Interdisciplinary Applied Mathematics

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u(r)



1 — r



Ъ


+ 2Д’


and correspondingly the nondimensional flowrate is


n (    4b


Q=8 +~R


Solving for the effective slip length, we obtain


b=*№_1


4 n


Therefore, by obtaining the flowrate for a given configuration, we can then obtain the effective slip length from above (Lauga and Stone, 2003).


For the longitudinal configuration, (with m no-shear bands with halfangle a) shown in Figure 10.24, an exact solution was obtained in (Philip, 1972) for the velocity distribution


u(r, в)



R2 dp 4p dr



1 — (r/R)2 + (4/m)9



1



cos (M) cos(ma/2)



M


where M = im/2 ln(re*e/R) and A implies imaginary part. The corresponding effective slip length, bl, for this case is


-f- = — In (sec(ma/2)) = — In (sec(#7r/2)).


R m    n


For the configuration with the transverse no-shear stripes the solution is a bit    more    complicated,    but    four asymptotic    limits    were    obtained    for    the


slip length bt in (Lauga and Stone, 2003), in terms of the slip percentage 6 and the separation between slip stripes L, as follows:


1. bt/R ж 6/4 for 6 ^ 0 and L fixed.


2. bt/R ж [4(1 — 6)]-1 for 6 ^ 1 and L fixed.


3. bt/R ж (L/(2n)) ln (sec(6n/2)) for L ^ 0 and 6 fixed.


4. bt/R ж [4(1 — 6)/6]-1 for L and 6 fixed.


For a small percentage of slip, the above limits suggest that the effective slip length decreases faster (quadratically) to zero for longitudinal stripes compared to linear decrease for transverse slip stripes. For a large percentage of slip the opposite is true. Also, for small separation between slip stripes (L ^ 0) we have that bl = 2bt. These two configurations represent the two extreme idealized cases, since in reality we expect a random distribution of no-shear pockets mixed with no-slip pockets. In addition, the dependence of the slip length of the shear rate can also be included in this model by assuming that the inhomogeneities (e.g., nanobubbles) are elongated at large values of shear rate, hence effectively increasing the relative no-shear to no-slip regions.

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