Impact of memory and long-range interaction in a two-dimensional semi-infinite solid cylinder

Year 2025, Volume: 11 Issue: 1, 240 - 253, 31.01.2025

Navneet Kumar Lamba

Abstract

Space fractional differential operators are used to study long-range interactions, and time differential operators handle memory effects. A semi-infinite circular cylinder is taken into consideration to analyse both effects in a two-dimensional thermoelastic situation where heat conduction is influenced by internal heat generation. A prescribed jump function is applied to the bottom of the semi-infinite circular cylinder, and the time-dependent heat flux happens at the curved edge of the cylinder. The transformative approach of Laplace, Fourier, and Hankel was used to solve the governing equation of heat transfer with Caputo and the finite fractional derivatives of Riesz. The outcomes are expressed in terms of the Bessel function series. The numerical calculations are performed with the material properties of pure copper, and the graphical representations of the thermal distributions are successfully plotted.

Keywords

Caputo Fractional Derivative, Integral Transformation, Mittag-Leffler Function, Riesz Fractional Derivative, Solid Circular CylinderTemperature, Thermal Stresses, Two Dimensional

References

• [1] Oldham KB, Spanier J. The Fractional Calculus. New York: Academic Press; 1974.
• [2] Miller KS, Ross B. An Introduction to the Fractional Calculus and Fractional Differential Equations. New York: Wiley; 1993.
• [3] Samko SG, Kilbas AA, Marichev OI. Fractional Integrals and Derivatives Theory and Applications. Amsterdam: Gordon and Breach; 1993.
• [4] Podlubny I. Fractional Differential Equations. San Diego: Academic Press; 1999.
• [5] Povstenko Y. Fractional Thermoelasticity. In: Hetnarski RB, editor. Encyclopedia of Thermal Stresses. Dordrecht: Springer; 2014. [CrossRef]
• [6] Kilbas AA, Srivastava HM, Trujillo JJ. Theory and Applications of Fractional Differential Equations. Amsterdam: Elsevier; 2006.
• [7] Diethelm K. The Analysis of Fractional Differential Equations. Berlin: Springer; 2010. [CrossRef]
• [8] Povstenko YZ. Two-dimensional axisymmetric stresses exerted by instantaneous pulses and sources of diffusion in an infinite space in a case of time-fractional diffusion equation. Int J Solids Struct 2007;44:2324–2348. [CrossRef]
• [9] Povstenko YZ. Theory of thermoelasticity based on the space-time-fractional heat conduction equation. Phys Scr T 2009;136:014017. [CrossRef]
• [10] Povstenko YZ. Fractional radial diffusion in a cylinder. J Mol Liq 2008;137:46–50. [CrossRef]
• [11] Povstenko YZ. Non-axisymmetric solutions to time-fractional diffusion-wave equation in an infinite cylinder. Frac Calc Appl Anal 2011;14:418–435. [CrossRef]
• [12] Povstenko YZ. Time-fractional radial heat conduction in a cylinder and associated thermal stresses. Arch Appl Mech 2012;82:345–362. [CrossRef]
• [13] Ezzat MA, EL-Karamany AS, EL-Bary AA. On thermo-viscoelasticity with variable thermal conductivity and fractional-order heat transfer. Int J Thermophys 2015;36:1684–1697. [CrossRef]
• [14] Ezzat MA, EL-Karamany AS, EL-Bary AA. Thermo-viscoelastic materials with fractional relaxation operators. Appl Math Modell 2015;39:7499–7512. [CrossRef]
• [15] Ezzat MA, EL-Bary AA. Unified fractional derivative models of magneto-thermo-viscoelasticity theory. Arch Mech 2016;68:285–308.
• [16] Caputo M, Mainardi F. A new dissipation model based on memory mechanism. Pure Appl Geophys 1971;91:134–147. [CrossRef]
• [17] Caputo M, Mainardi F. Linear model of dissipation in elastic solids. Rivista Nuovo Cimento 1971;1:161–1981. [CrossRef]
• [18] Said SM, Abd-Elaziz EM, Othman MIA. A two-temperature model and fractional order derivative in a rotating thick hollow cylinder with the magnetic field. Indian J Phys 2009;97:3057–3064. [CrossRef]
• [19] Sherief HH, Raslan WE. A problem in fractional order thermoelasticity theory for an infinitely long cylinder composed of 3 layers of different materials. J Therm Stress 2022;45:234–244. [CrossRef]
• [20] Sherief HH, Raslan WE. Fundamental solution for a line source of heat in the fractional order theory of thermoelasticity using the new Caputo definition. J Therm Stress 2019;42:18–28. [CrossRef]
• [21] Lamba NK. Thermosensitive response of a functionally graded cylinder with fractional order derivative. Int J Appl Mech Eng 2022;27:107–124. [CrossRef]
