Thermal stratification of rotational second-grade fluid through fractional differential operators
Tóm tắt
This manuscript predicts the change in temperature at a different epilimnion to the change in temperature at a different hypolimnion. The fractional analysis on rotational second-grade fluid with sinusoidal boundary conditions is performed for knowing thermal stratification. The mathematical modeling is also proposed by means of modern fractional differential operators, namely Caputo–Fabrizio and Atangana–Baleanu derivatives, for rotational second-grade fluid. Most of authors have proposed the classical solutions of rotational second-grade fluid, which are obtained by the Laplace transform only. Our fractionalized mathematical model of rotational second-grade fluid has been solved via Fourier sine and Laplace transform techniques simultaneously. The solutions of velocity and temperature have been investigated and expressed in the format of Mittag–Leffler and Fox-H functions. Both fractional solutions are presented for comparison of velocity and temperature through Caputo–Fabrizio and Atangana–Baleanu derivatives. Finally, our results showed that the fractional solutions investigated for the velocity and temperature via Fourier sine and Laplace transform methods are stable and rapid than classical solutions.
Tài liệu tham khảo
Rajagopal KR, Kaloni PN. Continuum mechanics and its applications. Washington, DC: Hemisphere Press; 1989.
Bandelli R, Rajagopal KR. Start-up flows of second grade fluids in domains with one finite dimension. Int J Non-Linear Mech. 1995;30:817–39.
Abro KA. porous effects on second grade fluid in oscillating plate. J Appl Environ Biol Sci. 2016;6(5):71–82.
Khan M, Nadeem S, Hayat T, Siddiqui AM. Unsteady motions of a generalized second grade fluid. Math Comput Model. 2005;41:629–37.
Abro KA, Solangi MA. Heat transfer in magnetohydrodynamic second grade fluid with porous impacts using Caputo–Fabrizoi fractional derivatives, Punjab University. J Math. 2017;49(2):113–25.
Khan A, Abro KA, Tassaddiq A, Khan I. Atangana-Baleanu and Caputo Fabrizio analysis of fractional derivatives for heat and mass transfer of second grade fluids over a vertical plate: a comparative study. Entropy. 2017;19(8):1–12.
Podlubny I. Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications. San Diego: Academic Press; 1999.
Muzaffar HL, Kashif AA, Asif AS. Helical flows of fractional viscoelastic fluid in a circular pipe. Int J Adv Appl Sci. 2017;4(10):97–105. https://doi.org/10.21833/ijaas.2017.010.014.
Kashif AA, Ilyas K, Jose FGA. A mathematical analysis of a circular pipe in rate type fluid via Hankel transform. Eur Phys J Plus. 2018;133:397. https://doi.org/10.1140/epjp/i2018-12186-7.
Caputo M, Fabrizio MA. New definition of fractional derivative without singular kernel. Prog. Fract. Diff. Appl. 2015;1:73–85.
Abro KA, Khan I, Aguilar JFG. Thermal effects of magnetohydrodynamic micropolar fluid embedded in porous medium with Fourier sine transform technique. J Braz Soc Mech Sci Eng. 2019;41:174–81. https://doi.org/10.1007/s40430-019-1671-5.
Abro KA, Memon AA, Abro SH, Khan I, Tlili I. Enhancement of heat transfer rate of solar energy via rotating Jeffrey nanofluids using Caputo–Fabrizio fractional operator: an application to solar energy. Energy Rep. 2019;5:41–9. https://doi.org/10.1016/j.egyr.2018.09.009.
Atangana A, Baleanu D. New fractional derivative with non local and non-singular kernel: theory and application to heat transfer model. Therm Sci. 2016;20:763–9.
Abro KA, Mirbhar MN, Gomez-Aguilar JF. Functional application of Fourier sine transform in radiating gas flow with non-singular and non-local kernel. J Braz Soc Mech Sci Eng. 2019;41:400. https://doi.org/10.1007/s40430-019-1899-0.
Abro KA, Rashidi MM, Khan I, Abro IA, Tassadiq A. Analysis of stokes’ second problem for nanofluids using modern fractional derivatives. J Nanofluids. 2018;7:738–47.
Hayat T, Nadeem S, Siddiqui AM, Asghar S. An oscillating hydromagnetic non-Newtonian flow in a rotating system. Appl Math Lett. 2004;17:609–14.
Hayat T, Fetecau C, Sajid M. Analytic solution for MHD transient rotating flow of a second grade fluid in a porous space. Nonlinear Anal Real World Appl. 2008;9:1619–27.
Ismail Z, Khan I, Mohamad AQ, Sharidan S. Second grade fluid for rotating MHD of an unsteady free convection flow in a porous medium. Defect Diffus. Forum. 2015;362:100–7.
Shah NA, Khan I. Heat transfer analysis in a second grade fluid over and oscillating vertical plate using fractional Caputo–Fabrizio derivatives. Eur Phys J C. 2016. https://doi.org/10.1140/epjc/s10052-016-4209-3.
