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A numerical method based on rational Gegenbauer functions for solving boundary layer flow of a Powell–Eyring non-Newtonian fluid

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Abstract

In this paper, the boundary layer flow of a Powell–Eyring non-Newtonian fluid over a stretching sheet is considered which is lucrative in the production of many beneficial materials in the industry. Rational Gegenbauer (RG) functions are used to find the better solution comparing to other current works. The drawback of nonlinearity is met using a linearization method, namely, the quasi-linearization method (QLM). As comping on a semi-infinite domain, an approximation is considered to satisfy the infinity condition using the algebraic mapping of \(\frac{\xi -L}{\xi +L}\), where L is a positive arbitrary numerical parameter and a suitable value is calculated for it. Using the QLM, the equation is converted into a sequence of linear ordinary differential equations (ODE); then, these ODEs are solved using the RG collocation method. Finally, the numerical results are presented and the proposed method is compared with the state-of-the-art methods.

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References

  • Abramowitz M, Stegun IA (1972) Handbook of mathematical functions with formulas, graphs, and mathematical tables. The United States

  • Agbaje TM, Mondal S, Motsa SS, Sibanda P (2017) A numerical study of unsteady non-Newtonian Powell–Eyring nanofluid flow over a shrinking sheet with heat generation and thermal radiation. Alex Eng J 56(1):81–91

    Article  Google Scholar 

  • Assari P, Adibi H, Dehghan M (2014) A meshless discrete Galerkin (MDG) method for the numerical solution of integral equations with logarithmic kernels. J Comput Appl Math 30(267):160–81

    Article  MathSciNet  MATH  Google Scholar 

  • Bellman RE, Kalaba RE (1965) Quasilinearization and nonlinear boundary-value problems

  • Bhrawy AH, Abdelkawy MA (2015) A fully spectral collocation approximation for multi-dimensional fractional Schrdinger equations. J Comput Phys 1(294):462–483

    Article  MATH  Google Scholar 

  • Bhrawy AH, Alofi AS (2012) A Jacobi–Gauss collocation method for solving nonlinear Lane–Emden type equations. Commun Nonlinear Sci Numer Simul 17(1):62–70

    Article  MathSciNet  MATH  Google Scholar 

  • Bhrawy AH, Zaky MA (2017) An improved collocation method for multi-dimensional space-time variable-order fractional Schrdinger equations. Appl Numer Math 31(111):197–218

    Article  MATH  Google Scholar 

  • Boyd JP (2001) Chebyshev and Fourier spectral methods. Courier Corporation, Chelmsford

    MATH  Google Scholar 

  • Canuto C, Hussaini MY, Quarteroni A, Thomas A Jr (2012) Spectral methods in fluid dynamics. Springer, Berlin

    MATH  Google Scholar 

  • Dehghan M (2006) Finite difference procedures for solving a problem arising in modeling and design of certain optoelectronic devices. Math Comput Simul 71(1):16–30

    Article  MathSciNet  MATH  Google Scholar 

  • Doha EH, Bhrawy AH, Baleanu D, Hafez RM (2013) Efficient Jacobi–Gauss collocation method for solving initial value problems of Bratu type. Comp Math Math Phys 53(9):1292–1302

    Article  MathSciNet  MATH  Google Scholar 

  • Doha EH, Bhrawy AH, Abdelkawy MA, Van Gorder RA (2014) Jacobi–Gauss–Lobatto collocation method for the numerical solution of 1 + 1 nonlinear Schrdinger equations. J Comput Phys 15(261):244–255

    Article  MATH  Google Scholar 

  • Ellahi R, Shivanian E, Abbasbandy S, Hayat T (2016) Numerical study of magnetohydrodynamics generalized Couette flow of Eyring–Powell fluid with heat transfer and slip condition. Int J Numer Method H 26(5):1433–1445

    Article  MathSciNet  MATH  Google Scholar 

  • Funaro D (2008) Polynomial approximation of differential equations. Springer, Berlin

    MATH  Google Scholar 

  • Ghadikolaei SS, Yassari M, Sadeghi H, Hosseinzadeh K, Ganji DD (2017 Dec 27) Analytical solution of viscoelastic non-Newtonian Second-grade fluid flow on a stretching sheet. Therm Sci Eng Progress

  • Ghadikolaei SS, Hosseinzadeh K, Ganji DD (2017) Analysis of unsteady MHD Eyring–Powell squeezing flow in stretching channel with considering thermal radiation and Joule heating effect using AGM. Case Stud Therm Eng 1(10):579–94

    Article  Google Scholar 

  • Guo BY, Yan JP (2009) Legendre–Gauss collocation method for initial value problems of second order ordinary differential equations. Appl Numer Math 59(6):1386–1408

