Abstract
In this study, the effects of magnetic field and nanoparticle on the Jeffery-Hamel flow are studied using two powerful analytical methods, Homotopy Perturbation Method (HPM) and a simple and innovative approach which we have named it Akbari-Ganji’s Method(AGM). Comparisons have been made between HPM, AGM and Numerical Method and the acquired results show that these methods have high accuracy for different values of α, Hartmann numbers, and Reynolds numbers. The flow field inside the divergent channel is studied for various values of Hartmann number and angle of channel. The effect of nanoparticle volume fraction in the absence of magnetic field is investigated.
It is necessary to represent some of the advantages of choosing the new method, AGM, for solving nonlinear differential equations as follows: AGM is a very suitable computational process and is applicable for solving various nonlinear differential equations. Moreover, in AGM by solving a set of algebraic equations, complicated nonlinear equations can easily be solved and without any mathematical operations such as integration, the solution of the problem can be obtained very simply and easily. It is notable that this solution procedure, AGM, can help students with intermediate mathematical knowledge to solve a broad range of complicated nonlinear differential equations.
[1] Jeffery G. B., The two-dimensional steady motion of a viscous fluid, Phil. Mag., 6, 1915, 455–465 http://dx.doi.org/10.1080/1478644040863532710.1080/14786440408635327Search in Google Scholar
[2] Hamel G., Spiralformige Bewgungen zaher Flussigkeiten, Jahresber, Deutsch. Math. Verein, 25, 1916, 34–60 Search in Google Scholar
[3] Bansal L., Magnetofluiddynamics of Viscous Fluids, Jaipur Publishing House, Jaipur, India, 1994 Search in Google Scholar
[4] Cha J. E., Ahn Y. C., and Kim M. H., Flow measurement with an electromagnetic flowmeter in two-phase bubbly and slug flow regimes, Flow Measurement and Instrumentation, 12(5–6), 2002, 329–339 http://dx.doi.org/10.1016/S0955-5986(02)00007-910.1016/S0955-5986(02)00007-9Search in Google Scholar
[5] Tendler M., Confinement and related transport in extrap geometry, Nuclear Instruments and Methods in Physics Research, 207(1–2), 1983, 233–240 http://dx.doi.org/10.1016/0167-5087(83)90240-510.1016/0167-5087(83)90240-5Search in Google Scholar
[6] Makinde O. D. and Motsa S. S., Hydromagnetic stability of plane Poiseuille flow using Chebyshev spectral collocation method, J. Ins. Math. Comput. Sci., 12(2), 2001, 175–183 Search in Google Scholar
[7] Makinde O. D., Magneto-hydrodynamic stability of plane-Poiseuille flow using multi-deck asymptotic technique, Math. Comput. Model., 37(3–4), 2003, 251–259 http://dx.doi.org/10.1016/S0895-7177(03)00004-910.1016/S0895-7177(03)00004-9Search in Google Scholar
[8] Anwari M., Harada N., and Takahashi S., Performance of a magnetohydrodynamic accelerator using air-plasma as working gas, Energy Conversion Management, 4, 2005, 2605–2613 http://dx.doi.org/10.1016/j.enconman.2004.12.00310.1016/j.enconman.2004.12.003Search in Google Scholar
[9] Homsy A., Koster S., Eijkel J. C. T., Ven der Berg A., Lucklum F., Verpoorte E., and de Rooij N. F., A high current density DC magnetohydrodynamic (MHD) micropump, Royal Society of Chemistry’s Lab on a Chip, 5, 2005, 466–471 http://dx.doi.org/10.1039/b417892k10.1039/b417892kSearch in Google Scholar
[10] Kakac S. and Pramuanjaroenkij A., Review of convective heat transfer enhancement with nanofloids, International Communications in Heat and Mass Transfer, 52(13–14), 2009, 3187–3196 http://dx.doi.org/10.1016/j.ijheatmasstransfer.2009.02.00610.1016/j.ijheatmasstransfer.2009.02.006Search in Google Scholar