Itrat AM, Muhammad SA, Imran S. Flows of a generalized second grade fluid in a cylinder due to a velocity shock. Chin J Phys. 2019;1:720–30.
Kashif AA, Ilyas K. Analysis of the heat and mass transfer in the MHD flow of a generalized casson fluid in a porous space via non-integer order derivatives without a singular Kernel. Chin J Phys. 2017. https://doi.org/10.1016/j.cjph.2017.05.012.
Nauman R, Abdullah M, Asma RB, Aziz A, Ehsan H. Flow of a second grade fluid with fractional derivatives due to a quadratic time dependent shear stress. Alex Eng J. 2018;133:1963–9.
Kashif AA, Sumera D, Mirza MB. Effects of transverse magnetic field on oscillating plate of second grade fluid. Sindh Univ Res J. 2016;48(3):605–10.
Mohamad AQ, Khan I, Ismail Z, Sharidan S. The unsteady free convection flow of rotating second grade fluid over an oscillating vertical plate. J Teknol. 2016;78:57–63.
Kashif AA, Asif AS, Sanuallah D. Exact Solutions on the oscillating plate of maxwell fluids. Mehran Univ Res J Eng Technol. 2016;35(1):157–62.
Khan I, Farhad A, Sharidan S, Norzieha M. Exact solutions for accelerated flows a rotating second grade fluid in a porous medium. World Appl Sci J. 2010;9:55–68.
Abro KA, Mukarrum H, Mirza MB. Analytical solution of MHD generalized Burger’s fluid embedded with porosity. Int J Adv Appl Sci. 2017;4(7):80–9. https://doi.org/10.21833/ijaas.2017.07.012.
Azhar AZ, Vieru D, Shahraz A. Magnetohydrodynamics of rotating fractional second grade fluid in a porous medium. J Prime Res Math. 2015;10:45–58.
Kashif AA, Mukarrum H, Mirza MB. A mathematical analysis of magnetohydrodynamic generalized burger fluid for permeable oscillating plate. Punjab Univ J Math. 2018;50(2):97–111.
Ambreen S, Abro A, Muhammad AS. Thermodynamics of magnetohydrodynamic Brinkman fluid in porous medium: applications to thermal science. J Therm Anal Calorim. 2018;136(6):2295–304. https://doi.org/10.1007/s10973-018-7897-0.
Kashif AA, Ilyas K. Effects of CNTs on magnetohydrodynamic flow of methanol based nanofluids via Atangana–Baleanu and Caputo–Fabrizio fractional derivatives. Therm Sci. 2018;23:165. https://doi.org/10.2298/TSCI180116165A.
Imran MA, Imran M, Fetecau C. MHD oscillating flows of rotating second grade fluids in a porous medium. Commun Numer Anal. 2014;1:1–12.
Abro KA, Ali DC, Irfan AA, Ilyas K. Dual thermal analysis of magnetohydrodynamic flow of nanofluids via modern approaches of Caputo–Fabrizio and Atangana–Baleanu fractional derivatives embedded in porous medium. J Therm Anal Calorim. 2018;135(4):2197–207. https://doi.org/10.1007/s10973-018-7302-z.
Faisal S, Aziz ZA, Ching DLC. On accelerated MHD flows of second grade fluid in a porous medium and rotating frame. Int J Appl Math. 2013;43:1–8.
Kashif AA, Mukarrum H, Mirza MB. An analytic study of molybdenum disulfide nanofluids using modern approach of Atangana–Baleanu fractional derivatives. Eur Phys J Plus. 2017;132:439. https://doi.org/10.1140/epjp/i2017-11689-y.
Abro KA, Khan I, Tassadiqq A. Application of Atangana–Baleanu fractional derivative to convection flow of MHD Maxwell fluid in a porous medium over a vertical plate. Math Model Nat Phenom. 2018;13:1. https://doi.org/10.1051/mmnp/2018007.
Ahmad QM, Ilyas K, Lim Y, Sharidan S, Zaiton M, Zulkhibri I. Heat transfer on rotating second grade fluid through an accelerated plate. Malays J Fund Appl Sci. 2017;13(3):218–22.
Abro KA, Jose FGA. A comparison of heat and mass transfer on a Walter’s-B fluid via Caputo–Fabrizio versus Atangana–Baleanu fractional derivatives using the Fox-H function. Eur Phys J Plus. 2019;134:101. https://doi.org/10.1140/epjp/i2019-12507-4.
Kashif AA, Irfan AA, Sikandar MA, Ilyas K. On the thermal analysis of magnetohydrodynamic jeffery fluid via modern non integer order derivative. J King Saud Univ Sci. 2018. https://doi.org/10.1016/j.jksus.2018.07.012.
