V03 Fluid Mechanics I WS 23.pdf

V03 V03 Fluid Mechanics I Description of flow behavior in small channels Lecture „Microfluidic Systems - Bio-MEMS“ – V03 Fluid Mechanics I Prof. Dr.-Ing. Uwe Schnakenberg | Institute of Materials in Electrical Engineering 1 | WS 23 1 Contents V03 Contents 3.1 Dynamic Viscosity η 3.2 Navier-Stokes Eq...

V03 V03 Fluid Mechanics I Description of flow behavior in small channels Lecture „Microfluidic Systems - Bio-MEMS“ – V03 Fluid Mechanics I Prof. Dr.-Ing. Uwe Schnakenberg | Institute of Materials in Electrical Engineering 1 | WS 23 1 Contents V03 Contents 3.1 Dynamic Viscosity η 3.2 Navier-Stokes Equation 3.2.1 Euler Equation 3.2.2 Bernoulli Equation 3.3 Reynolds Number Re 3.4 Hydraulic Diameter Dh 3.5 Simplification of Navier-Stokes Equation for Microfluidics Stokes Equation Lecture „Microfluidic Systems - Bio-MEMS“ – V03 Fluid Mechanics I Prof. Dr.-Ing. Uwe Schnakenberg | Institute of Materials in Electrical Engineering 1 | WS 23 2 Learning Targets V03 Learning Targets  Interpretation of Navier-Stokes, Stokes and Bernoulli equation  Navier-Stokes equation can be simplified to Stokes equation for small Reynolds numbers Re and steady-state flow  Small Re number (Re < 1) in combination with steady-state flow means  Friction dominates over inertial forces  Laminar flow  Pressure driven flow shows a parabolic low profile (Poiseuille/Stokes flow) in microchannels Lecture „Microfluidic Systems - Bio-MEMS“ – V03 Fluid Mechanics I Prof. Dr.-Ing. Uwe Schnakenberg | Institute of Materials in Electrical Engineering 1 | WS 23 3 V03 3.1 Dynamic Viscosity Lecture „Microfluidic Systems - Bio-MEMS“ – V03 Fluid Mechanics I Prof. Dr.-Ing. Uwe Schnakenberg | Institute of Materials in Electrical Engineering 1 | WS 23 4 3.1 Dynamic Viscosity V03 5 Viscosity Viscosity is the measure of the resistance of a fluid to deform Dynamic (Absolute) Viscosity η  Is the measure of the resistance of a fluid to deform Kinematic Viscosity ν  Ratio of dynamic viscosity to density under shear forces  Information force  Information how fast the fluid is needed to make the fluid flow moving when a certain force is at a certain rate applied on the Lecture „Microfluidic Systems - Bio-MEMS“ – V03 Fluid Mechanics I Prof. Dr.-Ing. Uwe Schnakenberg | Institute of Materials in Electrical Engineering 1 | WS 23 3.1 Dynamic Viscosity V03 6 Dynamic Viscosity η The dynamic (absolute) viscosity of a fluid expresses its resistance  to shear forces, where adjacent layers move parallel to each other with different speeds  to movement of one layer of a fluid over another https://en.wikipedia.org/wiki/Viscosity, April 26, 2017 Lecture „Microfluidic Systems - Bio-MEMS“ – V03 Fluid Mechanics I Prof. Dr.-Ing. Uwe Schnakenberg | Institute of Materials in Electrical Engineering 1 | WS 23 3.1 Dynamic Viscosity V03 7  Viscous flow between two parallel plates moving relatively to another  Each fluid layer moves faster than the one just below it  Viscous friction between the layers give rise to a force resisting their relative motion https://en.wikipedia.org/wiki/Viscosity, 2017-04-26 Lecture „Microfluidic Systems - Bio-MEMS“ – V03 Fluid Mechanics I Prof. Dr.