Problem 5: Microfluidics Math in Industry Workshop Student Mini-Camp CGU 2009

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Problem 5: Microfluidics Math in Industry Workshop Student Mini-Camp CGU 2009. Abouali, Mohammad (SDSU) Chan, Ian (UBC) Kominiarczuk , Jakub (UCB) Matusik , Katie (UCSD) Salazar, Daniel (UCSB). Advisor: Michael Gratton. Part I:. Introduction. - PowerPoint PPT Presentation

Transcript of Problem 5: Microfluidics Math in Industry Workshop Student Mini-Camp CGU 2009

  • Problem 5: MicrofluidicsMath in Industry WorkshopStudent Mini-CampCGU 2009Abouali, Mohammad (SDSU)Chan, Ian (UBC)Kominiarczuk, Jakub (UCB)Matusik, Katie (UCSD)Salazar, Daniel (UCSB)Advisor: Michael Gratton

  • IntroductionMicro-fluidics is the study of a thin layer of fluid, of the order of 100m, at very low Reynolds number (Re
  • Micro-fluidics in Drug Studies

  • Problems and MotivationsDue to diffusion and the cell reaction, the concentration of the analyte is changing across and along the channelProblems:Maximize the number of the cell colonies placed along the channelsWhat are the locations where the analyte concentrations are constant?

  • Width: 1 cmLength: 10 cmHeight: 100 mDimensions of Channel and Taylor Dispersion

    Sheet1

    width0.02

    length0.1

    height1.00E-04

    123456

    1.00E-061.00E-071.00E-081.00E-091.00E-101.00E-11

    11.00E-062.00E-022.00E-012.00E+002.00E+012.00E+022.00E+03

    25.00E-061.00E-011.00E+001.00E+011.00E+021.00E+031.00E+04

    35.00E-051.00E+001.00E+011.00E+021.00E+031.00E+041.00E+05

    41.00E-042.00E+002.00E+012.00E+022.00E+032.00E+042.00E+05

    51.00E-032.00E+012.00E+022.00E+032.00E+042.00E+052.00E+06

    65.00E-031.00E+021.00E+031.00E+041.00E+051.00E+061.00E+07

    123456

    10.000000010.0000010.00010.011100

    20.000000250.0000250.00250.25252500

    30.0000250.00250.25252500250000

    40.00010.011100100001000000

    50.011100100001000000100000000

    60.25252500250000250000002500000000

    Diffusion (m2/s)

    1.00E-061.00E-071.00E-081.00E-091.00E-101.00E-11

    Velocity (m/s)1.00E-06OKOKOKOKOKOK

    5.00E-06OKOKOKOKOKNot OK

    5.00E-05OKOKOKOKNot OKNot OK

    1.00E-04OKOKOKOKNot OKNot OK

    1.00E-03OKOKOKNot OKNot OKNot OK

    5.00E-03OKOKNot OKNot OKNot OKNot OK

    Chart2

    Chart1

  • Depth-wise Averaged Equation

  • Two ChannelsConcentrationVelocityVorticity

  • Two Channel x=0mm

  • Two Channel x=25mm

  • Two Channel x=50mm

  • Two Channel x=75mm

  • Two Channel x=100mm

  • Three ChannelsConcentrationVelocityVorticity

  • Three Channel x=0mm

  • Three Channel x=25mm

  • Three Channel x=50mm

  • Three Channel x=75mm

  • Three Channel x=100mm

  • Width Changes Along the Channel

  • ModelEquation:Uptake is assumed to be at a constant rate over the cell patch.The reaction rate is chosen to be the maximum over the range of concentrations used

  • Defining Non-dimensionalize equation:Boundary Conditions:

  • Analytical solutionAn analytical solution can be found via Fourier transform:Transformed equation:

    Solutions:

  • - Demand continuity and differentiability across boundary, and apply boundary conditions.- Apply inverse Fourier transform

  • - We are interested the wake far away from the cell patch:- The integral can be evaluated via Laplaces method:Taylor ExpansionFor large x:>>

  • Restoration is defined as Restoration length:Larger flow velocity enhances recovery??

  • Numerical wake computationAdvection-Diffusion-Reaction equation with reaction of type C0Domain size 10 x 60 to avoid effects of outflow boundaryDirichlet boundary condition at inflow boundary, homogeneous Neuman at sides and outflowSolved using Higher Order Compact Finite Difference Method (Kominiarczuk & Spotz)Grid generated using TRIANGLE

  • Numerical wake computationChoose a set of neighborsCompute optimal finite difference stencil for the PDESolve the problem implicitly using SuperLUMethod of 1 - 3 order, reduce locally due to C0 solution

  • Conclusions from numerical experimentsDiffusion is largely irrelevant as typical Peclet numbers are way above 1

    Depth of the wake depends on the relative strength of advection and reaction terms

    Because reaction rates vary wildly, we cannot conclude that it is safe to stack colonies along the lane given the constraints of the design

  • Outstanding Issues:Will vertically averaging fail for small diffusivity?What are the limitations of the vertically averaging?Taylor dispersion?Pattern of colony placements?Realistic Reaction Model?Effect of Boundaries along the device?

  • ReferencesY.C. Lam, X. Chen, C. Yang (2005) Depthwise averaging approach to cross-stream mixing in a pressure-driven michrochannel flow Microfluid Nanofluid 1: 218-226R.A. Vijayendran, F.S. Ligler, D.E. Leckband (1999) A Computational Reaction-Diffusion Model for the Analysis of Transport-Limited Kinetics Anal. Chem. 71, 5405-5412