Creeping motion of a sphere through a Bingham plastic

Journal of Fluid Mechanics - Tập 158 - Trang 219-244 - 1985
Antony N. Beris1, John Tsamopoulos1, R. C. Armstrong1, Robert A. Brown1
1Department of Chemical Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139.

Tóm tắt

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 stressYgbelow which the material acts as a solid. The computed flow fields differ appreciably from Stokes’ solution. The sphere will fall only whenYgis 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 ofYgthere 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.

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Tài liệu tham khảo

Baird, M. H. I. & Hamilelec A. E. 1962 Forced convection transfer around spheres at intermediate Reynolds numbers.Can. J. Chem. Engng 40,119–121.

Prager W. 1954 On slow viscoplastic flow. In Studies in Mathematics and Mechanics: R. von Mises Presentation Volume ,pp.208–216.Academic.

Oldroyd J. G. 1947b Two-dimensional plastic flow of a Bingham solid. A plastic boundary-layer theory for slow motion.Proc. Camb. Phil. Soc. 43,383–395.

Slater R. A. 1977 Engineering Plasticity ,pp.182–236.Wiley.

Valentic, L. & Whitmore R. L. 1965 The terminal velocity of spheres in Bingham plastics.Brit. J. Appl. Phys. 16,1197–1203.

Oldroyd J. G. 1947a A rational formulation of the equations of plastic flow for a Bingham solid.Proc. Camb. Phil. Soc. 43,100–105.

Bingham E. C. 1922 Fluidity and Plasticity ,pp.215–218.McGraw-Hill.

Hill R. 1950 The Mathematical Theory of Plasticity .Oxford University Press.

dy Plessis M. P. & Ansley, R. W. 1967 Setting the parameters in solids pipelining.J. Pipeline Div. ASCE 93,1–17.

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.30,1–40.

Bercovier, M. & Engelman M. 1979 A finite element method for the numerical solution of viscous incompressible flows.J. Comp. Phys. 30,181–201.

Van Dyke M. 1964 Perturbation Methods in Fluid Mechanics .Academic.

Symonds P. S. 1949 On the general equations of problems of axial symmetry in the theory of plasticity.Q. Appl. Maths 6,448–452.

Bird R. B. , Dai, G. C. & Yarusso B. J. 1983 The rheology of flow of viscoplastic materials.Rev. Chem. Engng 1,1–70.

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. 2,237–262.

Lipscomb, G. G. & Denn M. M. 1984 Flow of Bingham fluids in complex geometries.J. Non-Newt. Fluid Mech. 14,337–346.

Hood P. 1976 Frontal solution program for unsymmetric matrices.Intl J. Num. Meth. Engng 10,379–399.

Bhavaraju S. M. , Mashelkar, R. A. & Blanch H. W. 1978 Bubble motion and mass transfer in non-Newtonian fluids.AIChE J. 24,1063–1069.

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.289–307.SIAM.

Adachi, K. & Yoshioka N. 1973 On the creeping flow of a viscoplastic fluid past a circular cylinder.Chem. Engng Sci. 28,215–226.

Yoshioka, N. & Adachi K. 1971 On variational principles for a non-Newtonian fluid.J. Chem. Engng Japan 4,217–220.

Glowinski R. , Lions, J. L. & Tremoliers R. 1981 Numerical Analysis of Variational Inequalities .North-Holland.

Ansley, R. W. & Smith T. N. 1967 Motion of spherical particles in a Bingham plastic.AIChE J. 13,1193–1196.

Yoshioka N. , Adachi, K. & Ishimura H. 1971 On creeping flow of a viscoplastic fluid past a sphere.Kagaku Kogaku 10,1144–1152.

Ito, S. & Kajiachi T. 1969 Drag force on a sphere moving in a plastic solid.J. Chem. Engng Japan 2,19–24.

Bercovier, M. & Engleman M. 1980 A finite element method for the incompressible non-Newtonian flows.J. Comp. Phys. 36,313–326.

Bird R. B. , Armstrong, R. C. & Hassager O. 1977 Dynamics of Polymeric Liquids , vol. 1.Wiley.

Ettouney, H. M. & Brown R. A. 1983 Finite element methods for steady solidification problems.J. Comp. Phys. 49,118–150.

Andres V. T. 1960 Equilibrium and motion of a sphere in a viscoplastic fluid.Dokl. Akad. Nauk SSSR 133,777–780.

Volarovich, M. P. & Gutkin A. M. 1953 Theory of flow in a viscoplastic medium.Colloid J. 15,153–159.

Duvaut, G. & Lions J. L. 1976 Inequalities in Mechanics and Physics .Springer.

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Thông tin xuất bản

Journal of Fluid Mechanics

Tập 158

219-244

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