-Ing. Uwe Schnakenberg | Institute of Materials in Electrical Engineering 1 | WS 23 Maurice M.A. Couette (1858 – 1943) chemistryworld.com Couette Flow 3.1 Dynamic Viscosity V03 8 Couette Flow  An external shear (friction) force Fr is required to keep the top plate moving at constant speed u or v* Fr    A  dv dy * Velocity is described by “u” or “v” - depends on the reference https://en.wikipedia.org/wiki/Viscosity, 2017-04-26 Lecture „Microfluidic Systems - Bio-MEMS“ – V03 Fluid Mechanics I Prof. Dr.-Ing. Uwe Schnakenberg | Institute of Materials in Electrical Engineering 1 | WS 23 η … Dynamic viscosity, [η] = Pa·s A … Area v,u.. Velocity 3.1 Dynamic Viscosity Newtonian Law of Viscosity V03 Fr    A  dv dy Shear rate Shear velocity Shear/Friction force Fr related to contact area Shear Stress τ  Fr dv   A dy Lecture „Microfluidic Systems - Bio-MEMS“ – V03 Fluid Mechanics I Prof. Dr.-Ing. Uwe Schnakenberg | Institute of Materials in Electrical Engineering 1 | WS 23 9 3.1 Dynamic Viscosity V03 10 Newtonian und Non-Newtonian Fluids Bingham-plastic   Cellulose, grease, soap, paint Newton  Water, gasoline, motor oil www.wikipedia.com Pseudoplastic Shear stress τ  Thermoplastics, loam, tar Dilatant  Sand Shear rate dv/dy Lecture „Microfluidic Systems - Bio-MEMS“ – V03 Fluid Mechanics I Prof. Dr.-Ing. Uwe Schnakenberg | Institute of Materials in Electrical Engineering 1 | WS 23 Fr dv   A dy V03 11 3.2 Navier-Stokes Equation Lecture „Microfluidic Systems - Bio-MEMS“ – V03 Fluid Mechanics I Prof. Dr.-Ing. Uwe Schnakenberg | Institute of Materials in Electrical Engineering 1 | WS 23 3.2 Navier-Stokes Equation V03 12 Navier-Stokes Equation (NSE) Is THE Pivotal Equation of Fluid Mechanics  NSE is a partial differential equation which describes the motion of viscous fluids  NSE expresses the momentum balance (2nd Newtonian law) for Newtonian fluids  Used to model, e.g.,  weather  ocean currents  pollution  water flow in a pipe  blood flow  Flow around an airfloil  … Lecture „Microfluidic Systems - Bio-MEMS“ – V03 Fluid Mechanics I Prof. Dr.-Ing. Uwe Schnakenberg | Institute of Materials in Electrical Engineering 1 | WS 23 3.2 Navier-Stokes Equation V03 13 Navier-Stokes Equation was Independently Developed by Claude Navier (1785 – 1836) Siméon Denis Poisson (1781 – 1840) Adhémar Jean Claude Barré de Saint-Venant (1797 – 1886) Lecture „Microfluidic Systems - Bio-MEMS“ – V03 Fluid Mechanics I Prof. Dr.-Ing. Uwe Schnakenberg | Institute of Materials in Electrical Engineering 1 | WS 23 de.wikipedia.org (1845) de.wikipedia.org (1843) de.wikipedia.org (1831) http://quickshott.com/ (1827) Sir George Gabriel Stokes (1819 – 1903) 3.2 Navier-Stokes Equation V03 14 Basic Mathematical Knowledge (Especially for Students of Biotechnology) Scalar quantity p , T ,  , m,... independent on direction it is a single number Vector quantity F , v, a,... dependent on direction consists of several numbers, e.g. Fx, Fy, Fz Lecture „Microfluidic Systems - Bio-MEMS“ – V03 Fluid Mechanics I Prof. Dr.-Ing. Uwe Schnakenberg | Institute of Materials in Electrical Engineering 1 | WS 23 3.2 Navier-Stokes Equation V03 15 Scalar multiplied by a vector → Vector  bx   a  bx      a  b  a   by    a  b y   b   a b  z   z  Vector multiplied with a vector → Scalar  ax   bx      a  b   a y    by   axbx  a y by  az bz a  b   z  z Lecture „Microfluidic Systems - Bio-MEMS“ – V03 Fluid Mechanics I Prof. Dr.-Ing. Uwe Schnakenberg | Institute of Materials in Electrical Engineering 1 | WS 23 3.2 Navier-Stokes Equation Nabla operator V03 16    ( , , ) x y z  p     x  p p p  p  p  ( , , )    Nabla operator on a scalar → Gradient x y z y    p     z     v  Nabla operator on a vector → Divergence    x  x         vx v y vz div v    v     v y      y x y z            vz   z  Lecture „Microfluidic Systems - Bio-MEMS“ – V03 Fluid Mechanics I Prof. Dr.