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Xu, Kuan; Booty, Michael; Siegel, Michael (2013)
Languages: English
Types: Article
Subjects: QA297, QA901

Classified by OpenAIRE into

arxiv: Physics::Fluid Dynamics
A hybrid method is used to determine the influence of surfactant solubility on two-phase flow by solution of a reduced transition layer equation near a fluid interface in the limit of large\ud bulk Peclet number. The method is applied to finding the evolution of a drop of arbitrary viscosity that is deformed by an imposed linear strain or simple shear flow. A semi-analytical solution of the transition layer equation is given that expresses exchange of surfactant between its bulk and interfacial forms in terms of a convolution integral in time. Results of this semi-analytical solution are compared with the results of a spatially spectrally accurate numerical solution. Although both the hybrid method and its semi-analytical solution are valid in three dimensions, the two-dimensional context of this study allows additional validation of results by comparison with those of conformal mapping techniques applied to inviscid bubbles.
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    • [1] D. A. Edwards, H. Brenner, and D. T. Wasan, Interfacial Transport Processes and Rheology, Butterworth-Heinemann, Boston, 1991.
    • [2] M. R. Booty and M. Siegel, A hybrid numerical method for interfacial fluid flow with soluble surfactant, J. Comput. Phys., 229 (2010), pp. 3864-3883.
    • [3] G. I. Taylor, The formation of emulsions in definable fields of flow, Proc. R. Soc. Lond. A, 146 (1934), pp. 501-523.
    • [4] H. P. Grace, Dispersion phenomena in high viscosity immiscible fluid systems and application of static mixers as dispersion devices in such systems, Chem. Engrg. Commun., 14 (1982), pp. 225-277.
    • [5] B. J. Bentley and L. G. Leal, A computer-controlled four-roll mill for investigations of particle and drop dynamics in two-dimensional linear shear flows, J. Fluid Mech., 167 (1986), pp. 219-240.
    • [6] R. A. de Bruijn, Tipstreaming of drops in simple shear flows, Chem. Engrg. Sci., 48 (1993), pp. 277-284.
    • [7] J. J. M. Janssen, A. Boon, and W. G. M. Agterof, Influence of dynamic interfacial properties on droplet breakup in simple shear flow, AIChE J., 40 (1994), pp. 1929-1939.
    • [8] J. J. M. Janssen, A. Boon, and W. G. M. Agterof, Influence of dynamic interfacial properties on droplet breakup in plane hyperbolic flow, AIChE J., 43 (1997), pp. 1436-1447.
    • [9] H. A. Stone and L. G. Leal, The effects of surfactants on drop deformation and breakup, J. Fluid Mech., 220 (1990), pp. 161-186.
    • [10] W. J. Milliken, H. A. Stone, and L. G. Leal, The effect of surfactant on the transient motion of Newtonian drops, Phys. Fluids, 5 (1993), pp. 69-79.
    • [11] I. B. Bazhlekov, P. D. Anderson, and H. E. H. Meijer, Numerical investigation of the effect of insoluble surfactants on drop deformation and breakup in simple shear flow, J. Colloid Interface Sci., 298 (2006), pp. 369-394.
    • [12] H. A. Stone, Dynamics of drop deformation and breakup in viscous fluids, in Annu. Rev. Fluid Mech. 26, Annual Reviews, Palo Alto, CA, 1994, pp. 65-102.
    • [13] J. Eggers, Nonlinear dynamics and breakup of free-surface flows, Rev. Modern Phys., 69 (1997), pp. 865-929.
    • [14] D. Qu´er´e, Fluid coating on a fiber, in Annu. Rev. Fluid Mech. 31, Annual Reviews, Palo Alto, CA, 1999, pp. 347-384.
    • [15] O. A. Basaran, Small-scale free surface flows with breakup: Drop formation and emerging applications, AIChE J., 48 (2002), pp. 1842-1848.
    • [16] C. D. Eggleton, T.-M. Tsai, and K. J. Stebe, Tip streaming from a drop in the presence of surfactants, Phys. Rev. Lett., 87 (2001), 048302.
    • [17] S. L. Anna and H. C. Mayer, Microscale tipstreaming in a microfluidic flow focusing device, Phys. Fluids, 18 (2006), 121512.
    • [18] S. Tanveer and G. Vasconcelos, Time-evolving bubbles in two-dimensional Stokes flow, J. Fluid Mech., 301 (1995), pp. 325-344.
    • [19] M. Siegel, Influence of surfactant on rounded and pointed bubbles in two-dimensional Stokes flow, SIAM J. Appl. Math., 59 (1999), pp. 1998-2027.
    • [20] N. I. Muskhelishvili, Some Basic Problems of the Mathematical Theory of Elasticity, P. Noordhoff, Groningen, The Netherlands, 1963.
    • [21] S. G. Mikhlin, Integral Equations and Their Applications to Certain Problems in Mechanics, Mathematical Physics and Technology, MacMillan, New York, 1964.
    • [22] L. G. Leal, Laminar Flow and Convective Transport Processes, Butterworth-Heinemann, Newton, MA, 1992.
    • [23] C.-H. Chang and E. I. Franses, Adsorption dynamics of surfactants at the air/water interface: A critical review of mathematical models, data, and mechanisms, Colloids Surfaces A, 100 (1995), pp. 1-45.
