V04 Fluid Mechanics II WS 23.pdf

V04 V04 Fluid Mechanics II Lecture „Microfluidic Systems - Bio-MEMS“ – V04 Fluid Mechanics II Prof. Dr.-Ing. Uwe Schnakenberg | Institute of Materials in Electrical Engineering 1 | WS 23 1 Contents V04 Contents 4.1 Flow Profile of Pressure-Driven Flow in a Microchannel 4.2 Hagen-Poiseuille equation...

V04 V04 Fluid Mechanics II Lecture „Microfluidic Systems - Bio-MEMS“ – V04 Fluid Mechanics II Prof. Dr.-Ing. Uwe Schnakenberg | Institute of Materials in Electrical Engineering 1 | WS 23 1 Contents V04 Contents 4.1 Flow Profile of Pressure-Driven Flow in a Microchannel 4.2 Hagen-Poiseuille equation 4.3 Laminar Flow: Hydrodynamic Particle Focusing 4.4 Fluidic Circuit Analysis Lecture „Microfluidic Systems - Bio-MEMS“ – V04 Fluid Mechanics II Prof. Dr.-Ing. Uwe Schnakenberg | Institute of Materials in Electrical Engineering 1 | WS 23 2 Learning Targets V04 Learning Targets  Pressure-driven flow profile @ Re number < 1 in a channel is parabolic  Hagen-Poiseuille equation  Fluidic circuit analysis  Hydrodynamic particle focusing Lecture „Microfluidic Systems - Bio-MEMS“ – V04 Fluid Mechanics II Prof. Dr.-Ing. Uwe Schnakenberg | Institute of Materials in Electrical Engineering 1 | WS 23 3 V04 4.1 Flow Profile of Pressure-Driven Flow in a Microchannel Lecture „Microfluidic Systems - Bio-MEMS“ – V04 Fluid Mechanics II Prof. Dr.-Ing. Uwe Schnakenberg | Institute of Materials in Electrical Engineering 1 | WS 23 4 4.1 Flow Profile of Pressure-Driven Flow in a Microchannel Flow Profile of Pressure-Driven Flow in a Microchannel (incompressible fluid, laminar and steady-state flow) p1 > p2 y p1 d/2 x p2 L We are interested in the velocity component vx in direction of y https://zeus.plmsc.psu.edu/~manias/MatSE447/04_FlowInVariedGeometries.pdf, 2023-06-01 Lecture „Microfluidic Systems - Bio-MEMS“ – V04 Fluid Mechanics II Prof. Dr.-Ing. Uwe Schnakenberg | Institute of Materials in Electrical Engineering 1 | WS 23 V04 5 4.1 Flow Profile of Pressure-Driven Flow in a Microchannel Stokes equation V04 6 p    2 v v  (vx , 0, 0)  Pressure drop is only in x-direction  Only velocity component vx in direction of y   2 vx  2 vx  2 vx   2  2  2  y z   x   2v  2v y  2v y  y Δv   2  2  2  y z   x  2  2 2  v  v  v z z z   2  2   x 2 y z     / x   p / x      p    / y  p   0    / z   0      Stokes equation  2 vx p   2 x y https://zeus.plmsc.psu.edu/~manias/MatSE447/04_FlowInVariedGeometries.pdf, 2023-06-01 Lecture „Microfluidic Systems - Bio-MEMS“ – V04 Fluid Mechanics II Prof. Dr.-Ing. Uwe Schnakenberg | Institute of Materials in Electrical Engineering 1 | WS 23 4.1 Flow Profile of Pressure-Driven Flow in a Microchannel Stokes equation Simplified NSE for Re number < 1 V04 7  2 vx p   2 x y How can f(x) = h(y) ? → Each function must be constant Left side p  C1 x  Boundary conditions Left side p  C1 x  C2 p ( x  0)  p1  C2  p1 p ( x  L )  p2  C1   p1  p2 p  L L p p  p1  x L https://zeus.plmsc.psu.edu/~manias/MatSE447/04_FlowInVariedGeometries.pdf, 2023-06-01 Lecture „Microfluidic Systems - Bio-MEMS“ – V04 Fluid Mechanics II Prof. Dr.-Ing. Uwe Schnakenberg | Institute of Materials in Electrical Engineering 1 | WS 23 4.1 Flow Profile of Pressure-Driven Flow in a Microchannel Stokes equation Right side  2 vx p   2 x y  2 vx p   2  C1   y L  2 vx 1 p  2 y  L 1st integration 2nd integration vx p  y  C3 y L vx   p 2 y  C3  y  C4 2  L Lecture „Microfluidic Systems - Bio-MEMS“ – V04 Fluid Mechanics II Prof. Dr.-Ing. Uwe Schnakenberg | Institute of Materials in Electrical Engineering 1 | WS 23 V04 8 4.1 Flow Profile of Pressure-Driven Flow in a Microchannel V04 p 2 vx   y  C3  y  C4 2  L Top plate Boundary conditions (no slip) Bottom plate p 2 d d  C3   C4 8  L 2 d vx ( y  )  0 2 0 d vx ( y   )  0 2 p 2 d 0 d  C3   C4 8  L 2 C3  0 C4  p 2 d 8  L p  d 2 2 vx ( y )   y   2  L  4  Lecture „Microfluidic Systems - Bio-MEMS“ – V04 Fluid Mechanics II Prof. Dr.-Ing. Uwe Schnakenberg | Institute of Materials in Electrical Engineering 1 | WS 23 Poiseuille flow Stokes flow 9 4.1 Flow Profile of Pressure-Driven Flow in a Microchannel V04 10 p  d 2 2 vx ( y )  y   2  L  4  In a microchannel, the flow profile for laminar and for pressure-driven flow is parabolic Shear stress τ = dv/dy y vmax 0 vx(y) x vm Bohl et al.: Technische Strömungslehre, ISBN: 10-3-8343-3029-9 Lecture „Microfluidic Systems - Bio-MEMS“ – V04 Fluid Mechanics II Prof. Dr.-Ing. Uwe Schnakenberg | Institute of Materials in Electrical Engineering 1 | WS 23 4.1 Flow Profile of Pressure-Driven Flow in a Microchannel vmax p  d 2     2  L  4  vm  vmax 2 V04 11 Maximum velocity at channel axis (y = 0) Mean velocity https://www.engr.colostate.edu/CBE101/_images/parabolic_flow_profile.gif, 2023-09-20 Lecture „Microfluidic Systems - Bio-MEMS“ – V04 Fluid Mechanics II Prof. Dr.-Ing. Uwe Schnakenberg | Institute of Materials in Electrical Engineering 1 | WS 23 http://faculty.washington.edu/yagerp/microfluidicstutorial/ basicconcepts/basicconcepts.htm, 2023-09-20 4.1 Flow Profile of Pressure-Driven Flow in a Microchannel V04 12 Parabolic Flow Velocity Profile Also Occurs in Rectangular Channels Parabolic flow velocity profile means  Velocity depends on position in tube  Resting time high in area of side walls  Resting time low in area of tube axis J.P. Brody et al: Biophys. J 71 3430-3441 (1996) Lecture „Microfluidic Systems - Bio-MEMS“ – V04 Fluid Mechanics II Prof. Dr.-Ing. Uwe Schnakenberg | Institute of Materials in Electrical Engineering 1 | WS 23 4.1 Flow Profile of Pressure-Driven Flow in a Microchannel Black glycerin Transparent glycerin https://www.youtube.com/watch?v=jQy-t_6LPDE 2023/09-20 Lecture „Microfluidic Systems - Bio-MEMS“ – V04 Fluid Mechanics II Prof. Dr.-Ing. Uwe Schnakenberg | Institute of Materials in Electrical Engineering 1 | WS 23 V04 13 4.1 Flow Profile of Pressure-Driven Flow in a Microchannel V04 14 Parabolic flow profile appears after a distinct distance, the entrance length Le  A fluid with constant flow velocity enters a channel  “No Slip” condition on the wall: the velocity is zero on the wall  A layer is build where the velocity builds up slowly from zero at wall to the uniform velocity towards the center of the channel (friction) https://www.accessengineeringlibrary.com https://www.youtube.com/watch?v=FWaO17n2pIc Lecture „Microfluidic Systems - Bio-MEMS“ – V04 Fluid Mechanics II Prof. Dr.-Ing. Uwe Schnakenberg | Institute of Materials in Electrical Engineering 1 | WS 23 4.1 Flow Profile of Pressure-Driven Flow in a Microchannel V04 15  This layer is called “Boundary layer”  The part with the uniform velocity is called “Inviscid Core”  The boundary layer grows: its thickness increases as the fluid moves downstream  At the entrance length, Le, the inviscid core terminates  The flow is now called a “Fully developed flow”. The velocity profile becomes parabolic. https://www.accessengineeringlibrary.com https://www.youtube.com/watch?v=FWaO17n2pIc Lecture „Microfluidic Systems - Bio-MEMS“ – V04 Fluid Mechanics II Prof. Dr.-Ing. Uwe Schnakenberg | Institute of Materials in Electrical Engineering 1 | WS 23 4.1 Flow Profile of Pressure-Driven Flow in a Microchannel Example V04 16 Le  0.06  Re l  Water  Characteristic length (channel diameter): 100 µm  Flow velocity: 0.2 m/s Le  300 µm l …. Characteristic length Le …Entrance length for establishing parabolic flow profile Re.. Reynolds number M.A. Van Dilla: Flow Cytometry: instrumentation and data analysis, Michigan, Academic Press (1985) https://www.youtube.com/watch?v=FWaO17n2pIc Lecture „Microfluidic Systems - Bio-MEMS“ – V04 Fluid Mechanics II Prof. Dr.-Ing. Uwe Schnakenberg | Institute of Materials in Electrical Engineering 1 | WS 23 V04 17 4.2 Hagen-Poiseuille Equation Lecture „Microfluidic Systems - Bio-MEMS“ – V04 Fluid Mechanics II Prof. Dr.-Ing. Uwe Schnakenberg | Institute of Materials in Electrical Engineering 1 | WS 23 4.2 Hagen-Poiseuille Equation Poiseuille / Stokes flow V04 18 p  d 2 2 vx ( y )  y   2  L  4  Volume flow rate results from integration of vx(y) over the cylindrical cross-section Gotthilf Hagen (1797-1884) d… L… r0 … v …. ∆p.. η …. Channel diameter Channel length Channel radius Velocity Pressure difference over L Viscosity Lecture „Microfluidic Systems - Bio-MEMS“ – V04 Fluid Mechanics II Prof. Dr.-Ing. Uwe Schnakenberg | Institute of Materials in Electrical Engineering 1 | WS 23 Hagen-Poiseuille equation for a cylindrical channel de.wikipedia.org/ de.wikipedia.org/ Flow rate   r04 p Q  8 L Jean Louis Marie Poiseuille (1797–1869) 4.2 Hagen-Poiseuille Equation Rfl U  RI Tolerances in channel geometries due to fabrication processes (lithography, etching) have a severe influence on Rfl !!! Hagen-Poiseuille equation rearranged to ∆p for a cylindrical channel de.wikipedia.org/ 8   L p  Q 4   r0 V04 19 Rfl … Fluid resistance Analogy to Ohm’s Law Georg Simon Ohm (1789 – 1854) 8   L R fl    r04 Decrease of r0 by a factor of 10 → Increase of Rfl by a factor of 104 Lecture „Microfluidic Systems - Bio-MEMS“ – V04 Fluid Mechanics II Prof. Dr.-Ing. Uwe Schnakenberg | Institute of Materials in Electrical Engineering 1 | WS 23 4.2 Hagen-Poiseuille Equation V04 20 Fluid Resistance Rfl  Cylindrical channel R fl  8   L   r04  Rectangular channel with w ≈ h 12   L  h  192  1  n    w   R fl   1    tanh  2  h    w  h3  w  w n 1,3,5 n 5  1  Slit-type channel (w >> h or w