# Microfluidics 1 Basics, Laminar flow, shear and flow profiles

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1 MT Microfluidics and BioMEMS Microfluidics 1 Basics, Laminar flow, shear and flow profiles Ville Jokinen

2 Outline of the next 3 weeks: Today: Microfluidics 1: Laminar flow, flow profiles, basic components. Next week, 18.1: Microfluidics 2: Scaling, Surface tension, contact angle, capillary flow. Exercise 1: Dimensions and orders of magnitude - On the spot exercise is done at class, 4 exercise points are awarded for being present and participating. Exercise is from 14:15 to 15:00 The week after, 25.1: Surfaces: Adsorption, Surface charging, Surface modification. Exercise 2: Microfluidics - On the spot exercise is done at class, 4 exercise points are awarded for being present and participating. Exercise is from 14:15 to 15:00

3 Fluid, liquid and gas Fluid: a substance that shows continuous shear deformation in response to an applied shear force. Fluids are either liquids or gases (or plasmas). Liquids: viscosity, surface tension, miscible or immiscible in other liquids, conserves volume Gases: viscosity, NO surface tension, always mixes with other gasses, expands to fill container Microfluidics on this course will be primarily about liquids. Flow mechanics, viscosity and diffusion are equally applicable to liquids and gases. Surface tension effects only apply for liquids. 04OL_LecNotes_Ch100/02_EnergyStateMatter/203_StMatter/203_StMatterIMF.htm

4 Shear and viscosity Defining quality of a fluid: continuous flow in response to an applied force. Contrast to a solid: fixed deformation as a response to an applied force (Hookes Law) Shear stress τ = force / area Newtonian fluid: shear rate is directly proportional to shear stress F/A = τ = μ δu/δy The constant of proportionality is called viscosity, μ High viscosity = thick liquid that flows slowly (like honey) Low viscosity = thin liquid that flows easily (like water)

5 Viscosity Shear viscosity, or dynamic viscosity μ, unit is Pa*s Kinematic viscosity ν (= μ/ρ), unit is m 2 /s Non-Newtonian fluids common: shear thinning and shear thickening. Water and air are close to Newtonian. Ketchup is shear thinning. Mixture of potato starch and water is shear thickening. Water viscosity

6 Dimensionless numbers A dimensionless number compares the relative magnitudes of two different phenomena to find out which one is dominant at a given size scale. In microfluidics, commonly different forces are dominant compared to a macroscopic systems: Weak: inertial forces, gravity and other body forces Strong: surface forces, viscous forces Dimensionless numbers are approximations that do not take into account the fine details of the physical system. Nevertheless they are very useful if they show that one or the other force is orders of magnitude larger than the other. Reynolds number: inertial forces vs viscous forces Bond number: body forces vs surface forces (e.g. gravity vs suface tension) Capillary number: viscous forces vs surface forces (Bond and Capillary numbers will be discussed next week)

7 Reynolds number: Laminar of turbulent? Reynolds number: inertial force / viscous force Inertial force: F = ρv 2 A = ρv 2 L 2 Viscous force: F = μ δu/δy A = μv/l * L 2 Re = ρvl µ ρ = density v = velocity L = characteristic length scale μ = dynamic viscosity If Re < 10, the flow is very laminar, If Re > 2000, the flow is turbulent Microfluidics is almost always laminar.

8 Laminar and turbulent Flow direction Flow direction Re = ρvl µ Micro: small L (and v) Macro: large L (and v)

9 Hydraulic radius and diameter What is the characteristic length scale of a system? Hydraulic radius, r H = 2A/p (2 x area / perimeter of the channel cross section) For Reynolds number, use Hydraulic diameter d H = 2r H Examples: 40µm 10µm Rectangle 10 x 40 µm r H = 8 µm, d H = 16 µm Circle, diameter 16 µm r H = 8 µm, d H = 16 µm

10 Pressure driven laminar flow: Poiseuille flow Assumptions: Newtonian and non compressible fluid, laminar flow, cylindrical channel Velocity profile inside a microchannel is parabolic. Ideally, the velocity at channel walls is 0. (called no-slip boundary condition) P 1 P 2 Parabolic velocity profile: u = P L ( R 4µ r ) ΔP= pressure difference = P 1 -P 2 L = length of the channel R = the radius of the channel

11 Hagen-Poiseuille s Law If a pressure difference of ΔP is applied over a cylindrical channel, what is the volumetric flow rate Q (μl/min)? Length L, radius r, viscosity μ 8µ L P = Q 4 πr = R H Q R H = hydraulic resistance Notice the L 1 and inverse R 4 dependency. Why inverse R 4 dependency? One R 2 comes from the area of the cross section a The other R 2 comes from the average velocity of parabolic flow profile. (Q=A*V ave ) The smaller the channel, the more pressure it takes to drive liquid through it. If using a pressure pump: use Hagen-Poiseuille to calculate the flow rate If using a syringe pump: use Hagen-Poiseuille to calculate the back pressure

12 Fluidic circuits Analogous to electric circuits, Hydraulic resistances in series and parallel sum exactly as electrical resistors Analogies to Kirchhoffs laws also exist: volume is conserved and pressure drop over a loop is 0. Series: Parallel: R total 1 R total = = R 1 1 R 1 + R R R N 1 R N Fluidic circuit: 1. calculate R of each component, 2. calculate R total, 3. insert R total into Hagen-Poiseuille

