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

Outline of the next 3 weeks: Today: Microfluidics 1: Laminar flow, flow profiles, basic components. Next week, 20.1: Microfluidics 2: Scaling, Surface tension, contact angle, capillary flow. On the spot Exercise 1: Dimensions and orders of magnitude -On the spot exercise is done at class, max 2 points for being present and participating -You will receive home exercise 1 with one week time to complete. The week after, 27.1: Surfaces: Adsorption, Surface charging, Surface modification. Home exercise 1: Microfluidics -Home exercise is done home and returned prior to the lectures, max 8 points

Definition of a fluid Fluid: a substance that shows continuous shear deformation in response to an applied shear force. Fluids are either gases or liquids (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. However, gas phase microfluidics is still important. Topics related to flow mechanics, viscosity and diffusion are applicable for both liquids and gases. Topics related to surface tension effects only apply for liquids.

Navier stokes equations

Shear and viscosity Shear stress = force / area Newtonian fluid: shear rate is directly proportional to shear stress The constant of proportionality is called viscosity, μ For a Newtonian fluid in laminar flow, there is a linear relation: τ=μ δu/δy (contrast to a spring and Hookes law F=-kx)

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. 0 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

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

Reynolds number: Laminar of turbulent? Reynolds number: inertial forces / viscous forces 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 10 < Re < 2000 the flow is in between, laminar flow with increasingly turbulent characteristics. Microfluidic systems are almost always laminar. Flows with the same geometry, same Reynolds and same Euler numbers ( Eu = (p upstream – p downstream )/(ρV 2 ) ) are similar.

Hydraulic radius What is the “characteristic length scale” of a system? Hydraulic radius, r H = 2A/p (2 x area / perimeter of the channel cross section) Hydraulic radius has inverse correlation with flow resistance. For Reynolds number, use Hydraulic diameter d H = 2r H Examples: Circle, diameter 16 µm r H = 8 µm 10µm 40µm Rectangle 10 x 40 µm r H = 8 µm

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) Parabolic velocity profile: P1P1 P2P2 ΔP= pressure difference = P 1 -P 2 L = length of the channel R = the radius of the channel

Hagen-Poiseuille’s Law Length L, radius r, viscosity μ If a pressure difference of ΔP is applied over a cylindrical channel, what is the volumetric flow rate Q (μl/min)? R H = hydraulic resistance, contains all the geometrical parameters 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 )

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: Fluidic circuit: 1. calculate R of each component, 2. calculate R total, 3. insert R total into Hagen-Poiseuille

An example: P P=0 R1, Q1, ΔP1 R2, Q2, ΔP2 R3, Q3, ΔP3 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

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: Scales most strongly with the smallest dimension of the channel

Diffusion, Brownian motion Diffusion is stochastic random drift of molecules based on thermal fluctuations, collisions, etc. 1D: =2Dt3D: =6Dt D = diffusion constant, t = time and 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 molecules in question and the flow rate, diffusion can either be negligible or very significant in microfluidics.

Parallel laminar flows Lava flow:laminar flow in macroscale Two parallel streams do not mix (except by diffusion)

Taylor dispersion What happens to a liquid element under a parabolic flow profile ? Distortion of a plug profile Increased diffusivity

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.

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,

H filter Laminar flow and differences in diffusion constant lead to a possibility for filtering. Brody, J. P., and P. Yager, 1997, Sens. Actuators, A 58,

Valving Hundreds of different microvalves reported... Pneumatic, magnetic, electric, thermal, piezo, phase change, etc... Few examples: pH sensitive hydrogen actuates a flexible membrane one shot temperature operated valve

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) 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

Review Fluids, shear, viscosity 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. 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: