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Equation of motion for a sphere in non-uniform compressible flows

Published online by Cambridge University Press:  11 April 2012

M. Parmar ,
A. Haselbacher  and
S. Balachandar
M. Parmar
Affiliation:
Department of Mechanical and Aerospace Engineering, University of Florida, Gainesville, FL 32611-6250, USA
A. Haselbacher
Affiliation:
Department of Mechanical and Aerospace Engineering, University of Florida, Gainesville, FL 32611-6250, USA
S. Balachandar*
Affiliation:
Department of Mechanical and Aerospace Engineering, University of Florida, Gainesville, FL 32611-6250, USA
*
†Email address for correspondence: [email protected]
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Abstract

Linearized viscous compressible Navier–Stokes equations are solved for the transient force on a spherical particle undergoing unsteady motion in an inhomogeneous unsteady ambient flow. The problem is formulated in a reference frame attached to the particle and the force contributions from the undisturbed ambient flow and the perturbation flow are separated. Using a density-weighted velocity transformation and reciprocal relation, the total force is first obtained in the Laplace domain and then transformed to the time domain. The total force is separated into the quasi-steady, inviscid unsteady, and viscous unsteady contributions. The above rigorously derived particle equation of motion can be considered as the compressible extension of the Maxey–Riley–Gatignol equation of motion and it incorporates interesting physics that arises from the combined effects of inhomogeneity and compressibility.

JFM classification

Compressible Flows: Shock waves
Type
Papers
Information
Journal of Fluid Mechanics , Volume 699 , 25 May 2012 , pp. 352 - 375
Copyright
Copyright © Cambridge University Press 2012

