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doi: 10.3934/naco.2021018
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## Analysis of Rayleigh Taylor instability in nanofluids with rotation

 1 Research Scholar, Deparment of Mathematics, NIILM University, Kaithal, Haryana, India 2 Department of Mathematics, Post Graduate Government College, Sector-11, Chandigarh, India 3 Department of Mathematics, NIILM University, Kaithal, Haryana, India

* Corresponding author: jyotiahuja1985@gmail.com

Received  December 2020 Revised  April 2021 Early access May 2021

This article focuses on the hidden insights about the Rayleigh-Taylor instability of two superimposed horizontal layers of nanofluids having different densities in the presence of rotation factor. Conservation equations are subjected to linear perturbations and further analyzed by using the Normal Mode technique. A dispersion relation incorporating the effects of surface tension, Atwood number, rotation factor and volume fraction of nanoparticles is obtained. Using Routh-Hurtwitz criterion the stable and unstable modes of Rayleigh-Taylor instability are discussed in the presence/absence of nanoparticles and presented through graphs. It is observed that in the absence/presence of nanoparticles, surface tension helps to stabilize the system and Atwood number has a destabilizing impact without the consideration of rotation factor. But if rotation parameter is considered (in the absence/presence of nanoparticles) then surface tension destabilizes the system while Atwood number has a stabilization effect (for a particular range of wave number). The volume fraction of nanoparticles destabilizes the system in the absence of rotation but in the presence of rotation the stability of the system is significantly stimulated by the nanoparticles.

Citation: Pooja Girotra, Jyoti Ahuja, Dinesh Verma. Analysis of Rayleigh Taylor instability in nanofluids with rotation. Numerical Algebra, Control & Optimization, doi: 10.3934/naco.2021018
##### References:
 [1] J. Ahuja and U. Gupta, Magneto convection in rotating nanofluid layer: Local thermal non-equilibrium model, American Scientific Publisher, 8 (2019), 1-9.   Google Scholar [2] J. Ahuja and P. Girotra, Analytical and numerical investigation of Rayleigh-Taylor instability in nanofluids, Pramana - J. Phys., 95 (2021), 25. Google Scholar [3] K. A. Baldwin, M. M. Scase and R. J. A. Hill, The inhibition of Rayleigh Taylor instability by rotation, Sci. Rep., 5 (2015), 11706. Google Scholar [4] B. S. Bhaduria, A. Kumar, J. Kumar, N. C. Sacheti and P. Chandran, Natural convection in a rotating anisotropic porous layer with internal heat generation, Transport in Porous Media, 90 (2011), 687–705. doi: 10.1007/s11242-011-9811-0.  Google Scholar [5] P. K. Bhatia, Rayleigh-Taylor instability of a viscous compressible plasma of variable density, Astrophysics and Space Science, 26 (1974), 319-325.   Google Scholar [6] J. Buongiorno, Convective transport in nanofluids, ASME J. 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Tzou, Thermal instability of nanofluids in natural convection, Int. J. of Heat Mass Transf., 51 (2008), 2967-2979.   Google Scholar [28] A. Volk and C. J. Khaler, Density model for aqueous glycerol solutions, Experiments in Fluids, 59 (2018), 75 (1-4). Google Scholar

