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



H3(H + 26(1 + к)) 2(H + 6(2 + к))

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