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Entrainment in multifluid systems, and rotation induced occurrences. (English) Zbl 1500.76096

Summary: Fluid entrainment is a complex and deeply attractive phenomenon that is frequently encountered in both industry and nature. In general, entrainment deals with the basic understanding of penetration one fluid into another which leads to occurrence of complex interfacial structure. The present article addresses a detailed description of the recent studies on entrainment dynamics caused by the external rotational flux across the interfaces is elaborated. Despite of considerable progress in understanding rotation induced entrainment, there are still several questions unanswered due to its difficulty, unpredictability, three-dimensional effects, and chaotic nature. Therefore, here we explained numerous numerical, experimental, and analytical investigations with their relevance in engineering applications wherein the increase of interfacial area causes enhancement of heat and mass transfer. The description also includes various patterns of entrained fluid entities due to rotations induced across the interface. Furthermore, the entrainment behaviors have been critically analyzed to draw out the distinct categorization of rollers relying on parameters such as submergence ratio with different gas-liquid pairs.

MSC:

76T10 Liquid-gas two-phase flows, bubbly flows
76U05 General theory of rotating fluids
76M99 Basic methods in fluid mechanics
76-02 Research exposition (monographs, survey articles) pertaining to fluid mechanics
Full Text: DOI

References:

[1] Burley, R.; Jolly, R. P., Entrainment of air into liquids by a high speed continuous solid surface, Chem. Eng. Sci., 39, 1357-1372 (1984)
[2] Haehnel, R.; Dade, W. B., Physics of Particle Entrainment under the Influence of an Impinging Jet (2008), COLD REGIONS RESEARCH AND ENG. LAB HANOVER NH.
[3] Dimotakis, P. E., Two-dimensional shear-layer entrainment, AIAA J., 24, 1791-1796 (1986)
[4] Balakhrisna, T.; Ghosh, S.; Das, G.; Das, P. K., Oil-water flows through sudden contraction and expansion in a horizontal pipe-Phase distribution and pressure drop, Int. J. Multiph. Flow., 36, 13-24 (2010)
[5] Pradhan, C., Entrainment Through Rotation in Multiphase System (2016), (Doctoral dissertation)
[6] Chaini, U., Entrainment using Rotary Motion (2007), (Doctoral dissertation)
[7] Mazumdar, D.; Dhandapani, P.; Sarvanakumar, R., Modeling and optimisation of gas stirred ladle systems, ISIJ Int., 57, 286-295 (2017)
[8] Morales, R. D.; Calderon-Hurtado, F. A.; Chattopadhyay, K., Demystifying underlying fluid mechanics of gas stirred ladle systems with top slag layer using physical modeling and mathematical modeling, ISIJ Int. (2019)
[9] Lamarre, E.; Melville, W. K., Air entrainment and dissipation in breaking waves, Nature, 351, 469-472 (1991)
[10] Loewen, M. R.; O’Dor, M. A.; Skafel, M. G., Bubbles entrained by mechanically generated breaking waves, J. Geophys. Res.: Oceans, 101, 20759-20769 (1996)
[11] Terrill, E. J.; Melville, W. K.; Stramski, D., Bubble entrainment by breaking waves and their influence on optical scattering in the upper ocean, J. Geophys. Res.: Oceans, 106, 16815-16823 (2001)
[12] Xie, Z., Numerical study of breaking waves by a two phase flow model, Int. J. Numer. Methods Fluids, 70, 246-268 (2012) · Zbl 1412.76029
[13] Deike, L.; Melville, W. K.; Popinet, S., Air entrainment and bubble statistics in breaking waves, J. Fluid Mech., 801, 91-129 (2016) · Zbl 1462.76040
[14] Oguz, H. N.; Prosperetti, A., Bubble entrainment by the impact of drops on liquid surfaces, J. Fluid Mech., 219, 143-179 (1990)
[15] Thoroddsen, S. T.; Etoh, T. G.; Takehara, K., Air entrapment under an impacting drop, J. Fluid Mech., 478, 125 (2003) · Zbl 1032.76507
[16] Rein, M.; Delplanque, J. P., The role of air entrainment on the outcome of drop impact on a solid surface, Acta Mech., 201, 105-118 (2008) · Zbl 1155.76332
[17] Tran, T.; de Maleprade, H.; Sun, C.; Lohse, D., Air entrainment during impact of droplets on liquid surfaces, J. Fluid Mech., 726 (2013) · Zbl 1287.76222
[18] Leng, L. J., Splash formation by spherical drops, J. Fluid Mech., 427, 2001, 73 (2001) · Zbl 0968.76500
