# Interdisciplinary Applied Mathematics

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In this section, we study microfluidic transport mechanisms based on capillary phenomena, taking advantage of the relative importance and sen-

FIGURE 8.1. Droplet breakup by extensional (i.e., stretched) flow in T-junctions. (Courtesy of D.A. Weitz.)

sitivity of surface tension in microscales. In particular, we will study how temperature, electric potential, and light can affect the value and possibly the sign of surface tension, which is the dominant driving force in the cases we consider in this section. Its strong dependence on the type of surface material can be exploited to design scalable and controllable wet circuits for diverse functions in microdomains. A related topic, important both for engineering and biomedical applications, is bubble transport in microducts; we study this topic in some detail in the last section of this chapter. For background reading on the fundamental concepts presented in this chapter and for a more mathematical treatment we recommend to the interested reader    the    book    (deGennes    et    al.,    2004),    and    also    the    classical    book    on

physicochemical hydrodynamics of (Levich, 1962) (Chapters 7 to 12).

##### 8.1 Basic Concepts

We review some basic concepts from fluid mechanics that we will use in the following sections. We need to know how droplets are created, how they are transported, and the governing equations for droplets and thin films in general, under various conditions.

First, we discuss droplet formation, since in most applications we need to create and transport droplets. A long cylinder of liquid in air (e.g., water jet emanating from a tap) is unstable and breaks up into small droplets. This instability was first studied by Lord Rayleigh, who found that infinitesimal axisymmetric disturbances grow exponentially if their wavelength A exceeds a critical value, which is equal to the circumference of the undisturbed jet (radius R0), i.e., if

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