-Ing. Uwe Schnakenberg | Institute of Materials in Electrical Engineering 1 | WS 23 3.2 Navier-Stokes Equation Laplacian operator V03 17 2 2 2    Δ   2     2  2  2 x y z Laplacian operator on a scalar 2 2 2  p  p  p Δp   2 p  2  2  2 x y z   2 vx  2 vx  2 vx   2  2  2  y z   x   2vy  2vy  2vy  Laplacian operator on a vector Δv   2  2  2  y z   x  2  2 2  v  v  v z   2z  2z  Lecture „Microfluidic Systems - Bio-MEMS“ – V03 Fluid Mechanics I 2  23x   y  z Prof. Dr.-Ing. Uwe Schnakenberg | Institute of Materials in Electrical Engineering 1 | WS   3.2 Navier-Stokes Equation V03 18 How to Describe Flow in a Channel? v Flow Assumption 1 Incompressible fluid with constant density ρ Assumption 2 Conservation of mass (inlet flow = outlet flow) means no sources and sinks   const.       v   0 t When density (mass) is changed in a control volume then the mass must dissipate or be added to the volume v  0 Continuity equation Lecture „Microfluidic Systems - Bio-MEMS“ – V03 Fluid Mechanics I Prof. Dr.-Ing. Uwe Schnakenberg | Institute of Materials in Electrical Engineering 1 | WS 23 3.2 Navier-Stokes Equation V03 19 How to Describe Flow in a Channel? v Flow Assumption 3 Newtonian fluid Linear dependency between shear stress and shear rate Fr dv     A dy Lecture „Microfluidic Systems - Bio-MEMS“ – V03 Fluid Mechanics I Prof. Dr.-Ing. Uwe Schnakenberg | Institute of Materials in Electrical Engineering 1 | WS 23 η… A… v …. Fr... Dynamic viscosity Area Velocity Friction/Shear force 3.2 Navier-Stokes Equation V03 20 v Flow Conservation of momentum defined by 2nd Newtonian law 2nd Newtonian law related to a volume element m … Mass F … Force a … Acceleration V … Volume dv ma  m   Fj dt j Fj ma dv   a       fj V dt j V j v … Velocity ρ … Density Lecture „Microfluidic Systems - Bio-MEMS“ – V03 Fluid Mechanics I Prof. Dr.-Ing. Uwe Schnakenberg | Institute of Materials in Electrical Engineering 1 | WS 23 t … Time f … Volume force 3.2 Navier-Stokes Equation V03 21 In Fluid Mechanics: The Coordination System is Changed  Typically One uses a fixed coordination system origin and describes the behavior of, e.g., a parcel or a volume element with respect to this fixed point  In Fluid Mechanics We follow the parcel (volume element) along its pathline and look how a quantity P changes* *P … Macroscopic quantity (e.g. energy, momentum, temperature, velocity, …) Lecture „Microfluidic Systems - Bio-MEMS“ – V03 Fluid Mechanics I Prof. Dr.-Ing. Uwe Schnakenberg | Institute of Materials in Electrical Engineering 1 | WS 23 3.2 Navier-Stokes Equation V03 22 The rate of change of a quantity P by a volume element that is moving along with → the flow with velocity v is mathematically D P   v  P Dt t described by → We follow a volume element with velocity v in a flow field and look how the parameter P is changed locally on its way. In addition, the intrinsic variation of the parameter have to be taken into account https://www.youtube.com/watch?v=mdN8OOkx2ko Lecture „Microfluidic Systems - Bio-MEMS“ – V03 Fluid Mechanics I Prof. Dr.