Problem 5: Microfluidics Math in Industry Workshop Student Mini-Camp CGU 2009 Abouali, Mohammad...
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Problem 5: MicrofluidicsMath in Industry Workshop
Student Mini-CampCGU 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
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?
D
wuPePeclet Number:
2102
w
PeHTaylor-Aris Dispersion Condition:
Width: 1 cm
Length: 10 cm
Height: 100 µm
Dimensions of Channel and Taylor Dispersion
Depth-wise Averaged Equation
2
2
2
2
xD
yD
xu eff
Governing Equation:
20,0|),( 0
wyxy y
€
∂ϕ∂x y,x=L
= 0,∂ϕ
∂yy=0,x
= 0,∂ϕ
∂yy=w,x
= 0
wyw
xy ox 2,|),( 0
Boundary Conditions:
22
2101
w
HPeDDeffwhere
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
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 Laplace’s method:
Taylor Expansion
For large x:
>>
φ
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