Migration of a Drop in Simple Shear Flow at Finite Reynolds Numbers: Size and Viscosity Ratio Effects
The migration of a deformable drop in simple shear
flow at finite Reynolds numbers is investigated numerically by
solving the full Navier-Stokes equations using a finite
difference/front tracking method. The objectives of this study are to
examine the effectiveness of the present approach to predict the
migration of a drop in a shear flow and to investigate the behavior of
the drop migration with different drop sizes and non-unity viscosity
ratios. It is shown that the drop deformation depends strongly on the
capillary number, so that; the proper non-dimensional number for the
interfacial tension is the capillary number. The rate of migration
increased with increasing the drop radius. In other words, the
required time for drop migration to the centreline decreases. As the
viscosity ratio increases, the drop rotates more slowly and the
lubrication force becomes stronger. The increased lubrication force
makes it easier for the drop to migrate to the centre of the channel.
The migration velocity of the drop vanishes as the drop reaches the
centreline under viscosity ratio of one and non-unity viscosity ratios.
To validate the present calculations, some typical results are
compared with available experimental and theoretical data.
[1] Taylor, G.I., The deformation of emulsions in definable fields of flow,
Proc,Ray.Soc. (London), 1934, A146, 501-523.
[2] Karnis,A. and Mason,S.G., Particle motions in sheared suspensions.
XXΙΙΙ. Wall migration of fluid drops, J.Colloid and Inerface. Science,
1967, 24, 164-169.
[3] Halow,J.S., and Willis,G.B., Radial migration of spherical particles in
Couette system, AICHE J., 1970, 16, 281-286.
[4] Rallison, J.M., The deformation of small viscous drops and bubbles in
shear flows, Annu, Rev. Fluid Mech., 1984, 16, 45-66.
[5] Magna, M. and Stone, H.A., Buoyancy-driven interactions between two
deformable viscous drops, J.Fluid Mech, 1993, 256, 647-683.
[6] Zhou, H. and Pozrikidis,C., The flow of suspensions in channels: single
files of drops, Phys. Fluids, 1993, A5(2), 311-324.
[7] Feng, J., Hu, H.H., and Joseph, D.D., Direct simulation of initial value
problems for the motion of solid bodies in a Newtonian fluid. Part2.
Couette and Poiseuille flows, J.Fluid Mech, 1994, 277, 271-301.
[8] Li, X., Zhou H. and Pozrikidis,C., A numerical study of the shearing
motion of emulsions and foams, J.Fluid Mech, 1995, 286, 374-404.
[9] Loewenberg, M. and Hinch, E., Numerical simulation of a concentrated
emulsion in shear flow, J.Fluid Mech., 1996, 321, 395-419.
[10] Esmaeeli, A., and Tryggvason, G., Direct numerical simulations of
bubbly flows Part1. Low Reynolds number arrays, J.Fluid Mech, 1998,
377, 313-345.
[11] Mortazavi, S.S. and Tryggvasson, G., A numerical study of the motion
of drop in Poiseuille flow, part1: lateral migration of one drop, J.Fluid
Mech, 1999, 411, 325-350.
[12] Esmaeeli, A., and Tryggvason, G., Direct numerical simulations of
bubbly flows Part2. Low Reynolds number arrays, J.Fluid Mech, 1999,
385, 325-358.
[13] Balabel A., Binninger B., Herrmann M. and Peters N., Calculation of
droplet deformation by surface tension effects using the Level Set
method, J. Combustion Science and Technology, 2002, 174, 257-278.
[14] [Crowdy D.G., Compressible bubbles in Stokes flow, J. Fluid Mech.,
2003, 476, 345-356.
[15] Yoon Y., Borrell M., Park C.C., and Leal G., Viscosity ratio effects on
the coalescence of two equal-sized drops in a two-dimensional linear
flow, J. Fluid Mech., 2005, 525, 355-379.
[16] Norman, J.T., Nayak, H.V., and Bonnecaze T.B., Migration of buoyant
particles in low-Reynolds-number pressure-driven flows, J. Fluid Mech.,
2005, 523, 1-35.
[17] Yang B.H., Wang J., Hu H.H., Pan T.W., and Glowinski R., Migration
of a sphere in tube flow, J. Fluid Mech., 2005, 540, 109-131.
[18] Sibillo, V., Pasquariello, G., Simeone, M., Cristini, V., and Guido, S.,
Drop deformation in micro confined shear flow, PhysRevLett, 2007, 97,
pp. 2-4.
[19] Zhao, X., Drop break up in dilute Newtonian emulsions in simple shear
flow: new drop break up mechanism. J. Rheology, 2007, 51, 367-192.
[20] Unverdi, S.O., and Tryggvason, G., Computations of multi-fluid flows,
J. Physics, 1992, D60, 70-83.
[21] Ho, B. P., and Leal, L. G., Inertial migration of rigid spheres in twodimensional
unidirectional flows, J. Fluid Mech., 1974, 65, 365-383.
[22] Vasseur, P., and Cox, R.G., The lateral migration of a spherical particle
in two-dimensional shear flow, J. Fluid Mech., 1976, 78, 385-402.
[23] Janssen, P.J.A., and Anderson, P.D., A boundary integral model for drop
deformation between two parallel plates with non-unit viscosity ratio
drops, J. of Computational Physics, 2008, 227, 8807-8819.
[1] Taylor, G.I., The deformation of emulsions in definable fields of flow,
Proc,Ray.Soc. (London), 1934, A146, 501-523.