[11] He J. H., Homotopy perturbation technique, Comput. Methods Appl. Mech. Eng., 178, 1999, 257–262 http://dx.doi.org/10.1016/S0045-7825(99)00018-310.1016/S0045-7825(99)00018-3Search in Google Scholar
[12] He J. H., Homotopy perturbation method: a new nonlinear analytical technique, Appl. Math. Comput., 135, 2003, 73–79 http://dx.doi.org/10.1016/S0096-3003(01)00312-510.1016/S0096-3003(01)00312-5Search in Google Scholar
[13] He J. H., Application of homotopy perturbation method to nonlinear wave equations, Chaos Solitons Fractals 26, 2005, 695–700 http://dx.doi.org/10.1016/j.chaos.2005.03.00610.1016/j.chaos.2005.03.006Search in Google Scholar
[14] He J. H., Homotopy perturbation method for bifurcation of nonlinear problems, Int. J. Nonlinear Sci. Numer. Simul., 6, 2005, 207–208 10.1515/IJNSNS.2005.6.2.207Search in Google Scholar
[15] Roozi A., Alibeiki E., Hosseini S. S., Shafiof S. M., Ebrahimi M., Homotopy perturbation method for special nonlinear partial differential equations, J. King Saud Univ. (Sci.), 23, 2011, 99–103 http://dx.doi.org/10.1016/j.jksus.2010.06.01410.1016/j.jksus.2010.06.014Search in Google Scholar
[16] Ganji D. D., Sadighi A., Application of homotopyperturbation and variational iteration methods to nonlinear heat transfer and porous media equations, J. Comput. Appl. Math., 207, 2007, 24–34 http://dx.doi.org/10.1016/j.cam.2006.07.03010.1016/j.cam.2006.07.030Search in Google Scholar
[17] Sadighi A., Ganji D. D., Analytic treatment of linear and nonlinear Schrodinger equations: a study with homotopy-perturbation and Adomian decomposition methods equations, Phys. Lett. A, 372, 2008, 465–469 http://dx.doi.org/10.1016/j.physleta.2007.07.06510.1016/j.physleta.2007.07.065Search in Google Scholar
[18] Sadighi A., Ganji D. D., Exact solutions of Laplace equation by homotopy-perturbation and Adomian decomposition methods, Phys. Lett. A, 367, 2007, 83–87 http://dx.doi.org/10.1016/j.physleta.2007.02.08210.1016/j.physleta.2007.02.082Search in Google Scholar
[19] Ziabakhsh Z., Domairry G., Analytic solution of natural convection flow of a non-Newtonian fluid between two vertical flat plates using homotopy analysis method, Commun. Nonlinear Sci. Numer. Simul., 14, 2009, 1868–1880 http://dx.doi.org/10.1016/j.cnsns.2008.09.02210.1016/j.cnsns.2008.09.022Search in Google Scholar
[20] Ganji S. S., Ganji D. D., Sfahani M. G., Karimpour S., Application of AFF and HPM to the systems of strongly nonlinear oscillation, Curr. Appl. Phys., 10, 2010, 1317–1325 http://dx.doi.org/10.1016/j.cap.2010.03.01510.1016/j.cap.2010.03.015Search in Google Scholar
[21] Khavaji A., Ganji D.D., Roshan N., Moheimani R., Hatami M., A. Hasanpour, Slope variation effect on large deflection of compliant beam using analytical approach, Struct. Eng. Mech., 44(3), 2012, 405–416 http://dx.doi.org/10.12989/sem.2012.44.3.40510.12989/sem.2012.44.3.405Search in Google Scholar
[22] Moghimi S. M., Ganji D. D., Bararnia H., Hosseini M., and Jalaal M., Homotopy perturbation method for nonlinear MHD Jeffery-Hamel problem, Computers and Mathematics with Applications, 61(8), 2011, 2213–2216 http://dx.doi.org/10.1016/j.camwa.2010.09.01810.1016/j.camwa.2010.09.018Search in Google Scholar
[23] Aminossadati S. M. and Ghasemi B. Natural convection cooling of a localized heat source at the bottom of a nanofluid-filled enclosure, Eur. J. Mech. B/Fluids, 28, 2009, 630–640 http://dx.doi.org/10.1016/j.euromechflu.2009.05.00610.1016/j.euromechflu.2009.05.006Search in Google Scholar
© 2014 Versita Warsaw
This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.
