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Capillarity and Dynamic Wetting Andreas Carlson Capillarity and dynamic wetting by Andreas Carlson March 2012 Technical Reports Royal Institute of Technology Department of Mechanics SE-100 44 Stockholm, Sweden Akademisk avhandling som med tillst˚and av Kungliga Tekniska H¨ogskolan i Stockholm framl¨agges till offentlig granskning f¨or avl¨aggande av teknologie doktorsexamen fredagen den 23 mars 2012 kl 10.00 i Salongen, Biblioteket, Kungliga Tekniska H¨ogskolan, Osquars Backe 25, Stockholm. c Andreas Carlson 2012 E-PRINT, Stockholm 2012 iii Capillarity and dynamic wetting Andreas Carlson 2012 Linn´eFLOW Centre, KTH Mechanics SE-100 44 Stockholm, Sweden Abstract In this thesis capillary dominated two{phase flow is studied by means of nu- merical simulations and experiments. The theoretical basis for the simulations consists of a phase field model, which is derived from the system's thermody- namics, and coupled with the Navier Stokes equations. Two types of interfacial flow are investigated, droplet dynamics in a bifurcating channel and sponta- neous capillary driven spreading of drops. Microfluidic and biomedical applications often rely on a precise control of droplets as they traverse through complicated networks of bifurcating channels. Three{dimensional simulations of droplet dynamics in a bifurcating channel are performed for a set of parameters, to describe their influence on the resulting droplet dynamics. Two distinct flow regimes are identified as the droplet in- teracts with the tip of the channel junction, namely, droplet splitting and non- splitting. A flow map based on droplet size and Capillary number is proposed to predict whether the droplet splits or not in such a geometry. A commonly occurring flow is the dynamic wetting of a dry solid substrate. Both experiments and numerical simulations of the spreading of a drop are presented here. A direct comparison of the two identifies a new parameter in the phase field model that is required to accurately predict the experimental spreading behavior. This parameter µf [P a s], is interpreted as a friction factor at the moving contact line. Comparison· of simulations and experiments for different liquids and surface wetting properties enabled a measurement of the contact line friction factor for a wide parameter space. Values for the contact line friction factor from phase field theory are reported here for the first time. To identify the physical mechanism that governs the droplet spreading, the different contributions to the flow are measured from the simulations. An im- portant part of the dissipation may arise from a friction related to the motion of the contact line itself, and this is found to be dominating both inertia and viscous friction adjacent to the contact line. A scaling law based on the con- tact line friction factor collapses the experimental data, whereas a conventional inertial or viscous scaling fails to rationalize the experimental observation, sup- porting the numerical finding. Descriptors: Phase field theory, finite element simulations, experiments, two{ phase flow, dynamic wetting, contact line physics, capillarity. iv Preface Research results presented in this thesis were obtained by the author during the period between December 2007 and March 2012. The work was performed at the Department of Mechanics at The Royal Institute of Technology. Paper 1. Carlson, A., Do-Quang, M., & Amberg, G. 2010 Droplet dynamics in a bifurcating channel. International Journal of Multiphase Flow 36 (5), 397{405. Paper 2. Carlson, A., Do-Quang, M., & Amberg, G. 2009 Modeling of dynamic wetting far from equilibrium. Physics of Fluids 21 (12), 121701-1{4. Paper 3. Carlson, A., Do-Quang, M., & Amberg, G. 2011 Dissipa- tion in rapid dynamic wetting. Journal of Fluid Mechanics 682, 213{240. Paper 4. Carlson, A., Bellani, G., & Amberg, G. 2012 Contact line dissipation in short-time dynamic wetting. EPL 97, 44004-1{6. Paper 5. Carlson, A., Bellani, G., & Amberg, G. 2012 Universality in dynamic wetting dominated by contact line friction. Submitted to Rapid Communications in Physical Review E. v Below is a list of related work that has been performed by the author in the same period of time, but not included in this thesis. International archival journals Laurila, T., Carlson, A., Do-Quang, M, Ala-Nissila,¨ T. & Amberg, G. 2012 Thermo-hydrodynamics of boiling in a van der Waals fluid. Accepted in Physical Review E. Do-Quang, M., Carlson, A., & Amberg, G. 2011 The Impact of Ink-Jet Droplets on a Paper-Like Structure. Fluid Dynamics & Materials Processing 7 4, 389{402. Lin, Y., Skjetne, P. & Carlson, A. 2011 A phase field model for multiphase electro{hydrodynamic flow. Submitted to International Journal of Multiphase Flow Refereed conference proceedings Lin, Y., Skjetne, P. & Carlson, A. 2010 A phase field method for mul- tiphase electro-hydrodynamic flow. International Conference on Multiphase