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Effects of elasticity number and time constant ratio on breakup and droplet formation of viscoelastic planar liquid sheet co-flowing with gases of equal velocities

Published online by Cambridge University Press:  04 June 2021

Debayan Dasgupta Open the ORCID record for Debayan Dasgupta [Opens in a new window] ,
Saurabh Sharma Open the ORCID record for Saurabh Sharma [Opens in a new window] ,
Sujit Nath Open the ORCID record for Sujit Nath [Opens in a new window]  and
Dipankar Bhanja Open the ORCID record for Dipankar Bhanja [Opens in a new window]
Debayan Dasgupta
Affiliation:
Department of Mechanical Engineering, National Institute of Technology Silchar, Assam788 010, India
Saurabh Sharma
Affiliation:
Department of Mechanical Engineering, National Institute of Technology Silchar, Assam788 010, India
Sujit Nath*
Affiliation:
Department of Mechanical Engineering, National Institute of Technology Silchar, Assam788 010, India
Dipankar Bhanja
Affiliation:
Department of Mechanical Engineering, National Institute of Technology Silchar, Assam788 010, India
*
† Email address for correspondence: sujitnath@mech.nits.ac.in, sujitnath2008@gmail.com
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Abstract

A weakly nonlinear investigation of sinuous instabilities in a planar viscoelastic liquid sheet having corotational Jeffrey's rheological model is performed. Analysis predicts that viscoelastic properties may exhibit a non-monotonic dual effect depending upon their range and velocity ratios. At velocity ratios of 2 and 2.15, an increase in elasticity stabilizes the sheet for elasticity number ranging from 0.1 to 4 and from 0.1 to 1, respectively. Beyond this range, elasticity produces a destabilizing effect on the sheet. However, at higher velocity ratios of 2.50 and 2.75, an increase in elasticity only destabilizes the liquid sheet. The effect of time constant ratio at different velocity ratios is opposite to that of elasticity number. An increase in time constant ratio destabilizes the sheet at velocity ratio of 2, whereas it stabilizes the sheet for relatively higher velocity ratios of 2.50 and 2.75. At intermediate velocity ratio of 2.15, two regimes of time constant ratio are identified in the range 0.1 to 0.4 and 0.4 to 0.9, representing stabilizing and destabilizing effect of time constant ratio, respectively. The nonlinear interaction between the viscoelastic sheet and surrounding gases may enhance or dampen the second-order amplitude. The contribution of second-order amplitude to sheet breakup is much higher than that of linear growth rate and is responsible for the dual effect of viscoelastic properties. Finally, the size distribution of droplets formed after primary breakup is investigated using maximum entropy formulation. Results reveal that an increase in elasticity number and time constant ratio produces finer and larger droplets, respectively.

JFM classification

Drops and Bubbles: Breakup/coalescence Instability: Nonlinear instability Non-Newtonian Flows: Viscoelasticity
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 920 , 10 August 2021 , A7
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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References