[19] Chanson, H., Air bubble entrainment in hydraulic jumps: Similitude and scale effects (2006)
[20] Chanson, H.; Guiltier, C., Similitude and scale effects of air entrainment in hydraulic jumps, J. Hydraul. Res., 46, 35-44 (2008)
[21] Wang, H., Turbulence and air entrainment in hydraulic jumps (2014)
[22] Wang, H.; Chanson, H., Air entrainment and turbulent fluctuations in hydraulic jumps, Urban Water J., 12, 502-518 (2015)
[23] Valentine, G. A.; Groves, K. R., Entrainment of country rock during basaltic eruptions of the lucero volcanic field, New Mexico, J. Geol., 104, 71-90 (1996)
[24] Suzuki, Y. J.; Koyaguchi, T., Numerical determination of the efficiency of entrainment in volcanic eruption columns, Geophys. Res. Lett., 37, 5 (2010)
[25] Suzuki, Y. J.; Koyaguchi, T., Effects of wind on entrainment efficiency in volcanic plumes, J. Geophys. Res.: Solid Earth, 120, 6122-6140 (2015)
[26] Zhao, H. D.; Wang, F.; Li, Y. Y.; Xia, W., Experimental and numerical analysis of gas entrapment defects in plate ADC12 die castings, J. Mater. Process. Technol., 209, 4537-4542 (2009)
[27] McHardy, C.; Rudolph, A.; Panckow, R.; Kostova, J.; Wegener, M. M.; Rauh, C., Morphological characterization of foams during the filling of non-carbonated beverages, (Proceedings der 26. GALA-Fachtagung ExperimentelleStrömungsmechanik, 4.-6. September 2018 Rostock (2018), Deutsche Gesellschaft Für Laser-Anemometrie-German Asso. for Laser Anemometry GALA EV)
[28] Deotale, S.; Dutta, S.; Moses, J. A.; Balasubramaniam, V. M.; Anandharamakrishnan, C., Foaming characteristics of beverages and its relevance to food processing, Food Eng. Rev., 1-22 (2020)
[29] Shotton, E. E.; Habeeb, A. F.S. A., The entrainment of liquid during distillation, J. Pharm. Pharmacol., 6, 1023-1036 (1954)
[30] Nguyen-Chi, H.; Groll, M., Entrainment or flooding limit in a closed two-phase thermosyphon, J. Heat Recovery Syst., 1, 1981, 275-286 (1981)
[31] Zhang, T.; Yan, Z. W.; Wang, L. Y.; Zheng, W. J.; Su, Y. H.; Zheng, R. C., Droplet entrainment within the evaporator to the suitable volume-filling ratio of a vertical two-phase closed thermosyphon, Int. J. Therm. Sci., 159, Article 106576 pp. (2021)
[32] Kondur, R.; Shaw, J. M., The behaviour of large gas bubbles at a liquid-liquid interface. Part-1: the entrainment of liquid drops, (Materials Handling in Pyrometallurgy (1990), Pergamon), 14-24
[33] Lovick, J.; Angeli, P., Droplet size and velocity profiles in liquid-liquid horizontal flows, Chem. Eng. Sci., 59, 3105-3115 (2004)
[34] Ge, W.; Wang, L.; Wang, W.; Gao, J.; Zhang, J.; Zhao, H.; Kwauk, H. M., Multi-scale modeling, Chem. Eng. Chem. Process. Technol., 43-78 (2010)
[35] Colombo, M.; Fairweather, M.; Walker, S. P.; Kumar, M.; Moharana, A.; Nayak, A. K.; Giteshkumar, P., CFD model development for two-phase flows, (Advances of Comp. Fluid Dyna. in Nuclear Reactor Design and Safety Assessment (2019), Woodhead Publishing), 239-335
[36] Yeoh, G. H.; Tu, J., Computational Techniques for Multiphase Flows (2019), Butterworth-Heinemann
[37] Hajisharifi, A.; Marchioli, C.; Soldati, A., Particle capture by drops in turbulent flow, Phys. Rev. Fluids, 6, Article 024303 pp. (2021)
[38] Lakehal, D., Advanced simulation of transient multiphase flow & flow assurance in the oil & gas industry, Can. J. Chem. Eng., 91, 1201-1214 (2013)
[39] Sun, X.; Sakai, M., Three-dimensional simulation of gas-solid-liquid flows using the DEM-VOF method, Chem. Eng. Sci., 134, 531-548 (2015)
[40] Mer, S.; Praud, O.; Neau, H.; Merigoux, N.; Magnaudet, J.; Roig, V., The emptying of a bottle as a test case for assessing interfacial momentum exchange models for Euler-Euler simulations of multi-scale gas-liquid flows, Int. J. Multiph. Flow., 106, 109-124 (2018)
[41] Davis, R. F., The physical aspect of steam generation at high pressures and the problem of steam contamination, Proc. Inst. Mech. Eng., 144, 198-216 (1940)
[42] Cheng, S. I.; Teller, A. J., Free entrainment behaviour in sieve trays, AIChE J., 7, 282-287 (1961)
[43] Dolna, O.; Mikielewicz, J.; Rolka, P., Analytical studies on deposition and entrainment present in the Venturi nozzle two-phase flow, Int. J. Energy Environ. Eng., 1-16 (2021)