    Article  MathSciNet  MATH  Google Scholar 

  • Hayat T, Qasim M, Abbas Z (2010) Radiation and mass transfer effects on the magnetohydrodynamic unsteady flow induced by a stretching sheet. Z Nat A 65(3):231–239

    Google Scholar 

  • Hayat T, Iqbal Z, Qasim M, Obaidat S (2012) Steady flow of an Eyring Powell fluid over a moving surface with convective boundary conditions. Int J Heat Mass Trans 55(7):1817–1822

    Article  Google Scholar 

  • Hayat T, Awais M, Asghar S (2013) Radiative effects in a three-dimensional flow of MHD Eyring–Powell fluid. J. Egypt Math Soc 21(3):379–384

    Article  MathSciNet  MATH  Google Scholar 

  • Hayat T, Asad S, Mustafa M, Alsaedi A (2014) Radiation effects on the flow of Powell–Eyring fluid past an unsteady inclined stretching sheet with non-uniform heat source/sink. PLoS One 9(7):e103214

    Article  Google Scholar 

  • Hughes TJ, Franca LP, Hulbert GM (1989) A new finite element formulation for computational fluid dynamics: VIII. The Galerkin/least-squares method for advective–diffusive equations. Comput Method Appl Mech 73(2):173–189

    Article  MathSciNet  MATH  Google Scholar 

  • Ishak A, Nazar R, M Arifin N, Pop I (2007) Mixed convection of the stagnation-point flow towards a stretching vertical permeable sheet. Malays J Math Sci 1(2):217–226

    Google Scholar 

  • Jalil M, Asghar S, Imran SM (2013) Self similar solutions for the flow and heat transfer of Powell–Eyring fluid over a moving surface in a parallel free stream. Int J Heat Mass Trans 31(65):73–79

    Article  Google Scholar 

  • Kalaba R (1959) On nonlinear differential equations, the maximum operation, and monotone convergence. J Math Mech 8(4):519–574

    MathSciNet  MATH  Google Scholar 

  • Kamali F, Saeedi H (2018 Feb 15) Generalized fractional-order Jacobi functions for solving a nonlinear systems of fractional partial differential equations numerically. Math Method Appl Sci. https://doi.org/10.1002/mma.4808

  • Krivec R, Mandelzweig VB (2001) Numerical investigation of quasilinearization method in quantum mechanics. Comput Phys Commun 138(1):69–79

    Article  MATH  Google Scholar 

  • Malik MY, Khan I, Hussain A, Salahuddin T (2015) Mixed convection flow of MHD Eyring–Powell nanofluid over a stretching sheet: a numerical study. AIP Adv 5:11. https://doi.org/10.1063/1.4935639

    Article  Google Scholar 

  • Mandelzweig VB, Tabakin F (2001) Quasilinearization approach to nonlinear problems in physics with application to nonlinear ODEs. Comput Phys Commun 141(2):268–281

    Article  MathSciNet  MATH  Google Scholar 

  • Nadeem S, Saleem S (2015) Series solution of unsteady Eyring Powell nanofluid flow on a rotating cone. Indian J Pure Appl Phys 52(11):725–737

    Google Scholar 

  • Nadeem S, Haq RU, Khan ZH (2014) Numerical study of MHD boundary layer flow of a Maxwell fluid past a stretching sheet in the presence of nanoparticles. J Taiwan I Chem Eng 45(1):121–126

    Article  Google Scholar 

  • Parand K, Delkhosh M (2017) Accurate solution of the Thomas–Fermi equation using the fractional order of rational Chebyshev functions. J Comput Appl Math 30(317):624–642

    Article  MathSciNet  MATH  Google Scholar 

  • Parand K, Delkhosh M (2018) Systems of nonlinear Volterra integro-differential equations of arbitrary order. Bol Soc Paran Mat 36(4):33–54

    Article  MathSciNet  Google Scholar 

  • Parand K, Delkhosh M (2018) An accurate numerical method for solving unsteady isothermal flow of a gas through a semi-infinite porous medium. J Comput Nonlinear Dyn 13(1):011007

    Article  Google Scholar 

  • Parand K, Hajimohammadi Z (2018) Using modified generalized Laguerre functions, QLM and collocation method for solving an Eyring–Powell problem. J Braz Soc Mech Sci Eng 40(4):182

    Article  Google Scholar 

  • Parand K, Hemami M (2017) Application of meshfree method based on compactly supported radial basis function for solving unsteady isothermal gas through a micro-nano porous medium. Iran J Sci Technol Trans A Sci 41(3):677–684

