# Interdisciplinary Applied Mathematics

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the liquid.    The    origin    of    this    film    may    be    due    to external    dissolved gases

up to metastable concentrations. According to (deGennes, 2002), this gas film nucleates bubbles preferentially near the wall at contact angles greater than 90°, i.e., on hydrophobic surfaces. This mechanism can take place above a threshold in shear rate, a fact consistent with the experimental observations. Evidence of nanobubbles on a hydrophobic glass surface in water using an atomic force microscope was reported in (Tyrrell and At-tard, 2002). Another possibility, consistent with MD simulations that reveal a depletion of the first layer of molecules, is that a flat vapor bubble is generated at    the    solid-liquid    interface    due    to    thermal    fluctuations.    In    either

case, the gaseous film is assumed to be small, e.g., less than 0.5 nm.

A simple mathematical model was proposed by (deGennes, 2002) for this case. He assumed that the gas in the gap is in the molecular regime (since the mean free path satisfies A ^ h, where h is the film thickness), and thus the only    collisions    are    with    the    wall.    Correspondingly,    a    molecule    leaving

the liquid has a Gaussian velocity distribution for the tangential velocity component with the corresponding peak at the slip velocity vs. Denoting by p, m the density and molecular mass of the gas, respectively, the average momentum transmitted to the solid by the gas is mvs, and thus the shear stress a at the wall is

P —    —

a = mvsvy = pvsvz,

m

where (p/m)vz    is the    average    number of    collisions    with    the    wall,    and    the

normal to the surface average velocity vz is

vv

z

with vth

л/кТ jm.

On the other hand, a = pdv/dz = pvs/b, and thus by comparing with the above expression, we obtain the slip length

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