Skip to main content Accessibility help
×
Home

Effect of transverse temperature gradient on the migration of a deformable droplet in a Poiseuille flow

  • Sayan Das (a1), Shubhadeep Mandal (a1) and Suman Chakraborty (a1)

Abstract

Intricate manipulation of droplets in fluidic confinements may turn out to be critically important for achieving their controlled transverse distributions. Here, we study the migration characteristics of a suspended deformable droplet in a parallel plate channel under the combined influence of a constant temperature gradient in the transverse direction and an imposed pressure driven flow. An outstanding question concerning the resultant non-trivial dynamical features that we address here pertains to the nonlinearity that results as a consequence of the shape deformation, which does not permit us to analyse the combined transport as a mere linear superposition of the results for the thermocapillary and imposed flow driven droplet migration in an effort to obtain the final solution. For the analytical solution, an asymptotic approach is used, where we neglect any effect of inertia or thermal convection of the fluid in either of the phases. To obtain a numerical solution, we use the conservative level set method. We perform numerical simulations over a wide range of governing parameters and obtain the dependence of the transverse steady position of the droplet on different parameters. In order to address practical microfluidic set-ups, the influence of a bounding wall as well as the effect of thermal convection and finite shape deformation on the cross-stream migration of the droplet is investigated through numerical simulations. Increase in the thermal Marangoni stress shifts the steady-state transverse position of the droplet further away from the channel centreline, for any particular value of the capillary number (which signifies the ratio of the viscous force to the surface tension force). The confinement ratio, which is the ratio of the droplet radius to the channel height, plays an important role in predicting the transverse position of the droplet and thus has immense consequences for the design of droplet-based microfluidic devices with enhanced functionalities. A large confinement ratio drives the droplet towards the channel centre, whereas a smaller confinement ratio causes the droplet to move towards the wall. Moreover, for a fixed droplet radius and constant imposed temperature gradient, an increase in the channel height results in an increase in the time required for the droplet to reach the steady-state position. However, the final steady-state position of the droplet is independent of its initial position but at the same time dependent on the droplet phase thermal conductivity. A larger droplet thermal conductivity compared with the carrier phase results in a steady-state droplet position closer to the channel centreline. A higher fluid inertia, on the other hand, shifts the steady-state position towards the channel wall.

