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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 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 Micro-fluidics in Drug Studies 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 concentrations are constant? where the analyte Dimensions of Channel and Taylor Dispersion Width: 1 cm Peclet Number: Length: 10 cm uw Pe D Height: 100 µm 2 Taylor-Aris Dispersion Condition: PeH 210 w Depth-wise Averaged Equation Governing Equation: 2 2 u D D eff 2 2 x y x 2 2 Pe H D 1 where D eff w 210 Boundary Conditions: w ( y , x ) | 0 ,0 y 2 y 0 w ( y , x ) | , y w 2 x 0 o 0, 0, x y,x L y y 0,x y 0 y w,x Two Channels Concentration Velocity Vorticity Two Channel x=0mm Two Channel x=25mm Two Channel x=50mm Two Channel x=75mm Two Channel x=100mm Three Channels Concentration Velocity Vorticity 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 Acceptable Width for Cell Colony Placement 10 9 Acceptable width 8 7 6 d Lk x c L 10.0 k 0.3681 c 0.67712 5 4 3 2 1 0 0 10 20 30 40 50 60 Distance from the entrance of the channel 70 80 90 100 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 Defining Non-dimensionalize equation: Boundary Conditions: Analytical solution An 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 Laplace’s method: Taylor Expansion For large x: φ >> Restoration is defined as Restoration length: Larger flow velocity enhances recovery?? 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 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 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 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? 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