Interdisciplinary Applied Mathematics

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A/(2nRo) > 1.

Similarly, if    the    same    column    of    liquid    resides    on    a solid    surface    with    a

contact angle в between the liquid and the surface, a similar hydrodynamic instability appears if

A/(27ri?o) > Vl — cos в.

Another situation, often encountered in microfluidics is the case of a liquid column trapped between two parallel plates. Unlike the two aforementioned cases, this case is hydrodynamically stable irrespective of the value of the wavelength A. This, in turn, implies that it is very difficult to create droplets in a confined environment unlike the open surfaces where droplets are created spontaneously by hydrodynamic instabilities. Details of the theoretical derivation for these cases can be found in (Isenberg, 1992). In applications, if we transport microdroplets over an open area, evaporation may be very high, so    often a    cover    glass is    used to prevent    this.    However,    due    to    the

aforementioned enhanced hydrodynamic stability, creating droplets from a liquid column sandwiched between two plates requires large amounts of energy.

We now discuss some useful concepts and the governing laws related to surface tension. Surface tension, denoted by 7, is force per unit length. For a    system    at    equilibrium    in    the presence    of    an    interface    of    area A,    it

contributes jA to the system’s energy. Correspondingly, the work done on the system to change the interface (for relatively small changes in the system dynamics) is 76A. Surface tension can have either positive or negative values, reflecting    the    fact    that    it    is associated    with    the free energy    of    the

system, which can increase or decrease. When a liquid is in contact with a gas, the surface tension is positive. For example, the surface tension between water and air at 20° is 72.8 dyn/cm, or 72.8 mN/m in SI units; it depends strongly on    impurities    and    on    the    temperature.    The    surface    tension    of a

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