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Thermoelectric Viscoelastic Fluid with Fractional Integral and Derivative Heat Transfer

Published online by Cambridge University Press:  29 May 2015

Magdy A. Ezzat ,
A. S. Sabbah ,
A. A. El-Bary  and
S. M. Ezzat
Magdy A. Ezzat*
Affiliation:
Department of Mathematics, Faculty of Education, Alexandria University, Alexandria, Egypt
A. S. Sabbah
Affiliation:
Department of Mathematics, Faculty of Science, Zagazig University, Zagazig, Egypt
A. A. El-Bary
Affiliation:
Arab Academy for Science and Technology, P.O. Box 1029, Alexandria, Egypt
S. M. Ezzat
Affiliation:
Department of Mathematics, Faculty of Education, Alexandria University, Alexandria, Egypt
*
*Corresponding author. Email: [email protected] (M. A. Ezzat)
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Abstract

A new mathematical model of magnetohydrodynamic (MHD) theory has been constructed in the context of a new consideration of heat conduction with a time-fractional derivative of order 0 < α ≤ 1 and a time-fractional integral of order 0 < γ ≤ 2. This model is applied to one-dimensional problems for a thermoelectric viscoelastic fluid flow in the absence or presence of heat sources. Laplace transforms and state-space techniques will be used to obtain the general solution for any set of boundary conditions. According to the numerical results and its graphs, conclusion about the new theory has been constructed. Some comparisons have been shown in figures to estimate the effects of the fractional order parameters on all the studied fields.

Keywords

65M1078A48Viscoelastic fluidmodified Fourier’s and Ohm’s lawsmodified Riemann-Liouville fractional integralnew fractional Taylor’s seriesstate space approachfractional calculus
Type
Research Article
Information
Advances in Applied Mathematics and Mechanics , Volume 7 , Issue 4 , August 2015 , pp. 528 - 548
Copyright
Copyright © Global-Science Press 2015 

