Use of Atangana-Baleanu fractional derivative in helical flow of a circular pipe. (English) Zbl 1508.35055

Summary: There is no denying fact that helically moving pipe/cylinder has versatile utilization in industries; as it has multi-purposes, such as foundation helical piers, drilling of rigs, hydraulic simultaneous lift system, foundation helical brackets and many others. This paper incorporates the new analysis based on modern fractional differentiation on infinite helically moving pipe. The mathematical modeling of infinite helically moving pipe results in governing equations involving partial differential equations of integer order. In order to highlight the effects of fractional differentiation, namely, Atangana-Baleanu on the governing partial differential equations, the Laplace and Hankel transforms are invoked for finding the angular and oscillating velocities corresponding to applied shear stresses. Our investigated general solutions involve the gamma functions of linear expressions. For eliminating the gamma functions of linear expressions, the solutions of angular and oscillating velocities corresponding to applied shear stresses are communicated in terms of Fox-H function. At last, various embedded rheological parameters such as friction and viscous factor, curvature diameter of the helical pipe, dynamic analogies of relaxation and retardation time and comparison of viscoelastic fluid models (Burger, Oldroyd-B, Maxwell and Newtonian) have significant discrepancies and semblances based on helically moving pipe.


35Q35 PDEs in connection with fluid mechanics
35Q74 PDEs in connection with mechanics of deformable solids
76A10 Viscoelastic fluids
74D05 Linear constitutive equations for materials with memory
74M10 Friction in solid mechanics
44A10 Laplace transform
35A20 Analyticity in context of PDEs
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
33E12 Mittag-Leffler functions and generalizations
26A33 Fractional derivatives and integrals
35R11 Fractional partial differential equations
Full Text: DOI


[1] Ferry, J. D., Viscoelastic Properties of Polymers, 3rd edn. (John Wiley & Sons, New York, 1980).
[2] Abro, K. A., Khan, I. and Gómez-Aguilar, J. F., A mathematical analysis of a circular pipe in rate type fluid via Hankel transform, Eur. Phys. J. Plus133 (2018) 397. https://doi.org/10.1140/epjp/i2018-12186-7
[3] Koeller, R. C., Applications of fractional calculus to the theory of viscoelasticity, J. Appl. Mech.51 (1984) 299-307. · Zbl 0544.73052
[4] Abro, K. A., Khan, I. and 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. Phenomena13 (2018) 1, https://doi.org/10.1051/mmnp/2018007. · Zbl 1405.76065
[5] Lakes, R., Viscoelastic Materials (Cambridge University Press, Cambridge, 2009). · Zbl 1049.74012
[6] Abro, K. A., Shaikh, A. A. and Dehraj, S., Exact solutions on the oscillating plate of Maxwell fluids, Mehran Univ. Res. J. Eng. Technol.35(1) (2016) 157-162.
[7] Fetecau, C., Starting solutions for the motion of a second grade fluid due to longitudinal and torsional oscillations of a circular cylinder, Int. J. Eng. Sci.44 (2006) 788-796. · Zbl 1213.76014
[8] Fetecau, C., Corina, F., Jamil, M. and Mahmood, A., Flow of fractional Maxwell fluid between coaxial cylinders, Arch. Appl. Mech.81 (2011) 1153-1163. · Zbl 1271.76021
[9] Shah, N. A., Fetecau, C. and Vieru, D., First general solutions for unsteady unidirectional motions of rate type fluids in cylindrical domains, Alex. Eng. J. (2017), http://dx.doi.org/10.1016/j.aej.2017.03.014.
[10] Nauman, R., Abdullah, M., Asma, R. B., Awan, A. and Ehsan, H., Flow of a second grade fluid with fractional derivatives due to a quadratic time dependent shear stress, Alex. Eng. J. (2017), http://dx.doi.org/10.1016/j.aej.2017.04.004.
[11] Rabia, S., Imran, M. and Masood, K. C., Time-dependent flow model of a generalized Burgers’ fluid with fractional derivatives through a cylindrical domain: An exact and numerical approach, Results Phys.9 (2018) 237-245.
[12] Demartino, C. and Ricciardelli, F., Aerodynamics of nominally circular cylinders: A review of experimental results for Civil Engineering applications, Eng. Struct.137 (2017) 76-114, http://dx.doi.org/10.1016/j.engstruct.2017.01.023.
