Abstract
The generalized Couette flow of Jeffrey nanofluid through porous medium, subjected to the oscillating pressure gradient and mixed convection, is numerically simulated using variable-order fractional calculus. The effect of several involving parameters such as chemical reactions, heat generation, thermophoresis, radiation, channel inclination, and Soret effect is also considered. To the best of the authors’ knowledge, the described general form of the Couette flow problem is not tackled by the researchers yet. The non-dimensional form of heat, mass, and momentum equations is solved as a coupled set. The effect of several parameters such as Grashof, Hartmann, Prandtl, Soret, and Schmidt numbers in addition to oscillation frequency, retardation time, radiation, heat absorption, and reaction rate are determined and presented graphically. An operational matrix method based on the second kind shifted Chebyshev polynomials is proposed to investigate the behavior of the interested problem. In fact, regarding the established method, the unknown solutions are expanded by the mentioned basis polynomials. Then, the operational matrix of the variable-order fractional derivative is utilized to transfer the problem into solving an algebraic system of equations. According to the obtained results, the growth of fractional order from 0 to 1 changes the skin friction coefficient, Nu and flow rate by \(-18.1\), 35.5, and \(10\%\), respectively.
Similar content being viewed by others
References
Whitesides GM (2006) The origins and the future of microfluidics. Nature 441:368–373
Kiyasatfar M (2018) Convective heat transfer and entropy generation analysis of non-newtonian power-law fluid flows in parallel-plate and circular microchannels under slip boundary conditions. Int J Therm Sci 128:15–27
Abbasi FM, Shehzad SA, Alsaedi A, Hayat T, Obid MA (2015) Influence of heat and mass flux conditions in hydromagnetic flow of jeffrey nanofluid. AIP Adv 5:1–12
Roohi R, Emdad H, Jafarpur K, Mahmoudi M (2018) Determination of magnetic nanoparticles injection characteristics for optimal hyperthermia treatment of an arbitrary cancerous cells distribution. J Test Eval 126:1–17
Roohi R, Emdad H, Jafarpur K (2019) A comprehensive study and optimization of magnetic nanoparticle drug delivery to cancerous tissues via external magnetic field. J Test Eval 127:1–23
Hasona WM, El-Shekhipy AA, Ibrahim MG (2018) Combined effects of magnetohydrodynamic and temperature dependent viscosity on peristaltic flow of jeffrey nanofluid through a porous medium: Applications to oil refinement. Int J Heat Mass Transf 126:700–714
Ramesh K (2018) Effects of viscous dissipation and joule heating on the couette and poiseuille flows of a jeffrey fluid with slip boundary conditions. Propuls Power Res 7(4):329–341
Ahmed S, Zueco J, López-González L (2017) Numerical and analytical solutions for magneto-hydrodynamic 3D flow through two parallel porous plates. Int J Heat Mass Transf 108:322–331
Raju RS, Reddy GJ, Rao JA, Rashidi MM (2016) Thermal diffusion and diffusion thermo effects on an unsteady heat and mass transfer magnetohydrodynamic natural convection couette flow using fem. J Comput Des Eng 3:349–362
Samko SG, Ross B (1993) Integration and differentiation to a variable fractional order. Integral Transform Spec Funct 1:277–300
Samko SG (1995) Fractional integration and differentiation of variable order. Anal Math 21:213–236
Samko SG (2013) Fractional integration and differentiation of variable order: an overview. Nonlinear Dyn 71:653–662
Atangana A, Gómez-Aguilar JF (2018) Decolonisation of fractional calculus rules: breaking commutativity and associativity to capture more natural phenomena. Eur Phys J Plus 133:166
Atangana A, Gómez-Aguilar JF (2018) Fractional derivatives with no-index law property: application to chaos and statistics. Chaos Solitons Fractals 114:516–535
Atangana A (2016) Derivative with two fractional orders: a new avenue of investigation toward revolution in fractional calculus. Eur Phys J Plus 131:373
Ramirez LES, Coimbra CFM (2011) On the variable order dynamics of the nonlinear wake caused by a sedimenting particle. Phys D 240:1111–1118
Sun HG, Chen W, Wei H, Chen YQ (2011) A comparative study of constant-order and variable-order fractional models in characterizing memory property of systems. Eur Phys J Spec Top 193:185
Shyu J-J, Pei S-C, Chan C-H (2009) An iterative method for the design of variable fractional-order fir differintegrators. Signal Process 89(3):320–327
Coimbra C (2003) Mechanics with variable-order differential operators. Ann Phys 12:692–703
Chechkin AV, Gorenflo R, Sokolov IM (2005) Fractional diffusion in inhomogeneous media. J Phys A Math Gen 38:679–684