13 An example: Inlet, Pressure P The red and blue channels are otherwise the same except the red channel is twice as long. A syringe pump is used to Pump 3 µl / min of water. How much of this comes from outlet 2 and how much from outlet 3? R1, Q1, ΔP1 1. Conservation of volume: Q 1 =Q 2 + Q 3 2.Pressure drop across both paths is the same: P= ΔP 1 + ΔP 2 = ΔP 1 +ΔP 3 Ratio of flow in channels 2 and 3: ΔP 2 =ΔP 3 -> apply Hagen Poiseuille s law R 2 Q 2 = R 3 Q 3 Q 2 /Q 3 = R 3 /R 2 R2, Q2, ΔP2 Outlet 2, P=0 R3, Q3, ΔP3 Outlet 3, P=0

14 Example, continued Inlet, Pressure P The red and blue channels are otherwise the same except the red channel is twice as long. A syringe pump is used to Pump 3 µl / min of water. How much of this comes from outlet 2 and how much from outlet 3? R1, Q1 = 3µl, ΔP1 R 2 = 2*R 3 (Flow resistance of a cylinder) Q 2 /Q 3 = R 3 /R 2, so Q 2 =0.5 *Q 3 Q 1 =Q 2 + Q 3 = 1.5 * Q 3 = 3 µl /min Q 2 = 1 µl /min Q 3 = 2 µl /min R2, Q2, ΔP2 R3, Q3, ΔP3 Outlet 3, P=0 Outlet 2, P=0

15 Non-circular cross section Hagen-Poiseuille s Law is correct only for channels with circular cross section In microfluidics, circular cross sections are not typical. How to calculate hydraulic resistance? Very rough approximation, use hydraulic radius: Literature is full of all kinds of more accurate approximations for rectangular channels. For example, when channel height < width: R H 8µ L πr 4 H R H wh 3 12µ L ( h / w) Scales most strongly with the smallest dimension of the channel

16 Diffusion, Brownian motion Diffusion is stochastic random drift of molecules based on thermal fluctuations, collisions, etc. 1D: <x 2 >=2Dt 3D: <x 2 >=6Dt D = diffusion constant, t = time and <x 2 > is the expectation for the square of the distance traveled. D depends on the size o the molecule, the temperature and the viscosity. Typical diffusion constants (in water, room temperature): Small molecules: >500 µm 2 /s Small proteins: 100 µm 2 /s Large proteins 10 µm 2 /s Large DNA molecules <1 µm 2 /s Depending on the microfluidic dimensions, the diffusion constant and the flow rate, diffusion can either be negligible or very significant in microfluidics.

17 Mixing by diffusion Microfluidic parallel laminar flows (2 dyes in water) Danube (left) joins another river Low reynolds number: mixing only by diffusion, no turbulent mixing Diffusion distances compared to flow speed / resident time

18 Microfluidic mixing Sometimes pure diffusive mixing Often diffusive mixing is not enough and assistance from special geometries is used. Diffusive mixing Enhanced mixing by herringbone structures Ismagilov et al., Appl. Phys. Lett. 76, Stroock et al., 2002, Science 295,

19 H filter Laminar flow and differences in diffusion constant lead to a possibility for filtering. Red circles = bigger particles (smaller diffusion constant) Blue liquid = contains smaller particles (higher diffusion constant) Brody, J. P., and P. Yager, 1997, Sens. Actuators, A 58,

20 Taylor dispersion What happens to a liquid element under pressure driven flow with also diffusion taken into account? Distortion of a plug profile Increased diffusivity

21 Microfluidic components A very quick primer on basic components used in microfluidics: 1. Channels 2. Reservoirs 3. Pressure sources (often off chip) 4. Mixers 5. Filters 6. Valves There will be a separate lecture on microfluidic components.

22 Flow actuation (pumping) The most common flow types microfluidics: 1. Pressure driven flow (this lecture) 2. Capillary flow (next lecture) 3. Electro-osmotic flow (later lectures) Pressure drive flow: - With set flow rate (e.g. syringe pump) or - With set pressure Other alternatives: centrifugal force, electrowetting, etc.. Methods differ in flow profile, possible flow rates, required equipment and interconnections Different methods suited for different applications, sometimes combinations are used Electro-osmotic flow

23 Navier stokes equations -The governing equations for fluid flow. In vector form: ρ δuu δtt = + µ 2 uu + g ρ = density, μ = viscosity, u = velocity (vector), P=pressure, g = gravity - Momentum conservaton (Newtons second law, ma = F) - Fluid element affected by 3 forces: pressure gradient, viscous forces, and gravity. - Also, mass conservation: δρ + ρuu = 0 δtt

24 Shear driven laminar flow: Couette Flow Simplest case of laminar Newtonian flow: Couette flow Liquid (depth h) between two solid plates, one of which is moving with velocity u 0. ρ δuu δtt = + µ 2 uu + g Velocity independent of x and z due to symmetry, steady state, negligible gravity, no pressure gradient -> 0 μδ 2 u/δy 2 = 0 integrate to get: u(y)=ay + B apply boundary conditions u(0)=0 (no slip) and u(h)=u 0 for the final solution: u(y)=(u 0 /h)* y The viscosity does not appear in the final solution since we specified the velocity u 0 and not the shear stress. Viscosity does affect how hard we need to pull the top layer of liquid to move it at velocity u 0. τ=μ δu/δy = (μ*u 0 )/h

25 Fluids, shear, viscosity Review Reynolds number, laminar flow, flow profiles, consequences for microfluidics Basic physics of pressure driven laminar flow: flow resistance calculation, Hagen-Poiseuille s Law, fluidic circuits. Diffusion in microfluidics Reading material For Fluidics 1 and 2 lectures, the reading material is: Squires and Quake, Microfluidics: Fluid physics at the nanoliter scale, Rev. Mod. Phys. 77, Pages Link:

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