The lubrication force between spherical drops, bubbles and rigid particles in a viscous fluid

Abstract A theoretical prediction of the lubrication force resisting the close approach of two spherical drops of arbitrary radii and viscosities separated by a thin viscous fluid layer is presented. The hydrodynamic resistance is predicted to be weaker than that for two colliding rigid spheres in near contact due to the mobility of the drop interfaces. Solutions are also presented for the limiting cases of (a) a drop or bubble approaching a rigid sphere or flat plate, (b) a drop approaching a bubble or a flat free surface and (c) a rigid sphere approaching a flat free surface. When one of the interfaces is completely mobile, as for a bubble or a free surface, the ratio of the lubrication force to that for two rigid spheres with the same relative velocity and reduced radius is predicted to be less than one-fourth; when one interface is completely rigid or immobile, this ratio is predicted to be greater than one-fourth. The results are shown to be in good agreement with previous solutions based on the method of bispherical coordinates when the gap is much smaller than the radius of the smallest drop.

[1]  G. Hetsroni,et al.  Low reynolds number motion of two drops submerged in an unbounded arbitrary velocity field , 1978 .

[2]  George Keith Batchelor,et al.  An Introduction to Fluid Dynamics. , 1969 .

[3]  Y. O. Fuentes,et al.  Mobility functions for two unequal viscous drops in Stokes flow. I. Axisymmetric motions , 1988 .

[4]  E. Wacholder,et al.  Slow motion of a fluid sphere in the vicinity of another sphere or a plane boundary , 1972 .

[5]  A. Acrivos,et al.  A numerical study of the deformation and burst of a viscous drop in an extensional flow , 1978, Journal of Fluid Mechanics.

[6]  I. Ivanov,et al.  Slow motion of two droplets and a droplet towards a fluid or solid interface , 1978 .

[7]  E. Rushton,et al.  The slow unsteady settling of two fluid spheres along their line of centres , 1973 .

[8]  E. Rushton,et al.  The slow motion of two spherical particles along their line of centres , 1978 .

[9]  O. Reynolds.  IV. On the theory of lubrication and its application to Mr. Beauchamp tower’s experiments, including an experimental determination of the viscosity of olive oil , 1886, Philosophical Transactions of the Royal Society of London.

[10]  H. Brenner.  The slow motion of a sphere through a viscous fluid towards a plane surface , 1961 .

[11]  A. Solan,et al.  On the low reynolds number motion of two droplets , 1973 .

[12]  D. Dimitrov,et al.  Hydrodynamics of thin liquid films , 1974 .

[13]  Robert H. Davis,et al.  The lubrication force between two viscous drops , 1989 .