Rahmanzadeh, M., Asadi, T., Atashafrooz, M. (2020). The Development and Application of the RCW Method for the Solution of the Blasius Problem. Journal of Applied and Computational Mechanics, 6(Issue 1 (In progress)), 105-111. doi: 10.22055/jacm.2019.28250.1469

Mostafa Rahmanzadeh; Tahereh Asadi; Meysam Atashafrooz. "The Development and Application of the RCW Method for the Solution of the Blasius Problem". Journal of Applied and Computational Mechanics, 6, Issue 1 (In progress), 2020, 105-111. doi: 10.22055/jacm.2019.28250.1469

Rahmanzadeh, M., Asadi, T., Atashafrooz, M. (2020). 'The Development and Application of the RCW Method for the Solution of the Blasius Problem', Journal of Applied and Computational Mechanics, 6(Issue 1 (In progress)), pp. 105-111. doi: 10.22055/jacm.2019.28250.1469

Rahmanzadeh, M., Asadi, T., Atashafrooz, M. The Development and Application of the RCW Method for the Solution of the Blasius Problem. Journal of Applied and Computational Mechanics, 2020; 6(Issue 1 (In progress)): 105-111. doi: 10.22055/jacm.2019.28250.1469

The Development and Application of the RCW Method for the Solution of the Blasius Problem

^{1}Department of Chemical Engineering, Sirjan University of Technology, Sirjan, Iran

^{2}Department of Mechanical Engineering, Sirjan University of Technology, Sirjan, Iran

Abstract

In this research, a numerical algorithm is employed to investigate the classical Blasius equation which is the governing equation of boundary layer problem. The base of this algorithm is on the development of RCW (Rahmanzadeh-Cai-White) method. In fact, in the current work, an attempt is made to solve the Blasius equation by using the sum of Taylor and Fourier series. While, in the most common numerical methods, the answer is considered only as a Taylor series. It should be noted that in these algorithms which use Taylor expansion, the values of the truncation error are considerable. However, adding the Fourier series to the Taylor series leads to reduce the amount of the truncation error. Nevertheless, the results of this research show the RCW method has the ability to achieve the accuracy of analytical solution. Moreover, it is well illustrated that the accuracy of RCW method is higher than the Runge-Kutta one.

[1] Ahmadi, G., Self-similar solution of incompressible micropolar boundary layer flow over a semi-infinite plate, International Journal of Engineering Science, 14(7), 1976, 639-646.

[2] Fang, T., A note on the unsteady boundary layers over a flat plate, International Journal of Non-Linear Mechanics, 43(9), 2008, 1007-1011.

[3] Aziz, A., A similarity solution for laminar thermal boundary layer over a flat plate with a convective surface boundary condition, Communications in Nonlinear Science Numerical Simulation, 14(4), 2009, 1064-1068.

[4] Li, Y., Rao, Y., Wang, D., Zhang, P. and Wu, X., Heat transfer and pressure loss of turbulent flow in channels with miniature structured ribs on one wall, International Journal of Heat and Mass Transfer, 131, 2019, 584-593.

[5] Ushida, A., Shuichi Ogawa, S., Narumi, T., Sato, T. and Hasegawa T., Pseudo-laminarization effect of dilute and ultra-dilute polymer solutions on flows in narrow pipes, Experimental Thermal Fluid Science, 99, 2018, 233-241.

[6] Najafi, E., Numerical quasilinearization scheme for the integral equation form of the Blasius equation, Computational Methods for Differential Equations, 6(2), 2018, 141-156.

[7] Sewell, G., The numerical solution of ordinary and partial differential equations, John Wiley & Sons, New York, 2005.

[8] Parand, K., Dehghan, M. and Pirkhedri, A., Sinc-collocation method for solving the Blasius equation, Physics Letters A, 373(44), 2009, 4060-4065.

[9] Iacono, R. and Boyd, J. P., Simple analytic approximations for the Blasius problem, Physica D: Nonlinear Phenomena, 310, 2015, 72-78.

[10] Cortell, R., Numerical solutions of the classical Blasius flat-plate problem, Applied Mathematics and Computation, 170(1), 2005, 706-710.

[11] Chavaraddi, K. B. and Page, M. H., Solution of Blasius equation by adomian decomposition Mmethod and differential transform method, International Journal of Mathematics and its Applications, 55, 2018, 219–1226

[12] Jafarimoghaddam, A. and Aberoumand, S., Exact approximations for skin friction coefficient and convective heat transfer coefficient for a class of power law fluids flow over a semi-infinite plate: Results from similarity solutions, Engineering Science and Technology, An International Journal, 20(3), 2017, 1115-1121.

[13] Benlahsen, M., Guedda, M. and Kersner, R., The generalized Blasius equation revisited, Mathematical and Computer Modelling, 47(9-10), 2008, 1063-1076.

[14] Wang, L., A new algorithm for solving classical Blasius equation, Applied Mathematics and Computation, 157(1), 2004, 1-9.

[15] Munson, B. R., Okiishi, T. H., Huebsch, I. W. W., Rothmayer, A. P., Fundamentals of fluid mechanics, Wiley Singapore, 2013.

[16] White, F. M., Fluid mechanics, McGraw-hill, 1986.

[17] Brugnano, L. and Magherini, C., Blended implementation of block implicit methods for ODEs, Applied Numerical Mathematics, 42(1-3), 2002, 29-45.

[18] Ibáñez, J. J., Hernández, V., Ruiz, P. A. and Arias, E., A piecewise-linearized algorithm based on the Krylov subspace for solving stiff ODEs, Journal of Computational Applied Mathematics, 235(7), 2011, 1798-1804.

[19] Brugnano, L., Magherini, C. and Mugnai, F., Blended implicit methods for the numerical solution of DAE problems, Journal of Computational Applied Mathematics, 189(1-2), 2006, 34-50.

[20] Fazio, R., A novel approach to the numerical solution of boundary value problems on infinite intervals, SIAM Journal on Numerical Analysis, 33(4), 1996, 1473-1483.

[21] Fang, T., Guo, F. and Chia-fon, F. L., A note on the extended Blasius equation, Applied Mathematics Letters, 19(7), 2006, 613-617.

[22] Catal, S., Some of semi analytical methods for Blasius problem, Applied Mathematics, 3(7), 2012, 724-728.

[23] Rahmanzadeh, M., Cai, L., and White, R. E., A new method for solving initial value problems, Computers and Chemical Engineering, 58, 2013, 33-39.

[24] Nelder, J. A. and Mead, R., A simplex method for function minimization, The Computer Journal, 7(4), 1965, 308-313.

[25] Bock, H. G., Diehl, M. M., Leineweber, D. B., Schlöder, J. P., Nonlinear Model Predictive Control, Birkhiiuser Verlag Basel, Switzerland, 2000.

[26] Rahmanzadeh, M. and Barfeie, M., An explicit time-stepping method based on error minimization for solving stiff system of ordinary differential equations, Malaysian Journal of Mathematical Sciences, 12(2), 2018, 267-283.

[27] Ahmad, F. and Al-Barakati, W. H., An approximate analytic solution of the Blasius problem, Communications in Nonlinear Science Numerical Simulation, 14(4), 2009, 1021-1024.