Alsharif, A.M. 2019 Temporal and spatial instability of viscoelastic compound jets. Eur. J. Mech. (B / Fluids) 74, 320–330.CrossRefGoogle Scholar
Asare, H.R., Takahashi, R.K. & Hoffman, M.A. 1981 Liquid sheet jet experiments: comparison with linear theory. J. Fluids Engng 103, 595–603.CrossRefGoogle Scholar
Atalık, K. & Keunings, R. 2002 Non-linear temporal stability analysis of viscoelastic plane channel flows using a fully-spectral method. J. Non-Newtonian Fluid Mech. 102, 299–319.CrossRefGoogle Scholar
Bhatia, J., Dominick, J., Drust, F. & Tropea, C. 1988 Phase-doppler-anemometry and the log-hyperbolic ditribution applied to liquid sprays. Part. Part. Syst. Charact. 5, 153–164.CrossRefGoogle Scholar
Brenn, G., Liu, Z. & Durst, F. 2000 Linear analysis of the temporal instability of axisymmetrical non-Newtonian liquid jets. Intl J. Multiphase Flow 26, 1621–1644.CrossRefGoogle Scholar
Brenn, G., Liu, Z. & Durst, F. 2001 Three dimensional temporal instability of non-Newtonian liquid sheets. Atomiz. Sprays 11, 49–84.CrossRefGoogle Scholar
Carroll, C.P. & Joo, Y.L. 2006 Electrospinning of viscoelastic Boger fluids: modeling and experiments. Phys. Fluids 18, 053102.CrossRefGoogle Scholar
Chin, L.P., Larose, P.G., Tankin, R.S., Jackson, T., Stutrud, J. & Switzer, G. 1991 Droplet distributions from the breakup of a cylindrical liquid jet. Phys. Fluids 3, 1897–1906.CrossRefGoogle Scholar
Christanti, Y. & Walker, L.M. 2001 Surface tension driven jet break up of strain-hardening polymer solutions. J. Non-Newtonian Fluid Mech. 100, 9–26.CrossRefGoogle Scholar
Clark, C.J. & Dombrowski, N. 1972 Aerodynamic instability and disintegration of inviscid liquid sheets. Proc. R. Soc. 329, 467–478.Google Scholar
Cooper, J., Fagan, J., Tirtaatmadja, V., Lester, D. & Boger, D. 2002 Drop formation dynamics of constant low-viscosity, elastic fluids. J. Non-Newtonian Fluid Mech. 106, 29–59.CrossRefGoogle Scholar
Dasgupta, D., Nath, S. & Bhanja, D. 2018 Dual-mode nonlinear instability analysis of a confined planar liquid sheet sandwiched between two gas streams of unequal velocities and prediction of droplet size and velocity distribution using maximum entropy formulation. Phys. Fluid 30, 044104.CrossRefGoogle Scholar
Dasgupta, D., Nath, S. & Bhanja, D. 2019 A study on dual role of viscosity on the stability of a viscous planar liquid sheet surrounded by inviscid gas streams of equal velocities, and prediction of resulting droplet distribution using maximum entropy formulation. Phys. Fluids 31, 074103.CrossRefGoogle Scholar
Deblais, A., Velikov, K.P. & Bonn, D. 2018 Pearling instabilities of a viscoelastic thread. Phys. Rev. Lett. 120, 194501.CrossRefGoogle ScholarPubMed
Denn, M.M. 1990 Issues in viscoelastic fluid mechanics. Annu. Rev. Fluid Mech. 22, 13–32.CrossRefGoogle Scholar
Denn, M.M. 2004 Fifty years of non-Newtonian fluid dynamics. Fluid Mech. Transp. Phenom. 50, 2335–2345.Google Scholar
Ibrahim, A.A. & Jog, M.A. 2008 Nonlinear instability of an annular liquid sheet exposed to gas flow. Intl J. Multiphase Flow 34, 647–664.CrossRefGoogle Scholar
James, D.F. 2009 Boger fluids. Annu. Rev. Fluid Mech. 41, 129–142.CrossRefGoogle Scholar
Jazayeri, S.A. & Li, X. 2000 Nonlinear instability of plane liquid sheets. J. Fluid Mech. 406, 281–308.CrossRefGoogle Scholar
Keshavarz, B., Houze, E.C., Moore, J.R., Koerner, M.R. & McKinley, G.H. 2016 Ligament mediated fragmentation of viscoelastic liquids. Phys. Rev. Lett. 117, 1–6.CrossRefGoogle ScholarPubMed
Kooij, S., Sijs, R., Denn, M.M., Villermaux, E. & Bonn, D. 2018 What determines the drop size in sprays? Phys. Rev. X 8, 031019.Google Scholar
Langlois, W. & Rivlin, R. 1959 Steady Flow of Slightly Viscoelastic Fuids. Division of Applied Mathematics, Brown University.Google Scholar
Langlois, W. & Rivlin, R. 1963 Slow steady-state flow of viscoelastic fluids through non-circular tubes. Rend. di Mat. 22, 169–185.Google Scholar
Larson, R. 1992 Instabilities in viscoelastic flows. Rheol. Acta 31, 213–263.CrossRefGoogle Scholar
Li, M. & Li, X. 2006 A second-order Newton-Raphson method for improved numerical stability in the determination of droplet size distributions in sprays. Atomiz. Sprays 16, 71–82.CrossRefGoogle Scholar
Li, X. & Tankin, R.S. 1988 Derivation of droplet size distribution in sprays by using information theory. Combust. Sci. Technol. 60, 345–357.Google Scholar
Li, X. & Tankin, R.S. 1991 On the temporal instability of a two-dimensional viscous liquid sheet. J. Fluid Mech. 226, 425–443.CrossRefGoogle Scholar
Li, X. & Tankin, R.S. 1992 On the prediction of droplet size and velocity distribution in sprays through maximum entropy principle. Part. Part. Syst. Charact. 68, 147–155.Google Scholar
Li, X., Tankin, R.S. & Renksizbulut, M. 1990 Calculated characteristics of droplet size and velocity distributions in liquid sprays. Part. Part. Syst. Charact. 7, 54–59.CrossRefGoogle Scholar
