Hostname: page-component-7c8c6479df-995ml Total loading time: 0 Render date: 2024-03-27T17:54:28.441Z Has data issue: false hasContentIssue false

Creeping motion of a sphere through a Bingham plastic

Published online by Cambridge University Press:  20 April 2006

A. N. Beris
Affiliation:
Department of Chemical Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139
J. A. Tsamopoulos
Affiliation:
Department of Chemical Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139
R. C. Armstrong
Affiliation:
Department of Chemical Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139
R. A. Brown
Affiliation:
Department of Chemical Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139

Abstract

A solid sphere falling through a Bingham plastic moves in a small envelope of fluid with shape that depends on the yield stress. A finite-element/Newton method is presented for solving the free-boundary problem composed of the velocity and pressure fields and the yield surfaces for creeping flow. Besides the outer surface, solid occurs as caps at the front and back of the sphere because of the stagnation points in the flow. The accuracy of solutions is ascertained by mesh refinement and by calculation of the integrals corresponding to the maximum and minimum variational principles for the problem. Large differences from the Newtonian values in the flow pattern around the sphere and in the drag coefficient are predicted, depending on the dimensionless value of the critical yield stress Yg below which the material acts as a solid. The computed flow fields differ appreciably from Stokes’ solution. The sphere will fall only when Yg is below 0.143 For yield stresses near this value, a plastic boundary layer forms next to the sphere. Boundary-layer scalings give the correct forms of the dependence of the drag coefficient and mass-transfer coefficient on yield stress for values near the critical one. The Stokes limit of zero yield stress is singular in the sense that for any small value of Yg there is a region of the flow away from the sphere where the plastic portion of the viscosity is at least as important as the Newtonian part. Calculations For the approach of the flow field to the Stokes result are in good agreement with the scalings derived from the matched asymptotic expansion valid in this limit.

Type
Research Article
Copyright
© 1985 Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Adachi, K. & Yoshioka N. 1973 On the creeping flow of a viscoplastic fluid past a circular cylinder. Chem. Engng Sci. 28, 215226.Google Scholar
Andres V. T. 1960 Equilibrium and motion of a sphere in a viscoplastic fluid. Dokl. Akad. Nauk SSSR 133, 777780.Google Scholar
Ansley, R. W. & Smith T. N. 1967 Motion of spherical particles in a Bingham plastic. AIChE J. 13, 11931196.Google Scholar
Baird, M. H. I. & Hamilelec A. E. 1962 Forced convection transfer around spheres at intermediate Reynolds numbers. Can. J. Chem. Engng 40, 119121.Google Scholar
Bercovier, M. & Engelman M. 1979 A finite element method for the numerical solution of viscous incompressible flows. J. Comp. Phys. 30, 181201.Google Scholar
Bercovier, M. & Engleman M. 1980 A finite element method for the incompressible non-Newtonian flows. J. Comp. Phys. 36, 313326.Google Scholar
Bhavaraju S. M., Mashelkar, R. A. & Blanch H. W. 1978 Bubble motion and mass transfer in non-Newtonian fluids. AIChE J. 24, 10631069.Google Scholar
Bingham E. C. 1922 Fluidity and Plasticity, pp. 215218. McGraw-Hill.
Bird R. B., Armstrong, R. C. & Hassager O. 1977 Dynamics of Polymeric Liquids, vol. 1. Wiley.
Bird R. B., Dai, G. C. & Yarusso B. J. 1983 The rheology of flow of viscoplastic materials. Rev. Chem. Engng 1, 170.Google Scholar
Brown R. A., Scriven, L. E. & Silliman W. J. 1980 Computer-aided analysis of nonlinear problems in transport phenomena. In New Methods in Nonlinear Dynamics (ed. P. Holmes), pp. 289307. SIAM.
dy Plessis M. P. & Ansley, R. W. 1967 Setting the parameters in solids pipelining. J. Pipeline Div. ASCE 93, 117.Google Scholar
Duvaut, G. & Lions J. L. 1976 Inequalities in Mechanics and Physics. Springer.
Ettouney, H. M. & Brown R. A. 1983 Finite element methods for steady solidification problems. J. Comp. Phys. 49, 118150.Google Scholar
Glowinski R., Lions, J. L. & Tremoliers R. 1981 Numerical Analysis of Variational Inequalities. North-Holland.
Hill R. 1950 The Mathematical Theory of Plasticity. Oxford University Press.
Hood P. 1976 Frontal solution program for unsymmetric matrices. Intl J. Num. Meth. Engng 10, 379399.Google Scholar
Hughes T. J. R., Liu, W. K. & Brooks A. 1979 Finite element analysis of incompressible viscous flows by the penalty function formulation. J. Comp. Phys. 30, 140.Google Scholar
Ito, S. & Kajiachi T. 1969 Drag force on a sphere moving in a plastic solid. J. Chem. Engng Japan 2, 1924.Google Scholar
Lipscomb, G. G. & Denn M. M. 1984 Flow of Bingham fluids in complex geometries. J. Non-Newt. Fluid Mech. 14, 337346.Google Scholar
Oldroyd J. G. 1947a A rational formulation of the equations of plastic flow for a Bingham solid. Proc. Camb. Phil. Soc. 43, 100105.Google Scholar
Oldroyd J. G. 1947b Two-dimensional plastic flow of a Bingham solid. A plastic boundary-layer theory for slow motion. Proc. Camb. Phil. Soc. 43, 383395.Google Scholar
Prager W. 1954 On slow viscoplastic flow. In Studies in Mathematics and Mechanics: R. von Mises Presentation Volume, pp. 208216. Academic.
Proudman, I. & Pearson J. R. A. 1957 Expansions at small Reynolds number for the flow past a sphere and a circular cylinder. J. Fluid Mech. 2, 237262.Google Scholar
Slater R. A. 1977 Engineering Plasticity, pp. 182236. Wiley.
Symonds P. S. 1949 On the general equations of problems of axial symmetry in the theory of plasticity. Q. Appl. Maths 6, 448452.Google Scholar
Valentic, L. & Whitmore R. L. 1965 The terminal velocity of spheres in Bingham plastics. Brit. J. Appl. Phys. 16, 11971203.Google Scholar
Van Dyke M. 1964 Perturbation Methods in Fluid Mechanics. Academic.
Volarovich, M. P. & Gutkin A. M. 1953 Theory of flow in a viscoplastic medium. Colloid J. 15, 153159.Google Scholar
Yoshioka, N. & Adachi K. 1971 On variational principles for a non-Newtonian fluid. J. Chem. Engng Japan 4, 217220.Google Scholar
Yoshioka N., Adachi, K. & Ishimura H. 1971 On creeping flow of a viscoplastic fluid past a sphere. Kagaku Kogaku 10, 11441152.Google Scholar