Navier-Stokes Equation Convert surface integrals to volume integrals dS = 2 U + ( U ) dV.. S V 3 . Pn dS = P dV. S V. U( U n ) dS = U( U ) dV. S V. EE C245. 15. U. Srinivasan . Navier-Stokes Differential Form . P + g + U + ( U ) dV =. 2. V 3.
6 U dV + U( U ) dV.. V t V. DU dV.. V Dt . DU . = P + g + 2 U + ( U ). EE C245. Dt 3. 16. U. Srinivasan . 8. Incompressible Laminar Flow Incompressible fluid U = 0. DU . = P + g + 2 U + ( U ). Dt 3. DU. = P + g + 2 U. Dt EE C245. 17. U. Srinivasan . Cartesian Coordinates DU. = P + g + 2 U. Dt x direction U x U x U x U x . + Ux +U y + Uz =. t x y z . Px 2U x 2U x 2U x . + g x + 2 + 2 + . EE C245. x x y z 2 . 18. U. Srinivasan . 9. Dimensional Analysis DU P 2 U. = +g+. Dt . Each term has dimension L/t2 so ratio of any two gives dimensionless group inertia/viscous = = Reynolds number pressure /inertia = P/ U2 = Euler number flow v/sound v = = Mach number EE C245.
7 In geometrically similar systems, if dimensionless numbers are equal, systems are dynamically similar 19. U. Srinivasan . Two Laminar Flow Cases U. w High h w Low h w P w P. y y EE C245. Ux Ux U Umax 20. U. Srinivasan . 10. Couette Flow Couette flow is steady viscous flow U = 0 U = Uxix between parallel plates, where top plate is moving parallel to bottom plate U = U x ( y )i x No-slip boundary conditions at plates DU x 1 P 2U x 2U x . = + gx + + . Dt x x 2 y 2 . 2U x U x = 0 at y = 0. = 0 , and y 2 U x = U at y = h U y w EE C245. h Ux Ux =. w U. Srinivasan . U 21. 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 w = =.
8 Y y =h U. w w A A. h w RCouette = =. U h PCouette = RU 2. 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 DU x 1 P 2U x 2U x . = + gx + + . Dt x x 2. y 2.. P P 2U x P. = = , U x = 0 at y = 0, h x L y 2. L . Ux =. y w EE C245. Highh Low P w P Ux U max =. U. Srinivasan . Umax 23. Poiseuille Flow Volumetric flow rate Q ~ [h3]. Q = W 0 U x dy =. h Shear stress on plates, $, is dissipative Force balance Net force on fluid and plates is zero U x Ph since they are not accelerating w = =. Fluid pressure force PWh (+x) y y =h 2L.
9 Balanced by shear force at the walls of 2$WL (-x). Wall exerts shear force (-x), so fluid Q. must exert equal and opposite force U=. on walls, provided by external force Wh h P 2. 2. U= = 3 U max EE C245. w w Ux 12 L. W 24. U. Srinivasan . 12. Poiseuille Flow Flow in channels of circular cross section Ux =. (r o 2. r 2 ) P Q=. ro 4 P. 4 L 32 L. Flow in channels of arbitrary cross section 4 Area L. Dh = P = f D (12 U 2 ) f D Re D = dimensionless Perimeter Dh constant Lumped element model for Poiseuille flow P 12 L. EE C245. R pois = =. Q Wh 3. 25. U. Srinivasan . 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 Development length Distance before flow assume steady-state profile EE C245.
10 26. U. Srinivasan . 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 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 ).