Abstract
Within the framework of the model of two interpenetrating continua, a horizontal laminar dilute-suspension flow in a vertical Hele-Shaw cell is investigated. Using the method of matched asymptotic expansions, an asymptotic model of the transverse migration of sedimenting particles is constructed. The particle migration in the horizontal section of the cell is caused by an inertial lateral force induced by the particle sedimentation and the shear flow of the carrier phase. A characteristic longitudinal length scale is determined, on which the particles migrate across the slot through a distance of the order of the slot half-width. The evolution of the particle number concentration and velocity fields along the channel is studied using the full Lagrangian method. Depending on the particle inertia parameter, different particle migration regimes (with and without crossing of the channel central plane by the particles) are detected. A critical value of the particle inertia parameter corresponding to the change in migration regime is found analytically. The possibility of intersection of the particle trajectories and the formation of singularities in the particle number concentration is demonstrated.
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References
H. Lamb, Hydrodynamics (Cambridge Univ. Press, Cambridge, 1932).
P.S. Hammond, “Settling and Slumping in a Newtonian Slurry, and Implications for Proppant Placement During Hydraulic Fracturing of Gas Wells”, Chem. Eng. Sci. 50(20), 3247–3260 (1995).
S.J. McCaffery, L. Elliott, and D.B. Ingham, Enhanced Sedimentation in Inclined Fracture Channels. Topics in Engng. 32 (CD-ROM ISBN 1-85312-546-6, 1997).
M.J. Economides and K.G. Notle, Reservoir Stimulation (John Wiley, New York, 2000).
G. Segre and A. Silberger, “Behavior of Macroscopic Rigid Spheres in Poiseuille Flow. Pt. 2. Experimental Results and Interpretation,” J. Fluid Mech. 14, 136–157 (1962).
R.C. Jeffrey and J.R.A. Pearson, “Particle Motion in Laminar Vertical Tube Flow,” J. Fluid Mech. 22, 721–735 (1965).
J.P. Matas, J.F. Morris, and E. Guazelli, “Inertial Migration of Rigid Spherical Particles in Poiseuille Flow,” J. Fluid Mech. 515, 171–195 (2004).
P.G. Saffman, “The Lift on a Small Sphere in a Slow Shear Flow,” J. Fluid Mech. 22, 385–400 (1965); Corrigendum. J. Fluid Mech. 31, 625 (1968).
E.S. Asmolov, “Dynamics of a Spherical Particle in a Laminar Boundary Layer,” Fluid Dynamics 25(6), 886 (1990).
J.B. McLaughlin, “The Lift on a Small Sphere in Wall-Bounded Linear Shear Flows,” J. Fluid Mech. 246, 249–265 (1993).
P. Vasseur and R.G. Cox, “The Lateral Migration of a Spherical Particle in Two-Dimensional Shear Flows,” J. Fluid Mech. 78, 385–413 (1976).
A.J. Hogg, “The Inertial Migration of Neutrally-Buoyant Spherical Particles in Two-Dimensional Shear Flows,” J. Fluid Mech. 272, 285–318 (1994).
E.S. Asmolov, “The Inertial Lift on a Spherical Particle in a Plane Poiseuille Flow at Large Channel Reynolds Number,” J. Fluid Mech. 381, 63–87 (1999).
E.S. Asmolov and A.A. Osiptsov, “The Inertial Lift on a Spherical Particle Settling in a Horizontal Viscous Flow Through a Vertical Slot,” Phys. Fluids, (2009) (under consideration).
F.E. Marble, “Dynamics of Dusty Gases,” Annu. Rev. Fluid Mech. 2, 397–446 (1971).
A.N. Osiptsov, “Mathematical Modeling of Dusty-Gas Boundary Layers,” Appl. Mech. Rev. 50, 357–370 (1997).
A.N. Osiptsov, “Motion of a Dusty Gas at the Entrance to a Flat Channel and a Circular Pipe,” Fluid Dynamics 23, 867–874 (1988).
M.R. Foster, P.W. Duck, and R.E. Hewitt, “Boundary layers in a Dilute Particle Suspension,” Proc. Roy. Soc. A 462(2068), 1145–1166 (2006).
A.N. Osiptsov “Investigation of Regions of Unbounded Growth of the Particle Concentration in Disperse Flows,” Fluid Dynamics 19, 378–384 (1984).
A.A. Osiptsov and E.S. Asmolov, “Asymptotic Model of the Inertial Migration of Particles in a Dilute Suspension Flow Through the Entry Region of a Channel,” Phys. Fluids 20(1–2), 123301 (2008).
A.N. Osiptsov, “Lagrangian Modeling of Dust Admixture in Gas Flows,” Astrophys. Space Sci. 274(1–2), 377–386 (2000).
S.A. Slater and J.B. Young, “The Calculation of Particle Transport in Dilute Gas-Particle Flows,” Int. J.Multiphase Flow 27, 61–87 (2001).
M.R. Maxey and J.R. Riley, “Equation of Motion of a Small Rigid Sphere in a Non-Uniform Flow,” Phys. Fluids 26, 883–889 (1983).
S.I. Chernyshenko, “Mean Distance between Particles in Dusty Gas in the Presence of Singularities in “Smeared” Particle Density,” Vestn. MSU, Ser. 1, Matem. Mekh. (1), 69–70 (1984).
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Original Russian Text © E.S. Asmolov, N.A. Lebedeva, A.A. Osiptsov, 2009, published in Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, 2009, Vol. 44, No. 3, pp. 85–101.
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Asmolov, E.S., Lebedeva, N.A. & Osiptsov, A.A. Inertial migration of sedimenting particles in a suspension flow through a Hele-Shaw cell. Fluid Dyn 44, 405–418 (2009). https://doi.org/10.1134/S0015462809030089
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DOI: https://doi.org/10.1134/S0015462809030089