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Equation (10) there represents an approximate form of the present Eq. (1)
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(a) For example, the limiting behavior for small d is given by the asymptotic analysis for a solid interface in Eq. (2.48) of Ref. 80 as [6 π ηav ] (1 + (a/d) + 1/5 In(a/d)+k as d/a→0, where k ≅0.97.
-
-
-
-
101
-
-
84976040821
-
-
(b) This problem is related, by symmetry, to that of two equal spheres approaching each other, since the free surface boundary condition in the original problem is obeyed on the plane bisecting their line of centers. The latter problem is singular and so the former is also, behaving as 1/d. The asymptotic expression for the force for the two-sphere problem is given by various authors, e.g., R. E. Hansford, Mathematika 17, 250 (1970); D. J. Jeffrey, ibid. 29, 58 (1982); M. D. A. Cooley and M. E. O'Neill, ibid. 16, 37 (1969).
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102
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84971897176
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(b) This problem is related, by symmetry, to that of two equal spheres approaching each other, since the free surface boundary condition in the original problem is obeyed on the plane bisecting their line of centers. The latter problem is singular and so the former is also, behaving as 1/d. The asymptotic expression for the force for the two-sphere problem is given by various authors, e.g., R. E. Hansford, Mathematika 17, 250 (1970); D. J. Jeffrey, ibid. 29, 58 (1982); M. D. A. Cooley and M. E. O'Neill, ibid. 16, 37 (1969).
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Jeffrey, D.J.1
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103
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0002586183
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(b) This problem is related, by symmetry, to that of two equal spheres approaching each other, since the free surface boundary condition in the original problem is obeyed on the plane bisecting their line of centers. The latter problem is singular and so the former is also, behaving as 1/d. The asymptotic expression for the force for the two-sphere problem is given by various authors, e.g., R. E. Hansford, Mathematika 17, 250 (1970); D. J. Jeffrey, ibid. 29, 58 (1982); M. D. A. Cooley and M. E. O'Neill, ibid. 16, 37 (1969).
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0007077053
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note
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108
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0007027108
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In Ref. 1(h), 156
-
In Ref. 1(h), p. 156.
-
-
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109
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0003669009
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Wiley, New York
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We use here the hard sphere radial distribution function at contact of about 5. R. O. Watts and I. J. McGee. Liquid State Chemical Physics (Wiley, New York, 1976). p. 137.
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21844485649
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there is a confirmation at the molecular level of the Stokes' "stick" boundary conditions at the surface of the solute; this recent computer simulation of a hard sphere solute (radius a) in a solvent of hard spheres (radius b) in the range a/b of 1-4.5 confirmed the stick hydrodynamic boundary condition for that system. However, the difference between diffusion constants for a Stokes' "stick" and "slip" boundary conditions is only a factor of 1.5, reflecting the long range nature of hydrodynamic interactions in the bulk fluid streamline potential, having terms which vary as 1/r. In contrast, the cage effect or the hydrodynamic effect close to surfaces, are both very large
-
This caging effect should be distinguished from effects which lead to "stick" vs "slip" boundary conditions in the motion of an ion. In L. Bocquet, J.-P. Hansen, and J. Piasecki, J. Stat. Phys. 76, 527 (1994), there is a confirmation at the molecular level of the Stokes' "stick" boundary conditions at the surface of the solute; this recent computer simulation of a hard sphere solute (radius a) in a solvent of hard spheres (radius b) in the range a/b of 1-4.5 confirmed the stick hydrodynamic boundary condition for that system. However, the difference between diffusion constants for a Stokes' "stick" and "slip" boundary conditions is only a factor of 1.5, reflecting the long range nature of hydrodynamic interactions in the bulk fluid streamline potential, having terms which vary as 1/r. In contrast, the cage effect or the hydrodynamic effect close to surfaces, are both very large.
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Bocquet, L.1
Hansen, J.-P.2
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111
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0007024191
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See also Fig. 4 in Ref. 63
-
See also Fig. 4 in Ref. 63.
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