Abstract
In this study, the effects of magnetic field and nanoparticle on the Jeffery-Hamel flow are studied using a powerful analytical method called the Adomian decomposition method (ADM). The traditional Navier-Stokes equation of fluid mechanics and Maxwell’s electromagnetism governing equations are reduced to nonlinear ordinary differential equations to model the problem. The obtained results are well agreed with that of the Runge-Kutta method. The present plots confirm that the method has high accuracy for different α, Ha, and Re 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.
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Abbreviations
- A*:
-
constant parameter
- B 0 :
-
magnetic field (wb·m−2)
- F :
-
general nonlinear operator
- Lu :
-
linear term
- Ru :
-
remained of linear operator
- A :
-
Adomian polynominal
- f(η):
-
dimensionless velocity
- Ha :
-
Hartmann number
- P :
-
pressure term
- Re :
-
Reynolds number
- r,θ :
-
cylindrical coordinates
- U max :
-
maximum value of velocity
- u,v :
-
velocity components along x and y axes, respectively
- α :
-
angle of channel
- η :
-
dimensionless angle
- θ :
-
any angle
- ρ :
-
density
- ϕ :
-
nanoparticle volume fraction
- µ :
-
dynamic viscosity
- υ :
-
kinematic viscosity
- β :
-
constant
- ∞ :
-
condition at infinity
- nf:
-
nanofluid
- f:
-
base fluid
- s:
-
nano-solid-particles
References
Jeffery, G. B. The two-dimensional steady motion of a viscous fluid. Phil. Mag., 6, 455–465 (1915)
Hamel, G. Spiralförmige Bewgungen zäher Flüssigkeiten, Jahresber. Deutsch. Math. Verein., 25, 34–60 (1916)
Bansal, L. Magnetofluiddynamics of Viscous Fluids, Jaipur Publishing House, Jaipur, India (1994)
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), 329–339 (2002)
Tendler, M. Confinement and related transport in extrap geometry. Nuclear Instruments and Methods in Physics Research, 207(1–2), 233–240 (1983)
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), 175–183 (2001)
Makinde, O. D. Magneto-hydrodynamic stability of plane-Poiseuille flow using multi-deck asymptotic technique. Math. Comput. Model., 37(3–4), 251–259 (2003)
Anwari, M., Harada, N., and Takahashi, S. Performance of a magnetohydrodynamic accelerator using air-plasma as working gas. Energy Conversion Management, 4, 2605–2613 (2005)
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, 466–471 (2005)
Kakaç, S. and Pramuanjaroenkij, A. Review of convective heat transfer enhancement with nanofluids. International Communications in Heat and Mass Transfer, 52(13–14), 3187–3196 (2009)
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, 630–640 (2009)
Yacob, N., Ishak, A., Nazar, R., and Pop, I. Falkner-Skan problem for a static and moving wedge with prescribed surface heat flux in a nanofluid. International Communications in Heat and Mass Transfer, 38, 149–153 (2011)
Adomian, G. A review of the decomposition method in applied mathematics. Journal of Mathematical Analysis and Applications, 135(2), 501–544 (1988)
Ghosh, S., Roy, A., and Roy, D. An adaptation of Adomian decomposition for numeric-analytic integration of strongly nonlinear and chaotic oscillators. Comput. Meth. Appl. Mech. Engrg., 196, 1133–1153 (2007)
Jafari, H. and Daftardar-Gejji, V. Revised Adomian decomposition method for solving a system of non-linear equations. Appl. Math. Comput., 175, 1–7 (2006)
Allan, F. M. and Syam, M. I. On the analytic solutions of the nonhomogeneous Blasius problem. J. Comput. Appl. Math., 182, 362–371 (2005)
Hashim, I. Adomian decomposition method for solving BVPs for fourth-order integro-differential equations. J. Comput. Appl. Math., 193, 658–664 (2006)
Hashim, I. Comments on a new algorithm for solving classical Blasius equation. J. Comput. Appl. Math., 182, 362–371 (2005)
Kechil, S. A. and Hashim, I. Non-perturbative solution of free-convective boundary-layer equation by Adomian decomposition method. Phys. Lett. A., 363, 110–114 (2007)
Arslanturk, C. A decomposition method for fins efficiency of convective straight fins with temperature-dependent thermal conductivity. International Communications in Heat and Mass Transfer, 32, 831–841 (2005)
Pamuk, S. Solution of the porous media equation by Adomian’s decomposition method. Phys. Lett. A, 344, 184–188 (2005)
Daftardar-Gejji, V. and Jafari, H. An iterative method for solving nonlinear functional equations. J. Math. Anal. Appl., 316, 753–763 (2006)
