Interdisciplinary Applied Mathematics

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2y cos в r


and thus the rise of liquid is inversely proportional to the micropipe radius. For example, for a 10-micron glass pipe, there may be a rise of water of up to about 100 microns.


As we    have    seen in the    example of    Figure    8.2,    the    contact    angle    в is a


significant factor in capillarity. When a liquid droplet is in contact with a solid surface, the value of the contact angle reflects the type of interaction between the    liquid    and    the    solid.    The    shape    of    a small    liquid    droplet    on


a homogeneous substrate is usually a spherical cap. The contact angle is approximately constant if there are no impurities, temperature gradients, or surface defects, and it is determined by the Laplace-Young equation (8.2).    The    point    (or line)    of contact,    as shown    in    the    sketch    of Figure    8.3,


is subject to three different surface tensions. Because of equilibrium at the contact point (or line), we have


(8.6)


o


o


Ysg = Yis + Ygicos в,


where Ysg denotes the surface tension between solid and gas, Yls between liquid    and    solid, and    Ygl    between gas and    liquid.    This    equation    is    often


referred to as Young’s law. 11


Substance


Air


Water


dyn/cm


dyn/cm


°C


7 or


7 or


mN/m


mN/m


Water


10


74.2


20


72.8


30


71.2


40


69.6


Benzene


20


28.9


35


Ethyl ether


20


17


10.7


Glycerol


20


63.4


< 0


Mercury


20


487


375

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