Abstract
The present work presents an experimental study on a free-falling rigid sphere in a quiescent incompressible newtonian fluid, placed in an oscillating frame. The goal of this investigation is to examine the effect of the history force acting on the sphere at small Reynolds numbers (Re≤2.5) and finite Strouhal numbers (1≤Sl≤20). The particle trajectory is measured by using a high-speed video camera and modern techniques of image processing. The average terminal velocity, the oscillation magnitude, and the phase shift with the oscillating frame are measured and compared with those obtained from theoretical approaches. The comparison is made by solving the equation of motion of the sphere with and without the history force. In addition to the significant role that this force plays in the momentum balance, it was found that the correction of the added mass force and the history force by the empirical coefficients of Odar and Hamilton (J Fluid Mech 18:302–314, 1964; J Fluid Mech 25:591–592, 1966) are not necessary in our Re and Sl ranges. The added mass is the same as that obtained by the potential flow theory and the history force is well predicted by the Basset expression (Treatise on hydrodynamics, 1888).
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References
Auton TR, Hunt JRC, Prud’homme M (1988) The force exerted on a body in inviscid unsteady non-uniform rotational flow. J Fluid Mech 197:241–257
Basset AB (1888) Treatise on hydrodynamics. Deighton Bell, London
Boussinesq VJ (1885) Sur la résistance qu’oppose un liquide indéfini en repos. C R Acad Sci 100:935–937
Chang EJ, Maxey MR (1994) Unsteady flow about a sphere at low to moderate Reynolds number. Part 1. Oscillatory motion. J Fluid Mech 277:347–379
Chang EJ, Maxey MR (1995) Unsteady flow about a sphere at low to moderate Reynolds number. Part 2. Accelerated motion. J Fluid Mech 303:133–153
Chester W, Breach DR (1969) On the flow past a sphere at low Reynolds number. J Fluid Mech 37:751–760
Clift R, Grace JR, Weber ME (1978) Bubbles, drops, and particles. Academic Press, New York
Dennis SCR, Walker JDA (1972) Calculation of the steady flow past a sphere at low and moderate Reynolds numbers. J Fluid Mech 27(4):771–789
Karanfilian SK, Katos TJ (1978) Drag on a sphere in unsteady motion in a liquid at rest. J Fluid Mech 87(1):85–96
Kim I, Eghobashi S, Sirigano WA (1998) On the equation for spherical-particle motion: effect of Reynolds and acceleration numbers. J Fluid Mech 367:221–253
Lovalenti PM, Brady JF (1993) The force on a bubble, drop or particle in arbitrary time-dependent motion at small Reynolds number. Phys Fluids A 5(9):2104–2116
Maxey MR, Riley JJ (1983) Equation of motion for a small rigid sphere in a nonuniform flow. Phys Fluids 26(4):883–889
Maxworthy T (1965) Accurate measurements of sphere drag at low Reynolds numbers. J Fluid Mech 23:369–372
Mei R (1994) Flow due to an oscillating sphere and an expression for unsteady drag on the sphere at finite Reynolds number. J Fluid Mech 270:133–174
Mei R, Adrian RJ (1992) Flow past a sphere with an oscillation in the free-stream velocity and unsteady drag at finite Reynolds number. J Fluid Mech 237:323–341
Mei R, Lawrence CJ, Adrian RJ (1991) Unsteady drag on a sphere at finite Reynolds number with small fluctuations in the free-stream velocity. J Fluid Mech 233:613–631
Michaelides EE (2003) Hydrodynamic force and heat/mass transfer from particles, bubbles, and drops – The Freeman Scholar Lecture. J Fluids Eng 125:209–238
Mordant N, Pinton JF (2000) Velocity measurement of a settling sphere. Eur Phys J B 18:343–352
Ockendon JR, Evans GA (1972) The drag on a sphere in low Reynolds number flow. J Aeros Sci 3(4):237–242
Odar F, Hamilton WS (1964) Forces on a sphere accelerating in a viscous fluid. J Fluid Mech 18:302–314
Odar F, Hamilton WS (1966) Verification of the proposed equation for calculation of the forces on a sphere accelerating in a viscous fluid. J Fluid Mech 25:591–592
Oseen C (1910) Uber die Stokes’sche formel und uber eine verwandte aufabe in der hydrodynamik. Ark Mat Aston Fys 6(69)
Oseen C (1927) Hydrodynamik. Akademische, Leipzig
Proudman I, Pearson JRA (1957) Expansion at small Reynolds number for the flow past sphere and circular cylinder. J Fluid Mech 2:237–262
Rivero M, Magnaudet J, Fabre J (1991) Quelques résultats nouveaux concernant les forces exercées sur une inclusion sphérique en écoulement accéléré. C R Acad Sci, Ser II 312:1499–1506
Roos FW, Willmarth WW (1971) Some experimental results on sphere and disk drag. AIAA J 9(2):285–291
Schiller L, Nauman A (1933) Uber die drundlegende berechnung bei der schwekraftaufbereitung. Ver Deutch Ing 44:318–320; Multiphase Flows 2:307–317
Schoeneborn PR (1975) The interaction between a single sphere and a oscillating fluid. Int J Multiphase Flows 2:307–317
Stokes GG (1851) On the effect of the internal friction of fluids on the motion of a pendulum. Trans Camb Phil Soci 9:8–106
Tsuji Y, Kato N, Tanaka T (1991) Experiment on the unsteady drag and wake of a sphere at high Reynolds numbers. Int J Multiphase Flows 17(3):343–354
Van Dyke M (1970) Extension of Goldstein’s series for the Oseen drag of a sphere. J Fluid Mech 44(2):365–72
Vojir DJ, Michaelides EE (1994) Effect of the history term on the motion of rigid spheres in a viscous fluid. Int J Multiphase Flow 20(3):547–556
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Abbad, M., Souhar, M. Effects of the history force on an oscillating rigid sphere at low Reynolds number. Exp Fluids 36, 775–782 (2004). https://doi.org/10.1007/s00348-003-0759-x
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DOI: https://doi.org/10.1007/s00348-003-0759-x