Abstract
A boundary element method is used to simulate the unsteady motion of a sphere falling under gravity along the centreline of a cylindrical tube containing a viscoelastic fluid. The fluid is modelled by the upper-convected Maxwell constitutive equation. Results show that the viscoelasticity of the liquid leads to a damped oscillation in sphere velocity about its terminal value. The maximum sphere velocity, which occurs in the first overshoot, is approximately proportional to the square root of the Weissenberg number when the ratio of the sphere radius to the tube radius is sufficiently small. Particular attention is also paid to the wall effects. It is shown that a closer wall reduces the oscillatory amplitude of the sphere velocity but increases its frequency. The results suggest that the falling-ball technique, which is now widely used for viscosity measurement, might also be used for the determination of a relaxation time for a viscoelastic fluid.
Similar content being viewed by others
References
Banerjee PK, Butterfield R (1981) Boundary element methods in engineering science. McGraw-Hill, London
Bird RB, Armstrong RC, Hassager O (1987) Dynamics of polymeric liquids: vol I. Fluid Mechanics, 2nd ed. Wiley, New York
Brebbia CA (1980) The boundary element method for engineers. Pentech Press, London
Bush MB, Tanner RI (1983) Numerical solution of viscous flows using integral equation methods. Int J Num Meth Fluids 3:71–92
Carew E, Townsend P (1988) Non-Newtonian flow past a sphere in a long cylindrical tube. Rheol Acta 27:125–129
Caswell B, Schwarz WH (1962) The creeping motion of a non-Newtonian flow past a sphere. J Fluid Mech 13:417–426
Chilcott MD, Rallison JM (1988) Creeping flow of dilute solution past cylinder and sphere. J Non-Newt Fluid Mech 29:381–432
Crochet MJ (1988) Numerical simulation of highly viscoelastic flows. 10th Int Congr on Rheol, Sydney 1:19–24
Debbaut B, Crochet MJ (1988) Extensional effects in complex flows. Int J Num Meth Fluids 30:169–184
Fielder R, Thomas RH (1967) The unsteady motion of a lamina in an elastico-viscous liquid. Rheol Acta 6:306–311
Happel J, Brenner H (1973) Low Reynolds number hydrodynamics. Noordhoff, Leiden
Harlen OG (1990) High-Deborah-number flow of a dilute polymer solution past a sphere falling along the axis of a cylindrical tube. J Non-Newt Fluid Mech 37:157–173
Hassager O, Bisgaard C (1983) A Langrangian finite element method for the simulation of flow of non-Newtonian liquids. J Non-Newt Fluid Mech 12:153–164
Jin H, Phan-Thien N, Tanner RI (1991) A finite element analysis of the flow past a sphere in a cylindrical tube. Computational Mechanics 8:409–422
King MJ, Waters ND (1972) The unsteady motion of a sphere in an elastico-viscous liquid. J Phys D: Appl Phys 5:141–150
Lai RYS (1974) Drag on a sphere accelerating rectilinearly in a Maxwell fluid. Int J Engng Sci 12:645–655
Lai RYS, Fan CP (1978) Drag on a sphere accelerating rectilinearly in an elastico-viscous fluid. Int J Engng Sci 16:303–311
Landau LD, Lifshitz EM (1982) Fluid mechanics. Pergamon, New York
Leal LG, Denn MM, Keunings R (1988) Proceedings of the Lake Arrowhead Workshop — Introduction. J Non-Newt Fluid Mech 29:1–8
Leslie FM (1961) The slow flow of a visco-elastic liquid past a sphere. Quart J Mech Appl Math 14:36–48 (with an appendix by R.I. Tanner)
Lunsmann WJ, Armstrong RC, Brown RA (1989) Paper presented at 6th Workshop on Numerical Methods in Non-Newtonian Flows. Denmark
Lunsmann WJ, Northey PJ, Armstrong RC, Brown RA (1991) Recent computations for viscoelastic flow around a sphere moving in a cylindrical tube. Paper presented at 63rd Rheology Society Annual Meeting, New York, October 1991
Luo XL, Tanner RI (1986) A streamline element scheme for solving viscoelastic flow problem, part 1: Differential constitutive equations. J Non-Newt Fluid Mech 21:179–199
Sugeng F, Tanner RI (1986) The drag on spheres in viscoelastic fluids with significant wall effects. J Non-Newt Fluid Mech 20:281–292
Tanner RI (1988) Engineering theology. Revised edition. Oxford University Press, Oxford
Thomas RH, Walters K (1966) The unsteady motion of a sphere in an elastico-viscous liquid. Rheol Acta 5:23–27
Tiefenbruck G, Leal LG (1982) A numerical study of the motion of a viscoelastic fluid past rigid spheres and spherical bubbles. J Non-Newt Fluid Mech 10:115–155
Ultman JS, Denn MM (1971) Slow viscoelastic flow past submerged objects. Chem Eng J 2:81–89
Zheng R (1991) Boundary element method for some problems in fluid mechanics and theology. PhD Thesis, The University of Sydney
Zheng R, Coleman CJ, Phan-Thien N (1991a) A boundary element approach for non-homogeneous potential problems. Computational Mechanics 7:729–288
Zheng R, Phan-Thien N, Coleman CJ (1991b) A boundary element approach for non-linear boundary value problems. Computational Mechanics 8:71–86
Zheng R, Phan-Thien N, Tanner RI (1990a) On the flow past a sphere in a cylindrical tube: Limiting Weissenberg number. J Non-Newt Fluid Mech 36:27–49
Zheng R, Phan-Thien N, Tanner RI (1991c) The flow past a sphere in a cylindrical tube: effects of inertia, shearthinning and elasticity. Rheol Acta 30:499–510
Zheng R, Phan-Thien N, Tanner RI, Bush MB (1990b) Numerical analysis of viscoelastic flow through a sinusoidally corrugated tube using a boundary element method. J Rheol 34:79–102
Zheng R, Tanner RI (1988) A numerical analysis of calendering. J Non-Newt Fluid Mech 28:149–170
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Zheng, R., Phan-Thien, N. A boundary element simulation of the unsteady motion of a sphere in a cylindrical tube containing a viscoelastic fluid. Rheola Acta 31, 323–332 (1992). https://doi.org/10.1007/BF00418329
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/BF00418329