-Ing. Uwe Schnakenberg | Institute of Materials in Electrical Engineering 1 | WS 23 Substantial Derivative 3.2 Navier-Stokes Equation V03 23 D P   v  P Dt t Substantial Derivative or Material Derivate* In Fluid Mechanics  We are looking to the change of parameter v along the pathlines of the fluid element in the flow field Here: Pv Dv  v   v  v dt t * The substantial derivate arrives at by applying the chain rule to quantity P(x(t),t)). It is dependent on position and time (see Annex 1 and 2) Lecture „Microfluidic Systems - Bio-MEMS“ – V03 Fluid Mechanics I Prof. Dr.-Ing. Uwe Schnakenberg | Institute of Materials in Electrical Engineering 1 | WS 23 3.2 Navier-Stokes Equation V03 24 Example: Microfluidic Channel Dv  v   v  v dt t  The velocity in the channel is  Floating along with the flow on fluid determined by the force (pressure) element with velocity v , the velocity acting on the fluid gradient is added  Steady-state:  Pulsatile: v 0 t v  0 (e.g. aortic vessel) t  For laminar flow: v  0  For inertial forces acting: v  0 https://www.youtube.com/watch?v=mdN8OOkx2ko Lecture „Microfluidic Systems - Bio-MEMS“ – V03 Fluid Mechanics I Prof. Dr.-Ing. Uwe Schnakenberg | Institute of Materials in Electrical Engineering 1 | WS 23 3.2 Navier-Stokes Equation 2nd Newtonian law V03 25 dv    fj dt j Changing of the coordination system (We are sitting on the volume element with velocity v and looking to the change of parameter v) → Apply the chain rule (see Annex 1 and 2) 2nd Newtonian law d v( x, y, z , t ) Dv v   v  v  dt Dt t   v v    Lecture „Microfluidic Systems - Bio-MEMS“ – V03 Fluid Mechanics I Prof. Dr.-Ing. Uwe Schnakenberg | Institute of Materials in Electrical Engineering 1 | WS 23  v   fj t j https://www.spandidos-publications.com/etm/9/1/154?text=fulltext 3.2 Navier-Stokes Equation f j  f pressure f V03 26 friction  f volume Applied Forces j f Pump pressure p   p p2  p1 L  Pressure gradient causes fluid movement  Movement can be initiated by, e.g.: Laplace pressure p  r σ … Surface tension r … Curvature radius of meniscus L … Length of capillary / channel Lecture „Microfluidic Systems - Bio-MEMS“ – V03 Fluid Mechanics I Prof. Dr.-Ing. Uwe Schnakenberg | Institute of Materials in Electrical Engineering 1 | WS 23 3.2 Navier-Stokes Equation f j  f V03 27 pressure f friction  f volume j ffriction is Described by Newtonian Law of Viscosity One dimensional Three dimensional, related to a volume element * F friction f friction dv    A dy  * See Annex 3 Lecture „Microfluidic Systems - Bio-MEMS“ – V03 Fluid Mechanics I Prof. Dr.-Ing. Uwe Schnakenberg | Institute of Materials in Electrical Engineering 1 | WS 23 F friction    2 v V 3.2 Navier-Stokes Equation V03 28 f j  f pressure f friction  f volume j External volume forces acting on the fluid Examples of Volume Forces  Centrifugal force f volume, centrif.     2  r  Gravitation f volume, grav.    g  Electrical forces* f volume, electrical x   q x  E x  ω … Rotation velocity r … Radius ρ … Density g … Acceleration of gravity   * Will be discussed in detail in V09 - V10 Electrokinetics I - II Lecture „Microfluidic Systems - Bio-MEMS“ – V03 Fluid Mechanics I Prof. Dr.-Ing. Uwe Schnakenberg | Institute of Materials in Electrical Engineering 1 | WS 23 E … Electrical field strength ρq... Charge density 3.2 Navier-Stokes Equation V03 29 Example Gravitation Hydrostatic Pressure of Water in a Microfluidic Channel (flat position)  ρH2O = 1000 kg/m3  g = 9.81 m/s2  h = 100 µm (channel height) p    g h Hydrostatic pressure p  0.981 Pa  9.81106 bar  The hydrostatic pressure resulting from the fluid height in a channel is negligible  Hydrostatic pressure can act on the fluid in the channel from inlets and outlets (e.g., from vertically connected tubes) Lecture „Microfluidic Systems - Bio-MEMS“ – V03 Fluid Mechanics I Prof. Dr.