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The rise of Newtonian drops in a nematic liquid crystal

Published online by Cambridge University Press:  23 November 2007

CHUNFENG ZHOU ,
PENGTAO YUE  and
JAMES J. FENG
CHUNFENG ZHOU
Affiliation:
Department of Chemical and Biological Engineering, University of British Columbia, Vancouver, BC V6T 1Z3, Canada
PENGTAO YUE
Affiliation:
Department of Chemical and Biological Engineering, University of British Columbia, Vancouver, BC V6T 1Z3, Canada Department of Mathematics, University of British Columbia, Vancouver, BC V6T 1Z2, Canada
JAMES J. FENG
Affiliation:
Department of Chemical and Biological Engineering, University of British Columbia, Vancouver, BC V6T 1Z3, Canada Department of Mathematics, University of British Columbia, Vancouver, BC V6T 1Z2, Canada
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Abstract

We simulate the rise of Newtonian drops in a nematic liquid crystal parallel to the far-field molecular orientation. The moving interface is computed in a diffuse-interface framework, and the anisotropic rheology of the liquid crystal is represented by the Leslie–Ericksen theory, regularized to permit topological defects. Results reveal interesting coupling between the flow field and the orientational field surrounding the drop, especially the defect configuration. The flow generally sweeps the point and ring defects downstream, and may transform a ring defect into a point defect. The stability of these defects and their transformation are depicted in a phase diagram in terms of the Ericksen number and the ratio between surface anchoring and bulk elastic energies. The nematic orientation affects the flow field in return. Drops with planar anchoring on the surface rise faster than those with homeotropic anchoring, and the former features a vortex ring in the wake. These are attributed to the viscous anisotropy of the nematic. With homeotropic anchoring, the drop rising velocity experiences an overshoot, owing to the transformation of the initial surface ring defect to a satellite point defect. With both types of anchoring, the drag coefficient of the drop decreases with increasing Ericksen number as the flow-alignment of the nematic orientation reduces the effective viscosity of the liquid crystal.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 593 , 25 December 2007 , pp. 385 - 404
Copyright
Copyright © Cambridge University Press 2007