• [22] Lamba NK, Deshmukh KC. Hygrothermoelastic response of a finite solid circular cylinder. Multidiscip Model Mater Struct 2020;16:37–52. [CrossRef]
• [23] Lamba NK, Deshmukh KC. Hygrothermoelastic response of a finite hollow circular cylinder. Waves Random Complex Media 2022;2030501:1–16. [CrossRef]
• [24] Kumar N, Kamdi DB. Thermal behavior of a finite hollow cylinder in context of fractional thermoelasticity with convection boundary conditions. J Therm Stress 2020;43:1189–1204. [CrossRef]
• [25] Verma J, Lamba NK, Deshmukh KC. Memory impact of hygrothermal effect in a hollow cylinder by theory of uncoupled-coupled heat and moisture. Multidiscip Model Mater Struct 2022;18:826–844. [CrossRef]
• [26] Youssef HM. Theory of fractional order generalized thermoelasticity. J Heat Transf 2010;132:1–7. [CrossRef]
• [27] Abouelregal AE. The effect of temperature-dependent physical properties and fractional thermoelasticity on non-local nanobeams. J Math Theor Phys 2018;1:49–58. [CrossRef]
• [28] Rashad AM, Chamkha AJ, Mallikarjuna B, Abdou MMM. Mixed bioconvection flow of a nanofluid containing gyrotactic microorganisms past a vertical slender cylinder. Front Heat Mass Transf 2018;10:10–21. [CrossRef]
• [29] Rashad AM, Khan WA, EL-Kabeir SMM, EL-Hakiem AMA. Mixed convective flow of micropolar nanofluid across a horizontal cylinder in saturated porous medium. Appl Sci 2019;9:5241. [CrossRef]
• [30] Rashad AM, Abbasbandy S, Chamkha AJ. Non-Darcy natural convection from a vertical cylinder embedded in a thermally stratified and nano-fluid-saturated porous media. J Heat Transf 2014;136:HT-13-1078. [CrossRef]
• [31] Murali G, Paul A, Babu NN. Numerical study of chemical reaction effects on unsteady MHD fluid flow past an infinite vertical plate embedded in a porous medium with variable suction. Electron J Math Anal Appl 2015;3:179–192. [CrossRef]
• [32] Deepa G, Murali G. Effects of viscous dissipation on unsteady MHD free convective flow with thermophoresis past a radiate inclined permeable plate. Iran J Sci 2014;38:379–388.
• [33] El-Sayed AMA, Gaber M. On the finite Caputo and finite Riesz derivatives. Electron J Theor Phys 2006;3:81–95.
• [34] Fil’Shtinskii LA, Kirichok TA, Kushnir DV. One-dimensional fractional quasi-static thermoelasticity problem for a half-space. WSEAS Trans Heat Mass Transf 2013;2:31–36.
• [35] Ciesielski M, Leszczynski J. Numerical solutions to boundary value problem for anomalous diffusion equation with Riesz-Feller fractional operator. J Theor Appl Mech 2006;44:393–403.
• [36] Momani Sh. General solutions for the space and time-fractional diffusion-wave equation. J Phys Sci 2006;10:30–43.
• [37] Povstenko Y. Theories of thermal stresses based on space–time-fractional telegraph equations. Comput Math Appl 2012;64:3321–3328. [CrossRef]
• [38] Assiri TA. Time-space variable-order fractional nonlinear system of thermoelasticity: numerical treatment. Adv Differ Equ 2020;288. [CrossRef]
• [39] Ozdemir N, Avci D, Iskender DB. The numerical solutions of a two-dimensional space-time Riesz-Caputo fractional diffusion equation. Int J Optim Control 2011;1:17–26. [CrossRef]
• [40] Atta D. Thermal diffusion responses in an infinite medium with a spherical cavity using the Atangana-Baleanu fractional operator. J Appl Comput Mech 2022;8:1358–1369.
• [41] Abouelregal AE, Moustapha MV, Nofal TA, Rashid S, Ahmad H. Generalized thermoelasticity based on higher-order memory-dependent derivative with time delay. Results Phys 2021;20:103705. [CrossRef]
• [42] Abouelregal AE, Sedighi HM, Faghidian SA, Shirazi AH. Temperature-dependent physical characteristics of the rotating nonlocal nanobeams subject to a varying heat source and a dynamic load. Facta Univ 2021;19:633–656. [CrossRef]
• [43] Abouelregal AE, Atta D, Sedighi HM. Vibrational behavior of thermoelastic rotating nanobeams with variable thermal properties based on memory-dependent derivative of heat conduction model. Arch Appl Mech 2023;93:197–220. [CrossRef]
• [44] Abouelregal AE, Askar SS, Marin M, Badahiould M. The theory of thermoelasticity with a memory-dependent dynamic response for a thermo-piezoelectric functionally graded rotating rod. Sci Rep 2023;13:9052. [CrossRef]
• [45] Lamba NK, Verma J, Deshmukh KC. A brief note on space-time fractional order thermoelastic response in a layer. Appl Appl Math Int J 2023;18:1–9.
• [46] Ozisik MN. Boundary value problems of heat conductions. Pennsylvania: Scranton; 1986.