[44] Cioncolini, A.; Thome, J. R.J. R., Prediction of the entrained liquid fraction in vertical annular gas-liquid two-phase flow, Int. J. Multiph. Flow., 36, 2010, 293-302 (2010)
[45] Cioncolini, A.; Thome, J. R., Entrained liquid fraction prediction in adiabatic and evaporating annular two-phase flow, Nucl. Eng. Des., 243, 2022, 200-213 (2012)
[46] Taylor, G. I., Flow induced by jets, J. Aerosp. Sci., 25, 464-465 (1958)
[47] Ishii, M.; Mishima, K., Droplet entrainment correlation in annular two-phase flow, I, J. Heat Mass Transfer, 32, 1835-1846 (1989)
[48] Bagul, R. K.; Pilkhwal, D. S.; Vijayan, P. K.; Joshi, J. B., Entrainment phenomenon in gas-liquid two-phase flow: A review, Sadhana, 38, 1173-1217 (2013) · Zbl 1322.76063
[49] Li, Z.; Spaulding, M. L.; French-McCay, D., An algorithm for modeling entrainment and naturally and chemically dispersed oil droplet size distribution under surface breaking wave conditions, Mar. Pollut. Bull., 119, 145-152 (2017)
[50] Sanjay, V.; Das, A. K., On air entrainment in a water pool by impingement of a jet, AIChE J., 63, 5169-5181 (2017)
[51] Rohsenow, W., A method of correlating heat transfer data for surface boiling of liquids, Trans. ASME, 74, 969-975 (1952)
[52] Zuber, N., Nucleate boiling. The region of isolated bubbles and the similarity with natural convection, Int. J. Heat Mass Transfer, 6, 53-78 (1963)
[53] Cooper, M. G.; Lloyd, A. J.P., The microlayer in nucleate pool boiling, Int. J. Heat Mass Transfer, 12, 895-913 (1969)
[54] Unal, H. C., Maximum bubble diameter, maximum bubble-growth time and bubble-growth rate during the subcooled nucleate flow boiling of water up to 17.7 MN/m 2, Int. J. Heat Mass Transfer, 19, 643-649 (1976)
[55] Son, G.; Dhir, V. K.; Ramanujapu, N., Dynamics and heat transfer associated with a single bubble during nucleate boiling on a horizontal surface, J. Heat Transfer, 121, 623-631 (1999)
[56] Banerjee, D.; Dhir, V. K., Study of subcooled film boiling on a horizontal disc: part 2-experiments, J. Heat Transfer, 123, 285-293 (2001)
[57] Kandlikar, S. G., Fundamental issues related to flow boiling in minichannels and microchannels, Exp. Therm. Fluid Sci., 26, 389-407 (2002)
[58] Das, S. K.; Putra, N.; Roetzel, W., Pool boiling characteristics of nano-fluids, Int. J. Heat Mass Transfer, 46, 851-862 (2003)
[59] You, S. M.; Kim, J. H.; Kim, K. H., Effect of nanoparticles on critical heat flux of water in pool boiling heat transfer, Appl. Phys. Lett., 83, 3374-3376 (2003)
[60] Das, A. K.; Das, P. K.; Saha, P., Heat transfer during pool boiling based on evaporation from micro and macrolayer, Int. J. Heat Mass Transfer, 49, 3487-3499 (2006) · Zbl 1189.76624
[61] Das, A. K.; Das, P. K.; Bhattacharyya, S.; Saha, P., Nucleate boiling heat transfer from a structured surface-Effect of liquid intake, Int. J. Heat Mass Transfer, 50, 1577-1591 (2007) · Zbl 1160.80303
[62] Peng, X., Micro Transport Phenomena During Boiling (2011), Springer Sci. & Business Media · Zbl 1221.80001
[63] Miglani, A.; Joo, D.; Basu, S.; Kumar, R., Nucleation dynamics and pool boiling characteristics of high pressure refrigerant using thermochromic liquid crystals, Int. J. Heat Mass Transfer, 60, 188-200 (2013)
[64] Katiyar, G.; Karagadde, S.; Saha, S. K.; Sharma, A., Numerical modelling of bubble growth in microchannel using Level Set Method, Int. J. Heat Mass Transfer, 101, 719-732 (2016)
[65] Deep, A.; Swaroop Meena, C.; Kumar Das, A., Interaction of asymmetric films around boiling cylinder array: Homogeneous interface to chaotic phenomenon, J. Heat Transfer, 139, 4 (2017)
[66] Meena, C. S.; Deep, A.; Das, A. K., Understanding of interactions for bubbles generated at neighboring nucleation sites, Heat Transfer Eng., 39, 885-900 (2018)
[67] Beal, C., The viscosity of air, water, natural gas, crude oil and its associated gases at oil field temperatures and pressures, Trans. AIME, 165, 94-115 (1946)
[68] Taitel, Y.; Dukler, A. E., A model for predicting flow regime transitions in horizontal and near horizontal gas-liquid flow, AIChE J., 22, 47-55 (1976)
[69] Taitel, Y.; Barnea, D., A consistent approach for calculating pressure drop in inclined slug flow, Chem. Eng. Sci., 45, 1199-1206 (1990)