    Article  MathSciNet  MATH  Google Scholar 

  • Parand K, Baharifard F, Bayat BF (2012) Comparison between rational gegenbauer and modified generalized laguerre functions collocation methods for solving the case of heat transfer equations arising in porous medium. Int J Ind Math 4(2):107–122

    Google Scholar 

  • Parand K, Dehghan M, Baharifard F (2013) Solving a laminar boundary layer equation with the rational Gegenbauer functions. Appl Math Model 37(3):851–863

    Article  MathSciNet  MATH  Google Scholar 

  • Parand K, Hossayni SA, Rad JA (2016) Operation matrix method based on Bernstein polynomials for the Riccati differential equation and Volterra population model. Appl Math Model 40(2):993–1011

    Article  MathSciNet  Google Scholar 

  • Parand K, Ghaderi A, Yousefi H, Delkhosh M (2016) A new approach for solving nonlinear Thomas–Fermi equation based on fractional order of rational Bessel functions. Electron J Differ Equ 2016:331

    MathSciNet  MATH  Google Scholar 

  • Parand K, Lotfi Y, Rad JA (2017a) An accurate numerical analysis of the laminar two-dimensional flow of an incompressible Eyring–Powell fluid over a linear stretching sheet. Eur Phys J Plus 132(9):397

    Article  Google Scholar 

  • Parand K, Moayeri MM, Latifi S, Delkhosh M (2017b) A numerical investigation of the boundary layer flow of an Eyring–Powell fluid over a stretching sheet via rational Chebyshev functions. Eur Phys J Plus 132(7):325

    Article  Google Scholar 

  • Parand K, Latifi S, Moayeri MM (2017 Oct 23) Shifted Lagrangian Jacobi collocation scheme for numerical solution of a model of HIV infection. SeMA J. https://doi.org/10.1007/s40324-017-0138-9

  • Rad JA, Parand K (2017) Numerical pricing of American options under two stochastic factor models with jumps using a meshless local Petrov–Galerkin method. Appl Numer Math 31(115):252–274

    MathSciNet  MATH  Google Scholar 

  • Rahimi J, Ganji DD, Khaki M, Hosseinzadeh K (2016 Nov 29) Solution of the boundary layer flow of an Eyring-Powell non-Newtonian fluid over a linear stretching sheet by collocation method. Alex Eng J. https://doi.org/10.1016/j.aej.2016.11.006

  • Ralston A, Rabinowitz P (2001) A first course in numerical analysis. Courier Corporation, Chelmsford

    MATH  Google Scholar 

  • Saeedi HA (2017) fractional-order operational method for numerical treatment of multi-order fractional partial differential equation with variable coefficients. SeMA J. https://doi.org/10.1007/s40324-017-0141-1

    Article  Google Scholar 

  • San Kim D, Kim T, Rim SH (2012) Some identities involving Gegenbauer polynomials. Adv Differ Equ 2012(1):219

    Article  MathSciNet  MATH  Google Scholar 

  • Sheikholeslami M, Ganji DD (2014) Ferrohydrodynamic and magnetohydrodynamic effects on ferrofluid flow and convective heat transfer. Energy 1(75):400–410

    Article  Google Scholar 

  • Steger JL, Warming RF (1981) Flux vector splitting of the inviscid gasdynamic equations with application to finite-difference methods. J Comput Phys 40(2):263–293

    Article  MathSciNet  MATH  Google Scholar 

  • Szego G (1975) Orthogonal polynomials, vol. 23 of Amer. InMath. Soc. Colloq. Publ., Amer. Math. Soc., Providence, RI

  • Zaman H, Shah MA, Ibrahim M (2013) Unsteady incompressible Couette flow problem for the Eyring–Powell model with porous walls. Am J Comput Math 3(4):313–325

    Article  Google Scholar 

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Authors and Affiliations

  1. Department of Computer Sciences, Shahid Beheshti University, G.C., Tehran, Iran

    Kourosh Parand, Arman Bahramnezhad & Hadi Farahani

  2. Department of Cognitive Modeling, Institute for Cognitive and Brain Sciences, Shahid Beheshti University, G.C., Tehran, Iran

    Kourosh Parand

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  1. Kourosh Parand
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  2. Arman Bahramnezhad
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  3. Hadi Farahani
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Correspondence to Kourosh Parand.

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Communicated by Corina Giurgea.

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Parand, K., Bahramnezhad, A. & Farahani, H. A numerical method based on rational Gegenbauer functions for solving boundary layer flow of a Powell–Eyring non-Newtonian fluid. Comp. Appl. Math. 37, 6053–6075 (2018). https://doi.org/10.1007/s40314-018-0679-2

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  • DOI: https://doi.org/10.1007/s40314-018-0679-2

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