In this paper three oscillating viscous flows are studied by applying the Adomian decomposition method (ADM). Major improvement is on the choice of the assignment of the first term of the decomposition series. Different from past studies in which the initial velocity profile of the whole domain is assigned as the first term of the decomposition series, the assignment in present study is simply the boundary velocity for Stokes’ second problem and the pressure gradient for pulsatile flows. This improvement demonstrates and implies that ADM is not only good in approaching the known exact solution, but also possesses the practicability in treating realistic problems. The derived approximate solutions accurate up to any order can be obtained after two key parameters are determined. Present results show an excellent agreement with those calculated by the exact solutions. Based on the present results, more periodic problems can be analyzed by ADM with the help of Fourier analysis.
Application of the Adomian Decomposition Method to Oscillating Viscous Flows, Applied and Computational Mathematics.
Vol. 5, No. 3,
2016, pp. 121-132.
G. A. Adomian, “A review of the decomposition method in applied mathematics,” Journal of Mathematical Analysis and Applications, Vol. 135, No. 2, pp. 501-544, 1988.
G. A. Adomian, Solving frontier problems of physics: the decomposition method, Dordrecht, Kluwer Academic Pub., 1994.
A. M. Wazwaz, “The numerical solution of fifth-order boundary value problems by the decomposition method,” Journal of Computational and Applied Mathematics, Vol. 136, pp. 259-270, 2001.
A. M. Wazwaz and S. M. El-Sayed, “A new modification of the Adomian decomposition method for linear and nonlinear operators,” Applied Mathematics and Computation, Vol. 122, pp. 393-405, 2001.
M. Dehghan, “Application of the Adomian decomposition method for two-dimensional parabolic equation subject to nonstandard boundary specifications,” Applied Mathematics and Computation, Vol. 157, pp. 549-560, 2004.
M. Dehghan, “The use of Adomian decomposition method for solving the one-dimensional parabolic equation with non-local boundary specifications,” International Journal of Computer Mathematics, Vol. 81, pp. 25-34, 2004.
M. M. Hosseini, “Adomian decomposition method for solution of differential-algebraic equations,” Journal of Computational and Applied Mathematics, Vol. 197, pp. 495-501, 2006.
I. Hashim, “Adomian decomposition method for solving BVPs for fourth-order integro-differential equations,” Journal of Computational and Applied Mathematics, Vol. 193, pp. 658–664, 2006.
A. M. Wazwaz, “A comparison between the variational iteration method and Adomian decomposition method,” Journal of Computational and Applied Mathematics, Vol. 207, pp. 129-136, 2007.
A. H. Bokhari, G. Mohammad, M. T. Mustafa and F. D. Zaman, “Adomian decomposition method for a nonlinear heat equation with temperature dependent thermal properties,” Mathematical Problems in Engineering, Vol. 2009, 926086, 2009.
J. S. Duan, “An efficient algorithm for the multivariable Adomian polynomials,” Applied Mathematics and Computation, Vol. 217, pp. 2456-2467, 2010.
A. Wazwaz, “The modified decomposition method and Padé approximation for solving the Thomas-Fermi equation,” Applied Mathematics and Computation, Vol. 102, pp. 11-19, 1999.
M. Dehghan, A. Hamidi and M. Shakourifar, “The solution of coupled Burgers’ equations using Adomian-Padé technique,” Applied Mathematics and Computation, Vol. 189, pp. 1034-1047, 2007.
T. A. Abassy, M. A. El-Tawil and H. K. Saleh, “The solution of Burgers’ and good Boussinesq equations using ADM-Padé technique,” Chaos, Solitons and Fractals, Vol. 32, pp. 1008-1026, 2007.
Y. Cherruault and G. A. Adomian, “Decomposition method: a new proof of convergence,” Mathematical and Computer Modelling, Vol. 18, pp. 103-106, 1993.
D. Lesnic, “Convergence of Adomian’ decomposition method: periodic temperatures,” Computers & Mathematics with Applications, Vol. 44, pp. 13-24, 2002.
C. M. Liu, “On the study of oscillating viscous flows by using the Adomian-Padé approximation,” Journal of Applied Mathematics, Vol. 2015, 864190, 2015.
M. E. Erdogan, “A note on an unsteady flow of a viscous fluid due to an oscillating plane wall,” International Journal of Non-Linear Mechanics, Vol. 35, pp. 1-6, 2000.
C. M. Liu and I. C. Liu, “A note on the transient solution of Stokes’ second problem with arbitrary initial phase,” Journal of Mechanics, Vol. 22, No. 4, pp. 349-354, 2006.
F. Farkhadnia, R. Kamrani and D. D. Ganji, “Analytical investigation for fluid behavior over a float plate with oscillating motion and wall transpiration,” New Trends in Mathematical Sciences, Vol. 2, pp. 178-189, 2014.
C. M. Liu, “Extended Stokes’ problems of relatively moving porous half-planes,” Mathematical Problems in Engineering, Vol. 2009, 185965, 2009.
D. O. de Almeida Cruz and E. F. Lins, “The unsteady flow generated by an oscillating wall with transpiration,” International Journal of Non-linear Mechanics, Vol. 45, pp. 453-457, 2010.
G. G. Stokes, “On the effect of the internal friction of fluids on the motion of pendulums,” Transactions of the Cambridge Philosophical Society, Vol. 9, pp. 8-106, 1851.
L. Rayleigh, “On the motion of solid bodies through viscous liquid,” Philosophical Magazine, Vol. 6, pp. 697-711, 1911.
R. Panton, “The transient for Stokes’ oscillating plate: a solution in terms of tabulated functions,” Journal of Fluid Mechanics, Vol. 31, pp. 819-825, 1968.
C. M. Liu, “Complete solutions to extended Stokes’ problems,” Mathematical Problems in Engineering, Vol. 2008, 754262, 2008.
L. Ai and K. Vafai, “An investigation of Stokes’ second problem for non-Newtonian fluids,” Numerical Heat Transfer, Part A, Vol. 47, pp. 955-980, 2005.
D. Pritchard, C. R. McArdle and S. K. Wilson, “The Stokes boundary layer for a power-law fluid,” Journal of Non-Newtonian Fluid Mechanics, Vol. 166, pp. 745-753, 2011.
S. P. Hu, C. M. Fan, C. W. Chen and D. L. Young, “Method of fundamental solutions for Stokes’ first and second problems,” Journal of Mechanics, Vol. 21, pp. 25-13, 2005.
I. G. Currie, Fundamental Mechanics of Fluids, 2nd ed., McGraw-Hill, New York, 1993.
Y. C. Fung, Biomechanics: motion, flow, stress, and growth, Springer, New York, 1990.