Interdisciplinary Applied Mathematics

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gently initially. The driving force for the assembly is due to the minimization of free energy of the solder-water interface. Figure 1.22 shows a fully addressable cylindrical display, but other planar arrays were also fabricated with this approach on a much larger area. The one shown in the figure has eight columns of eight receptors interleaved with seven columns of seven receptors on the bottom electrode that connect to 15 rows of crossing copper wires on the top electrode. To test the functionality of the display with all its interconnects, a potential of 2 V was applied between the top and substrate electrodes. Solder-based assembly is appropriate for connecting components electrically and also provides mechanical strength.

FIGURE 1.22. Scanning electron micrographs (SEM) of a cylindrical display. (A) Images    of    LEDs    that    assembled    into a    regular    array.    (B)    Display    after

self-alignment of the top electrode. (C to E) Photographs of the operating display performing different functions. (Courtesy of G.M. Whitesides.)

1.4 Modeling of Microflows

We consider in this book gas, liquid, and particulate microflows and present governing equations, simplified models, and efficient numerical methods that can be used in simulations. Specifically, modeling of microflows is discussed in Chapters 2 to 9 based on continuum or quasi-continuum approximations. Because of the unfamiliar physics involved, detailed simulations are required, and this necessitates either using atomistic (particle-based) simulations or adding correction terms to the macroscopic full simulation methods. For example, in the slip flow regime it is reasonable to employ the Navier-Stokes equations modified at the wall surface with appropriate velocity slip and temperature jump conditions.

In Chapter 2 we present continuum-based velocity slip and temperature jump models for gas microflows, in conjunction with the appropriate governing equations for the slip and transitional flow regimes.

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