[2] Karnis,A. and Mason,S.G., Particle motions in sheared suspensions.
XXΙΙΙ. Wall migration of fluid drops, J.Colloid and Inerface. Science,
1967, 24, 164-169.
[3] Halow,J.S., and Willis,G.B., Radial migration of spherical particles in
Couette system, AICHE J., 1970, 16, 281-286.
[4] Rallison, J.M., The deformation of small viscous drops and bubbles in
shear flows, Annu, Rev. Fluid Mech., 1984, 16, 45-66.
[5] Magna, M. and Stone, H.A., Buoyancy-driven interactions between two
deformable viscous drops, J.Fluid Mech, 1993, 256, 647-683.
[6] Zhou, H. and Pozrikidis,C., The flow of suspensions in channels: single
files of drops, Phys. Fluids, 1993, A5(2), 311-324.
[7] Feng, J., Hu, H.H., and Joseph, D.D., Direct simulation of initial value
problems for the motion of solid bodies in a Newtonian fluid. Part2.
Couette and Poiseuille flows, J.Fluid Mech, 1994, 277, 271-301.
[8] Li, X., Zhou H. and Pozrikidis,C., A numerical study of the shearing
motion of emulsions and foams, J.Fluid Mech, 1995, 286, 374-404.
[9] Loewenberg, M. and Hinch, E., Numerical simulation of a concentrated
emulsion in shear flow, J.Fluid Mech., 1996, 321, 395-419.
[10] Esmaeeli, A., and Tryggvason, G., Direct numerical simulations of
bubbly flows Part1. Low Reynolds number arrays, J.Fluid Mech, 1998,
377, 313-345.
[11] Mortazavi, S.S. and Tryggvasson, G., A numerical study of the motion
of drop in Poiseuille flow, part1: lateral migration of one drop, J.Fluid
Mech, 1999, 411, 325-350.
[12] Esmaeeli, A., and Tryggvason, G., Direct numerical simulations of
bubbly flows Part2. Low Reynolds number arrays, J.Fluid Mech, 1999,
385, 325-358.
[13] Balabel A., Binninger B., Herrmann M. and Peters N., Calculation of
droplet deformation by surface tension effects using the Level Set
method, J. Combustion Science and Technology, 2002, 174, 257-278.
[14] [Crowdy D.G., Compressible bubbles in Stokes flow, J. Fluid Mech.,
2003, 476, 345-356.
[15] Yoon Y., Borrell M., Park C.C., and Leal G., Viscosity ratio effects on
the coalescence of two equal-sized drops in a two-dimensional linear
flow, J. Fluid Mech., 2005, 525, 355-379.
[16] Norman, J.T., Nayak, H.V., and Bonnecaze T.B., Migration of buoyant
particles in low-Reynolds-number pressure-driven flows, J. Fluid Mech.,
2005, 523, 1-35.
[17] Yang B.H., Wang J., Hu H.H., Pan T.W., and Glowinski R., Migration
of a sphere in tube flow, J. Fluid Mech., 2005, 540, 109-131.
[18] Sibillo, V., Pasquariello, G., Simeone, M., Cristini, V., and Guido, S.,
Drop deformation in micro confined shear flow, PhysRevLett, 2007, 97,
pp. 2-4.
[19] Zhao, X., Drop break up in dilute Newtonian emulsions in simple shear
flow: new drop break up mechanism. J. Rheology, 2007, 51, 367-192.
[20] Unverdi, S.O., and Tryggvason, G., Computations of multi-fluid flows,
J. Physics, 1992, D60, 70-83.
[21] Ho, B. P., and Leal, L. G., Inertial migration of rigid spheres in twodimensional
unidirectional flows, J. Fluid Mech., 1974, 65, 365-383.
[22] Vasseur, P., and Cox, R.G., The lateral migration of a spherical particle
in two-dimensional shear flow, J. Fluid Mech., 1976, 78, 385-402.
[23] Janssen, P.J.A., and Anderson, P.D., A boundary integral model for drop
deformation between two parallel plates with non-unit viscosity ratio
drops, J. of Computational Physics, 2008, 227, 8807-8819.
@article{"International Journal of Engineering, Mathematical and Physical Sciences:64636", author = "M. Bayareh and S. Mortazavi", title = "Migration of a Drop in Simple Shear Flow at Finite Reynolds Numbers: Size and Viscosity Ratio Effects", abstract = "The migration of a deformable drop in simple shear
flow at finite Reynolds numbers is investigated numerically by
solving the full Navier-Stokes equations using a finite
difference/front tracking method. The objectives of this study are to
examine the effectiveness of the present approach to predict the
migration of a drop in a shear flow and to investigate the behavior of
the drop migration with different drop sizes and non-unity viscosity
ratios. It is shown that the drop deformation depends strongly on the
capillary number, so that; the proper non-dimensional number for the
interfacial tension is the capillary number. The rate of migration
increased with increasing the drop radius. In other words, the
required time for drop migration to the centreline decreases. As the
viscosity ratio increases, the drop rotates more slowly and the
lubrication force becomes stronger. The increased lubrication force
makes it easier for the drop to migrate to the centre of the channel.
The migration velocity of the drop vanishes as the drop reaches the
centreline under viscosity ratio of one and non-unity viscosity ratios.
To validate the present calculations, some typical results are
compared with available experimental and theoretical data.", keywords = "drop migration, shear flow, front-tracking method,finite difference method.", volume = "4", number = "1", pages = "198-7", }
