# Interdisciplinary Applied Mathematics

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assume that bi = b(1 + к), where к characterizes the specific type of the interaction. Specifically,

• к = — l corresponds to a hydrophilic upper surface.

к = 0 corresponds to a hydrophobic upper surface.

• к ^ ж corresponds to a bubble upper surface.

In addition, к can take any other value between — l and ж to represent other types of interaction and surfaces. We note that contrast to the standard Reynolds problem, where the only length scale present is the gap h,

here we    have two additional    length    scales,    namely    b and b(1    +    k).    So    the

boundary conditions on the lower surface are

vz = 0 and vr = b-^-dz

and those on the upper surface are

Wr

vz ——— = —v and vr

z Z?    r

Re

b(1 +

k)

dvr dz ‘

The solution of the above Reynolds equation with the aforementioned boundary conditions yields

Vr (r,z )

1    dp Г 2H(H + 26(1 + к))

2 p dr    H + 6(2 + к)

ЪН(Н + 26(1 + к))’

Я + 6(2 + к) J ’

where H = h + r2/(2Re). The relative velocity v can be obtained from the continuity equation

dvz    1 d(rvr) _

dz    r dr    ’

which by integration yields

v

Id    dp 1 f H3

r dr    dr 2p 3