Interdisciplinary Applied Mathematics

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h 36 11V h )


which again shows the logarithmic dependence on the gap h.


The above formulation for two spheres has been extended to general curved hydrophobic surfaces by (Vinogradova, 1996). The results are very similar to those of the case of two spheres. For example, the resistance force is given by


рг = -ТГ7тГ    (1013)


hlW 12


where f* is defined by the same expression of equation (10.12), but the geometry is now described by the curvatures of the two surfaces as follows:


11


12



1111


+ vtx + vw +



R    R+



R—    R2 J



1



RR+



+



1



RR2



+ cos2ф



+ sin2ф


11


+



1



R+ R+



+



1



R R



RRi



R+R-


Here R+ and R denote the maximum and minimum principal radii of the surface, and    thus    I1    and    I2    are    the    mean    and    Gaussian    curvatures


of the effective surface, respectively. Also, ф defines the orientation of the two coordinate systems attached to the two surfaces. For example, we can consider the interaction of a sphere with a plane, a case typical in the surface force apparatus,    in    which case    we    obtain    I1    = 1/R and    I2    =    1/(4R2).


Similarly, we can model two crossed cylinders for which R+, R+ ^ ж and ф = п/2, so the two invariants are


Ii



2 U2~ + Ri



and I2



1


АЩВф ‘


The reader can    try    to determine    the    two    invariants    for    the case    of    two


hydrophobic cylinders with aligned axes for which ф = 0.

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