# Interdisciplinary Applied Mathematics

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liquid decreases linearly with temperature in the range from the melting point to the boiling point. For example, for the liquid polydimethylsiloxane (PDMS), a silicone oil, an empirical fit to available data was obtained in (Darhuber et al., 2003), as follows:

Y (T ) = 21.5 — 0.06T mN/m,

where T is in degrees Celsius. This indicates that the variation of surface tension with temperature is linear; also, the thermal coefficient dY/dT is constant in the range of 0° to 100° , unlike the dynamic viscosity of liquids, which also decreases with temperature, but nonlinearly.

Surface tension is created by intermolecular cohesive forces. Liquid molecules that are close to a liquid-gas interface and interact with gas molecules

experience a cohesive force directed toward the bulk of the liquid and away from the interface. This imbalance of forces creates an apparent contraction and a corrugation of the interface. For interactions with other media such as solids or other liquids, the surface tension sign depends on many other factors and can be either positive or negative. For immiscible liquids we have j > 0;    for    example,    for    an    oil-water    interface, 7 =    20    mN/m    or

(dyn/cm) at 20°. Typical values of surface tension for different substances interacting with air or water are listed in Table 8.1.

The presence of an interface creates a jump in the pressure, which can be easily computed for systems in equilibrium; it depends on the surface tension and the deformation of the interface. Following (Batchelor, 1998), we compute a tension force at a small segment dx on the interface lying in a plane (x-y) as follows:

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