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References

[1]Ezzat, M. A., State space approach to solids and fluids, Can. J. Phys. Rev., 86 (2008), pp. 1241.Google Scholar
[2]Caputo, M., Linear models of dissipation whose Q is almost independent, II. Geophy, J. Roy. Astronom., 13 (1967), pp. 529.CrossRefGoogle Scholar
[3]Kiryakova, V., Generalized fractional calculus and applications, in: Generalized Fractional Calculus and Applications, in: Pitman Res. Notes Math. Ser., Vol. 301, Longman-Wiley, New York, 1994.Google Scholar
[4]Mainrdi, F., Fractional calculus: some basic problems in continuum and statistical mechanics, in: Carpinteri, A. and Mainardi, F. (eds.), Fractals and Fractional Calculus in Continuum Mechanics, pp. 291–348, Springer, New York, 1997.Google Scholar
[5]Qi, H. and Jin, H., Nonlinear Analysis, Real World Applications, 10 (2009), pp. 2700.Google Scholar
[6]Miller, K. S. and Ross, B., An Introduction to the Fractional Integrals and Derivatives-Theory and Applications, John Wiley & Sons Inc, New York, 1993.Google Scholar
[7]Samko, S. G., Kilbas, A. A. and Marichev, O. I., Gordon and Breach, Longhorne, PA, 1993.Google Scholar
[8]Oldham, K. B. and Spanier, J., The Fractional Calculus, Academic Press, New York, 1974.Google Scholar
[9]Gorenflo, R. and Mainardi, F., Fractals and Fractional Calculus in Continuum Mechanics, Springer, Wien, 1997.Google Scholar
[10]El-Shahed, M., In the impulsive motion of flat plate in a generalized second grade fluid, Z. Natur-forsch., 59 (2003), pp. 829.Google Scholar
[11]Cattaneo, C., Sullacodizion del calore, Atti. Sem. Mat. Fis. Univ. Modena, 3 (1948).Google Scholar
[12]Puri, P. and Kythe, P. K., Non-classical thermal effects in Stoke’s second problem, Acta Mech., 112 (1995), pp. 1.Google Scholar
[13]Kimmich, R., Strange kinetics, porous media, and NMR, J. Chem. Phys., 284 (2002), pp. 243.Google Scholar
[14]Podlubny, I., Fractional Differential Equations, Academic Press, New York, 1999.Google Scholar
[15]Mainardi, F. and Gorenflo, R., J. Comput. Appl. Math., 118 (2000), pp. 283.Google Scholar
[16]Fujita, Y., Integrodifferential equation which interpolates the heat equation and wave equation (II), Osaka J. Math., 27 (1990), pp. 797.Google Scholar
[17]Povstenko, Y. Z., J. Therm. Stress., 28 (2005), pp. 83.Google Scholar
[18]Sherief, H., El-Said, A. and El-Latief, A. Abd, Int. J. Solid Struct., 47 (2010), pp. 269.CrossRefGoogle Scholar
[19]Lord, H. and Shulman, Y., A generalized dynamical theory of thermoelasticity, J. Mech. Phys. Solids, 15 (1967), pp. 299.CrossRefGoogle Scholar
[20]Lebon, G., Jou, D. and Casas-Vzquez, J., Understanding Non-Equilibrium Thermodynamics: Foundations, Applications, Frontiers, Springer-Verlag, Berlin, Heidelberg, 2008.Google Scholar
[21]Jou, D., Casas-Vzquez, J. and Lebon, G., Rep. Prog. Phys., 51 (1988), pp. 1105.Google Scholar
[22]Youssef, H., Theory of fractional order generalized thermoelasticity, J. Heat Transfer, 132 (2010), pp. 1.CrossRefGoogle Scholar
[23]Ezzat, M. A., Thermoelectric MHD non-Newtonian fluid with fractional derivative heat transfer, Phys. B, 405 (2010), pp. 4188.Google Scholar
[24]Ezzat, M. A., Magneto-thermoelasticity with thermoelectric properties and fractional derivative heat transfer, Phys. B, 406 (2011), pp. 30.Google Scholar
[25]Ezzat, M. A., Heat transfer of fractional order in thermoelectric MHD, Heat Mass Transfer, 48 (2012), pp. 71.Google Scholar
[26]Jumarie, G., Derivation and solutions of some fractional Black-Scholes equations in coarse-grained space and time, Application to Merton’s optimal portfolio, Comput. Math. Appl., 59 (2010), pp. 1142.CrossRefGoogle Scholar
[27]El-Karamany, A. S. and Ezzat, M. A., Convolutional variational principle, reciprocal and uniqueness theorems in linear fractional two-temperature thermoelasticity, J. Therm. Stress., 34 (2011), pp. 264.Google Scholar
[28]El-Karamany, A. S. and Ezzat, M. A., On the fractional thermoelasticity, Math. Mech. Solid, 16 (2011), pp. 334.Google Scholar
[29]Ezzat, M. A. and Fayik, M. A., Fractional order theory of thermoelastic diffusion, J. Therm. Stress., 34 (2011), pp. 851.Google Scholar
[30]Ezzat, M. A., El Karamany, A. S. and Fayik, M. A., Fractional order theory in thermoelastic solid with three-phase lag heat transfer, Arch. Appl. Mech., 82 (2012), pp. 557.Google Scholar
[31]Ezzat, M. A. and El Karamany, A. S., Fractional order theory of a perfect conducting ther-moelastic medium, Can. J. Phys., 89 (2011), pp. 311.Google Scholar
[32]Ezzat, M. A. and El-Karamany, A. S., Theory of fractional order in electro-thermoelasticity, Euro. J. Mech. A/Solids, 30 (2011), pp. 450.Google Scholar
[33]Ezzat, M. A. and El-Karamany, A. S., Fractional order heat conduction law in magneto-thermoelasticity involving two temperatures, Zamp, 62 (2011), pp. 937.Google Scholar
[34]Ezzat, M. A. and El-Karamany, A. S., Fractional thermoelectric viscoelastic materials, J. Appl. Poly. Sci., 124 (2012), pp. 2187.Google Scholar
[35]Sun, H. G., Chen, W., Li, C. and Chen, Y. Q., Fractional differential models for anomalous diffusion, Phys. A, 389 (2010), pp. 2719.Google Scholar
[36]Gracia, J. L. and Stynes, M., Central difference approximation of convection in Caputo fractional derivative two-point boundary value problems, J. Comput. Appl. Math., 273 (2015), pp. 103.Google Scholar
[37]Wang, G., Liu, S. and Zhang, L., Neutral fractional integro-differential equation with nonlinear term depending on lower order derivative, J. Comput. Appli. Math., 260 (2014), pp. 167.Google Scholar
[38]Gao, G., Sun, Z. and Zhang, H., A new fractional numerical differentiation formula to approximate the Caputo fractional derivative and its applications, J. Comput. Phys., 259 (2014), pp. 33.CrossRefGoogle Scholar
[39]Ezzat, M. A., Free convection effects on perfectly conducting fluid, int. J. Eng. Sci., 39 (2001), pp. 799.Google Scholar
[40]Honig, G. and Hirdes, U., A method for the numerical inversion of the Laplace Transform, J. Comput. Appl. Math., 10 (1984), pp. 113.Google Scholar
[41]Ezzat, M. A. and Zakaria, M., State space approach to viscoelastic fluid flow of hydromagnetic fluctuating boundary-layer through a porous medium, Zamm, 77 (1997), pp. 197.Google Scholar
[42]Ezzat, M. A., Zakaria, M., Shaker, O. and Barakat, F., State space formulation to viscoelas-tic fluid flow of magnetohydrodynamie free convection through a porous medium, Acta Mech., 199 (1996), pp. 147.Google Scholar
[43]Ezzat, M. A. and Zakaria, M., Free convection effects on a viscoelastic boundary layer flow with one relaxation time through a porous medium, J. Franklin Inst., 334 (1997), pp. 685.Google Scholar