-Ing. Uwe Schnakenberg | Institute of Materials in Electrical Engineering 1 | WS 23 3.2 Navier-Stokes Equation    v v    v  f t V03 30 pressure f friction  f volume Final Navier-Stokes Equation for Incompressible Newtonian Fluids (non-linear partial differential equation which specify the velocity distribution of a fluid in dependency of position and time)    v v    v  p    2 v  f volume t Conservation of Momentum Applied to a Fluid Element Lecture „Microfluidic Systems - Bio-MEMS“ – V03 Fluid Mechanics I Prof. Dr.-Ing. Uwe Schnakenberg | Institute of Materials in Electrical Engineering 1 | WS 23 3.2 Navier-Stokes Equation V03 31 Example: Calculation of Couette Flow Profile uH Couette flow  Parallel plates are located at y = ± h y=h y  Top plate moves with velocity uH  Bottom plate moves with velocity uL Couette flow has  No acceleration (means constant velocity) x y=-h uL v  (u, v, w)  No net pressure force  No net convective transport of momentum (means parallel pathlines) Lecture „Microfluidic Systems - Bio-MEMS“ – V03 Fluid Mechanics I Prof. Dr.-Ing. Uwe Schnakenberg | Institute of Materials in Electrical Engineering 1 | WS 23 3.2 Navier-Stokes Equation V03 32 Navier-Stokes equation without body force    v    v v  p    2 v t 1. Assumption  Fluid motion only in x-direction  u p  2u  2u  2u   u    u    2  2  2 t x x x y z 2. Assumption  Steady flow (no acceleration)  u p  2u  2u  2u   u    u    2  2  2 t x x x y z Lecture „Microfluidic Systems - Bio-MEMS“ – V03 Fluid Mechanics I Prof. Dr.-Ing. Uwe Schnakenberg | Institute of Materials in Electrical Engineering 1 | WS 23 uH y=h y x y=-h uL v  (u, v, w) 3.2 Navier-Stokes Equation V03 33 uH 3. Assumption  Velocity profile is independent of x and z  u p u u u   u    u    2  2  2 t x x x y z 2 2 y=h y x 2 4. Assumption  No pressure driven flow  u p  2u  2u  2u   u    u    2  2  2 t x x x y z  2u 0  2 y Lecture „Microfluidic Systems - Bio-MEMS“ – V03 Fluid Mechanics I Prof. Dr.-Ing. Uwe Schnakenberg | Institute of Materials in Electrical Engineering 1 | WS 23 y=-h uL 3.2 Navier-Stokes Equation V03 34  2u 0  2 y uH y=h y x Integration u  C1  y  C2 Boundary condition at top plate u ( y  h)  u H Boundary condition at bottom plate y=-h uL u ( y   h)  u L uH  uL uH  uL y u   2 2 h The velocity profile of a Couette flow is independent of the viscosity, but the force required to move the plates is Lecture „Microfluidic Systems - Bio-MEMS“ – V03 Fluid Mechanics I Prof. Dr.-Ing. Uwe Schnakenberg | Institute of Materials in Electrical Engineering 1 | WS 23 3.2.1 Euler Equation V03 35 3.2.1 Euler Equation* NSE for incompressible fluids with negligible friction Negligible friction (Inviscid flow)    v v    v  p    2v  f volume t No body forces    v v    v  p    2v  f volume t Euler equation   v v     v  p t de.wikipedia.org/ (and no body forces) Leonhard Euler (1707 – 1783) * In 1757, Euler was the first who published partial differential equations for describing fluid flows Lecture „Microfluidic Systems - Bio-MEMS“ – V03 Fluid Mechanics I Prof. Dr.-Ing. Uwe Schnakenberg | Institute of Materials in Electrical Engineering 1 | WS 23 3.2.2 Bernoulli Equation V03 36 3.2.2 Bernoulli Equation (1738) Euler equation for steady-state and laminar flow   v v     v  p t 1 2 v v  v  v  (  v ) 2  1 2     v  v  (  v )     v  p t 2  Lecture „Microfluidic Systems - Bio-MEMS“ – V03 Fluid Mechanics I Prof. Dr.