Copyright

Corresponding author

Email address for correspondence: suman@mech.iitkgp.ernet.in

References

Hide All
Ahn, K., Kerbage, C., Hunt, T. P., Westervelt, R. M., Link, D. R. & Weitz, D. A. 2006 Dielectrophoretic manipulation of drops for high-speed microfluidic sorting devices. Appl. Phys. Lett. 88 (2), 24104.
Anna, S. L. 2016 Droplets and bubbles in microfluidic devices. Annu. Rev. Fluid Mech. 48 (1), 285309.
Balasubramaniam, R. & Lavery, J. E. 1989 Numerical simulation of thermocapillary bubble migration under microgravity for large Reynolds and Marangoni numbers. Numer. Heat Transfer A 16 (2), 175187.
Balasubramaniam, R. & Subramaniam, R. S. 1996 Thermocapillary bubble migration – thermal boundary layers for large Marangoni numbers. Intl J. Multiphase Flow 22 (3), 593612.
Balasubramaniam, R. & Subramanian, R. S. 2004 Thermocapillary convection due to a stationary bubble. Phys. Fluids 16 (8), 31313137.
Balcázar, N., Oliva, A. & Rigola, J. 2016 A level-set method for thermal motion of bubbles and droplets. J. Phys.: Conf. Ser. 745, 32113.
Bandopadhyay, A., Mandal, S., Kishore, N. K. & Chakraborty, S. 2016 Uniform electric-field-induced lateral migration of a sedimenting drop. J. Fluid Mech. 792, 553589.
Baroud, C. N., Gallaire, F. & Dangla, R. 2010 Dynamics of microfluidic droplets. Lab on a Chip 10 (16), 20322045.
Barton, K. D. & Shankar Subramanian, R. 1990 Thermocapillary migration of a liquid drop normal to a plane surface. J. Colloid Interface Sci. 137 (1), 170182.
Barton, K. D. & Subramanian, R. S. 1991 Migration of liquid drops in a vertical temperature gradient – interaction effects near a horizontal surface. J. Colloid Interface Sci. 141 (1), 146156.
Casadevall i Solvas, X. & DeMello, A. J. 2011 Droplet microfluidics: recent developments and future applications. Chem. Commun. (Camb). 47 (7), 19361942.
Chan, P. C.-H. & Leal, L. G. 1979 The motion of a deformable drop in a second-order fluid. J. Fluid Mech. 92 (1), 131170.
Chen, S. H. 1999 Thermocapillary deposition of a fluid droplet normal to a planar surface. Langmuir 15 (8), 26742683.
Chen, S. H. 2003 Thermocapillary coagulations of a fluid sphere and a gas bubble. Langmuir 19 (11), 45824591.
Choudhuri, D. & Raja Sekhar, G. P. 2013 Thermocapillary drift on a spherical drop in a viscous fluid. Phys. Fluids 25 (4), 043104.
Das, S., Mandal, S., Som, S. K. & Chakraborty, S. 2017 Migration of a surfactant-laden droplet in non-isothermal Poiseuille flow. Phys. Fluids 29 (1), 12002.
Di Carlo, D., Irimia, D., Tompkins, R. G. & Toner, M. 2007 Continuous inertial focusing, ordering, and separation of particles in microchannels. Proc. Natl Acad. Sci. USA 104 (48), 1889218897.
Haj-Hariri, H., Nadim, A. & Borhan, A. 1990 Effect of inertia on the thermocapillary velocity of a drop. J. Colloid Interface Sci. 140 (1), 277286.
Haj-Hariri, H., Shi, Q. & Borhan, A. 1997 Thermocapillary motion of deformable drops at finite Reynolds and Marangoni numbers. Phys. Fluids 9 (4), 845855.
Hetsroni, G. & Haber, S. 1970 The flow in and around a droplet or bubble submerged in an unbound arbitrary velocity field. Rheol. Acta 9 (4), 488496.
Huebner, A., Sharma, S., Srisa-Art, M., Hollfelder, F., Edel, J. B. & Demello, A. J. 2008 Microdroplets: a sea of applications? Lab on a Chip 8 (8), 12441254.
Karbalaei, A., Kumar, R. & Cho, H. J. 2016 Thermocapillarity in microfluidics – a review. Micromachines 7 (1), 141.
Kim, H. S. & Subramanian, R. S. 1989 The thermocapillary migration of a droplet with insoluble surfactant: II. General case. J. Colloid Interface Sci. 130 (1), 112129.
Kim, J. H., Jeon, T. Y., Choi, T. M., Shim, T. S., Kim, S.-H. & Yang, S.-M. 2014 Droplet microfluidics for producing functional microparticles. Langmuir 30 (6), 14731488.
Kinoshita, H., Kaneda, S., Fujii, T. & Oshima, M. 2007 Three-dimensional measurement and visualization of internal flow of a moving droplet using confocal micro-PIV. Lab on a Chip 7 (3), 338346.
Leal, L. G. 1980 Particle motions in a viscous fluid. Annu. Rev. Fluid Mech. 12 (1), 435476.
Link, D. R., Grasland-Mongrain, E., Duri, A., Sarrazin, F., Cheng, Z., Cristobal, G., Marquez, M. & Weitz, D. A. 2006 Electric control of droplets in microfluidic devices. Angew. Chem. Intl Ed. Engl. 45 (16), 25562560.
Mandal, S., Bandopadhyay, A. & Chakraborty, S. 2015 Effect of interfacial slip on the cross-stream migration of a drop in an unbounded Poiseuille flow. Phys. Rev. E 92 (2), 23002.
