Skip to main content Accessibility help
×
Home
Hostname: page-component-7bb4899584-rm9q9 Total loading time: 0.331 Render date: 2023-01-27T00:27:41.467Z Has data issue: true Feature Flags: { "useRatesEcommerce": false } hasContentIssue true

Thermoelectric Viscoelastic Fluid with Fractional Integral and Derivative Heat Transfer

Published online by Cambridge University Press:  29 May 2015

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: maezzat2000@yahoo.com (M. A. Ezzat)
Get access

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.

Type
Research Article
Copyright
Copyright © Global-Science Press 2015 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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. 291348, Springer, New York, 1997.CrossRefGoogle 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.CrossRefGoogle 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.CrossRefGoogle 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.CrossRefGoogle 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.CrossRefGoogle Scholar
[21]Jou, D., Casas-Vzquez, J. and Lebon, G., Rep. Prog. Phys., 51 (1988), pp. 1105.CrossRefGoogle 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.CrossRefGoogle Scholar
[24]Ezzat, M. A., Magneto-thermoelasticity with thermoelectric properties and fractional derivative heat transfer, Phys. B, 406 (2011), pp. 30.CrossRefGoogle 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.CrossRefGoogle Scholar
[28]El-Karamany, A. S. and Ezzat, M. A., On the fractional thermoelasticity, Math. Mech. Solid, 16 (2011), pp. 334.CrossRefGoogle Scholar
[29]Ezzat, M. A. and Fayik, M. A., Fractional order theory of thermoelastic diffusion, J. Therm. Stress., 34 (2011), pp. 851.CrossRefGoogle 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.CrossRefGoogle 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.CrossRefGoogle 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.CrossRefGoogle 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.CrossRefGoogle 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.CrossRefGoogle 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.CrossRefGoogle 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.CrossRefGoogle 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.CrossRefGoogle 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.CrossRefGoogle 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.CrossRefGoogle 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.CrossRefGoogle 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.CrossRefGoogle Scholar
4
Cited by

Save article to Kindle

To save this article to your Kindle, first ensure coreplatform@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Thermoelectric Viscoelastic Fluid with Fractional Integral and Derivative Heat Transfer
Available formats
×

Save article to Dropbox

To save this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you used this feature, you will be asked to authorise Cambridge Core to connect with your Dropbox account. Find out more about saving content to Dropbox.

Thermoelectric Viscoelastic Fluid with Fractional Integral and Derivative Heat Transfer
Available formats
×

Save article to Google Drive

To save this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you used this feature, you will be asked to authorise Cambridge Core to connect with your Google Drive account. Find out more about saving content to Google Drive.

Thermoelectric Viscoelastic Fluid with Fractional Integral and Derivative Heat Transfer
Available formats
×
×

Reply to: Submit a response

Please enter your response.

Your details

Please enter a valid email address.

Conflicting interests

Do you have any conflicting interests? *