[70] Barnea, D.; Taitel, Y., Interfacial and structural stability of separated flow, Int. J. Multiph. Flow., 20, 387-414 (1994) · Zbl 1134.76488
[71] Kantarci, N.; Borak, F.; Ulgen, K. O., Bubble column reactors, Process Biochem., 40, 2005, 2263-2283 (2005)
[72] Shaikh, A.; Al-Dahhan, M. H., A review on flow regime transition in bubble columns, Int. J. Chem. React. Eng., 5, 1-68 (2007)
[73] Martín, M.; Montes, F. J.; Galán, M. A., Bubbling process in stirred tank reactors II: Agitator effect on the mass transfer rates, Chem. Eng. Sci., 63, 3223-3234 (2008)
[74] Albadawi, A.; Donoghue, D. B.; Robinson, A. J.; Murray, D. B.; Delaure, Y. M.C., On the analysis of bubble growth and detachment at low capillary and bond numbers using volume of fluid and level set methods, Chem. Eng. Sci., 90, 77-91 (2013)
[75] Nedeltchev, S., Theoretical prediction of mass transfer coefficients in both gas-liquid and slurry bubble columns, Chem. Eng. Sci., 157, 169-181 (2017)
[76] Rohilla, L.; Das, A. K., On transformation of a Taylor bubble to an asymmetric sectorial wrap in an annuli, Ind. Eng. Chem. Res., 56, 14384-14395 (2017)
[77] Rana, B. K.; Das, A. K.; Das, P. K., Mechanism of bursting Taylor bubbles at free surfaces, Langmuir, 31, 9870-9881 (2015)
[78] Rana, B. K.; Das, A. K.; Das, P. K., Asymmetric bursting of Taylor bubble in inclined tubes, Phys. Fluids, 28, Article 082106 pp. (2016)
[79] Multiphase Science and Technology: Volume 2, Vol. 2 (2013), Springer Science & Business Media
[80] Rana, B. K.; Das, A. K.; Das, P. K., Collapse of a Taylor bubble at a free surface, Multiphase Sci. Technol., 28, 2 (2016)
[81] Tsai, J. H.; Lin, L., A thermal-bubble-actuated micronozzle-diffuser pump, J. Microelectromech. Syst., 11, 665-671 (2002)
[82] Ajaev, V. S.; Homsy, G. M., Modeling shapes and dynamics of confined bubbles, Annu. Rev. Fluid Mech., 38, 277-307 (2006) · Zbl 1098.76069
[83] Garstecki, P.; Fuerstman, M. J.; Stone, H. A.; Whitesides, G. M., Formation of droplets and bubbles in a microfluidic T-junction—scaling and mechanism of break-up, Lab A Chip, 6, 437-446 (2006)
[84] Marmottant, P.; Raven, J. P.; Gardeniers, H. J.G. E.; Bomer, J. G.; Hilgenfeldt, S., Microfluidics with ultrasound-driven bubbles, J. Fluid Mech., 568, 109 (2006) · Zbl 1104.76013
[85] Whitesides, G. M., The origins and the future of microfluidics, Nature, 442, 368-373 (2006)
[86] Marmottant, P.; Raven, J. P., Microfluidics with foams, Soft Matter, 5, 3385-3388 (2009)
[87] Chakraborty, D.; Chakraborty, S., Erratum: Controlled microbubble generation on a compact disk, Appl. Phys. Lett., 98, Article 234103 pp. (2011)
[88] Abate, A. R.; Weitz, D. A., Air-bubble-triggered drop formation in microfluidics, Lab A Chip, 11, 1713-1716 (2011)
[89] Huerre, A.; Miralles, V.; Jullien, M. C., Bubbles and foams in microfluidics, Soft Matter, 10, 6888-6902 (2014)
[90] Iqbal, R.; Dhiman, S.; Sen, A. K.; Shen, A. Q., Dynamics of a water droplet over a sessile oil droplet: Compound droplets satisfying a Neumann condition, Langmuir, 33, 5713-5723 (2017)
[91] Jayaprakash, K. S.; Banerjee, U.; Sen, A. K., Dynamics of rigid microparticles at the interface of co-flowing immiscible liquids in a microchannel, J. Colloid Interface Sci., 493, 317-326 (2017)
[92] MacDonald, B. D.; Rowe, A. M., Experimental and numerical analysis of dynamic metal hydride hydrogen storage systems, J. Power Sources, 174, 282-293 (2007)
[93] Addamane, S. R.; Hajilou, M.; Belmont, E. L., Experimental and analytical study of a porous media reformer with passive air entrainment, Int. J. Hydrog. Energy, 41, 12738-12746 (2016)
[94] Heskestad, G., Fire plumes, flame height, and air entrainment, (SFPE Handbook of Fire Protection Engg. (2016), Springer: Springer New York, NY), 396-428
[95] Brauner, N.; Maron, D. M., Flow pattern transitions in two-phase liquid-liquid flow in horizontal tubes, Int. J. Multiph. Flow., 18, 123-140 (1992) · Zbl 1144.76355
[96] Angeli, P.; Hewitt, G. F., Pressure gradient in horizontal liquid-liquid flows, Int. J. Multiph. Flow., 24, 1183-1203 (1999) · Zbl 1121.76419
[97] Angeli, P.; Hewitt, G. F., Flow structure in horizontal oil-water flow, Int. J. Multiph. Flow., 26, 1117-1140 (2000) · Zbl 1137.76518