Lesnic, D. Decomposition methods for non-linear non-characteristic Cauchy heat problems. Commun. Nonlinear Sci. Numer. Simul., 10, 581–596 (2005)
Luo, X. G. A two-step Adomian decomposition method. Appl. Math. Comput., 170, 570–583 (2005)
Zhang, X. A modification of the Adomian decomposition method for a class of nonlinear singular boundary value problems. J. Comput. Appl. Math., 180, 377–389 (2005)
Kaya, D. and Yokus, A. A comparison of partial solutions in the decomposition method for linear and nonlinear partial differential equations. Math. Comput. Simul., 60, 507–512 (2002)
Ganji, Z. Z., Ganji, D. D., and Rostamiyan, Y. Solitary wave solutions for a time-fraction generalized Hirota-Satsuma coupled KdV equation by an analytical technique. Applied Mathematical Modelling, 33(7), 3107–3113 (2009)
Esmaeilpour, M. and Ganji, D. D. Solution of the Jeffery-Hamel flow problem by optimal homotopy asymptotic method. Computers and Mathematics with Applications, 59, 3405–3411 (2010)
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), 2213–2216 (2011)
Babazadeh, H., Ganji, D. D., and Akbarzade, M. He’s energy balance method to evaluate the effect of amplitude on the natural frequency in nonlinear vibration systems. Journal of Electromagnetic Waves and Applications (JEMWA) Progress in Electromagnetic Research, 4, 143–154 (2008)
Ganji, D. D., Babazadeh, H., Jalaei, M. H., and Tashakkorian, H. Application of He’s variational iteration methods for solving nonlinear BBMB equations and free vibrations of systems. Acta Appl. Math., 106(3), 359–367 (2009)
Si, X. H., Zheng, L. C., Zhang, X. X., and Chao, Y. Perturbation solution to unsteady flow in a porous channel with expanding or contracting walls in the presence of a transverse magnetic field. Appl. Math. Mech.-Engl. Ed., 31(2), 151–158 (2010) DOI 10.1007/s10483-010-0203-z
Ganji, D. D., Rokni, H. B., Sfahani, M. G., and Ganji, S. S. Approximate traveling wave solutions for coupled shallow water. Advances in Engineering Software, 41, 956–961 (2010)
Tari, H., Ganji, D. D., and Babazadeh, H. The application of He’s variational iteration method to nonlinear equations arising in heat transfer. Phys. Lett. A, 363, 213–217 (2007)
Ganji, S. S., Ganji, D. D., Babazadeh, H., and Sadoughi, N. Application of amplitude-frequency formulation to nonlinear oscillation system of the motion of a rigid rod rocking back. Mathematical Method in Applied Sciences, 10, 151–159 (2009)
Yuan, P. X. and Li, Y. Q. Primary resonance of multiple degree-of-freedom dynamic systems with strong non-linearity using the homotopy analysis method. Appl. Math. Mech.-Engl. Ed., 31(10), 1293–1304 (2010) DOI 10.1007/s10483-010-1362-6
Ganji, S. S., Ganji, D. D., Karimpour, S., and Babazadeh, H. Applications of He’s homotopy perturbation method to obtain second-order approximations of coupled two-degree-of-freedom system. International Journal of Nonlinear Science and Numerical Simulation, 10(3), 303–312 (2009)
Ganji, D. D., Rokni, H. B., Rafiee, M. H., Imani, A. A., Esfandyaripour, M., and Sheikholeslami, M. Reconstruction of variational iteration method for boundary value problems in structural engineering and fluid mechanics. International Journal of Nonlinear Dynamics in Engineering and Sciences, 3, 1–10 (2011)
Sheikholeslami, M., Ashorynejad, H. R., Ganji, D. D., and Kolahdooz, A. Investigation of rotating MHD viscous flow and heat transfer between stretching and porous surfaces using analytical method. Mathematical Problems in Engineering (2011) DOI 10.1155/2011/258734
Ganji., D. D., Nezhad, H. R. A., and Hasanpour, A. Effect of variable viscosity and viscous dissipation on the Hagen-Poiseuille flow and entropy generation. Numerical Methods for Partial Differential Equations, 27(3), 529–540 (2011)
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Sheikholeslami, M., Ganji, D.D., Ashorynejad, H.R. et al. Analytical investigation of Jeffery-Hamel flow with high magnetic field and nanoparticle by Adomian decomposition method. Appl. Math. Mech.-Engl. Ed. 33, 25–36 (2012). https://doi.org/10.1007/s10483-012-1531-7
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DOI: https://doi.org/10.1007/s10483-012-1531-7
Key words
- magnetohydrodynamic
- Jeffery-Hamel flow
- Adomian decomposition method
- nonlinear ordinary differential equation
- nanofluid