Fractional Cattaneo Heat Equation in a Multilayer Elliptic Ring Membrane and its Thermal Stresses

Subject Areas : Applied Mechanics

G Dhameja ¹ , L Khalsa ² , V Varghese ³

1 - Department of Mathematics, M.G. College, Armori, Gadchiroli, India
2 - Department of Mathematics, M.G. College, Armori, Gadchiroli, India
3 - Department of Mathematics, M.G. College, Armori, Gadchiroli, India

Received: 2023-01-27 Accepted : 2023-06-20 Published : 2024-09-21

Keywords: elliptic membrane, non-Fourier heat conduction, integral transform, Fractional Cattaneo-type equation, Fractional Calculus,

Abstract :

A fractional Cattaneo model from the generalized Cattaneo model with two fractional derivatives of different orders is considered for studying the thermoelastic response for a multilayer elliptic ring membrane with source function. The solution is obtained by applying an integral transform technique analogous to Vodicka's approach considering series expansion functions in terms of an eigenfunction to the generalized fractional Cattaneo-type heat conduction equation within an elliptic coordinates system. The analytical expressions of displacement and stress components employing Airy's stress function approach are investigated. The results are obtained as a series solution in terms of Mathieu functions and hold convergence test. The effects of fractional parameters on the temperature fields and their thermal stresses are also discussed. The findings are depicted graphically for different kinds of surface temperature gradients, and it is distinguished that the higher the fractional-order parameter, the higher the thermal response. Lastly, the generalized theory of thermoelasticity predicts an instantaneous response, but the fractional theory, which is currently under consideration, predicts a delayed response to physical stimuli, which is something that can be seen occurring in nature. This delayed response can be explained by the fact that fractional theories are currently being considered. This gives credibility to the motivation behind this topic of study in the research.

References:

[1] V. Vodicka, Warmeleitung in geschichteten Kugel-und Zylinderkorpern, Schweizer Archiv, Vol. 10, pp. 297-304, 1950.
[2] V. Vodicka, Eindimensionale Wärmeleitung in geschichteten. Körpern, Math. Nachr., Vol. 14, pp. 47–55, 1955.
[3] R. Chiba, An analytical solution for transient heat conduction in a composite slab with time-dependent heat transfer coefficient, Math. Probl. Eng., Vol. 2018, Article ID 4707860, 2018. DOI: 10.1155/2018/4707860
[4] S.M. Moghimi, M. Hosseini, and M. Ghanbarpour, Temperature distribution definition in one-dimensional transient cooling in a three-layer slab using orthogonal expansion technique, International Journal of Nonlinear Dynamics in Engineering and Sciences, Vol. 11, No. 2, pp. 21-30, 2019.
[5] T. Dhakate, V. Varghese and L. Khalsa, An analytical solution for transient asymmetric heat conduction in a multilayer elliptic annulus and its associated thermal stresses, Int. J. Math. And Appl., Vol. 6, No. 1, pp. 29-42, 2018.
[6] C.W. Tittle, Boundary value problems in composite media: quasi-orthogonal functions, J. Applied Physics, Vol. 36, No. 4, pp. 1486-1488, 1965.
[7] P.E. Bulavin and V.M. Kascheev, Solution of the non-homogeneous heat conduction equation for multilayered bodies, Int. Chemical Engineering, Vol. 1, No. 5, pp. 112-115, 1965.
[8] M.D. Mikhailov and M.N. Ozisik, Transient conduction in a three-dimensional composite slab, Int. J. Heat Mass Transfer, Vol. 29, pp. 340–342, 1986.
[9] X. Lu, P. Tervola, M. Viljanen, A new analytical method to solve heat equation for multi-dimensional composite slab, J. Phys. A: Math. Gen., Vol. 38, pp. 2873–2890, 2005.
[10] X. Lu, P. Tervola, M. Viljanen, Transient analytical solution to heat conduction in multi-dimensional composite cylinder slab, Int. J. Heat Mass Transfer, Vol. 49, pp. 1107–1114, 2006.
[11] H.S. Carslaw, J.C. Jaeger, Conduction of Heat in Solids, second ed., Oxford University Press, Oxford, 1959.
[12] Kevin D. Cole, A. Haji-Sheikh, James V. Beck, Bahman Litkouhi, Heat conduction using Green's function, Taylor and Francis Group, LLC, 2011.
[13] Haji-Sheikh, J.V. Beck, Temperature solution in multidimensional multilayer bodies, Int. J. Heat Mass Transfer, Vol. 45, pp. 1865–1877, 2002.
[14] T. Dhakate, V. Varghese, L. Khalsa, A Green's function approach for the thermoelastic analysis of an elliptical cylinder, Int. J. Adv. Appl. Math. and Mech., Vol. 5, no. 2, pp. 30-40, 2017.
[15] W. Heidemann, H. Mandel, E. Hahne, Computer aided determination of closed form solutions for linear transient heat conduction problems in inhomogeneous bodies, in: L.C. Wrobel, C.A. Brebbia, A.J. Nowak (Eds.), Advanced Computational Methods in Heat Transfer III, Computational Mechanics Publications, Boston, 1994, pp. 19-26.
[16] S. Verma, V.S. Kulkarni, K.C. Deshmukh, Finite element solution to transient asymmetric heat conduction in multilayer annulus, Int. J. Adv. Appl. Math. and Mech., Vol. 2, No. 3, pp. 119-125, 2015.
[17] P.P. Bhad, V. Varghese, and L. Khalsa, Heat source problem of thermoelasticity in an elliptic plate with thermal bending moments, J. Therm. Stresses, Vol. 40, no. 1, pp. 96-107, 2016.
[18] P.P. Bhad, V. Varghese, and L. Khalsa, A modified approach for the thermoelastic large deflection in the elliptical plate, Arch. Appl. Mech., Vol. 87, no. 4, pp. 767–781, 2016.
[19] I. Khan, L. Khalsa, and V. Varghese, Inverse quasi-static unsteady-state thermal stresses in a thick circular plate, Cogent Math., Vol. 4, Article ID 1283763, 2017.
[20] P.P. Bhad, V. Varghese, and L. Khalsa, Thermoelastic-induced vibrations on an elliptical disk with internal heat sources, J. Therm. Stresses, Vol. 40, no. 4, pp. 502-516, 2017.
[21] T. Dhakate, V. Varghese, and L. Khalsa, Integral transform approach for solving dynamic thermal vibrations in the elliptical disk, J. Therm. Stresses, Vol. 40, no. 9, pp. 1093-1110, 2017.
[22] J.C. Jaeger, Some problems involving line sources in conduction of heat, London Edinburgh Dub. Phil. Mag. J. Sci., Vol. 242, pp. 169–179, 1944.
[23] M. Li, A.C.K. Lai, Analytical model for short-time responses of borehole ground heat exchangers: model development and validation, Appl. Energy, Vol. 104, pp. 510–516, 2013.
[24] P.E. Bulavin and V.M. Kashcheev, Solution of nonhomogenous heat-conduction equation for multilayer bodies, Int. Chem. Eng., Vol. 5, no. 1, pp. 112–115, 1965.
[25] A. Agrawal, Higher-order continuum equation based heat conduction law, INAE Lett., pp. 1-5, 2016. DOI 10.1007/s41403-016-0007-3
[26] C. Cattaneo, Sulla conduzione del calore, Atti Sem. Mat. Fis. Univ. Modena, Vol. 3, pp. 83–101, 1948.