[13] Uddin, M. J., Numerical simulation on free convective heat transfer in a copper-water nanofluid filled semi-circular annulus with the magnetic field, Ital. J. Eng. Sci.: Tecnica Italiana (2018). https://doi.org/10.18280/ijes.620210
[14] Noroozi, M., Ghassemi, A., Atrian, A. and Vahabi, M., Multiple cylindrical interface cracks in FGM coated cylinders under torsional transient loading, Theoret. Appl. Fract. Mech. (2018). https://doi.org/10.1016/j.tafmec.2018.08.015
[15] Zafar, A. A., Shah, N. A. and Khan, I., Two phase flow of blood through a circular tube with magnetic properties, J. Magn. Magn. Mater. (2018). https://doi.org/10.1016/j.jmmm.2018.08.035
[16] Abro, K. A., Khan, I. and Gómez-Aguilar, J. F., A mathematical analysis of a circular pipe in rate type fluid via Hankel transform, Eur. Phys. J. Plus133 (2018) 397. https://doi.org/10.1140/epjp/i2018-12186-7
[17] Haitao, Q. and Hui, J., Unsteady helical flows of a generalized Oldroyd-B fluid with fractional derivative, Nonlinear Anal. Real World Appl.10 (2009) 2700-2708. · Zbl 1162.76006
[18] Laghari, M. H., Abro, K. A. and Shaikh, A. A., Helical flows of fractional viscoelastic fluid in a circular pipe, Int. J. Adv. Appl. Sci.4(10) (2017) 97-105.
[19] Chunrui, L., Zheng, L., Zhang, Y., Ma, L. and Zhang, X., Helical flows of a heated generalized Oldroyd-B fluid subject to a time-dependent shear stress in porous medium, Commun. Nonlinear Sci. Numer. Simul.17 (2012) 5026-5041. · Zbl 1302.76016
[20] Jamil, M., Abro, K. A. and Khan, N. A., Helices of fractionalized Maxwell fluid, Nonlinear Eng.4(4) (2015) 191-201.
[21] Imran, M. A., Tahir, M., Javaid, M. and Imran, M., Exact solutions for unsteady flow of a fractional Maxwell fluid through moving coaxial circular cylinders, J. Comput. Theor. Nanosci.13 (2016) 3405-3413.
[22] Zhao, J. H., Zheng, L. C., Zhang, X. X. and Liu, F. W., Unsteady natural convection boundary layer heat transfer of fractional Maxwell viscoelastic fluid over a vertical plate, Int. J. Heat Mass Transf.97 (2016) 760-766.
[23] Kashif, A. A., Muhammad, A. S. and Muzaffar, H. L., Influence of slippage in heat and mass transfer for fractionalized MHD flows in porous medium, Int. J. Adv. Appl. Math. Mech.4(4) (2017) 5-14. · Zbl 1393.76130
[24] Gomez-Aguilar, J. F., Morales-Delgado, V. F., Taneco-Hernandez, M. A., Baleanu, D., Jimenez, R. F. Escobar and Al Quarashi, M. M., Analytical solutions of the electrical RLC circuit via Liouville-Caputo operators with local and non-local kernels, Entropy18 (2016) 402. https://doi.org/10.3390/e18080402
[25] Abro, K. A., Hussain, M. and Baig, M. M., A mathematical analysis of magnetohydrodynamic generalized Burger fluid for permeable oscillating plate, Punjab Univ. J. Math.50(2) (2018) 97-111.
[26] Abro, K. A. and Solangi, M. Anwar, Heat transfer in magnetohydrodynamic second grade fluid with porous impacts using Caputo-Fabrizoi fractional derivatives, Punjab Univ. J. Math.49(2) (2017) 113-125. · Zbl 1381.76394
[27] Hristov, J., Derivatives with non-singular kernels from the Caputo-Fabrizio definition and beyond: Appraising analysis with emphasis on diffusion models, in Frontiers in Fractional Calculus (Bentham Science Publishers, Sharjah, 2017), pp. 235-295.
[28] Abro, K. A., Abro, I. Ali, Almani, S. M. and Khan, I., 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
[29] Ambreen, S., Kashif, A. A. and Muhammad, A. S., Thermodynamics of magnetohydrodynamic Brinkman fluid in porous medium: Applications to thermal science, J. Therm. Anal. Calorim. (2018). https://doi.org/10.1007/s10973-018-7897-0
[30] Abro, K. A., Memon, A. A., Abro, S. H., Khan, I. and 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.5 (2019) 41-49, https://doi.org/10.1016/j.egyr.2018.09.009.