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References

REFERENCES

Akers, B. & Belmonte, A. 2006 Impact dynamics of a solid sphere falling into a viscoelastic micellar fluid. J. Non-Newtonian Fluid Mech. 135, 97–108.CrossRefGoogle Scholar
Batchelor, G. K. 1980 An Introduction to Fluid Mechanics. Cambridge University Press.Google Scholar
Carlsson, T. 1984 Theoretical investigation of the shear flow of nematic liquid crystals with the Leslie viscosity α3 > 0: Hydrodynamic analogue of first order phase transitions. Mol. Cryst. Liq. Cryst. 104, 307–334.CrossRefGoogle Scholar
Chandrasekhar, S. 1992 Liquid Crystals, 2nd edition. Cambridge University Press.CrossRefGoogle Scholar
Cluzeau, P., Poulin, P., Joly, G. & Nguyen, H. T. 2001 Interactions between colloidal inclusions in two-dimensional smetic-c* films. Phys. Rev. E 63, 031702.Google ScholarPubMed
Dandy, D. S. & Leal, L. G. 1989 Buoyancy-driven motion of a deformable drop through a quiescent liquid at intermediate Reynolds numbers. J. Fluid Mech. 208, 161–192.CrossRefGoogle Scholar
Diogo, A. C. 1983 Friction drag on a sphere moving in a nematic liquid crystal. Mol. Cryst. Liq. Cryst. 100, 153–165.CrossRefGoogle Scholar
Feng, J. J., Liu, C., Shen, J. & Yue, P. 2005 An energetic variational formulation with phase field methods for interfacial dynamics of complex fluids: Advantages and challenges. In Modeling of Soft Matter (ed. Calderer, M.-C. T. & Terentjev, E.). Springer.Google Scholar
Feng, J. J., Sgalari, G. & Leal, L. G. 2000 A theory for flowing nematic polymers with orientational distortion. J. Rheol. 44, 1085–1101.CrossRefGoogle Scholar
Feng, J. J. & Zhou, C. 2004 Orientational defects near colloidal particles in a nematic liquid crystal. J. Colloid Interface Sci. 269, 72–78.CrossRefGoogle Scholar
Fukuda, J.-I., Stark, H., Yoneya, M. & Yokoyama, H. 2004 Dynamics of a nematic liquid crystal around a spherical particle. J. Phys.: Condens. Matter 16, S1957–S1968.Google Scholar
deGennes, P. G. Gennes, P. G. & Prost, J. 1993 The Physics of Liquid Crystals. Oxford University Press.Google Scholar
Grace, J. R., Wairegi, T. & Nguyen, T. H. 1976 Shapes and velocities of single drops and bubbles moving freely through immiscible liquids. Trans. Inst. Chem. Engrs 54, 167–173.Google Scholar
Gu, Y. & Abbott, N. L. 2000 Observation of Saturn-ring defects around solid microspheres in nematic liquid crystals. Phys. Rev. Lett. 85, 4719–4722.CrossRefGoogle ScholarPubMed
Jacqmin, D. 1999 Calculation of two-phase Navier–Stokes flows using phase-field modelling. J. Comput. Phys. 155, 96–127.CrossRefGoogle Scholar
Jadżyn, J. & Czechowski, G. 2001 The shear viscosity minimum of freely flowing nematic liquid crystals. J. Phys.: Condens. Matter 13, L261–L265.Google Scholar
Khullar, S., Zhou, C. & Feng, J. J. 2007 Dynamic evolution of topological defects around drops and bubbles rising in a nematic liquid crystal. Phys. Rev. Lett. (submitted).CrossRefGoogle Scholar
Kléman, M. 1983 Points, Lines and Walls: In Liquid Crystals, Magnetic Systems and Various Ordered Media. Wiley.Google Scholar
Kuksenok, O. V., Ruhwandl, R. W., Shiyanovskii, S. V. & Terentjev, E. M. 1996 Director structure around a colloid particle in a nematic liquid crystal. Phys. Rev. E 54, 5198–5204.Google Scholar
Kuss, E. 1978 pVT-data and viscosity-pressure behavior of MBBA and EBBA. Mol. Cryst. Liq. Cryst. 47, 71–83.CrossRefGoogle Scholar
Lavrentovich, O. D. 1998 Topological defects in dispersed liquid crystals, or words and worlds around liquid crystal drops. Liq. Cryst. 24, 117–125.CrossRefGoogle Scholar
Leslie, F. M. 1968 Some constitutive equations for liquid crystals. Arch. Rat. Mech. Anal. 28, 265–283.CrossRefGoogle Scholar
Liu, C. & Shen, J. 2003 A phase field model for the mixture of two incompressible fluids and its approximation by a Fourier–spectral method. Physica D 179, 211–228.Google Scholar
Liu, C. & Walkington, N. J. 2000 Approximation of liquid crystal flows. SIAM J. Numer. Anal. 37, 725–741.CrossRefGoogle Scholar
Loudet, J. C., Barois, P. & Poulin, P. 2000 Colloidal ordering from phase separation in a liquid crystalline continuous phase. Nature 407, 611–613.CrossRefGoogle Scholar
Loudet, J. C. & Poulin, P. 2001 Application of an electric field to colloidal particles suspended in a liquid-crystal solvent. Phys. Rev. Lett. 87, 165503.CrossRefGoogle Scholar
Lubensky, T. C., Pettey, D., Currier, N. & Stark, H. 1998 Topological defects and interactions in nematic emulsions. Phys. Rev. E 57, 610–625.Google Scholar
Mondain-Monval, O., Dedieu, J. C., Gulik-Krzywicki, T. & Poulin, P. 1999 Weak surface energy in nematic dispersions: Saturn ring defects and quadrupolar interactions. Eur. Phys. J. B 12, 167–170.CrossRefGoogle Scholar
Nastishin, Y. A., Liu, H., Schneider, T., Nazarenko, V., Vasyuta, R., Shiyanovskii, S. V. & Lavrentovich, O. D. 2005 Optical characterization of the nematic lyotropic chromonic liquid crystals: Light absorption, birefringence, and scalar order parameter. Phys. Rev. E 72, 041711.Google ScholarPubMed
Poulin, P., Cabuil, V. & Weitz, D. A. 1997 a Direct measurement of colloidal forces in an anisotropic solvent. Phys. Rev. Lett. 79, 4862–4865.CrossRefGoogle Scholar
Poulin, P., Stark, H., Lubensky, T. C. & Weitz, D. A. 1997 b Novel colloidal interactions in anisotropic fluids. Science 275, 1770–1773.CrossRefGoogle ScholarPubMed
Poulin, P. & Weitz, D. A. 1998 Inverted and multiple nematic emulsions. Phys. Rev. E 57, 626–637.Google Scholar
Rapini, A. & Papoular, M. 1969 Distortion d/une lamelle nematique sous champ magnetique: Conditions d/ancrage aux parois. J. Phys. (Paris) C 30, 54–56.CrossRefGoogle Scholar
Rey, A. D. & Tsuji, T. 1998 Recent advances in theoretical liquid crystal rheology. Macromol. Theory Simul. 7, 623–639.3.0.CO;2-E>CrossRefGoogle Scholar
Ruhwandl, R. W. & Terentjev, E. M. 1996 Friction drag on a particle moving in a nematic liquid crystal. Phys. Rev. E 54, 5204–5210.Google Scholar
Ruhwandl, R. W. & Terentjev, E. M. 1997 Monte Carlo simulation of topological defects in the nematic liquid crystal matrix around a spherical colloidal particle. Phys. Rev. E 56, 5561–5565.Google Scholar
Stark, H. 1999 Director field configurations around a spherical particle in a nematic liquid crystal. Eur. Phys. J. B 10, 311–321.CrossRefGoogle Scholar
Stark, H. 2001 Physics of colloidal dispersions in nematic liquid crystals. Phys. Rep. 35, 387–474.CrossRefGoogle Scholar
Stark, H. & Ventzki, D. 2001 Stokes drag of spherical particles in a nematic environment at low Ericksen number. Phys. Rev. E 64, 031711.Google Scholar
Stark, H. & Ventzki, D. 2002 Non-linear stokes drag of spherical particles in a nematic solvent. Europhys. Lett. 57, 60–66.CrossRefGoogle Scholar
Terentjev, E. M. 1995 Disclination loops, standing alone and around solid particles, in nematic liquid crystals. Phys. Rev. E 51, 1330–1337.Google ScholarPubMed
Tixier, T., Heppenstall-Butler, M. & Terentjev, E. M. 2006 Spontaneous size selection in cholesteric and nematic emulsions. Langmuir 22, 2365–2370.CrossRefGoogle ScholarPubMed
Trebin, H. R. 1982 The topology of nonuniform media in condensed matter physics. Adv. Phys. 31, 195–254.CrossRefGoogle Scholar
White, A. E., Cladis, P. E. & Torza, S. 1977 Study of liquid crystals in flow. Part I. conventional viscometry and density measurements. Mol. Cryst. Liq. Cryst. 43, 13–31.CrossRefGoogle Scholar
Yamamoto, R. 2001 Simulating particle dispersions in nematic liquid-crystal solvents. Phys. Rev. Lett. 87, 075502.CrossRefGoogle ScholarPubMed
Yoneya, M., Fukuda, J.-I., Yokoyama, H. & Stark, H. 2005 Effect of a hydrodynamic flow on the orientation profiles of a nematic liquid crystal around a spherical particle. Mol. Cryst. Liq. Cryst. 435, 75–85.CrossRefGoogle Scholar
Yue, P., Feng, J. J., Liu, C. & Shen, J. 2004 A diffuse-interface method for simulating two-phase flows of complex fluids. J. Fluid Mech. 515, 293–317.CrossRefGoogle Scholar
Yue, P., Feng, J. J., Liu, C. & Shen, J. 2005 a Diffuse-interface simulations of drop coalescence and retraction in viscoelastic fluids. J. Non-Newtonian Fluid Mech. 129, 163–176.CrossRefGoogle Scholar
Yue, P., Feng, J. J., Liu, C. & Shen, J. 2005 b Interfacial force and Marangoni flow on a nematic drop retracting in an isotropic fluid. J. Colloid Interface Sci. 290, 281–288.CrossRefGoogle Scholar
Yue, P., Feng, J. J., Liu, C. & Shen, J. 2005 c Transient drop deformation upon startup of shear in viscoelastic fluids. Phys. Fluids 17, 123101.CrossRefGoogle Scholar
Yue, P., Feng, J. J., Liu, C. & Shen, J. 2005 d Viscoelastic effects on drop deformation in steady shear. J. Fluid Mech. 540, 427–437.CrossRefGoogle Scholar
Yue, P., Zhou, C. & Feng, J. J. 2006 a A computational study of the coalescence between a drop and an interface in Newtonian and viscoelastic fluids. Phys. Fluids 18, 102102.CrossRefGoogle Scholar
Yue, P., Zhou, C. & Feng, J. J. 2007 Spontaneous shrinkage of drops and mass conservation in phase-field simulations. J. Comput. Phys. 223, 1–9.CrossRefGoogle Scholar
Yue, P., Zhou, C., Feng, J. J., Ollivier-Gooch, C. F. & Hu, H. H. 2006 b Phase-field simulations of interfacial dynamics in viscoelastic fluids using finite elements with adaptive meshing. J. Comput. Phys. 219, 47–67.CrossRefGoogle Scholar
Zhou, C., Yue, P. & Feng, J. J. 2006 Formation of simple and compound drops in microfluidics devices. Phys. Fluids 18, 092105.CrossRefGoogle Scholar
Zhou, C., Yue, P., Feng, J. J., Liu, C. & Shen, J. 2007 Heart-shaped bubbles rising in anisotropic liquids. Phys. Fluids 19, 041703.CrossRefGoogle Scholar