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

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.

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

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

  2. Part I:

  3. Introduction • Micro-fluidics is the study of a thin layer of fluid, of the order of 100μm, at very low Reynold’s number (Re<<1) flow • To drive the system, either electro-osmosis or a pressure gradient is used • This system is used to test the effects of certain analytes or chemicals on the cell colonies

  4. Micro-fluidics in Drug Studies

  5. Problems and Motivations • Due to diffusion and the cell reaction, the concentration of the analyte is changing across and along the channel • Problems: • Maximize the number of the cell colonies placed along the channels • What are the locations where the analyte concentrations are constant?

  6. Peclet Number: Dimensions of Channel and Taylor Dispersion Width: 1 cm Length: 10 cm Height: 100 µm Taylor-Aris Dispersion Condition:

  7. Governing Equation: where Boundary Conditions: Depth-wise Averaged Equation

  8. Two Channels Concentration Velocity Vorticity

  9. Two Channel x=0mm

  10. Two Channel x=25mm

  11. Two Channel x=50mm

  12. Two Channel x=75mm

  13. Two Channel x=100mm

  14. Three Channels Concentration Velocity Vorticity

  15. Three Channel x=0mm

  16. Three Channel x=25mm

  17. Three Channel x=50mm

  18. Three Channel x=75mm

  19. Three Channel x=100mm

  20. Width Changes Along the Channel

  21. Part II:

  22. Model Equation: 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

  23. Defining Non-dimensionalize equation: Boundary Conditions:

  24. Analytical solution An analytical solution can be found via Fourier transform: Transformed equation: Solutions:

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

  26. - We are interested the wake far away from the cell patch: - The integral can be evaluated via Laplace’s method: >> Taylor Expansion For large x: φ

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

  28. Part III:

  29. Numerical wake computation • Advection-Diffusion-Reaction equation with reaction of type C0 • Domain size 10 x 60 to avoid effects of outflow boundary • Dirichlet boundary condition at inflow boundary, homogeneous Neuman at sides and outflow • Solved using Higher Order Compact Finite Difference Method (Kominiarczuk & Spotz) • Grid generated using TRIANGLE

  30. Numerical wake computation • Choose a set of neighbors • Compute optimal finite difference stencil for the PDE • Solve the problem implicitly using SuperLU • Method of 1 - 3 order, reduce locally due to C0 solution

  31. Conclusions from numerical experiments • Diffusion 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

  32. 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?

  33. References • Y.C. Lam, X. Chen, C. Yang (2005) Depthwise averaging approach to cross-stream mixing in a pressure-driven michrochannel flow Microfluid Nanofluid 1: 218-226 • R.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

  34. Thank You

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