show all references

##### References:
 [1] J. Ahuja and U. Gupta, Magneto convection in rotating nanofluid layer: Local thermal non-equilibrium model, American Scientific Publisher, 8 (2019), 1-9.   Google Scholar [2] J. Ahuja and P. Girotra, Analytical and numerical investigation of Rayleigh-Taylor instability in nanofluids, Pramana - J. Phys., 95 (2021), 25. Google Scholar [3] K. A. Baldwin, M. M. Scase and R. J. A. Hill, The inhibition of Rayleigh Taylor instability by rotation, Sci. Rep., 5 (2015), 11706. Google Scholar [4] B. S. Bhaduria, A. Kumar, J. Kumar, N. C. Sacheti and P. Chandran, Natural convection in a rotating anisotropic porous layer with internal heat generation, Transport in Porous Media, 90 (2011), 687–705. doi: 10.1007/s11242-011-9811-0.  Google Scholar [5] P. K. Bhatia, Rayleigh-Taylor instability of a viscous compressible plasma of variable density, Astrophysics and Space Science, 26 (1974), 319-325.   Google Scholar [6] J. Buongiorno, Convective transport in nanofluids, ASME J. Heat Transf., 128 (2006), 240-250.   Google Scholar [7] B. B. Chakraborthy, A note on Rayleigh Taylor insability in presence of rotation, Z. Angew. Math. Mech., 59 (1959), 651-652.   Google Scholar [8] B. B. Chakraborthy, Hydromagnetic Rayligh Taylor instabilityof rotating stratified fluid, Phys. Fluids, 25 (1982), 743-747.  doi: 10.1063/1.863828.  Google Scholar [9] S. Chandrasekhar, The character of the equilibrium of an incompressible heavy viscous fluid of variable density, Proc. Cambridge Philos. Soc., 51 (1955), 162–178. doi: 10.1017/s0305004100030048.  Google Scholar [10] S. Chandrasekhar, Hydrodynamic and Hydromagnetic Stability, 2$^{nd}$ edition, Dover Publication, New York, 1981.  Google Scholar [11] S. Choi, Enhancing thermal conductivity of fluids with nanoparticles, In Development and Applications of Non-Newtonian flows, FED- 231/MD(eds. D. A. Siginer and H. P. Wang), ASME, 66 (1955), 99-105. Google Scholar [12] S. K. Das, N. Putra, P. Thiesen and W. Roetzel, Temperature dependence of thermal conductivity enhancement for nanofluids, ASME J. Heat Transfer, 25 (2003), 567-574.   Google Scholar [13] L. A. Dávalos-Orzoco, Rayleigh-Taylor instability of a continuously stratified fluid under a general rotation field, Phys. Fluids A, 1 (1989), 1192-1199.   Google Scholar [14] N. F. El-Ansary, G. A. Hoshoudy, A. S. Abd-Elrady and A. H. A. Ayyad, Effects of surface tension and rotation on Rayleigh Taylor instability, Phys. Chem. Chem. Phys., 4 (2002), 1464-1470.   Google Scholar [15] R. Hide, The character of the equilibrium of a heavy, viscous, incompressible, rotating fluid of variable density ii. two special cases, Q. J. Mech. Appl. Math., 9 (1956), 35-50.  doi: 10.1093/qjmam/9.1.35.  Google Scholar [16] P. Kumar, Rayleigh Taylor instability of rotating Oloroydian viscoelastic fluids in porous medium in presence of a variable magnetic field, Jnanabha, 24 (1977), 127-134.   Google Scholar [17] H. Masuda, A. Ebata, K. Teramae and N. Hishinuma, Alteration of thermal conductivity and viscosity of liquid by dispersing ultra fine particles, Netsu Bussei, 7 (1993), 227-233.   Google Scholar [18] D. A. Nield and A. V. Kuznetsov, Thermal instability in a porous medium layer saturated by nanofluid, Int. J. Heat Mass Transfer, 52 (2008), 5796-5801.  doi: 10.1007/s11242-009-9413-2.  Google Scholar [19] L. Rayleigh, Investigation of the character of equilibrium of an incompressible heavy fluid of variable density, Proc. Roy. Math. Soc., 14 (1883), 170-177.  doi: 10.1112/plms/s1-14.1.170.  Google Scholar [20] P. K. Sahrma, A. Tiwari and S. Argal, Effect of magnetic field on the Rayleigh Taylor instability of rotating and stratified plasma, IOP Conf. Series: Journal of Physics: Conf. Series, 836 (2017), 012009. Google Scholar [21] M. M. Scase, K. A. Baldwin and R. J. A. Hill, The rotating Rayleigh Taylor instability, Physical Review Fluids, 2 (2017), 024801 (1-21). Google Scholar [22] R. C. Sharma, MHD instability of rotating superposed fluids through porous medium, Acta Physica Academiae Scientiarium Hungaricace, 42 (1977), 21-28.  doi: 10.1007/BF03157196.  Google Scholar [23] R. C. Sharma, P. Kumar and S. Sharma, Rayleigh Taylor instability of Rivlin-Ericksen elastico-viscous fluid through porous medium, Indian Journal of Physics, 4 (2001), 337-340.   Google Scholar [24] R. C. Sharma and V. K. Bhardwaj, Rayleigh Taylor instability of Newtonian and Oldroydian viscoelastic fluids in porous medium, Z. Naturforsch, 49a (1994), 927-930.   Google Scholar [25] J. J. Tao, X. T. He, W. H. Ye and F. H. Busse, Nonlinear Rayleigh Taylor instability of inviscid fluids, Physical Review E, 87 (2013), 013001 (1-4). Google Scholar [26] G. I. Taylor, The instability of liquid surfaces when accelerated in a direction perpendicular to their planes, Proc. R. Soc. Ser. A., 201 (1950), 192-196.  doi: 10.1098/rspa.1950.0052.  Google Scholar [27] D. Y. Tzou, Thermal instability of nanofluids in natural convection, Int. J. of Heat Mass Transf., 51 (2008), 2967-2979.   Google Scholar [28] A. Volk and C. J. Khaler, Density model for aqueous glycerol solutions, Experiments in Fluids, 59 (2018), 75 (1-4). Google Scholar
Effect of surface tension on growth rate parameter of Rayleigh Taylor instability in the absence of nanoparticles with rotation and without rotation for varying T = 0.1, 0.2
Effect of Atwood number on growth rate parameter of Rayleigh Taylor instability in the absence of nanoparticles with rotation and without rotation for varying A = 0.11, 0.119
Effect of rotation parameter on growth rate parameter of Rayleigh Taylor instability in the absence of nanoparticles with rotation and without rotation for varying $\Omega$ = 0.1, 0.2, 0.3
Effect of surface tension on growth rate parameter of Rayleigh Taylor instability in the presence of same kind of nanoparticles with rotation and without rotation for varying T = 0.0001, 0.0002
Effect of Atwood number on growth rate parameter of Rayleigh Taylor instability in the presence of same kind of nanoparticles with rotation and without rotation for varying A = 0.11, 0.119
Effect of volume fraction of nanoparicles on growth rate parameter of Rayleigh Taylor instability in the presence of same kind of nanoparticles with rotation and without rotation for varying $\phi$ = 0.001, 0.005
Effect of rotation number on growth rate parameter of Rayleigh Taylor instability in the presence of same kind of nanoparticles $\Omega$ = 0.1, 0.2, 0.3
Effect of surface tension on growth rate parameter of Rayleigh Taylor instability in the presence of different kind of nanoparticles with rotation and without rotation for varying T = 0.0001, 0.0002
Effect of Atwood number on growth rate parameter of Rayleigh Taylor instability in the presence of same kind of nanoparticles with rotation and without rotation for varying A = 0.11, 0.119
Effect of volume fraction of nanoparicles on growth rate parameter of Rayleigh Taylor instability in the presence of different kind of nanoparticles with rotation and without rotation for varying $\phi$ = 0.001, 0.005
Effect of rotation number on growth rate parameter of Rayleigh Taylor instability in the presence of diffrent kind of nanoparticles $\Omega$ = 0.1, 0.2, 0.3
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