[31] Saqib, M., Farhad, A., Ilyas, K., Nadeem, A. S. and Sharidan, S., Convection in ethylene glycol based molybdenum disulfide nanofluid: Atangana-Baleanu frictional derivatives approach, J. Therm. Anal. Calorim. (2018), https://doi.org/10.1007/s10973-018-7054-9. · Zbl 1448.80007
[32] Abro, K. A., Khan, I. and 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. Phenomena13 (2018) 1, https://doi.org/10.1051/mmnp/2018007. · Zbl 1405.76065
[33] Coronel-Escamilla, A., Gómez-Aguilar, J. F., Torres, L. and Escobar, R. F., A numerical solution for a variable-order reaction-diffusion model by using fractional derivatives with non-local and non-singular kernel, Physica A (2018). https://doi.org/10.1016/j.physa.2017.09.014
[34] Abro, K. A., Rashidi, M. M., Khan, I., Abro, I. A. and Tassadiq, A., Analysis of Stokes’ second problem for nanofluids using modern fractional derivatives, J. Nanofluids7 (2018) 738-747.
[35] Abro, K. A., Hussain, M. and Baig, M. M., An analytic study of molybdenum disulfide nanofluids using modern approach of Atangana-Baleanu fractional derivatives, Eur. Phys. J. Plus132 (2017) 439. https://doi.org/10.1140/epjp/i2017-11689-y(2017).
[36] Abro, K. A., Memon, A. A. and Memon, A. A., Functionality of circuit via modern fractional differentiations, Analog Integr. Circ. Signal Process. (2018) 1-11, https://doi.org/10.1007/s10470-018-1371-6.
[37] Koca, I. and Atangana, A., Solutions of Cattaneo-Hristov model of elastic heat diffusion with Caputo-Fabrizio and Atangana-Baleanu fractional derivatives, Therm. Sci. (2017). https://doi.org/10.2298/TSCI160102102M
[38] Abro, K. A., Chandio, A. D., Abro, I. A. and Khan, I., 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) 1-11, https://doi.org/10.1007/s10973-018-7302-z.
[39] Sheikh, N. A., Ali, F., Khan, I., Gohar, M. and Saqib, M., On the applications of nanofluids to enhance the performance of solar collectors: A comparative analysis of Atangana-Baleanu and Caputo-Fabrizio fractional models, Eur. Phys. J. Plus132(12) (2017) 540.
[40] Abro, K. A., Memon, A. A. and Uqaili, M. A., A comparative mathematical analysis of RL and RC electrical circuits via Atangana-Baleanu and Caputo-Fabrizio fractional derivatives, Eur. Phys. J. Plus133 (2018) 113. https://doi.org/10.1140/epjp/i2018-11953-8
[41] Sheikh, N. A., Ali, F., Saqib, M., Khan, I., Jan, S. A. A., Alshomrani, A. S. and Alghamdi, M. S., Comparison and analysis of the Atangana-Baleanu and Caputo-Fabrizio fractional derivatives for generalized Casson fluid model with heat generation and chemical reaction, Res. Phys.7 (2017) 789-800.
[42] Khan, A., Abro, K. A., Tassaddiq, A. and 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, Entropy19(8) (2017) 1-12. · Zbl 1405.76065
[43] Atanganaa, A. and Kocab, I., On the new fractional derivative and application to nonlinear Baggs and Freedman model, J. Nonlinear Sci. Appl.9 (2016) 2467-2480. · Zbl 1335.34079
[44] Atangana, A., On the new fractional derivative and application to nonlinear Fisher’s reaction-diffusion equation, Appl. Math. Comput.273 (2016) 948-956. · Zbl 1410.35272
[45] Atangana, A. and Koca, I., Chaos in a simple nonlinear system with Atangana-Baleanu derivatives with fractional order, Chaos Solitons Fractals (2016) 1-8. · Zbl 1360.34150
[46] Al-Mdallal, Q., Abro, K. A. and Khan, I., Analytical solutions of fractional Walter’s-B fluid with applications, Complexity2018 (2018) 8918541. · Zbl 1398.76211
[47] Abro, K. A. and Khan, I., Analysis of heat and mass transfer in MHD flow of generalized Casson fluid in a porous space via non-integer order derivative without singular kernel, Chin. J. Phys.55(4) (2017) 1583-1595.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.