Liu, Z. & Durst, F. 1998 Linear analysis of the instability of two-dimensional non-Newtonian liquid sheets. J. Non-Newtonian Fluid Mech. 78, 133–166.CrossRefGoogle Scholar
Liu, Z. & Liu, Z. 2006 Linear analysis of three-dimensional instability of non-Newtonian liquid jets. J. Fluid Mech. 559, 451–459.CrossRefGoogle Scholar
Mitra, S.K., Li, X. & Renksizbulut, M. 2001 On the breakup of viscous liquid sheets by dual-mode linear analysis introduction. J. Propul. Power 17, 728–735.CrossRefGoogle Scholar
Mondal, D., Datta, A. & Sarkar, A. 2003 Theoretical prediction of drop size distribution in a spray from a pressure swirl atomizer using maximum entropy. Proc. Inst. Mech. Engrs C J. Mech. Engng Sci. 217, 831–838.CrossRefGoogle Scholar
Mujumdar, A.S., Huang, L.X. & Dong Chen, X. 2010 An overview of the recent advances in spray-drying. Dairy Sci. Technol. 90, 211–224.CrossRefGoogle Scholar
Mun, R.P., Byars, J.A. & Boger, D.A. 1998 The effects of polymer concentration and molecular weight onthe breakup of laminar capillary jets. J. Non-Newtonian Fluid Mech. 74, 285–297.CrossRefGoogle Scholar
Nath, S., Datta, A., Mukhopadhyay, A., Sen, S. & Tharakan, T.J. 2011 Prediction of size and velocity distributions in sprays formed by the breakup of planar liquid sheets using maximum entropy formalism. Atomiz. Sprays 21, 483–501.CrossRefGoogle Scholar
Nath, S., Mukhopadhyay, A. & Datta, A. 2014 Effect of confinement on breakup of planar liquid sheets sandwiched between two gas streams and resulting spray characteristics. Fluid Dyn. Res. 46, 015511.CrossRefGoogle Scholar
Nath, S., Mukhopadhyay, A., Sen, S. & Tharakan, T.J. 2010 Influence of gas velocity on break up of planar liquid sheets sandwiched between two gas streams. Atomiz. Sprays 20, 983–1003.CrossRefGoogle Scholar
Negeed, E.R. 2011 Experimental and analytical investigation of liquid sheet breakup characteristics. Intl J. Heat Fluid Flow 32, 95–106.CrossRefGoogle Scholar
Nukiyama, S. & Tanasawa, Y. 1939 On the droplet-size distribution in an atomized jet. Trans. Japan Soc. Mech. Engng 5, 62–67.Google Scholar
Rayleigh, Lord 1878 On the instability of jets. Proc. Lond. Math. Soc. 10, 4–13.CrossRefGoogle Scholar
Rizk, N.K. & Lefebvre, A.H. 1985 Droplet size distribution characteristics of spill return atomizers. J. Propul. Power 1, 16–22.CrossRefGoogle Scholar
Rosin, P. & Rammler, E. 1933 The laws governing the fineness of powedered coal. J. Inst. Fuel 31, 29–36.Google Scholar
Sattler, R., Wagner, C. & Eggers, J. 2008 Blistering pattern and formation of nanofibers in capillary thinning of polymer solutions. Phys. Rev. Lett. 100, 3–6.CrossRefGoogle ScholarPubMed
Sellens, R. & Brzustowski, T. 1985 A prediction of the drop size and velocity distribution in a spray from first principle. Atomiz. Spray Technol. 1, 195–201.Google Scholar
Sovani, S., Sojka, P.E. & Sivathanu, Y. 2000 Prediction of drop size distributions from first principles: joint-PDF effects. Atomiz. Sprays 10, 587–602.CrossRefGoogle Scholar
Thompson, J.C. & Rothstein, J.P. 2007 The atomization of viscoeleastic fluids in flat-fan and hollow cone spray nozzles. J. Non-Newtonian Fluid Mech. 147, 11–22.CrossRefGoogle Scholar
Villermaux, E., Marmottant, P. & Duplat, J. 2004 Ligament-mediated spray formation. Phys. Rev. Lett. 92, 1–4.CrossRefGoogle ScholarPubMed
Wagner, C., Amarouchene, Y., Bonn, D. & Eggers, J. 2005 Droplet detachment and satellite-bead formation in viscoelastic fluids. Phys. Rev. Lett. 95, 164504.CrossRefGoogle ScholarPubMed
Wang, C., Yang, L., Xie, L. & Chen, P. 2015 Weakly nonlinear instability of planar viscoelastic sheets. Phys. Fluid 27, 013103.CrossRefGoogle Scholar
Xie, L., Yang, L., Wang, J. & Qin, L. 2018 Weakly nonlinear instability of viscoelastic planar sheets with initial varicose disturbances. Aerosp. Sci. Technol. 79, 373–382.CrossRefGoogle Scholar
Yang, L., Chen, P. & Wang, C. 2014 Effect of gas velocity on the weakly nonlinear instability of a planar viscous sheet. Phys. Fluids 26, 74106.CrossRefGoogle Scholar
Yang, L., Qu, Y., Fu, Q. & Gu, B. 2010 Linear stability analysis of a non-Newtonian liquid sheet. J. Propul. Power 26, 1212–1224.CrossRefGoogle Scholar
Yang, L.J., Wang, C., Fu, Q., Du, M. & Tong, M. 2013 Weakly nonlinear instability of planar viscous sheets. J. Fluid Mech. 735, 249–287.CrossRefGoogle Scholar
Yang, L., Xu, B. & Fu, Q. 2012 Linear instability analysis of planar non-Newtonian liquid sheets in two gas streams of unequal velocities. J. Non-Newtonian Fluid Mech. 167–168, 50–58.CrossRefGoogle Scholar
Ye, H., Yang, L., Fu, Q., Ye, H., Yang, L. & Fu, Q. 2016 Instability of viscoelastic compound jets instability of viscoelastic compound jets. Phys. Fluids 28, 043101.CrossRefGoogle Scholar