-Ing. Uwe Schnakenberg | Institute of Materials in Electrical Engineering 1 | WS 23 Euler equation NSE for negligible friction Weber transformation Euler equation 3.2.2 Bernoulli Equation V03 37 1 2      v 2  v  (  v )      v  p t No friction means no turbulent flow* Irrotational flow means laminar flow  1 2     v  v  (  v )     v  p t 2  Steady-state flow  1 2     v  v  (  v )     v  p t 2   v2 p      0  2  * If the flow is turbulent, then the flow is viscous anyway Lecture „Microfluidic Systems - Bio-MEMS“ – V03 Fluid Mechanics I Prof. Dr.-Ing. Uwe Schnakenberg | Institute of Materials in Electrical Engineering 1 | WS 23 3.2.2 Bernoulli Equation  v2 p      0  2  Euler equation for steady-state and laminar flow Venturi Effect (by experiment) ↑ Flow velocity ↓ Pressure ↓ Flow velocity ↑ Pressure Lecture „Microfluidic Systems - Bio-MEMS“ – V03 Fluid Mechanics I Prof. Dr.-Ing. Uwe Schnakenberg | Institute of Materials in Electrical Engineering 1 | WS 23 Daniel Bernoulli (1700 – 1782) de.wikipedia.org/ Law of Bernoulli v2 p   const. 2  https://en.wikiquote.org/wiki/ V03 38 Giovanni Battista Venturi (1746 – 1822) 3.2.2 Bernoulli Equation (by experiment) v2 p   const. 2  Law of Bernoulli (by theory) www.lp.uni-goettingen.de Venturi Effect V03 39 Low pressure High flow velocity High pressure Low flow velocity High pressure Low flow velocity → → → → → Water flow → → → → → Lecture „Microfluidic Systems - Bio-MEMS“ – V03 Fluid Mechanics I Prof. Dr.-Ing. Uwe Schnakenberg | Institute of Materials in Electrical Engineering 1 | WS 23 Low pressure High flow velocity Example 1 Arterial Stenosis Law of Bernoulli High pressure Low flow velocity V03 40 High pressure Low flow velocity www.lp.uni-goettingen.de 3.2.2 Bernoulli Equation v2   const.  2 p Water flow At severe arterial stenosis the pressure Mild stenosis drop over the stenosis is too high that Moderate stenosis not enough blood can flow through Severe stenosis Lecture „Microfluidic Systems - Bio-MEMS“ – V03 Fluid Mechanics I Prof. Dr.-Ing. Uwe Schnakenberg | Institute of Materials in Electrical Engineering 1 | WS 23 3.2.2 Bernoulli Equation V03 41 Example 2: Aerodynamic Lift v2   const.  2 p  The shape of an airfoil is such that the air velocity above is faster than the velocity below the airfoil  There is less pressure above than below the airfoil, resulting in a lift (force) http://www.mpoweruk.com/flight_theory.htm, 2023-08-24 Lecture „Microfluidic Systems - Bio-MEMS“ – V03 Fluid Mechanics I Prof. Dr.-Ing. Uwe Schnakenberg | Institute of Materials in Electrical Engineering 1 | WS 23 en.wikipedia.org V03 42 3.3 Reynolds Number Re Osborne Reynolds (1842 - 1912) Lecture „Microfluidic Systems - Bio-MEMS“ – V03 Fluid Mechanics I Prof. Dr.-Ing. Uwe Schnakenberg | Institute of Materials in Electrical Engineering 1 | WS 23 3.3 Reynolds Number Re V03 43    v v    v  p    2 v  f volume t Friction force Viscous force Inertial force Reynolds number NS equation Re  inertial force friction force The Reynolds number as the ratio of inertial forces to viscous/friction forces is a means of describing whether the flow is dominated by the bulk motion of the fluid (inertial) or the interactions between different fluid elements (viscosity) Lecture „Microfluidic Systems - Bio-MEMS“ – V03 Fluid Mechanics I Prof. Dr.