Adesanya SO, Souayeh B, Rahimi-Gorji M, Khan MN, Adeyemi OG. Heat irreversibility analysis for a couple stress fluid flow in an inclined channel with isothermal boundaries. J Taiwan Inst Chem Eng. 2019;101:251–8.
Abro KA, Abro IA, Yildirim A. A comparative analysis of sulfate ion concentration via modern fractional derivatives: an industrial application to cooling system of power plant. Phys A Stat Mech Appl. 2019. https://doi.org/10.1016/j.physa.2019.123306.
Kumar KG, Rahimi-Gorji M, Reddy MG, Chamkha AJ, Alarifi IM. Enhancement of heat transfer in a convergent/divergent channel by using carbon nanotubes in the presence of a Darcy-Forchheimer medium. Microsyst Technol. 2019. https://doi.org/10.1007/s00542-019-04489-x.
Akermi M, Jaballah N, Alarifi IM, Rahimi-Gorji M, Chaabane RB, Ouada HB, Majdoub M. Synthesis and characterization of a novel hydride polymer P-DSBT/ZnO nano-composite for optoelectronic applications. J Mol Liquids. 2019;287:110963.
Abro KA, Yildirim A. Heat transfer on fractionalized micropolar nanofluid over oscillating plate via Caputo-Fabrizio fractional operator. Sci Iran Int J Sci Technol. 2019;45:50. https://doi.org/10.24200/sci.2019.52437.2717.
Morales-Delgado VF, Gomez-Aguilar JF, Escobar-Jimenez RF, Taneco-Hernandez MA. Fractional conformable derivatives of Liouville–Caputo type with low-fractionality. Phys A. 2018;48:5. https://doi.org/10.1016/j.physa.2018.03.018.
Abro KA, Aguilar JFG, Khan I, Nisar KS. Role of modern fractional derivatives in an armature-controlled DC servomotor. Eur Phys J Plus. 2019;134:553. https://doi.org/10.1140/epjp/i2019-12957-6.
Saad KM, Gomez-Aguilar JF. Analysis of reaction diffusion system via a new fractional derivative with non-singular kernel. Phys A. 2018. https://doi.org/10.1016/j.physa.2018.05.137.
Sheikholeslami M, Shehzad SA, Li Z, Shafee A. Numerical modeling for alumina nanofluid magnetohydrodynamic convective heat transfer in a permeable medium using Darcy law. Int J Heat Mass Transfer. 2018;127:614–22.
Sheikholeslami M, Li Z, Shafee A. Lorentz forces effect on NEPCM heat transfer during solidification in a porous energy storage system. Int J Heat Mass Transfer. 2018;127:665–74.
Sheikholeslami M, Jafaryar M, Saleem S, Li Z, Shafee A, Jiang Y. Nanofluid heat transfer augmentation and exergy loss inside a pipe equipped with innovative turbulators. Int J Heat Mass Transfer. 2018;126:156–63.
Sheikholeslami M, Ghasemi A, Li Z, Shafee A, Saleem S. Influence of CuO nanoparticles on heat transfer behavior of PCM in solidification process considering radiative source term. Int J Heat Mass Trans. 2018;126:1252–64.
Abro KA, Ahmet Y. Fractional treatment of vibration equation through modern analogy of fractional differentiations using integral transforms. Iran J Sci Technol Trans A Sci. 2019;43:1–8. https://doi.org/10.1007/s40995-019-00687-4.
Kashif AA, Ali AM, Anwer AM. Functionality of circuit via modern fractional differentiations, analog integrated circuits and signal processing. Int J. 2019;99(1):11–21. https://doi.org/10.1007/s10470-018-1371-6.
Abro KA, Anwar AM, Muhammad AU. A comparative mathematical analysis of RL and RC electrical circuits via Atangana–Baleanu and Caputo–Fabrizio fractional derivatives. Eur Phys J Plus. 2018;133:113. https://doi.org/10.1140/epjp/i2018-11953-8.
Abro KA, Ilyas K, Kottakkaran SN. Novel technique of Atangana and Baleanu for heat dissipation in transmission line of electrical circuit. Chaos Solitons Fractals. 2019;129:40–5. https://doi.org/10.1016/j.chaos.2019.08.001.
Gómez-Aguilar JF, Abro KA, Kolebaje O, Yildirim A. Chaos in a calcium oscillation model via Atangana–Baleanu operator with strong memory. Eur Phys J Plus. 2019;134:140. https://doi.org/10.1140/epjp/i2019-12550-1.
Dur MM, Kashif AA, Muhammad AS. Application of modern approach of Caputo–Fabrizio fractional derivative to MHD second grade fluid through oscillating porous plate with heat and mass transfer. Int J Adv Appl Sci. 2018;5(10):97–105.
Qasem AM, Kashif AA, Ilyas K. Analytical solutions of fractional walter’s-B fluid with applications. Complexity. 2018;2018:8918541.