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On Fractional Governing Equations of Spherical Particles Settling in Water
Current Issue
Volume 4, 2017
Issue 6 (November)
Pages: 105-109   |   Vol. 4, No. 6, November 2017   |   Follow on         
Paper in PDF Downloads: 62   Since Nov. 6, 2017 Views: 1110   Since Nov. 6, 2017
Authors
[1]
Kamran Ayub, Department of Mathematics, Riphah International University, Islamabad, Pakistan.
[2]
M. Yaqub Khan, Department of Mathematics, Riphah International University, Islamabad, Pakistan.
[3]
Qazi Mahmood Ul-Hassan, Department of Mathematics, University of Wah, Wah Cantt., Pakistan.
[4]
Memmona Yaqub, Department of Mathematics, Allama Iqbal Open University, Islamabad, Pakistan.
Abstract
This paper shows a structure to get the result to the uneven settle actions of few solid spherical particles declining in water as a Newtonian fluid by homotopy analysis method. The partial derivative is described in Modified Riemann liouville sense. This method performs very well in competence. Numerical results explain the whole consistency in used algorithm.
Keywords
Homotopy Analysis Method, Spherical Particles, Drag Coefficient, Fractional Calculus, Sedimentation Phenomenon, Modified Riemann-Liouville Fractional Derivative
Reference
[1]
I. Podlubny, Fractional Differential Equations, Academic Press, New York, 1999.
[2]
J. H. He, Nonlinear oscillation with fractional derivative and its applications, International Conference on Vibrating Engineering 98, Dalian, China, 1998, pp. 288-291.
[3]
J. H. He, some applications of nonlinear fractional differential equations and their Approximations, Bull. Sci. Technol., 15(2) (1999), 86-90.
[4]
J. H. He, approximate analytical solution for seepage flow with fractional derivatives in porous Media, Comput. Methods Appl. Mech. Engrg., 167 (1998), 57-68.
[5]
J. S. Bridge, S. J. Bennett, A model for the entrainment and transport of sediment grains of mixed sizes, shapes, and densities, Water Resour. Res. 28 (2) (1992) 337–363.
[6]
R. P. Chhabra, Bubbles, Drops and Particles in Non-Newtonian Fluids, CRC Press, Boca Raton, FL, 1993.
[7]
Haider, O. Levenspiel, Drag coefficients and terminal velocity of spherical and non-spherical particles, Powder Tech. 58 (1989) 63–70.
[8]
M. Jalaal, D. D. Ganji, G. Ahmadi, Analytical investigation on acceleration motion of a vertically falling spherical particle in incompressible Newtonian media, Adv. Powder Tech. 21 (2010) 298–304.
[9]
J. M. Ferreira, M. Duarte Naia, R. P. Chhabra, An analytical study of the transient motion of a dense rigid sphere in an incompressible Newtonian fluid, Chem. Eng. Commun. 168 (1) (1998).
[10]
S. Abbasbandy, Approximate solution for the nonlinear model of diffusion and reaction in porous catalysts by means of the homotopy analysis method. Chem. Eng. J. 2007. doi:10.1016/j.cej.2007.03.022.
[11]
A. M. Wazwaz, Blow-up for solutions of some linear wave equations with mixed nonlinear Boundary conditions. Appl Math Comput 2002; 131:517–29.
[12]
G. Jumarie, Table of some basic fractional calculus formulae derived from a modified Riemann–Liouville derivative for non-differentiable functions, Appl. Math. Lett. 22 (2009) 378–385.
[13]
B. J. West, M. Bolognab, P. Grigolini, Physics of Fractional Operators, Springer, New York, 2003.
[14]
K. S. Miller, B. Ross, an Introduction to the Fractional Calculus and Fractional Differential Equations, Wiley, New York, 1993.
[15]
S. G. Samko, A. A. Kilbas, O. I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach, Yverdon, 1993.
[16]
S. Momani, Z. Odibat, I. Hashim, Algorithms for nonlinear fractional partial differential equations: A selection of numerical methods, Topological Methods in Nonlinear Analysis 31 (2008) 211.
[17]
S. J. Liao The proposed homotopy analysis technique for the solution of nonlinear Problems. Ph.D. thesis, Shanghai Jiao Tong University; 1992.
[18]
S. J. Liao An approximate solution technique which does not depend upon small Parameters: a special example. Int J Nonlinear Mech 1995; 30:371–80.
[19]
S. J. Liao An approximate solution technique which does not depend upon small parameters (II): An application in fluid mechanics. Int. J. Nonlinear Mech. 1997; 32:815–22.
[20]
S. J. Liao An explicit, totally analytic approximation of Blasius viscous flow problems. Int. J. Nonlinear Mech. 1999; 34 (4):759–78.
[21]
S. J. Liao Beyond perturbation: introduction to the homotopy analysis method. Boca Raton: Chapman & Hall, CRC Press; 2003.
[22]
S. J. Liao On the homotopy analysis method for nonlinear problems. Appl. Math. Comput. 2004; 147:499–513.
[23]
S. J. Liao Campo A. Analytic solutions of the temperature distribution in Blasius viscous flow problems. J. Fluid Mech. 2002; 453:411–25.
[24]
M. Dehghan, J. Manaan, A. Saadatmandi Application of semi-analytic methods for the Fitzhugh–Nagumo equation which models the transmission of nerve impulses Math. Methods Appl. Sci., 33 (2010), pp. 1384–1398.
[25]
R. L. Fosdick, K. R. Rajagopal Thermodynamics and stability of fluids of third gradeProc. Roy. Soc. Lond. A, 339 (1980), pp. 351–377.
[26]
R. A. Van Gorder, K. Vajravelu On the selection of auxiliary functions operators and con-vergence control parameters in the application of the homotopy analysis method to nonlinear differential equations: a general approach Commun. Nonlinear Sci. Numer. Simul, 14 (2009), pp. 4078–4089.
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