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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 Lk 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