Flow, Tampa, USA. Do-Quang, M., Carlson, A. & Amberg, G. 2010 The impact of ink-jet droplets on a paper-like structure. International Conference on Multiphase Flow, Tampa, USA. Carlson, A., Do-Quang, M. & Amberg, G. 2010 Characterization of droplet dynamics in a bifurcating channel. International Conference on Multi- phase Flow, Tampa, USA. Carlson, A., Do-Quang, M. & Amberg, G. 2009 Spontaneously spread- ing liquid droplets in a dynamic wetting process. EUROTHERM-84, Namur, Belgium. Carlson, A., Lakehal, D. & Kudinov, P. 2009 A multiscale approach for thin-film slug flow. 7th World Conference on Experimental Heat Transfer, Fluid Mechanics and Thermodynamics, Krakow, Polen. Do-Quang, M., Carlson, A. & Amberg, G. 2008 Capillary force domi- nated impact of particle to free liquid surface Proc. of the 1st European Con- ference on Microfluidics - Microfluidics, Bologna, Italy. Carlson, A., Do-Quang, M. & Amberg, G. 2008 Droplet dynamics in a microfluidic bifurcation. Proc. of the 1st European Conference on Microfluidics - Microfluidics, Bologna, Italy. Carlson, A., Kudinov, P. & Narayanan, C. 2008 Prediction of two-phase flow in small tubes: A systematic comparison of state-of-the-art CMFD codes. 5th European Thermal-Sciences Conference, Eindhoven, Netherlands. vi Division of work between authors Paper 1. The code was implemented by Andreas Carlson (AC) and Minh Do-Quang (MDQ). AC performed the simulations and analyzed the data. The paper was written by AC, with feedback from MDQ and Gustav Amberg (GA). Paper 2. The code was implemented by AC and MDQ. AC performed the simulations and analyzed the data. The paper was written by AC, with feed- back from MDQ and GA. Paper 3. AC performed the simulations and analyzed the data. The pa- per was written by AC, with feedback from MDQ and GA. Paper 4. AC and Gabriele Bellani (GB) designed the initial experiments. The image processing tool was developed by GB. AC performed the exper- iments, the simulations and analyzed the data. AC wrote the article, with feedback from GB and GA. Paper 5. AC and GB designed the initial experiments. The image processing tool was developed by GB. AC performed the experiments, the simulations and analyzed the data. AC wrote the article, with feedback from GB and GA. vii viii ix Abstract iv Preface v Part I - Summary Chapter 1. Introduction 1 Chapter 2. Capillarity 5 2.1. Surface tension 5 2.2. Young{Laplace Law 5 2.3. Dynamic capillary flows and non{dimensional numbers 6 2.4. Wetting { Young's Law 7 Chapter 3. Dynamic wetting 9 3.1. Hydrodynamic theory 9 3.2. Microscopic model 11 3.3. Experiments of dynamic wetting 12 3.4. Simulations of dynamic wetting 16 3.4.1. Macroscopic free energy models 16 3.4.2. Molecular dynamics simulations 18 Chapter 4. Phase field theory 21 4.1. Free energy 21 4.2. Evolution of fluxes 22 4.2.1. Decrease of free energy 22 4.2.2. Cahn{Hilliard and Navier Stokes equations 23 4.3. Phase field interface and surface tension 24 4.4. Wetting boundary condition 25 4.4.1. Local equilibrium 25 4.4.2. Non{equilibrium 26 Chapter 5. Numerical methodology 29 5.1. Finite Element toolbox { femLego 29 5.2. Numerical schemes 29 5.2.1. Cahn{Hilliard equation 29 5.2.2. Navier Stokes equations 31 Chapter 6. Summary of results 33 Chapter 7. Concluding remarks 41 Acknowledgements 44 xi Bibliography 46 Part II - Papers Paper 1. Droplet dynamics in a bifurcating channel 63 Paper 2. Modeling of dynamic wetting far from equilibrium 87 Paper 3. Dissipation in rapid dynamic wetting 99 Paper 4. Contact line dissipation in short{time dynamic wetting 135 Paper 5. Universality in dynamic wetting dominated by contact line friction 151 Part I Summary CHAPTER 1 Introduction In our daily life we often observe beautiful two-phase flow phenomena; forma- tion of drops as the kitchen tap is turned on, how coffee spill stains the table linen, rain drops sliding on the windshield or cooking oil convecting towards the frying pans colder part. These common occurrences are all governed by the physics at the interface between the liquid and gas phase or at the contact line where the liquid-gas-solid phases meet. In other words they are dictated by the surface tension and the liquid-solid wettability. Nature has used surface tension to develop several ingenious designs for insect propulsion, water collection and capillary adhesion. For instance wa- ter striders are able to walk on water despite the fact that they are heavier than water, see fig. 1.1b. Their hairy legs prevent water from wetting them and instead of penetrating the surface and sink, the feet deform the interface generating a surface tension force that supports the insect. Various beetles use surface tension in a different way. Instead of running on water, they have developed a method based on surface tension to adhere onto solid substrates. This allows them to easily walk up a vertical wall, or to withstand a pulling force much greater than their own weight, fig.
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