    • [24] H. Wong, D. Rumschitzki, and C. Maldarelli, On the surfactant mass balance at a deforming fluid interface, Phys. Fluids, 8 (1996), pp. 3203-3204.
    • [25] B. Ambravaneswaran and O. A. Basaran, Effects of insoluble surfactants on the nonlinear deformation and breakup of stretching liquid bridges, Phys. Fluids, 11 (1999), pp. 997-1015.
    • [26] R. V. Craster, O. K. Matar, and D. T. Papageorgiou, Pinchoff and satellite formation in surfactant covered viscous threads, Phys. Fluids, 14 (2002), pp. 1364-1376.
    • [27] Y.-N. Young, M. R. Booty, M. Siegel, and J. Li, Influence of surfactant solubility on the deformation and breakup of a bubble or capillary jet in a viscous fluid, Phys. Fluids, 21 (2009), 072105.
    • [28] L. Greengard, M. C. Kropinski, and A. Mayo, Integral equation methods for Stokes flow and isotropic elasticity in the plane, J. Comput. Phys., 125 (1996), pp. 403-414.
    • [29] M. C. A. Kropinski, An efficient numerical method for studying interfacial motion in twodimensional creeping flows, J. Comput. Phys., 171 (2001), pp. 479-508.
    • [30] M. C. A. Kropinski, Numerical methods for multiple inviscid interfaces in creeping flows, J. Comput. Phys., 180 (2002), pp. 1-24.
    • [31] M. C. A. Kropinski and E. Lushi, Efficient numerical methods for multiple surfactant-coated bubbles in a two-dimensional Stokes flow, J. Comput. Phys., 230 (2011), pp. 4466-4487.
    • [32] G. F. Carrier, M. Krook, and C. E. Pearson, Functions of a Complex Variable: Theory and Technique, Classics in Appl. Math. 49, SIAM, Philadelphia, 2005.
    • [33] W. E. Langlois, Slow Viscous Flow, MacMillan, New York, 1964.
    • [34] T. Hou, J. Lowengrub, and M. Shelley, Removing the stiffness from interfacial flows with surface tension, J. Comput. Phys., 114 (1994), pp. 312-338.
    • [35] A. I. Van de Vooren, A numerical investigation of the rolling up of vortex sheets, Proc. Roy. Soc. London Ser. A, 373 (1980), pp. 67-91.
    • [36] G. Baker and A. Nachbin, Stable methods for vortex sheet motion in the presence of surface tension, SIAM J. Sci. Comput., 19 (1998), pp. 1737-1766.
    • [37] L. N. Trefethen, Spectral Methods in MATLAB, Software Environ. Tools 10, SIAM, Philadelphia, 2000.
    • [38] A. F. H. Ward and L. Tordai, Time-dependence of boundary tensions of solutions. I. The role of diffusion in time-effects, J. Chem. Phys., 14 (1946), pp. 453-461.
    • [39] Y.-C. Liao, E. I. Franses, and O. A. Basaran, Computation of dynamic adsorption with adaptive integral, finite difference, and finite element methods, J. Colloid Interface Sci., 258 (2003), pp. 310-321.
    • [40] I. Stakgold, Green's Functions and Boundary Value Problems, 2nd ed., Wiley, New York, 1998.
    • [41] K. Xu, Computational Methods for Two-Phase Flow with Soluble Surfactant, Ph.D. thesis, Mathematical Sciences, New Jersey Institute of Technology, Newark, NJ, 2010.
    • [42] J. Lee and C. Pozrikidis, Effect of surfactants on the deformation of drops and bubbles in Navier-Stokes flow, Comput. Fluids, 35 (2006), pp. 43-60.
    • [43] W. J. Milliken and L. G. Leal, The influence of surfactant on the deformation and breakup of a viscous drop: The effect of surfactant solubility, J. Colloid Interface Sci., 166 (1994), pp. 275-285.
    • [44] R. G. Cox, The deformation of a drop in a general time-dependent fluid flow, J. Fluid Mech., 37 (1969), pp. 601-623.
    • [45] J. M. Rallison, Note on the time-dependent deformation of a viscous drop which is almost spherical, J. Fluid Mech., 98 (1980), pp. 625-633.
    • [46] S. Torza, R. G. Cox, and S. G. Mason, Particle motions in sheared suspensions XXVII. Transient and steady deformation and burst of liquid drops, J. Colloid Interface Sci., 38 (1972), pp. 395-411.
    • [47] P. M. Vlahovska, Y.-N. Young, G. Danker, and C. Misbah, Dynamics of a non-spherical microcapsule with incompressible interface in shear flow, J. Fluid Mech., 678 (2011), pp. 221-247.
    • [48] L. Greengard and J. Strain, A fast algorithm for the evaluation of heat potentials, Comm. Pure Appl. Math., 43 (1990), pp. 949-963.
    • [49] S. K. Veerapaneni and G. Biros, A high-order solver for the heat equation in 1d domains with moving boundaries, SIAM J. Sci. Comput., 29 (2007), pp. 2581-2606.
    • [50] R. E. Ewing and H. Wang, A summary of numerical methods for time-dependent advectiondominated partial differential equations, J. Comput. Appl. Math., 128 (2001), pp. 423-445.
    • [51] T. Nakamura, R. Tanaka, T. Yabe, and K. Takizawa, Exactly conservative semi-Lagrangian scheme for multi-dimensional hyperbolic equations with directional splitting technique, J. Comput. Phys., 174 (2001), pp. 171-207.
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