[27] L.Q. Wang, X.S. Zhou, and X.H. Wei, Heat conduction, Springer, Berlin, 2008.
[28] I. Podlubny, Fractional differential equations, Academic Press, New York, 1999.
[29] R. Hilfer, Applications of fractional calculus in physics, World Scientific, Singapore, 2000.
[30] A.A. Kilbas, H.M. Srivastava, and J.J. Trujillo, Theory and applications of fractional differential equations, Elsevier, Amsterdam, 2006.
[31] R.L. Magin, Fractional calculus in bioengineering, Begell House Publishers, Connecticut, 2006.
[32] W. Chen, H.G. Sun, and X.C. Li, Fractional derivative modelling of mechanical and engineering problems, Science Press, Beijing, 2010.
[33] R. Herrmann, Fractional calculus: An introduction for physicists, World Scientific, Singapore, 2011.
[34] A. Compte and R. Metzler, The generalized Cattaneo equation for the description of anomalous transport processes, J. Phys. A: Math. Gen., Vol. 30, pp. 7277-7289, 1997.
[35] Y. Povstenko, Fractional Cattaneo-type equations and generalized thermoelasticity, J. Therm. Stresses, Vol. 34, No. 2, pp. 97-114, 2011. DOI: 10.1080/01495739.2010.511931
[36] Y. Povstenko, Fractional thermoelasticty, Springer, New York, 2015.
[37] G. Xu, and J. Wang, Analytical solution of time fractional Cattaneo heat equation for finite slab under pulse heat flux, Appl. Math. Mech., Vol. 39, pp. 1465–1476, 2018. DOI 10.1007/s10483-018-2375-8
[38] D. Maillet, A review of the models using the Cattaneo and Vernotte hyperbolic heat equation and their experimental validation, Int. J. Therm. Sci., Vol. 139, pp. 424-432, 2019.DOI 10.1016/j.ijthermalsci.2019.02.021
[39] H.T. Qi, H.Y. Xu, and X.W. Guo, The Cattaneo-type time fractional heat conduction equation for laser heating, Comput. Math. with Appl., Vol. 66, No. 5, pp. 824-831, 2013. DOI 10.1016/j.camwa.2012.11.021
[40] H.Y. Xu, H.T. Qi, and X.Y. Jiang, Fractional Cattaneo heat equation in a semi-infinite medium, Chin. Phys. B, Vol. 22, No. 1, 014401, 2013. DOI: 10.1088/1674-1056/22/1/014401
[41] J.H. Choi, S.H. Yoon, and S.G. Park, and S.H. Choi, Analytical solution of the Cattaneo-Vernotte equation (non-Fourier heat conduction), J. Korean Soc. of Marine Engineering, Vol. 40, No. 5 pp. 389~396, 2016. DOI 10.5916/jkosme.2016.40.5.389
[42] Y. Povstenko, Axisymmetric Solutions to Time-fractional heat conduction equation in a half-space under Robin boundary conditions, Int. J. Differ. Equ., Vol. 2012, pp. 1–13, 2012. DOI: 10.1155/2012/154085.
[43] Y. Povstenko, Axisymmetric solutions to fractional diffusion-wave equation in a cylinder under Robin boundary condition, Eur. Phys. J. Spec. Top., Vol. 222, No. 8, pp. 1767–1777, Sep. 2013. DOI: 10.1140/epjst/e2013-01962-4.
[44] Y. Povstenko, Fundamental solutions to the fractional heat conduction equation in a ball under Robin boundary condition, Centr. Eur. J. Math., Vol. 12, No. 4, pp. 611–622, 2014. DOI: 10.2478/s11533-013-0368-8.
[45] X.Y. Jiang and H.T. Qi, Thermal wave model of bioheat transfer with modified Riemann Liouville fractional derivative, J. Phys. A-Math. Theor., Vol. 45, No. 48, 485101, 2012.
[46] G.Y. Xu, J.B. Wang, and Z. Han, Study on the transient temperature field based on the fractional heat conduction equation for laser heating, Appl. Math. Mech., Vol. 36, pp. 844–849, 2015.
[47] H.Y. Xu, H.T. Qi, and X.Y. Jiang, Fractional Cattaneo heat equation on a semi-infinite medium, Chin. Phys. B, Vol. 22, 014401, 2013.
[48] H.T. Qi and X.W. Guo, Transient fractional heat conduction with generalized Cattaneo model, Int. J. Heat Mass Transfer, Vol. 76, pp. 535–539, 2014.
[49] T.N. Mishra and K.N. Rai, Numerical solution of FSPL heat conduction equation for analysis of thermal propagation, Appl. Math. Comput., Vol. 273, pp. 1006–1017, 2016.
[50] M. N. Özisik, Heat Conduction, John Wiley & Sons, New York, 1993.
[51] N.W. McLachlan, Theory and application of Mathieu functions, Oxford University Press, Oxford, 1947.
[52] T.M. Atanackovic, S. Pilipovic, and D. Zorica, A diffusion wave equation with two fractional derivatives of different order, J. Phys. A: Math. Theor., Vol. 40, pp. 5319–5333, 2007. DOI 10.1088/1751-8113/40/20/006
[53] V. Vodicka, Heat exchange in a three-layer plate of finite dimension, Arch. budowy maszyn, Vol. 3, No. 4, pp. 319-331, 1956.
[54] S. N. Li and B.Y. Cao, Fractional Boltzmann transport equation for anomalous heat transport and divergent thermal conductivity, Int. J. Heat Mass Transf., 137, 84-89, 2019.
[55] S. N. Li and B.Y. Cao, Fractional-order heat conduction models from generalized Boltzmann transport equation, Philos. Trans. R. Soc. A, 378, 20190280, 2020.