Interdisciplinary Applied Mathematics

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10.5.3 Reynolds-Vinogradova Theory for Hydrophobic Surfaces

In this section, we derive analytical solutions for the steady-state flow between two curved hydrophobic surfaces following the work of (Vinogradova, 1995; Vinogradova, 1996). This theory is an extension of Reynolds lubrication theory appropriate for slip surfaces.

The theory is    valid    for    general    curved    surfaces,    but    for    simplicity    here

we show the main steps in the solution for two spherical rigid bodies of radii R1 and R2. The distance h between the two bodies is small compared to the radii, and contact is allowed only at a single point. We also assume that a hydrophilic surface is characterized by b = 0. A schematic of the setup is shown in Figure 10.25; a cylindrical coordinate system is employed in deriving    the    solution.    The    relative    velocity is v =    v1    v2,    where    the

spheres move along the line connecting their centers with velocities v1 and V2.

The surfaces of the two bodies (upper and lower, respectively) can be described by paraboloid of revolution, i.e.,

1 r2    1 r2

h+2 Ж+°{Г4) and Z=~2R~2+°{riland introducing a shifted coordinate z = Z+r2/(2R2) and Re = RR2/(R+ R2), we can represent the two surfaces in a new coordinate system as

1 r2

z = h-———h (l(r4) and z = 0(rA).

2 Re

The governing equation is Reynolds’s lubrication equation, assuming that the characteristic length is the gap between the two particles, i.e.,

d2vr dp ^ dz2 dr

while in the z-direction we have that dp/dz = 0, which implies that the pressure is a function of r only. The boundary conditions correspond to slip on the lower    body,    characterized by b2    =    b,    while    on    the    upper surface    we

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