**26**, 3, pp. 407-426, 1978

### Infection of a Mixture of Small Particles Into a Flow, Treated as a Perturbation Problem

Small identical solid spherical particles are injected from a wall into a laminar flow of an incompressible viscous fluid. The concentration of particles is assumed to be low. The goal of this paper is to study the action of the particles on the velocity profile of the fluid. The equations used for the fluid-particles mixture reduce the effect of each particle to a Stokes force acting on its center. A perturbation method is applied to solve these equations: the non-dimensional number *S*=t_{p}/te, where to is a characteristic time for a particle and t_{e} a characteristic time of the flow, is required to be small. It is shown that this condition is compatible with the other assumptions, but that the equations for the mixture are valid only within a limited region close to the wall. The results prove that the velocity profile of the fluid is affected by the particles if the concentration of the particles at the wall is of order 1/Re *S* at least, where Re is the fluid flow Reynolds number. For the case where this condition is satisfied, velocity profiles are computed and written in closed form. Some typical profiles are shown for an uniform injection of particles.

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#### References

G. K. BATCHELOR, J. T. GREEN, J. Fluid Mech., 56, 2, 375-400, 1972.

H. FAXEN, Arkiv. Mat. Astron. Fys., 17, 27, 1923.

S. WAKIYA, Res. Rep. Eng. Niigata Univ. Japan, 9, 31, 1960.

J. HAPPEL, H. BRENNER, Low Reynolds number hydrodynamics, 327-330, Prentice-Hall Inc., Englewood Cliffs, New Jersey 1965.

P. G. SAFFMAN, J. Fluid Mech., 22, 2, 385-400, 1965.

A. B. BASSET, A treatise on hydrodynamics, II, 285-297, Deighton Bell and Co., Cambridge 1888.

H. VILLAT, Leçons sur les fluides visqueux, 178-213, Gauthier-Villars, Paris 1943.

S. L. SOO, Fluid dynamics of multiphase systems, 256-264, 16, Blaisdell Publishing Co. (a division of Ginn and Co), Waltham-Massachussetts-Toronto-London 1967.

H. SCHLICHTING, Boundary Layer theory, 125, 509, Mc Graw Hill, 1960.

M. VAN DYKE, Perturbation methods in fluid mechanics, Appl. Math. Mech., 8, Academic Press, New York-London 1964.

I. S. GRADSHTEYN, I. M. RYSHIK, Table of integrals series and products, Paragraphs 8-446, Academic Press, New York-Lonodn 1965.