Mandal, S., Bandopadhyay, A. & Chakraborty, S. 2016 Dielectrophoresis of a surfactant – laden viscous drop. Phys. Fluids 62006 (28), 62006.
Meyyappan, M. & Subramanian, R. S. 1987 Thermocapillary migration of a gas bubble in an arbitrary direction with respect to a plane surface. J. Colloid Interface Sci. 115 (1), 206219.
Mortazavi, S. & Tryggvason, G. 2000 A numerical study of the motion of drops in Poiseuille flow. Part 1. Lateral migration of one drop. J. Fluid Mech. 411, 325350.
Murr, L. E. & Johnson, W. L. 2017 3D metal droplet printing development and advanced materials additive manufacturing. J. Mater. Res. Technol. 6, 7789.
Nadim, A., Haj-Hariri, H. & Borhan, A. 1990 Thermocapillary migration of slightly deformed droplets. Part. Sci. Technol. 8 (3–4), 191198.
Nas, S. & Tryggvason, G. 2003 Thermocapillary interaction of two bubbles or drops. Intl J. Multiphase Flow 29 (7), 11171135.
Nguyen, H.-B. & Chen, J.-C. 2010 A numerical study of thermocapillary migration of a small liquid droplet on a horizontal solid surface. Phys. Fluids 22 (6), 62102.
Olsson, E. & Kreiss, G. 2005 A conservative level set method for two phase flow. J. Comput. Phys. 225 (1), 785807.
Pak, O. S., Feng, J. & Stone, H. A. 2014 Viscous Marangoni migration of a drop in a Poiseuille flow at low surface Péclet numbers. J. Fluid Mech. 753, 535552.
Robert de Saint Vincent, M., Wunenburger, R. & Delville, J. 2008 Laser switching and sorting for high speed digital microfluidics. Appl. Phys. Lett. 92, 154105.
Sajeesh, P. & Sen, A. K. 2014 Particle separation and sorting in microfluidic devices: a review. Microfluid Nanofluid 17 (1), 152.
Seemann, R., Brinkmann, M., Pfohl, T. & Herminghaus, S. 2012 Droplet based microfluidics. Rep. Prog. Phys. 75 (75), 1660116641.
Sethian, J. A. & Smereka, P. 2003 Level set methods for fluid interfaces. Annu. Rev. Fluid Mech. 35 (1), 341372.
Stan, C. A., Guglielmini, L., Ellerbee, A. K., Caviezel, D., Stone, H. A. & Whitesides, G. M. 2011 Sheathless hydrodynamic positioning of buoyant drops and bubbles inside microchannels. Phys. Rev. E 84 (3), 036302.
Stone, H. A., Stroock, A. D. & Ajdari, A. 2004 Engineering flows in small devices: microfluidics toward a lab-on-a-chip. Annu. Rev. Fluid Mech. 36 (1), 381411.
Subramanian, R. S. 1983 Thermocapillary migration of bubbles and droplets. Adv. Space Res. 3 (5), 145153.
Teh, S.-Y., Lin, R., Hung, L.-H. & Lee, A. P. 2008 Droplet microfluidics. Lab on a Chip 8 (2), 198220.
Uijttewaal, W. S. J. & Nijhof, E. J. 1995 The motion of a droplet subjected to linear shear flow including the presence of a plane wall. J. Fluid Mech. 302, 4563.
Uijttewaal, W. S. J., Nijhof, E.-J. & Heethaar, R. M. 1993 Droplet migration, deformation, and orientation in the presence of a plane wall: a numerical study compared with analytical theories. Phys. Fluids A 5 (4), 819.
Wang, J., Lu, P., Wang, Z., Yang, C. & Mao, Z.-S. 2008 Numerical simulation of unsteady mass transfer by the level set method. Chem. Engng Sci. 63 (12), 31413151.
Ward, T., Faivre, M., Abkarian, M. & Stone, H. A. 2005 Microfluidic flow focusing: drop size and scaling in pressure versus flow-rate-driven pumping. Electrophoresis 26 (19), 37163724.
Wu, Z.-B. & Hu, W.-R. 2011 Thermocapillary migration of a planar droplet at moderate and large Marangoni numbers. Acta Mech. 223 (3), 609626.
Yariv, E. & Shusser, M. 2006 On the paradox of thermocapillary flow about a stationary bubble. Phys. Fluids 18 (7), 072101.
Young, N. O., Goldstein, J. S. & Block, M. J. 1959 The motion of bubbles in a vertical temperature gradient. J. Fluid Mech. 6 (3), 350356.
Zhang, L., Subramanian, R. S. & Balasubramaniam, R. 2001 Motion of a drop in a vertical temperature gradient at small Marangoni number – the critical role of inertia. J. Fluid Mech. 448, 197211.
Zhou, H. & Pozrikidis, C. 1993 The flow of suspensions in channels: single files of drops. Phys. Fluids A 5 (2), 311324.
Zhou, H. & Pozrikidis, C. 1994 Pressure-driven flow of suspensions of liquid drops. Phys. Fluids 6 (1), 8094.
Zhu, Y. & Fang, Q. 2013 Analytical detection techniques for droplet microfluidics – a review. Anal. Chim. Acta 787, 2435.
MathJax
MathJax is a JavaScript display engine for mathematics. For more information see http://www.mathjax.org.

JFM classification

Type Description Title
UNKNOWN
Supplementary materials

Das et al. supplementary material
Das et al. supplementary material 1

 Unknown (32 KB)
32 KB

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed