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 where the analyte concentrations are constant?

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

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

Two Channels ConcentrationVelocityVorticity

Two Channel x=0 mm

Two Channel x=25 mm

Two Channel x=50 mm

Two Channel x=75 mm

Two Channel x=100 mm

Three Channels ConcentrationVelocityVorticity

Three Channel x=0 mm

Three Channel x=25 mm

Three Channel x=50 mm

Three Channel x=75 mm

Three Channel x=100 mm

Width Changes Along the Channel

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 C 0 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 order, reduce locally due to C 0 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: 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,