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The extensional viscosity of a dilute suspension of spherical particles at intermediate microscale Reynolds numbers

Published online by Cambridge University Press:  19 April 2006

G. Ryskin ,
G. Ryskin  and
J. M. Rallison
G. Ryskin
Affiliation:
Chemical Engineering 208–41, California Institute of Technology, Pasadena, California 91125
G. Ryskin
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge
J. M. Rallison
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge
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Abstract

The extensional viscosity of a dilute suspension of spherical particles (rigid spheres, viscous drops or gas bubbles) is computed for the case when the Reynolds number of the microscale disturbance motion R is not restricted to be small, as in the classical analysis of Einstein and Taylor. However, the present theory is restricted to steady axisymmetric pure straining flow (uniaxial extension). The rate of energy dissipation is expressed using the Bobyleff-Forsythe formula and then conditionally convergent integrals are removed explicitly. The problem is thereby reduced to a determination of the flow around a particle, subject to pure straining at infinity, followed (for rigid particles) by an evaluation of the volume integral of the vorticity squared. In the case of fluid particles, further integrals over the volume and surface of the particle are required. In the present paper, results are obtained numerically for 1 [les ] R [les ] 1000 for a rigid sphere, for a drop whose viscosity is equal to the viscosity of the ambient fluid, and for an inviscid drop (gas bubble). For the last case, limiting results are also obtained for R → ∞ using Levich's approach.

All of these results show a strain-thickening behaviour which increases with the viscosity of the particle. The possibility of experimental verification of the results, which is complicated by the inapplicability of the approximation of material frame-indifference in this case, is discussed.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 99 , Issue 3 , 14 August 1980 , pp. 513 - 529
Copyright
© 1980 Cambridge University Press

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References

Ames, W. F. 1969 Numerical Methods for Partial Differential Equations. New York: Barnes and Noble.
Astarita, G. & Marrucci, G. 1974 Principles of Non-Newtonian Fluid Mechanics. McGraw-Hill.
Barthés-Biesel, D. & Acrivos, A. 1973 The rheology of suspensions and its relation to phenomenological theories for non-Newtonian fluids. Int. J. Multiphase Flow 1, 1–24.Google Scholar
Batchelor, G. K. 1967 An Introduction to Fluid Dynamics. Cambridge University Press.
Batchelor, G. K. 1970 The stress system in a suspension of force-free particles. J. Fluid Mech. 41, 545–570.Google Scholar
Batchelor, G. K. 1971 The stress generated in a non-dilute suspension of elongated particles by pure straining motion. J. Fluid Mech. 46, 813–829.Google Scholar
Batchelor, G. K. 1974 Transport properties of two-phase materials with random structure. Ann. Rev. Fluid Mech. 6, 227–255.Google Scholar
Bird, R. B., Armstrong, R. C. & Hassager, O. 1977 Dynamics of Polymeric Liquids. Vol. 1. Fluid Mechanics. Wiley.
Brenner, H. & Condiff, D. W. 1974 Transport mechanics in systems of orientable particles. IV. Convective Transport. J. Colloid Interface Sci. 47, 199–264.Google Scholar
Dorodnitsyn, A. A. & Meller, N. A. 1968 Approaches to the solution of stationary Navier-Stokes equations. U.S.S.R. Comput. Math. & Math. Phys. 8 (2), 205–217.Google Scholar
Greenspan, H. P. 1968 The Theory of Rotating Fluids. Cambridge University Press.
Grosch, C. E. & Orszag, S. A. 1977 Numerical solution of problems in unbounded regions: coordinate transforms. J. Comp. Phys. 25, 273–296.Google Scholar
Hinch, E. J. & Leal, L. G. 1975 Constitutive equations in suspension mechanics. Part 1. General formulation. J. Fluid Mech. 71, 481–495.Google Scholar
Hinch, E. J. & Leal, L. G. 1976 Constitutive equations in suspension mechanics. Part 2. Approximate forms for a suspension of rigid particles affected by Brownian rotations. J. Fluid Mech. 76, 187–208.Google Scholar
Israeli, M. 1970 A fast implicit numerical method for time dependent viscous flows. Stud. Appl. Math. 49, 327–349.Google Scholar
Jeffrey, D. J. 1974 Group expansions for the bulk properties of a statistically homogeneous random suspension. Proc. Roy. Soc. A 338, 503–516.Google Scholar
Jeffrey, D. J. 1977 The physical significance of non-convergent integrals in expressions for effective transport properties. Proc. 2nd Int. Symp. on Continuum Models of Discrete Systems, pp. 653–674. University of Waterloo Press.
Jeffrey, D. J. & Acrivos, A. 1976 The rheological properties of suspensions of rigid particles. A.I.Ch.E. J. 22, 417–432.Google Scholar
Lin, C. J., Peery, J. H. & Schowalter, W. R. 1970 Simple shear flow round a rigid sphere: inertial effects and suspension rheology. J. Fluid Mech. 44, 1–17.Google Scholar
Lumley, J. L. 1970 Toward a turbulent constitutive relation. J. Fluid Mech. 41, 413–434.Google Scholar
McConnell, A. J. 1957 Applications of Tensor Analysis. Dover.
O'Brien, R. W. 1979 A method for the calculation of the effective transport properties of suspensions of interacting particles. J. Fluid Mech. 91, 17–39.Google Scholar
Pearson, J. R. A. 1959 The effect of uniform distortion on weak homogeneous turbulence. J. Fluid Mech. 5, 274–288.Google Scholar
Rivkind, V. Ya. & Ryskin, G. 1976 Flow structure in motion of a spherical drop in a fluid medium at intermediate Reynolds numbers. Fluid Dynamics 11, 5–12.Google Scholar
Rivkind, V. Ya., Ryskin, G. & Fishbein, G. A. 1976 Flow around a spherical drop at intermediate Reynolds numbers. Appl. Math. & Mech. (PMM) 40, 687–691.Google Scholar
Roache, P. J. 1972 Computational Fluid Dynamics. Albuquerque: Hermosa.
Schowalter, W. R. 1978 Mechanics of Non-Newtonian Fluids. Pergamon.
Serrin, J. 1959 Mathematical principles of classical fluid mechanics, Handbuch der Physik, vol. 8/1, pp. 125–263. Springer.
Taylor, G. I. 1934 The formation of emulsions in definable fields of flow. Proc. Roy. Soc. A 146, 501–523.Google Scholar
Truesdell, C. 1977 A First Course in Rational Continuum Mechanics, Vol. 1. General Concepts. Academic Press.
Truesdell, C. & Noll, W. 1965 The Non-Linear Field Theories of Mechanics, Handbuch der Physik, vol. 3/3. Springer.