Santamaria F, Wils S, De Schutter E, Augustine GJ (2006) Anomalous diffusion in purkinjecell dendrites caused by spines. Neuron 52:635–648
Atangana A, Shafiq A (2019) Differential and integral operators with constant fractional order and variable fractional dimension. Chaos Solitons Fractals 127:226–243
Ruiz LFÁ, Gómez-Aguilar JF, Atangana A, Owolabi KM (2019) On the dynamics of fractional maps with power-law, exponential decay and mittag-leffler memory. Chaos Solitons Fractals 127:364–388
Chen W, Zhang J, Zhang J (2013) A variable-order time-fractional derivative model for chloride ions sub-diffusion in concrete structures. Fract Calc Appl Anal 16(1):79–92
Lin R, Liu F, Anh V, Turner I (2009) Stability and convergence of a new explicit finite-difference approximation for the variable-order nonlinear fractional diffusion equation. Appl Math Comput 212:435–445
Bhrawy AH, Zaky MA (2016) Numerical simulation for two-dimensional variable-order fractional nonlinear cable equation. Nonlinear Dyn 80(1):101–116
Zayernouri M, Karniadakis GE (2015) Fractional spectral collocation methods for linear and nonlinear variable order FPDES. J Comput Phys 80(1):312–338
Li XY, Wu BY (2015) A numerical technique for variable fractional functional boundary value problems. Appl Math Lett 43:108–113
Heydari MH, Avazzadeh Z, Yang Y (2019) A computational method for solving variable-order fractional nonlinear diffusion-wave equation. Appl Math Comput 352:235–248
Heydari MH, Avazzadeh Z, Haromi MF (2019) A wavelet approach for solving multi-term variable-order time fractional diffusion-wave equation. Appl Math Comput 341:215–228
Hosseininia M, Heydari MH, Roohi R, Avazzadeh Z (2019) A computational wavelet method for variable-order fractional model of dual phase lag bioheat equation. J Comput Phys 395:1–18
Roohi R, Heydari MH, Aslami M, Mahmoudi MR (2018) A comprehensive numerical study of space-time fractional bioheat equation using fractional-order Legendre functions. Eur Phys J Plus 133:412–422
Heydari MH, Avazzadeh Z (2018) Legendre wavelets optimization method for variable-order fractional Poisson equation. Chaos Solitons Fractals 112:180–190
Hosseininia M, Heydari MH (2019) Meshfree moving least squares method for nonlinear variable-order time fractional 2D telegraph equation involving Mittag–Leffler non-singular kernel. Chaos Solitons Fractals 127:389–399
Sun HG, Chang A, Zhang Y, Chen W (2019) A review on variable-order fractional differential equations: mathematical foundations, physical models, numerical methods and applications. Eur Phys J Spec Top 22(1):27–59
Zhang H, Liu F, Phanikumar MS, Meerschaert MM (2013) A novel numerical method for the time variable fractional order mobile-immobile advection–dispersion model. Comput Math Appl 66(5):693–701
Heydari MH, Hooshmandasl MR, Ghaini FMM (2014) An efficient computational method for solving fractional biharmonic equation. Comput Math Appl 68(9):269–287
Heydari MH, Hooshmandasl MR, Loghmani GB, Cattani C (2016) Wavelets Galerkin method for solving stochastic heat equation. Int J Comput Math 93(9):1579–1596
Hooshmandasl MR, Heydari MH, Cattani C (2016) Numerical solution of fractional sub-diffusion and time-fractional diffusion-wave equations via fractional-order Legendre functions. Eur Phys J Plus 131:268
Heydari MH, Hooshmandasl MR, Shakiba A, Cattani C (2016) Legendre wavelets Galerkin method for solving nonlinear stochastic integral equations. Nonlinear Dyn 85(2):1185–1202
Canuto C, Hussaini M, Quarteroni A, Zang T (1988) Spectral methods in fluid dynamics. Springer, Berlin
Podlubny I (1999) Fractional differential equations. Academic Press, San Diego
Tripathi R, Seth GS, Mishra MK, El-Shekhipy AA, Ibrahim MG (2017) Double diffusive flow of a hydromagnetic nanofluid in a rotating channel with hall effect and viscous dissipation: active and passive control of nanoparticles. Adv Powder Technol 28:2630–2641
Shen S, Liu F, Chen J, Turner I, Anh V (2012) Numerical techniques for the variable order time fractional diffusion equation. Appl Math Comput 218:10861–10870
Chen Y, Liu L, Li B, Sun Y (2014) Numerical solution for the variable order linear cable equation with bernstein polynomials. Appl Math Comput 238:329–341
Doha EH, Bhrawy AH, Ezz-Eldien SS (2011) A Chebyshev spectral method based on operational matrix for initial and boundary value problems of fractional order. Comput Math Appl 62:2364–2373
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Roohi, R., Heydari, M.H., Bavi, O. et al. Chebyshev polynomials for generalized Couette flow of fractional Jeffrey nanofluid subjected to several thermochemical effects. Engineering with Computers 37, 579–595 (2021). https://doi.org/10.1007/s00366-019-00843-9
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00366-019-00843-9