[98] Al-Wahaibi, T.; Angeli, P., Transition between stratified and non-stratified horizontal oil-water flows. Part I: Stability analysis, Chem. Eng. Sci., 62, 2915-2928 (2007)
[99] Hanafizadeh, P.; Hojati, A.; Karimi, A., Experimental investigation of oil-water two phase flow regime in an inclined pipe, J. Pet. Sci. Eng., 136, 12-22 (2015)
[100] Al-Wahaibi, T.; Smith, M.; Angeli, P., Transition between stratified and non-stratified horizontal oil-water flows. Part II: Mechanism of drop formation, Chem. Eng. Sci., 62, 2007, 2929-2940 (2007)
[101] Chinaud, M.; Park, K. H.; Angeli, P., Flow pattern transition in liquid-liquid flows with a transverse cylinder, Int. J. Multiph. Flow., 90, 1-12 (2017)
[102] Simmons, M. J.H.; Azzopardi, B. J., Drop size distributions in dispersed liquid-liquid pipe flow, Int. J. Multiph. Flow., 27, 2001, 843-859 (2001) · Zbl 1137.76738
[103] Voulgaropoulos, V.; Angeli, P., Optical measurements in evolving dispersed pipe flows, Exp. Fluids, 58, 170 (2017)
[104] Al-Wahaibi, T.; Yusuf, N.; Al-Wahaibi, Y.; Al-Ajmi, A., Experimental study on the transition between stratified and non-stratified horizontal oil-water flow, Int. J. Multiph. Flow., 38, 126-135 (2012)
[105] Park, K. H.; Chinaud, M.; Angeli, P., Transition from stratified to non-stratified oil-water flows using a bluff body, Exp. Therm. Fluid Sci., 76, 175-184 (2016)
[106] Strang, E. J.; Fernando, H. J.S., Entrainment and mixing in stratified shear flows, J. Fluid Mech., 428, 349-386 (2001) · Zbl 0968.76501
[107] Jeffries, R. B.; Scott, D. S.; Rhodes, E., Structure of turbulence close to the interface in the liquid phase of a co-current stratified two-phase flow, (Proceedings of the Institution of Mechanical Engineers, Conference Proceedings, vol. 184 (1969), SAGE Publications: SAGE Publications Sage UK: London, England), 204-214
[108] Javadi, A.; Eggers, J.; Bonn, D.; Habibi, M.; Ribe, N. M., Delayed capillary breakup of falling viscous jets, Phys. Rev. Lett., 110, Article 144501 pp. (2013)
[109] Plateau, M., Statique des Liquids (1873), Gauthier-Villars: Gauthier-Villars Paris · JFM 06.0516.03
[110] Rayleigh, L., On the instability of jets, Proc. Lond. Math. Soc., 1, 4-13 (1878) · JFM 11.0685.01
[111] Kumar, P.; Das, A. K.; Mitra, S. K., Bending and growth of entrained air filament under converging and asymmetric rotational fields, Phys. Fluids, 29, Article 022101 pp. (2017)
[112] Kumar, P.; Das, A. K.; Mitra, S. K., Air entrainment driven by a converging rotational field in a viscous liquid, Phys. Fluids, 29, Article 102104 pp. (2017)
[113] Panda, S. K.; Rana, B. K.; Kumar, P., Competition of roller rotation and horizontal crossflow to control the free surface cusp-induced air entrainment, Phys. Fluids, 33, Article 112114 pp. (2021)
[114] Panda, S. K.; Rana, B. K., Numerical simulation and analytical prediction on development of entrained air filament caused by combined effect of rotational field and free stream flow, Ind. Eng. Chem. Res. (2022), (Just accepted)
[115] Xiangju, C.; Xuewei, C., Progress in numerical simulation of high entrained air-water two-phase flow, (2012 Third International Conference on Digital Manufacturing & Automation (2012), IEEE), 626-629, 626-629
[116] Chauhan, K.; Philip, J.; De Silva, C. M.; Hutchins, N.; Marusic, I., The turbulent/non-turbulent interface and entrainment in a boundary layer, J. Fluid Mech., 742, 119-151 (2014)
[117] Clanet, C.; Lasheras, J. C., Depth of penetration of bubbles entrained by a plunging water jet, Phys. Fluids, 9, 1864-1866 (1997)
[118] Eggers, J.; Villermaux, E., Physics of liquid jets, Rep. Prog. Phys., 71, Article 036601-1-19 (2008)
[119] Driessen, T.; Jeurissen, R.; Wijshoff, H.; Toschi, F.; Lohse, D., Stability of viscous long liquid filaments, Phys. Fluids, 25, Article 062109 pp. (2013)
[120] Rana, B. K.; Das, A. K.; Das, P. K., Numerical study of air entrainment and liquid film wrapping around a rotating cylinder, Ind. Eng. Chem. Res., 55, 11950-11960 (2016)
[121] Chaini, U.; Ghosh, S.; Samal, K.; Kumar, A., Rotary entrainment in two-phase stratified liquid-liquid layers, (Proceedings of the 24th National and 2nd International ISHMT-ASTFE Heat and Mass Transfer Conference (IHMTC-2017) (2017), Begel House Inc.)
