Novosibirsk, Russia, May, 30 – June, 4, 2011

# International Conference "Modern Problems of Applied Mathematics and Mechanics: Theory, Experiment and Applications", devoted to the 90th anniversary of professor Nikolai N. Yanenko

## Calculation of the interaction force of two drops in a plastic medium

Stebnovskii [1] considered the behavior of drops of various oils in an alcohol-water solution of uniform density. He found that, if the distance between two drops is of the order of their sizes, they approach each other until they merge into one drop, independent of the system scale. The experimental setup was insulated from external force and heat effects. In those experiments it was found that the approach is observed only if both drops possess surface tension.
The first stage of calculation of the interaction force in present work is solving of elasticity theory problem about definition of stress tensors and displacement vectors components that satisfied following conditions on the boundary of the drops: the jump of the normal stresses is proportional to the surface tension of the drop, as it is given in the classical hydrodynamics, and shear stresses, shear and normal displacements are continual. This problem is solved by alternating Schwarz method, which consists in reducing the problem in the non-canonical domain to infinite sequence of problems in the canonical domains (external and internal domains of a sphere). The force, acting on the drop from the side of the matrix in framework of elasticity theory is always zero by virtue of equilibrium equations. But it is showing further, that the shear stresses on the boundary of drop can not surpass the value of matrix yield point, whereas there are no constraints on the normal stresses. Therefor, the shear stresses on the boundary of the drop have to be corrected, while the normal stresses should be retained without any changes. As a results the force, acted on the drop, will be not equal to zero.
On a base of conducting by describing model calculations the graphics of dependance of the drops interaction force from the distance between the drops and other parameters of the problem are performed. Moreover, the modelling of the drop motion with taking into account of hydrodynamical flow around it on a base of Bingam model was made and a good agreement with experimental data was received.

1. Stebnovskii S.V. "Thermodynamic instability of disperse media isolated from axternal actions", J. Appl. Mech. Tech. Phys., 40, No 3, 407-411 (1999).

 Full text file: Pivovarov2.pdf

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