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A Legendre wavelet spectral collocation technique resolving anomalies associated with velocity in some boundary layer flows of Walter-B liquid

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Abstract

A Legendre wavelet spectral collocation method is proposed here to solve three boundary layer flow problems of Walter-B fluid namely the stagnation point flow, Blasius flow and Sakiadis flow. In the proposed method, we first transform the boundary value problems into initial value problems using shooting method. We then split the semi infinite domain into subintervals and the governing initial value problems are transformed to system of algebraic equations in each subinterval. The solutions of these algebraic equations yield an approximate solution of the differential equation in each subinterval. The overshoot in the velocity profile associated with the stagnation point and Blasius flows and undershoot in the Sakiadis flow is controlled. Physically realistic solutions are presented for both weakly and strongly viscoelastic parameters. The residual error validates the correctness, convergence and accuracy of the obtained solutions.

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References

  1. Beard DW, Walters K (1964) Elastico-viscous boundary-layer flows. I. Two-dimensional flow near a stagnation point. Proc Camb Phil Soc 60:667–674

    Article  ADS  MathSciNet  MATH  Google Scholar 

  2. Frater KR (1970) On the solution of some boundary value problems arising in elastico-viscous fluid mechanics. Z Angew Math Phys (ZAMP) 21:134–137

    Article  MATH  Google Scholar 

  3. Serth RW (1974) Solution of a viscoelastic boundary layer equation by orthogonal collocation. J Eng Math 8:89–92

    Article  MATH  Google Scholar 

  4. Ariel PD (1992) A hybrid method for computing the flow of viscoelastic fluids. Int J Numer Method Fluids 14:757–774

    Article  MATH  Google Scholar 

  5. Ariel PD (1997) Generalized Gear’s method for computing the flow of a viscoelastic fluid. Comput Methods Appl Mech Eng 142:111–121

    Article  ADS  MathSciNet  MATH  Google Scholar 

  6. Tonekaboni SAM, Abkar R, Khoeilar R (2012) On the study of viscoelastic Walters’ B fluid in boundary layer flows. Math Prob Eng 2012:861508

    MathSciNet  MATH  Google Scholar 

  7. Dizicheh AK, Ismail F, Kajani MT, Maleki M (2013) A Legendre wavelet collocation method for solving oscillatory initial value problems. J Appl Math 2013:591636

    Article  MathSciNet  MATH  Google Scholar 

  8. Boyd JP (2000) Chebyshev and Fourier spectral methods, 2nd edn. Dover, New York

    Google Scholar 

  9. Canuto C, Hussaini MY, Quarteroni A, Zang TA (1988) Spectral methods in fluid dynamics, Springer series in computation physics. Springer, New York

    Book  MATH  Google Scholar 

  10. Bhrawy AH (2014) An efficient Jacobi pseudospectral approximation for nonlinear complex generalized Zakharov system. Appl Math Comput 247:30–46

    MathSciNet  MATH  Google Scholar 

  11. Bhrawy AH, Zaky MA (2015) A method based on the Jacobi tau approximation for solving multi-term time-space fractional partial differential equations. J Comput Phys 281:876–895

    Article  ADS  MathSciNet  MATH  Google Scholar 

  12. Bhrawy AH (2015) A highly accurate collocation algorithm for 1 + 1 and 2 + 1 fractional percolation equations. J Vib Control. doi:10.1177/1077546315597815

    Google Scholar 

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

    Article  ADS  MathSciNet  MATH  Google Scholar 

  14. Bhrawy AH, Doha EH, Ezz-Eldien SS, Abdelkawy MA (2016) A numerical technique based on the shifted Legendre polynomials for solving the time fractional coupled KdV equation. Calcolo 53:1–17

    Article  MathSciNet  MATH  Google Scholar 

  15. Bhrawy AH, Zaky MA (2015) Shifted fractional-order Jacobi orthogonal functions: application to a system of fractional differential equations. Appl Math Model 40:832–845

    Article  MathSciNet  Google Scholar 

  16. Bhrawy AH, Zaky MA (2015) Numerical simulation for two-dimensional variable-order fractional nonlinear cable equation. Nonlinear Dyn 80:101–116

    Article  MathSciNet  MATH  Google Scholar 

  17. Beylkin G, Coifman R, Rokhlin V (1991) Fast wavelet transforms and numerical algorithms. I. Commun Pure Appl Math 44:141–183

    Article  MathSciNet  MATH  Google Scholar 

  18. Chen CF, Hsiao CH (1997) Haar wavelet method for solving lumped and distributed parameter systems. IEE Proc Control Theory Appl 144:87–94

    Article  MATH  Google Scholar 

  19. Razzaghi M, Yousefi S (2000) Legendre wavelets direct method for variational problems. Math Comput Simul 53:185–192

    Article  MathSciNet  Google Scholar 

  20. Islam S, Sarler B, Aziz I, Haq F (2011) Haar wavelet collocation method for the numerical solution of boundary layer fluid flow problems. Int J Therm Sci 50:686–697

    Article  Google Scholar 

  21. Aziz I, Islam S, Sarler B (2013) Wavelets collocation method for the numerical solution of elliptic BV problems. Appl Math Model 37:676–694

    Article  MathSciNet  MATH  Google Scholar 

  22. Na TY (1979) Computational methods in engineering boundary value problems. Academic Press, New York

    MATH  Google Scholar 

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Acknowledgments

The first author acknowledges the support provided by AS-ICTP.

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Author notes

  1. M. Sajid

    Present address: AS-ICTP, Costiera 11, Trieste, Italy

Authors and Affiliations

  1. Department of Mathematics and Statistics, International Islamic University, Islamabad, 44000, Pakistan

    M. Sajid & N. Ali

  2. Department of Basic Sciences, Riphah International University, Islamabad, 44000, Pakistan

    S. A. Iqbal

  3. Department of Mathematics, Quaid-i-Azam University, 45320, Islamabad, 44000, Pakistan

    T. Hayat

  4. Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah, 21589, Saudi Arabia

    T. Hayat

Authors
  1. M. Sajid
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  2. S. A. Iqbal
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  3. N. Ali
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  4. T. Hayat
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Correspondence to M. Sajid.

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Sajid, M., Iqbal, S.A., Ali, N. et al. A Legendre wavelet spectral collocation technique resolving anomalies associated with velocity in some boundary layer flows of Walter-B liquid. Meccanica 52, 877–887 (2017). https://doi.org/10.1007/s11012-016-0428-9

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  • DOI: https://doi.org/10.1007/s11012-016-0428-9

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