-Ing. Uwe Schnakenberg | Institute of Materials in Electrical Engineering 1 | WS 23 3.3 Reynolds Number Re V03 44 inertial force Re  friction force Re  Re  ρ… v... l… η… t… Density of fluid Velocity of fluid Characteristic length Dynamic viscosity Time mass  acceleration velocity vis cos ity   area dis tan ce   l3  v t v l   l2  v l Re   Re number is the most important dimensionless number in fluidic mechanics Lecture „Microfluidic Systems - Bio-MEMS“ – V03 Fluid Mechanics I Prof. Dr.-Ing. Uwe Schnakenberg | Institute of Materials in Electrical Engineering 1 | WS 23 3.3 Reynolds Number Re V03 45  v L Re   General Rule for Characteristic Numbers When scale models are used, the characteristic/dimensionless numbers must be constant for the scale as well as for the full-size model to obtain the same flow data (see Annex 4) Example: Place a small toy car in a wind tunnel and get exactly the same flow response as it's full-size version by increasing the speed of the flow * In the course, several characteristic/dimensionless numbers will be introduced https://www.physicsforums.com/threads/bernoulli-equation-questions.715701/, 2023-08-23 Lecture „Microfluidic Systems - Bio-MEMS“ – V03 Fluid Mechanics I Prof. Dr.-Ing. Uwe Schnakenberg | Institute of Materials in Electrical Engineering 1 | WS 23 3.3 Reynolds Number Re V03 46 1 Pa  s  Re number in Microfluidics N kg  s  m2 ms  Medium: water @ 20 °C (η = 1 10-3 Pa·s, ρ = 1000 kg/m3)  Typical flow velocities: 1 µm/s - 0.1 cm/s  Characteristic length l = Channel diameter: typical 1-100 µm → !!  v l Re   10 6... 1  inertial force Re  friction force → Friction (viscous) forces predominate in microfluidic channels compared to inertial forces → No inertial forces pushed the fluid element from its pathline → Particles flow side by side in fluid layers, pathlines are in parallel → Laminar flow Lecture „Microfluidic Systems - Bio-MEMS“ – V03 Fluid Mechanics I Prof. Dr.-Ing. Uwe Schnakenberg | Institute of Materials in Electrical Engineering 1 | WS 23 !! 3.3 Reynolds Number Re V03 47 Re  1 Laminar flow  Creep behavior  No convection  Parallel path lines 1  Re  2100 More or less laminar flow  Intermediate range  Viscous or “retarding” forces are dominating to suppress tendencies for the flow to gain so much inertia as to become “chaotic”  Turbulence increases Re  2100 www.ceb.cam.ac.uk/pages/ mass-transport.html      Turbulent flow Curling of velocity stream lines Turbulences Friction is low compared to kinetic energy Mixing between neighbored fluid layers Chaotic flow Lecture „Microfluidic Systems - Bio-MEMS“ – V03 Fluid Mechanics I Prof. Dr.-Ing. Uwe Schnakenberg | Institute of Materials in Electrical Engineering 1 | WS 23 3.3 Reynolds Number Re V03 48 In Microfluidics, Surface Roughness Rmax of Micro Channels Must Be Considered (because of high surface-to-volume ratio) ReµF , krit  500  1000 S.G. Kandlikar et al.: Bull Polish Ac. Techn. 53 (4) 343-349 (2005) Lecture „Microfluidic Systems - Bio-MEMS“ – V03 Fluid Mechanics I Prof. Dr.-Ing. Uwe Schnakenberg | Institute of Materials in Electrical Engineering 1 | WS 23 V03 49 3.4 Hydraulic Diameter Dh Lecture „Microfluidic Systems - Bio-MEMS“ – V03 Fluid Mechanics I Prof. Dr.