[122] Kumar, R.; Rohilla, L.; Das, A. K., Passage of a Taylor bubble through a stratified liquid-liquid interface, Ind. Eng. Chem. Res., 59, 3757-3771 (2019)
[123] Mandal, T. K.; Das, G.; Das, P. K., Motion of Taylor bubbles and Taylor drops in liquid- liquid systems, Ind. Eng. Chem. Res., 47, 7048-7057 (2008)
[124] Huang, X.; Guo, Q.; Qiu, B.; Feng, X., Prediction of air-entrained vortex in pump sump: Influence of turbulence models and interface-tracking methods, J. Hydraul. Eng., 146, Article 04020010 pp. (2020)
[125] Greene, G. A.; Chen, J. C.; Conlin, M. T., Onset of entrainment between immiscible liquid layers due to rising gas bubbles, Int. J. Heat Mass Transfer, 31, 1309-1317 (1988)
[126] Greene, G. A.; Chen, J. C.; Conlin, M. T., Bubble induced entrainment between stratified liquid layers, Int. J. Heat Mass Transfer, 34, 149-157 (1991)
[127] Wu, Q.; Kim, S.; Ishii, M.; Beus, S. G., One-group interfacial area transport in vertical bubbly flow, Int. J. Heat Mass Transfer, 41, 1103-1112 (1998) · Zbl 0940.76529
[128] Rana, B. K.; Das, A. K.; Das, P. K., Towards the understanding of bubble-bubble interaction upon formation at submerged orifices: A numerical approach, Chem. Eng. Sci., 161, 316-328 (2017)
[129] Rana, B. K.; Das, A. K.; Das, P. K., Study of interaction pattern between bubbles at three inline orifices in a submerged pool, Chem. Eng. Sci., 168, 41-54 (2017)
[130] Hesketh, R. P.; Etchells, A. W.; Russell, T. F., Bubble breakage in pipeline flow, Chem. Eng. Sci., 46, 1-9 (1991)
[131] Foroushan, H. K.; Jacobson, H. A., Experimental study of single bubble breakage in turbulent flow field: Evaluation of breakage models, Chem. Eng. Sci., 253, Article 117584 pp. (2022)
[132] Biń, A. K., Gas entrainment by plunging liquid jets, Chem. Eng. Sci., 48, 3585-3630 (1993)
[133] McKeogh, E. J.; Ervine, D. A., Air entrainment rate and diffusion pattern of plunging liquid jets, Chem. Eng. Sci., 36, 1161-1172 (1981)
[134] Durve, A. P.; Patwardhan, A. W., Numerical and experimental investigation of onset of gas entrainment phenomenon, Chem. Eng. Sci., 73, 140-150 (2012)
[135] Yu, X.; Hendrickson, K.; Campbell, B. K.; Yue, D. K., Numerical investigation of shear-flow free-surface turbulence and air entrainment at large Froude and Weber numbers, J. Fluid Mech., 880, 209-238 (2019) · Zbl 1430.76235
[136] Chanson, H., Air entrainment in two-dimensional turbulent shear flows with partially developed inflow conditions, Int. J. Multiphase Flow, 21, 1107-1121 (1995) · Zbl 1135.76379
[137] Narimousa, S.; Long, R. R.; Kitaigorodskii, S. A., Entrainment due to turbulent shear flow at the interface of a stably stratified fluid, Tellus A, 38, 76-87 (1986)
[138] Chu, V. H.; Vanvari, M. R., Experimental study of turbulent stratified shearing flow, J. Hydral. Div., 102, 691-706 (1976)
[139] Mehralizadeh, A.; Shabanian, S. R.; Bakeri, G., Effect of modified surfaces on bubble dynamics and pool boiling heat transfer enhancement: A review, Therm. Sci. Eng. Prog., 15, Article 100451 pp. (2020)
[140] Gupta, M. K.; Sharma, D. S.; Lakhera, V. J., Vapor bubble formation, forces, and induced vibration: A review, Appl. Mech. Rev., 68, 3 (2016)
[141] Luke, A., Interactions between bubble formation and heating surface in nucleate boiling, Exp. Therm. Fluid Sci., 35, 753-761 (2011)
[142] Luke, A., Active and potential bubble nucleation sites on different structured heated surfaces, Chem. Eng. Res. Des., 82, 462-470 (2004)
[143] Marston, J. O.; Vakarelski, I. U.; Thoroddsen, S. T., Bubble entrapment during sphere impact onto quiescent liquid surfaces, J. Fluid Mech., 680, 660-670 (2011) · Zbl 1241.76041
[144] Do-Quang, M.; Amberg, G., The splash of a solid sphere impacting on a liquid surface: numerical simulation of the influence of wetting, Phys. Fluids, 21, Article 022102 pp. (2009) · Zbl 1183.76183
[145] Grumstrup, T.; Keller, J. B.; Belmonte, A., Cavity ripples observed during the impact of solid objects into liquids, Phys. Rev. Lett., 99, Article 114502 pp. (2007)
[146] Moghisi, M.; Squire, P. T., An experimental investigation of the initial force of impact on a sphere striking a liquid surface, J. Fluid Mech., 108, 133-146 (1981)
[147] Sugimoto, T.; Kaneko, A.; Abe, Y.; Uchibori, A.; Kurihara, A.; Takata, T.; Ohshima, H., Droplet entrainment by high-speed gas jet into a liquid pool, Nucl. Eng. Des., 380, Article 111306 pp. (2021)
[148] Kolev, N. I., IVA2 verification: high-pressure gas injection in a liquid pool, Nucl. Technol., 83, 65-80 (1988)
