Senior Thesis

Department of Mathematics

Harvey Mudd College

Dan Beutel

Harvey Mudd College Mathematics 2003

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Thesis Title: Modeling Advection and Diffusion in Microchannels.

Thesis Advisor: Prof. A.J. Bernoff (HMC Math)

Second Reader: Prof Tom Donnelly (HMC Physics)



Modeling Advection and Diffusion in Microchannels.

Microchannels are channels of length scale on the order of microns (most experimental setups are channels 10 to 100 microns wide) which have a wide variety of potential applications in chemistry and biochemistry. For example, a potential application is testing or analyzing chemicals that are only available in small quantities (DNA would be one example) without wasting the limited supply. The potential uses have created a good deal of interest among fluid mechanics researchers.

Mixing which is caused by the properties of the channel (passive mixing) is the motivation for the research in this thesis. Specifically, the goal is to understand the flow of fluids in a microchannel of rectangular cross-section, especially how those fluids mix together. The two channels joining together below show the system being studied. Two distinct fluids flow from each side of the intersection and mix as they continue to flow downstream (with velocity u). The motion of a fluid is described by the simultaneous processes of advection (motion according to the velocity field) and diffusion (random spreading). The advection-diffusion equation describes the motion:
\begin{displaymath}
\frac{\partial c}{\partial t} + \vec u \cdot \nabla c = c \nabla^2 c
\end{displaymath} (1)
Equation 1 can be simulated computationally by a split-step particle method, which tracks the motion of one particle by separating the advection and diffusion.

The result of simulating mixing in the pipe shown above is that the two fluids spread together through diffusion but don't really mix. The below movie shows the concentration at a succession of cross-sections, each of which is slightly further downstream.
Diffusive Spreading in a Square Microchannel(Click for larger version)

To move beyond the spreading produce more rapid mixing, we introduce grooves on the bottom edge of the pipe perpendicular to the flow such as these. Finding the velocity field over these ridges is not only possible through a numerical solution of a simplified form of the Navier-Stokes equations, as detailed in chapters 3, 4 and 5 of the thesis. Suffice it to say that the result is flow upwards which should produce mixing. Analyzing the mixing which results from this field would be the logical next step for this research.

Vertical velocity as a function of position (x,z)

Horizontal velocity as a function of position (x,z)