-Ing. Uwe Schnakenberg | Institute of Materials in Electrical Engineering 1 | WS 23 3.4 Hydraulic Diameter Dh V03 50   v  lc Re   In channel-based fluid mechanics, the characteristic length lc is typically be represented by the hydraulic diameter Dh Hydraulic Diameter Dh ρ… v... lc … η… t… 4  cross-section area D = h wetted perimeter of cross-section Density of fluid Velocity of fluid Characteristic length Dynamic viscosity Time Lecture „Microfluidic Systems - Bio-MEMS“ – V03 Fluid Mechanics I Prof. Dr.-Ing. Uwe Schnakenberg | Institute of Materials in Electrical Engineering 1 | WS 23 3.4 Hydraulic Diameter Dh V03 51 4  cross-section area D = h wetted perimeter of cross-section Hydraulic Diameter Dh Examples Dh  4  d2 4  d 4d Dh  d 4d 2 d Dh  4(   d 2    d 2 4 4   d    d )  d  d Dh  4 wh 2 wh  2(h  w) (h  w) h w Lecture „Microfluidic Systems - Bio-MEMS“ – V03 Fluid Mechanics I Prof. Dr.-Ing. Uwe Schnakenberg | Institute of Materials in Electrical Engineering 1 | WS 23 V03 52 3.5 Simplification of Navier-Stokes Equation in Microfluidics - Stokes Equation Lecture „Microfluidic Systems - Bio-MEMS“ – V03 Fluid Mechanics I Prof. Dr.-Ing. Uwe Schnakenberg | Institute of Materials in Electrical Engineering 1 | WS 23 3.5 Simplification of Navier-Stokes Equation - Stokes Equation    v v    v  p    2 v  f volume t V03 53 NSE Small Re number means  No inertial forces means  No convective transport of momentum If, in addition, for steady-state flow   v v    No volume forces acting  v  p    2 v  f volume t    v v    v  p    2 v  f volume t p    2 v Stokes equation Lecture „Microfluidic Systems - Bio-MEMS“ – V03 Fluid Mechanics I Prof. Dr.-Ing. Uwe Schnakenberg | Institute of Materials in Electrical Engineering 1 | WS 23 3.5 Simplification of Navier-Stokes Equation - Stokes Equation p    v 2 Stokes equation NSE for steady-state flow with Re < 1 Stokes equation is time-reversible compared to Navier-Stokes equation  Flow is symmetric in time → Future and past situations are predictable  While no inertial forces are applied, the fluid element stays always on its pathline !!! Microfluidic is deterministic !!! Lecture „Microfluidic Systems - Bio-MEMS“ – V03 Fluid Mechanics I Prof. Dr.-Ing. Uwe Schnakenberg | Institute of Materials in Electrical Engineering 1 | WS 23 V03 54 4.4.1 Stokes Equation V03 55 Taylor-Couette Flow v l Re   http://www.youtube.com/watch?v=_dbnH-BBSNo G.I. Taylor: Phil. Trans. Royal Soc. London. Series A, 223 289–343 (1923) Lecture „Microfluidic Systems - Bio-MEMS“ – V03 Fluid Mechanics I Prof. Dr.-Ing. Uwe Schnakenberg | Institute of Materials in Electrical Engineering 1 | WS 23 Tip V03 56 Tip E.M. Purcell Live at Low Reynolds Number v l Re   E.M. Purcell: American Journal of Physics 45 (3) 3-11 (1977) Lecture „Microfluidic Systems - Bio-MEMS“ – V03 Fluid Mechanics I Prof. Dr.-Ing. Uwe Schnakenberg | Institute of Materials in Electrical Engineering 1 | WS 23 Conclusion V03 57 Conclusion Basic Equations for Fluid Flow Calculations  Navier-Stokes equation (NSE) derived here in this lecture for  Newtonian fluid (constant viscosity)  Incompressible fluid  Stokes equation  NSE for Re < 1 and steady-state flow  Euler equation  NSE for fluid with negligible friction (and without body forces)  Bernoulli equation  Euler equation for steady-state and laminar flow Lecture „Microfluidic Systems - Bio-MEMS“ – V03 Fluid Mechanics I Prof. Dr.-Ing. Uwe Schnakenberg | Institute of Materials in Electrical Engineering 1 | WS 23 Conclusion V03 58 Conclusion Reynolds Number Re  Most important dimensionless number in fluid mechanics  In microfluidics: Re