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Chaotic thermal convection of couple-stress fluid layer. (English) Zbl 1390.37141

Summary: In this work we study the pattern of bifurcations and intermittent-chaos of non-Newtonian couple-stress shallow fluid layer subject to heating from below. The couple-stress parameter delays onset of convection, synchronizes chaotic behavior, and decreases the heat transfer . Some global aspects of the dynamics such as homoclinic bifurcations and transition to chaos are explored. The effects of particle size on the intermittent-chaos regime at particular normalized Rayleigh number, say \(r=166.1\), are investigated. With the increase in couple-stress parameter, the present Lorenz-like system synchronizes to a steady state via a series of periodic solutions interspersed with intervals of chaotic behaviors.

MSC:

37N10 Dynamical systems in fluid mechanics, oceanography and meteorology
37G05 Normal forms for dynamical systems
37G10 Bifurcations of singular points in dynamical systems
34D06 Synchronization of solutions to ordinary differential equations
37C29 Homoclinic and heteroclinic orbits for dynamical systems
37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
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[1] Chandrasekhar, S.: Hydrodynamic and Hydromagnetic Stability. Oxford University Press, New York (1961) · Zbl 0142.44103
[2] Drazin, P.G., Reid, W.H.: Hydrodynamic Stability, 2nd edn. Cambridge University Press, Cambridge (2004) · Zbl 0449.76027
[3] Lorenz, EN, Deterministic nonperiodic flow, J. Atmos. Sci., 20, 130-141, (1963) · Zbl 1417.37129
[4] Ramanaiah, G; Sarkar, P, Slider bearings lubricated by fluids with couple stress, Wear, 52, 27-36, (1979)
[5] Sinha, P; Singh, C, Couple stresses in the lubrication of rolling contact bearings considering cavitation, Wear, 67, 85-98, (1981)
[6] Gupta, RS; Sharma, LG, Analysis of couple stress lubricant in hydrostatic thrust bearing, Wear, 125, 257-269, (1988)
[7] Mokhiamer, UM; Crosby, WA; El-Gamal, HA, A study of a journal bearing lubricated by fluids with couple stress considering the elasticity of the liner, Wear, 224, 194-201, (1999)
[8] Wang, XL; Zhu, KQ; Wen, SZ, On the performance of dynamically loaded journal bearings lubricated with couple stress fluids, Tribol. Int., 35, 185-191, (2002)
[9] Shehawey, EF; Mekheimer, KS, Couple-stresses in peristaltic transport of fluids, J. Phys. D: Appl. Phys., 27, 1163, (1994) · Zbl 0804.92010
[10] Maiti, S; Misra, JC, Peristaltic transport of a couple stress fluid: some applications to hemodynamics, J. Mech. Med. Biol., 12, 1250048, (2012)
[11] Mekheimer, KS, Peristaltic transport of a couple-stress fluid in a uniform and non-uniform channels, Biorheology, 39, 755-765, (2002)
[12] Mekheimer, KS; Abd elmaboud, Y, Peristaltic flow of a couple stress fluid in an annulus: application of an endoscope, Physica A, 387, 2403-2415, (2008) · Zbl 1310.76197
[13] Srivastava, LM, Flow of couple stress fluid through stenotic blood vessels, J. Biomech., 18, 479-485, (1985)
[14] Pralhad, RN; Schultz, DH, Modeling of arterial stenosis and its applications to blood diseases, Math. Biosci., 190, 203-220, (2004) · Zbl 1047.92028
[15] Eringen, AC, Simple microfluids, Int. J. Eng. Sci., 2, 205-217, (1964) · Zbl 0136.45003
[16] Eringen, AC, Theory of micropolar fluids, J. Math. Mech., 16, 1-18, (1966)
[17] Bhattacharyya, K; Mukhopadhyay, S; Layek, GC; Pop, I, Effects of thermal radiation on micropolar fluid flow and heat transfer over a porous shrinking sheet, Int. J. Heat Mass Transf., 55, 2945-2952, (2012)
[18] Stokes, VK, Couple stresses in fluids, Phys. Fluids, 9, 1709-1715, (1966)
[19] Ahmadi, G, Stability of a Cosserat fluid layer heated from below, Acta Mech., 31, 243-252, (1979) · Zbl 0441.76001
[20] Banyal, AS, The necessary condition for the onset of stationary convection in couple-stress fluid, Int. J. Fluid Mech. Res., 38, 450-457, (2011)
[21] Sunil, R., Devi., Mahajan, A.: Global stability for thermal convection in a couple-stress fluid. Int. Commun. Heat Mass Transf. 38, 938-942 (2011) · Zbl 1047.92028
[22] Jawdat, JM; Hashim, I; Bhadauria, BS; Momani, S, On onset of chaotic convection in couple-stress fluids, Math. Model. Anal., 19, 359-370, (2014)
[23] Layek, GC; Pati, NC, Bifurcations and chaos in convection taking non-Fourier heat-flux, Phys. Lett. A, 381, 3568-3575, (2017) · Zbl 1375.34064
[24] Wiggins, S.: Introduction to Applied Nonlinear Dynamical Systems and Chaos, vol. 2. Springer, New York (2003) · Zbl 1027.37002
[25] Layek, G.C.: An Introduction to Dynamical Systems and Chaos. Springer, India (2015) · Zbl 1354.34001
[26] Gang, H; Guo-jian, Y, Instability in injected-laser and optical-bistable systems, Phys. Rev. A, 38, 1979, (1988)
[27] Kuznetsov, Y.A.: Elements of Applied Bifurcation Theory, vol. 112, 3rd edn. Springer, New York (2004) · Zbl 1082.37002
[28] Hassard, B.D., Kazarinoff, N.D., Wan, Y.H.: Theory and Applications of Hopf Bifurcation. Cambridge University Press, Cambridge (1981) · Zbl 0474.34002
[29] Palis, J; Takens, F, Hyperbolicity and the creation of homoclinic orbits, Annals Math., 125, 337-374, (1987) · Zbl 0641.58029
[30] Sparrow, C.: The Lorenz Equations: Bifurcations, Chaos, and Strange Attractors. Springer, New York (1982) · Zbl 0504.58001
[31] Broer, H., Takens, F.: Dynamical Systems and Chaos. Springer, New York (2011) · Zbl 1218.37001
[32] Kaplan, JL; Yorke, JA, Preturbulence: a regime observed in a fluid flow model of Lorenz, Commun. Math. Phys., 67, 93-108, (1979) · Zbl 0443.76059
[33] Yorke, JA; Yorke, ED, Metastable chaos: the transition to sustained chaotic behavior in the Lorenz model, J. Stat. Phys., 21, 263-277, (1979)
[34] Bergé, P; Dubois, M; Manneville, P; Pomeau, Y, Intermittency in Rayleigh-Bénard convection, J. Phys. Lett., 41, 341-345, (1980)
[35] Pomeau, Y; Manneville, P, Intermittent transition to turbulence in dissipative dynamical systems, Commun. Math. Phys., 74, 189-197, (1980)
[36] Vadasz, P; Olek, S, Weak turbulence and chaos for low Prandtl number gravity driven convection in porous media, Trans. Porous Med., 37, 69-91, (1999)
[37] Wolf, A; Swift, JB; Swinney, HL; Vastano, JA, Determining Lyapunov exponents from a time series, Physica D, 16, 285-317, (1985) · Zbl 0585.58037
[38] Kaplan, JL; Yorke, JA; Peitgen, HO (ed.); Walther, HO (ed.), Chaotic behavior of multidimensional difference equations, No. 730, 204-227, (1979), Berlin
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