[149] Nakao, K.; Inazumi, S.; Takaue, T.; Tanaka, S.; Shinoi, T., Evaluation of discharging surplus soils for relative stirred deep mixing methods by mps-cae analysis, Sustainability, 14, 58 (2021)
[150] Das, A. K.; Das, P. K.; Saha, P., Formation of bubbles at submerged orifices-Experimental investigation and theoretical prediction, Exp. Therm. Fluid Sci., 35, 2011, 618-627 (2011)
[151] Martín, M.; García, J. M.; Montes, F. J.; Galán, M. A., On the effect of the orifice configuration on the coalescence of growing bubbles, Chem. Eng. Process.: Process Intensif., 47, 799-1809 (2008)
[152] Martín, M.; Montes, F. J.; Galán, M. A., Bubble coalescence at sieve plates: II. Effect of coalescence on mass transfer. Superficial area versus bubble oscillations, Chem. Eng. Sci., 62, 1741-1752 (2017)
[153] Ruzicka, M.; Drahoš, J.; Zahradnık, J.; Thomas, N. H., Structure of gas pressure signal at two-orifice bubbling from a common plenum, Chem. Eng. Sci., 55, 421-429 (2000)
[154] Shah, Y. T.; Kelkar, B. G.; Godbole, S. P.; Deckwer, W. D., Design parameters estimations for bubble column reactors, AICHE J., 28, 353-379 (1982)
[155] Notz, P. K.; Basaran, O. A., Dynamics and breakup of a contracting liquid filament, J. Fluid Mech., 512, 223-256 (2004) · Zbl 1163.76356
[156] Castrejón-Pita, A. A.; Castrejón-Pita, J. R.; Hutchings, I. M., Breakup of liquid filaments, Phys. Rev. Lett., 108, Article 074506 pp. (2012)
[157] Sun, T., A numerical study on dynamics behaviors of multi bubbles merger during nucleate boiling by lattice Boltzmann method, Int. J. Multiphase Flow, 118, 128-140 (2019)
[158] Mukherjee, A.; Dhir, V. K., Study of lateral merger of vapor bubbles during nucleate pool boiling, J. Heat Transfer, 126, 1023-1039 (2004)
[159] Chen, T.; Chung, J. N., Coalescence of bubbles in nucleate boiling on microheaters, Int. J. Heat Mass Transfer, 45, 2329-2341 (2002)
[160] Yang, Z. L.; Dinh, T. N.; Nourgaliev, R. R.; Sehgal, B. R., Numerical investigation of bubble growth and detachment by the lattice-Boltzmann method, Int. J. Heat Mass Transfer, 44, 195-206 (2001) · Zbl 1007.76064
[161] Sharma, Y.; Rana, B. K.; Das, A. K., Rotary entrainment in two phase stratified gas-liquid layers: an experimental study, Int. J. Chem. Mol. Nucl. Mater. Metall. Eng., 9, 1367-1372 (2015)
[162] Li, C.; Li, Y.; Wang, L., Configuration dependence and optimization of the entrainment performance for gas-gas and gas-liquid ejectors, Appl. Therm. Eng., 48, 237-248 (2012)
[163] Lin, T. J.; Donnelly, H. G., Gas bubble entrainment by plunging laminar liquid jets, AIChE J., 12, 563-571 (1966)
[164] Paul, E. L.; Midler, M.; Sun, Y., Mixing in the fine chemicals and pharmaceutical industries, (Handbook of Industrial Mixing: Sci. and Practice, 2004 (2004)), 1027-1069
[165] Turner, J. S., Turbulent entrainment: the development of the entrainment assumption, and its application to geophysical flows, J. Fluid Mech., 173, 431-471 (1986)
[166] Strang, E. J.; Fernando, H. J.S., Entrainment and mixing in stratified shear flows, J. Fluid Mech., 428, 349 (2001) · Zbl 0968.76501
[167] Li, S.; Chen, H.; Song, J.; He, H., Investigation of turbulent entrainment in ocean thermocline using large eddy simulation, (Geophysical Research Abstracts, Vol. 21 (2019))
[168] Weston, R. F., Studies in entrainment aeration, J. (Water Pollut. Control Fed.), 342-353 (1962)
[169] Yadav, A. A.; Sarkar, Kumar S., An experimental study to evaluate the efficacy of air entrainment holes on the throat of a venturi aeration system, Aquac. Int., 28, 1057-1068 (2020)
[170] Yu, S. H.; Lee, K. S.; Yook, S. J., Film flow around a fast rotating roller, Int. J. Heat Fluid Flow, 30, 796-803 (2009)
[171] Bonizzi, M.; Issa, R. L., A model for simulating gas bubble entrainment in two-phase horizontal slug flow, Int. J. Multiph. Flow., 29, 1685-1717 (2003) · Zbl 1136.76465
[172] Bartley, J. T.; Soliman, H. M.; Sims, G. E., Experimental investigation of the onsets of gas and liquid entrainment from a small branch mounted on an inclined wall, Int. J. Multiph. Flow., 34, 905-915 (2008)
[173] Bowden, R. C.; Hassan, I. G., The onset of gas entrainment from a flowing stratified gas-liquid regime in dual discharging branches: Part I: Flow visualization and related phenomena, Int. J. Multiph. Flow., 37, 1358-1370 (2011)
[174] Bowden, R. C.; Hassan, I. G., The onset of gas entrainment from a flowing stratified gas-liquid regime in dual discharging branches: Part II: Critical conditions at low to moderate branch Froude numbers, Int. J. Multiph. Flow., 37, 1371-1380 (2011)
[175] Kaluarachchi, J. J.; Parker, J. C., Multiphase flow with a simplified model for oil entrapment, Trans. Porous Media, 7, 1-14 (1992)
