Abstract
The problem of the rectilinear oscillations of two spherical particles along the line through their centers in an axi-symmetric, viscous, incompressible flow at low Reynolds number is considered. The particles oscillate with the same frequency and with different amplitudes. In addition, the particles may differ in their sizes. In order to solve the Stokes equations for the flow field, a general solution is constructed from the superposition of the basic solutions in the two spherical coordinate systems based at the centers of the particles. A collocation technique is used to satisfy the boundary conditions on the surfaces of the particles. The solution is valid for all values of the frequency parameter subject to the conditions that justify the use of the unsteady Stokes equations. Numerical results displaying the in phase and the out of phase force amplitudes acting on each particle are obtained with good convergence for various values of the physical parameters of the problem. The results are tabulated and represented graphically. Our results agree well with the existing solutions of the steady motion of two spherical particles and with the oscillations of single particle.
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References
Ardekani AM, Range RH (2006) Unsteady motion of two solid spheres in Stokes flow. Phys Fluids 18(10):103306
Basak S, Raman A (2007) Hydrodynamic coupling between micromechanical beams oscillating in viscous fluids. Phys Fluids 19:017105
Brenner H (1961) The slow motion of a sphere through a viscous fluid towards a plane surface. Chem Eng Sci 16:242–251
Chadwick RS, Liaot Z (2008) High-frequency oscillations of a sphere in a viscous fluid near a rigid plane. SIAM Rev Soc Ind Appl Math 50(2):313–322
Chen SH, Keh HJ (1995) axisymmetric motion of two spherical particles with slip surfaces. J Colloid Interface Sci 171:63–72
Clercx HJH (1997) Scaling of transient hydrodynamic interactions in hard sphere suspensions. Phys Rev E 56:2950
Clercx HJH, Schram PPJM (1991) Retarded hydrodynamic interactions in suspensions. Phys A 174:325–354
Cooley MDA, O’Neill ME (1969) On the slow motion generated in a viscous fluid by the approach of a sphere to a plane wall or stationary sphere. Mathematika 16:34–49
El-Sapa S, Saad EI, Faltas MS (2018) Axisymmetric motion of two spherical particles in a Brinkman medium with slip surfaces. Eur J Fluid Mech B Fluids 67:306–313
Felderhof BU (2012) Hydrodynamic force on a particle oscillating in a viscous fluid near a wall with dynamic partial-slip boundary condition. Phys Rev E 85:046303
Ganatos P, Weinbaum S, Pfefer R (1980) A strong interaction theory for the creeping motion of a sphere between plane parallel boundaries. Part 1. Perpendicular motion. J Fluid Mech 99:739–753
Happel J, Brenner H (1983) Low reynolds number hydrodynamics. Nijhoff, Dordrecht
Jeffrey DJ, Onishi Y (1984) Calculation of the resistance and mobility functions for two unequal rigid spheres in low-Reynolds-number flow. J Fluid Mech 139:261–290
Kanwal RP (1955) Rotatory and longitudinal oscillations of axi-symmetric bodies in a viscous fluid. Q J Mech Appl Math 8:146
Kanwal RP (1964) Drag on an axially symmetric body vibrating slowly along its axis in a viscous fluid. J Fluid Mech 19:631
Kim S, Karrila SJ (1991) Microhydrodynamics: principles and selected applications. Butterworth-Heinemann, Boston
Lawrence CJ, Weinbaum S (1988) The unsteady force on a body at low Reynolds number; the axisymmetric motion of a spheroid. J Fluid Mech 189:463
Loewenberg M (1993a) Stokes resistance, added mass, and Basset force for arbitrarily oriented, finite-length cylinders. Phys Fluids A 5:765
Loewenberg M (1993b) The unsteady Stokes resistance of arbitrarily oriented, finite-length cylinders. Phys Fluids A 5:3004
Loewenberg M (1994) Axisymmetric unsteady Stokes flow past an oscillating finite-length cylinder. J Fluid Mech 265:265
Lovalenti PM, Brady JF (1993) The force on a sphere in a uniform flow with small-amplitude oscillations at finite Reynolds number. J Fluid Mech 256:607
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
Payatakes AC, Dassios G (1987) Creeping flow around and through a permeable sphere moving with constant velocity towards a solid wall. Chem Eng Commun 58:119–138
Pozrikidis C (1989) A study of linearized oscillatory flow past particles by the boundary-integral method. J Fluid Mech 202:17
Schram PPJM, Usenko AS, Yakimenko IP (2006) Retarded many-sphere hydrodynamic interactions in a viscous fluid. J Phys A 39:12
Stokes GG (1922) On the effect of fluids on the motion of pendulums. Trans. Camb. Phil. (1851), Soc. 9, 8. Reprinted in mathematical and physical papers III, Cambridge University Press
Tabakova SS, Zapryanov ZD (1982a) On the hydrodynamic interaction of two spheres oscillating in a viscous fluid—I axisymmetrical case. ZAMP 33:344–357
Tabakova SS, Zapryanov ZD (1982b) On the hydrodynamic interaction of two spheres oscillating in a viscous fluid—II three dimensional case. ZAMP 33:487–502
Yap YW, Sader JE (2016) Sphere oscillating in a rarefied gas. J Fluid Mech 794:109–153
Zhang W, Stone HA (1998) Oscillatory motions of circular disks and nearly spherical particles in viscous flows. J Fluid Mech 367:329–358
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Faltas, M.S., El-Sapa, S. Rectilinear oscillations of two spherical particles embedded in an unbounded viscous fluid. Microsyst Technol 25, 39–49 (2019). https://doi.org/10.1007/s00542-018-3928-9
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DOI: https://doi.org/10.1007/s00542-018-3928-9