Abstract
Numerical simulations of a spherical particle sedimenting in circular, triangular and square conduits containing a viscous, inertialess, Newtonian fluid were investigated using the Boundary Element Method (BEM). Settling velocities and pressure drops for spheres falling along the centre-lines of the conduits were computed for a definitive range of sphere sizes. The numerical simulations for the settling velocities showed good agreement with existing experimental data. The most accurate analytic solution for a sphere settling along the axis of a circular conduit produced results which were almost indistinguishable from the present BEM calculations. For a sphere falling along the centre-line of a square conduit, the BEM calculations for small spheres agreed well with analytic results. No analytic results for a sphere falling along the axis of a triangular conduit were available for comparison. Extrapolation of the BEM predictions for the pressure drops, to infinitely small spheres, showed remarkable agreement with analytic results. For the circular conduit, the sphere's settling velocity and angular velocity were computed as a function of drop position for small, medium and large spheres. Excellent agreement with a reflection solution was achieved for the small sphere. In addition, end effects were investigated for centre-line drops and compared where possible with available experimental data and analytic results.
Similar content being viewed by others
References
Bacon LR (1936) J Franklin Inst 221:251
Banerjee PK, Butterfield R (1981) Boundary element methods in engineering sciences. McGraw-Hill, London
Bathe KJ (1982) Finite element procedures in engineering analysis. Prentice-Hall: Englewood Cliffs, New Jersey
Bohlin T (1959) On the drag on a rigid sphere moving in a viscous liquid inside a cylindrical tube. Trans R Inst Tech (Stockholm) No 155
Brebbia CA, Telles JFC, Wrobel LC (1984) Boundary element techniques. Springer, Berlin
Brenner H (1962) Dynamics of a particle in a viscous fluid. Chem Eng Sci 17:435–446
Brenner H (1966) Hydrodynamic resistance of particles at small Reynolds numbers, Chapter 5 in Advances in Chemical Engineering 6:340–396 (Drew TB, Hoopes JW, Vermeulen T, eds). Academic Press, New York
Bush MB, Tanner RI (1983) Numerical solutions of viscous flows using integral equation methods. Int J Num Meth Fluids 3:71–92
Bush BM, Tanner RI (1990) In: Boundary element methods in non-linear fluid dynamics 6:285–317. (Banerjee PK, Morino J, eds). Elsevier Applied Science, New York
Faxen H (1923) Die Bewegung einer starren Kugel längs der Achse eines mit zäher Flüssigkeit gefüllten Rohres ariv. Arkiv fur Matematik, Astronomi och Fysik No. 27:17
Fidleris V, Whitmore RL (1961) Experimental determination of the wall effect for spheres falling axially in cylindrical vessels. Brit J Appl Phys 12:490–494
Graham AL, Mondy LA, Miller JD, Wagner NJ, Cook WA (1989) Numerical simulations of eccentricity and end effects in falling ball rheometry. J Rheol 133:1107–1128
Haberman WL, Sayre RM (1958) David Taylor Model Basin Report No. 1143. US Navy Department, Washington DC
Happel J, Byrne BJ (1954) Motion of a sphere and fluid in a cylindrical tube. Ind Eng Chem No 27, 46:1181–1186
Happel J, Bart E (1974) The settling of a sphere along the axis of a long square duct at low Reynolds number. Appl Sci Res 29:241–258
Happel J, Brenner H (1983) Low Reynolds number hydrodynamics. Martinus Nijhoff Publishers, Boston
Hirschfeld BR, Brenner H, Falade A (1984) First- and second-order wall effects upon the slow viscous asymmetric motion of an arbitrarily-shaped, -positioned and -oriented particle within a circular cylinder. Physico Chemical Hydrodynamics, No 2, 5:99–133
Kim S, Karrila SJ (1991) Microhydrodynamics: Principles and selected applications. Butterworth-Heinemann, Boston
Lachat JC, Watson JO (1976) Effective numerical treatment of boundary integral equations: A formulation for three-dimensional elastostatics. Int J Num. Meth Eng 10:991–1005
Ladenburg R (1907) Über den Einfluß von Wänden auf die Bewegung einer Kugel in einer reibenden Flüissigkeit. Annalen der Physik 23:447–358
Miyamura A, Iwasaki S, Ishii T (1981) Experimental wall correction factors of single solid spheres in triangular and square cylinders and parallel plates. Int J Multiphase Flow 7:41–66
Smoluchowski M (1911) On the mutual action of spheres which move in a viscous liquid. Bull Acad Sci Cracovie A 1:28–39
Sonshine RM, Cox RG, Brenner H (1966) The Stokes translation of a particle of arbitrary shape along the axis of a circular cylinder filled to a finite depth with viscous liquid. I. Appl Sci Res 16:273–300
Stokes GG (1851) On the effect of the internal friction of fluids on the motion of pendulums. Trans Cambridge Philos Soc, Pt II, 9:8–27
Tanner RI (1963) End effects in falling-ball viscometry. J Fluid Mech 17:161–170
Tran-Cong T, Phan-Thien N (1988) Three dimensional study of extrusion processes by the Boundary Element Method. Part 1: an implementation of higher order elements and some Newtonian results. Rheol Acta 27:21–30
Tran-Cong T, Phan-Thien N (1989) ‘Stokes’ problems of multiparticle systems: A numerical method for arbitrary flows. Phys Fluids A 13:453–461
Youngren GK, Acrivos A (1975) Stokes flow past a particle of arbitrary shape: a numerical method of solution. J Fluid Mech 69:377–403
Author information
Authors and Affiliations
Additional information
Los Alamos National Laboratory, Los Alamos, New Mexico, USA.
Rights and permissions
About this article
Cite this article
Tullock, D.L., Phan Thien, N. & Graham, A.L. Boundary element simulations of spheres settling in circular, square and triangular conduits. Rheola Acta 31, 139–150 (1992). https://doi.org/10.1007/BF00373236
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF00373236