[176] Roy, A. K.; Maiti, B.; Das, P. K., Visualisation of air entrainment by a plunging jet, Procedia Eng., 56, 468-473 (2013)
[177] Joseph, D. D.; Nguyen, K.; Beavers, G. S., Non-uniqueness and stability of the configuration of flow of immiscible fluids with different viscosities, J. Fluid Mech., 141, 319-345 (1984) · Zbl 0544.76118
[178] Joseph, D. D.; Nelson, J.; Renardy, M.; Renardy, Y., Two-dimensional cusped interfaces, J. Fluid Mech., 223, 383-409 (1991)
[179] Joseph, D. D., Understanding cusped interfaces, J. Non-Newtonian Fluid Mech., 44, 127-148 (1992) · Zbl 0825.76046
[180] Jeong, J. T.; Moffatt, H. K., Free-surface cusps associated with flow at low Reynolds number, J. Fluid Mech., 241, 1-22 (1992) · Zbl 0780.76020
[181] Middleman, S., Free coating of viscous and viscoelastic liquids onto a partially submerged rotating roll, Polym. Eng. Sci., 18, 734-737 (1978)
[182] Tharmalingam, S.; Wilkinson, W. L., The coating of Newtonian liquids onto a roll rotating at low speeds, Polym. Eng. Sci., 18, 1155-1159 (1978)
[183] Bolton, B.; Middleman, S., Air entrainment in a roll coating system, Chem. Eng. Sci., 35, 597-601 (1980)
[184] Shul’Man, Z. P.; Baikov, V. I., Entrainment of a viscoplastic fluid by a rotating drum, J. Eng. Phys., 40, 605-608 (1981)
[185] Tekic, M. N.; Jovanovic, S., Liquid coating onto a rotating roll, Chem. Eng. Sci., 37, 1815-1817 (1982)
[186] Wilson, S. D., The drag-out problem in film coating theory, J. Eng. Math., 16, 209-221 (1982) · Zbl 0502.76053
[187] Landau, L.; Levich, B., Dragging of a liquid by a moving plate, (Dynamics of Curved Fronts (1988), Academic Press), 141-153
[188] Campanella, O. H.; Cerro, R. L., Viscous flow on the outside of a horizontal rotating cylinder: the roll coating regime with a single fluid, Chem. Eng. Sci., 39, 1443-1449 (1984)
[189] Campanella, O. H.; Cerro, R. L., Viscous flow on the outside of a horizontal rotating cylinder—III. Selective coating of two immiscible fluids, Chem. Eng. Sci., 41, 2715-2721 (1986)
[190] Lorenceau, E.; Restagno, F.; Quéré, D., Fracture of a viscous liquid, Phys. Rev. Lett., 90, Article 184501 pp. (2003)
[191] Rebouillat, S.; Steffenino, B.; Salvador, B., Hydrodynamics of high-speed fibre impregnation: the fluid layer formation from the meniscus region, Chem. Eng. Sci., 57, 3953-3966 (2002)
[192] Spiers, R. P.; Subbaraman, C. V.; Wilkinson, W. L., Free coating of a Newtonian liquid onto a vertical surface, Chem. Eng. Sci., 29, 389-396 (1974)
[193] Biń, A. K., Gas entrainment by plunging liquid jets, Chem. Eng. Sci., 48, 3585-3630 (1993)
[194] Kiger, K. T.; Duncan, J. H., Air-entrainment mechanisms in plunging jets and breaking waves, Annu. Rev. Fluid Mech., 44, 563-596 (2012) · Zbl 1352.76026
[195] Tharmalingam, S.; Wilkinson, W. L., The coating of Newtonian liquids onto a rotating roll, Chem. Eng. Sci., 33, 1481-1487 (1978)
[196] Decré, M.; Gailly, E.; Buchlin, J. M., Meniscus shape experiments in forward roll coating, Phys. Fluids, 7, 458-467 (1995)
[197] Carvalho, M. S.; Scriven, L. E., Three-dimensional stability analysis of free surface flows: application to forward deformable roll coating, J. Comput. Phys., 151, 534-562 (1999) · Zbl 0946.76028
[198] Peixinho, J.; Mirbod, P.; Morris, J. F.J. F., Free surface flow between two horizontal concentric cylinders, Eur. Phys. J. E, 35, 19 (2012)
[199] Skartlien, R.; Julshamn, J. A.; Lawrence, C. J.; Liu, L., A gas entrainment model for hydraulic jumps in near horizontal pipes, Int. J. Multiph. Flow., 43, 39-55 (2012)
[200] Gundlach, J.; Lübke, M.; Wünsch, O., Numerical investigation of the free-surface flow within an annulus, PAMM, 14, 681-682 (2014)
[201] Jeong, J. T., Formation of cusp on the free surface at low Reynolds number flow, Phys. Fluids, 11, 521-526 (1999) · Zbl 1147.76425
[202] Jeong, J. T., Two-dimensional Stokes flow due to a pair of vortices below the free surface, Phys. Fluids, 22, Article 082102 pp. (2010)
[203] Betelú, S.; Gratton, R.; Diez, J., Observation of cusps during the levelling of free surfaces in viscous flows, J. Fluid Mech., 377, 137-149 (1998) · Zbl 0941.76529
[204] Eggers, J., Air entrainment through free-surface cusps, Phys. Rev. Lett., 86, 4290 (2001)
[205] Lorenceau, É.; Quéré, D.; Eggers, J., Air entrainment by a viscous jet plunging into a bath, Phys. Rev. Lett., 93, Article 254501 pp. (2004)
[206] Kumar, P.; Prajapati, M.; Das, A. K.; Mitra, S. K., Vortex formation and subsequent air entrainment inside a liquid pool, Ind. Eng. Chem. Res., 57, 6538-6552 (2018)
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