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

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?

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

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∂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

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

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d = L − k x + c

L =10.0

k = 0.3681

c = −0.67712

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

Defining

Non-dimensionalize equation:

Boundary Conditions:

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

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