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References

1. Auton, T. R., Hunt, J. C. R. & Prud’homme, M. 1988 The force exerted on a body in inviscid unsteady non-uniform rotational flow. J. Fluid Mech. 197, 241–257.Google Scholar
2. Bagchi, P. & Balachandar, S. 2002 Steady planar straining flow past a rigid sphere at moderate Reynolds numbers. J. Fluid Mech. 466, 365–407.Google Scholar
3. Bagchi, P. & Balachandar, S. 2003 Inertial and viscous forces on a rigid sphere in straining flows at moderate Reynolds numbers. J. Fluid Mech. 481, 105–148.Google Scholar
4. Basset, A. B. 1888 A Treatise on Hydrodynamics. Deighton, Bell and Company.Google Scholar
5. Bedeaux, D. & Mazur, P. 1974 A generalization of Faxén theorem to nonsteady motion of a sphere through a compressible fluid in arbitrary flow. Physica 78, 505–515.CrossRefGoogle Scholar
6. Boussinesq, J. 1885 Sur la résistance qu’oppose un liquide indéfini au repos au mouvement varié d’une sphère solide. C. R. Acad. Sci. Paris 100, 935–937.Google Scholar
7. Chow, T. S. & Hermans, J. J. 1973 Brownian motion of a spherical particle in a compressible fluid. Physica 65, 156–162.Google Scholar
8. Cunha, F. R., Sousa, A. J. D. & Loewenberg, M. 2003 A mathematical formulation of the boundary integral equations for a compressible stokes flow. Comput. Appl. Math. 22, 53–73.Google Scholar
9. Eames, I. & Hunt, J. C. R. 2004 Forces on bodies moving unsteadily in rapidly compressed flows. J. Fluid Mech. 505, 349–364.Google Scholar
10. Faxén, H. 1924 Der Widerstand gegen die Bewegung einer starren Kugel in einer zähen Flüssigkeit, die zwischen zwei parallelen, ebenen Wänden eingeschlossen ist. Arkiv. Mat. Astron. Fys. 18, 3.Google Scholar
11. Felderhof, B. U. 2005 Backtracking of a sphere slowing down in a viscous compressible fluid. J. Chem. Phys. 123 (4), 044902.CrossRefGoogle Scholar
12. Felderhof, B. U. 2007 Effect of fluid compressibility on the flow caused by a sudden impulse applied to a sphere immersed in a viscous fluid. Phys. Fluids 19 (12), 126101.Google Scholar
13. Gatignol, R. 1983 The Faxén formulae for a rigid particle in an unsteady non-uniform Stokes flow. J. Méc. Théor. Appl. 1, 143–160.Google Scholar
14. Guz, A. N. 2009 Dynamics of Compressible Viscous Fluid. Cambridge University Press.Google Scholar
15. Happel, J. & Brenner, H. 1965 Low Reynolds Number Hydrodynamics. Prentice-Hall.Google Scholar
16. Kaneda, Y. 1980 A generalization of Faxén’s theorem to nonsteady motion of an almost spherical drop in an arbitrary flow of a compressible fluid. Physica A 101, 407–422.Google Scholar
17. Kim, I., Elghobashi, E. & Sirignano, W. A. 1998 On the equation for spherical-particle motion: effect of Reynolds and acceleration numbers. J. Fluid Mech. 267, 221–253.Google Scholar
18. Ling, Y., Haselbacher, A. & Balachandar, S. 2011a Importance of unsteady contributions to force and heating for particles in compressible flows. Part 1. Modelling and analysis for shock–particle interaction. Intl J. Multiphase Flow 37, 1026–1044.Google Scholar
19. Ling, Y., Haselbacher, A. & Balachandar, S. 2011b Importance of unsteady contributions to force and heating for particles in compressible flows. Part 2. Application to particle dispersal by blast wave. Intl J. Multiphase Flow 37, 1013–1025.CrossRefGoogle Scholar
20. Longhorn, A. L. 1952 The unsteady, subsonic motion of a sphere in a compressible inviscid fluid. Q. J. Mech. Appl. Maths 5, 64–81.Google Scholar
21. Lovalenti, P. M. & Brady, J. F. 1993 The force on a sphere in a uniform-flow with small-amplitude oscillations at finite Reynolds-number. J. Fluid Mech. 256, 607–614.CrossRefGoogle Scholar
22. Maxey, M. R. & Riley, J. J. 1983 Equation of motion for a small rigid sphere in a non-uniform flow. Phys. Fluids 26 (4), 883–889.Google Scholar
23. Mazur, P. & Bedeaux, D. 1974 A generalization of Faxén theorem to nonsteady motion of a sphere through an incompressible fluid in arbitrary flow. Physica 76, 235–246.Google Scholar
24. Mei, R. W. & Adrian, R. J. 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.Google Scholar
25. Metiu, H., Oxtoby, D. W. & Freed, K. F. 1977 Hydrodynamic theory for vibrational relaxation in liquids. Phys. Rev. A 15 (1), 361–371.Google Scholar
26. Oseen, C. W. 1927 Hydrodynamik. Akademische Verlagsgesellschaft.Google Scholar
27. Parmar, M. 2010 Unsteady forces on a particle in compressible flows, PhD thesis, University of Florida.CrossRefGoogle Scholar
28. Parmar, M., Haselbacher, A. & Balachandar, S. 2008 On the unsteady inviscid force on cylinders and spheres in subcritical compressible flow. Phil. Trans. R. Soc. Lond. 366 (1873), 2161–2175.Google ScholarPubMed
29. Parmar, M., Haselbacher, A. & Balachandar, S. 2009 Modelling of the unsteady force for shock–particle interaction. Shock Waves 19 (4), 317–329.Google Scholar
30. Parmar, M., Haselbacher, A. & Balachandar, S. 2010 Improved drag correlation for spheres and application to shock-tube experiments. AIAA J. 48 (6), 1273–1276.Google Scholar
31. Parmar, M., Haselbacher, A. & Balachandar, S. 2011 Generalized Basset–Boussinesq–Oseen equation for unsteady forces on a sphere in a compressible flow. Phys. Rev. Lett. 106, 084501.CrossRefGoogle Scholar
32. Rubinow, S. I. & Keller, J. B. 1961 The transverse force on a spinning sphere moving in a viscous fluid. J. Fluid Mech. 11, 447–459.Google Scholar
33. Skews, B. W., Bredin, M. S. & Efune, M. 2007 Drag measurements in unsteady compressible flow. Part 2: Shock wave loading of spheres and cones. R&D J. South African Inst. Mech. Engng 23, 13–19.Google Scholar
34. Stokes, G. G. 1851 On the effect of the internal friction of fluids on the motion of pendulums. Trans. Camb. Phil. Soc. 9, 8–106.Google Scholar
35. Temkin, S. & Leung, C. M. 1976 Velocity of a rigid sphere in a sound-wave. J. Sound Vib. 49 (1), 75–92.Google Scholar
36. Zwanzig, R. & Bixon, M. 1970 Hydrodynamic theory of the velocity correlation function. Phys. Rev. A 2 (5), 2005–2012.CrossRefGoogle Scholar