Abstract
Ferrofluid flow and heat transfer in the presence of an external variable magnetic field is studied. The inner cylinder is maintained at uniform heat flux and the outer cylinder has constant temperature. The Control Volume based Finite Element Method (CVFEM) is applied to solve the governing equations. Combined magnetohydrodynamic and ferrohydrodynamic effects have been taken into account. The effects of magnetic number, Hartmann number, Rayleigh number and nanoparticle volume fraction on hydrothermal behavior have been examined. Results show that the Nusselt number is an increasing function of Magnetic number, Rayleigh number and nanoparticle volume fraction while it is a decreasing function of the Hartmann number. Also, it can be concluded that the enhancement in heat transfer decreases with an increase in the Rayleigh number and magnetic number but it increases with an increase in the Hartmann number.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
Abbreviations
- B :
-
Magnetic induction (= η 0H)
- C p :
-
Specific heat at constant pressure
- Ec :
-
Eckert number (= (μ f α f )/[(ρC P ) f ΔT L 2])
- En :
-
Heat transfer enhancement
- Gr f :
-
Grashof number
- H x , H y :
-
Components of the magnetic field intensity
- H :
-
The magnetic field strength
- Ha :
-
Hartmann number \(\left( { = \mu _0 H_0 L\sqrt {\sigma _f /\mu _f } } \right)\)
- Mn F :
-
Magnetic number arising from FHD (=μ 0 H 20 ΔT L 2/(μ f α f ))
- M :
-
Magnetization \(\left( { = K'\bar H\left( {T'_c - T} \right)} \right)\)
- Nu loc :
-
Local Nusselt number
- Nu ave :
-
Average Nusselt number
- Pr :
-
Prandtl number (= ν f /α f )
- T :
-
Fluid temperature
- T′c :
-
Curie temperature
- u, v :
-
Velocity components in the x-direction and y-direction
- U, V :
-
Dimensionless velocity components in the x-direction and y-direction
- x, y :
-
Space coordinates nf Nanofluid
- X, Y :
-
Dimensionless space coordinates
- ζ :
-
Angle measured from the lower right plane
- α :
-
Thermal diffusivity
- ϕ :
-
Volume fraction
- γ :
-
Magnetic field strength at the source
- ε 1 :
-
Temperature number (= T 1/ΔT)
- σ :
-
Electrical conductivity
- μ :
-
Dynamic viscosity
- μ 0 :
-
Magnetic permeability of vacuum (=4 π × 10−7 Tm/A)
- ν :
-
Kinematic viscosity
- ψ and Ψ :
-
stream function and dimensionless stream function
- Θ :
-
dimensionless temperature
- ρ :
-
Fluid density
- β :
-
Thermal expansion coefficient
- c :
-
Cold
- h :
-
Hot
- nf :
-
Nanofluid
- f :
-
Base fluid
- r :
-
Non-dimensional radial distance
- k :
-
Thermal conductivity
- L :
-
Gap between inner and outer boundary of the enclosure L = r out − r in
- \(\overrightarrow g\) :
-
Gravitational acceleration vector
- q″:
-
Heat flux
- Ra :
-
Rayleigh number (= gβfΔT L3/αfνf)
- s :
-
Solid particles
- in :
-
Inner
- out :
-
Outer
References
B.R. Baliga, S.V. Patankar, Numer. Heat Transfer 6, 245 (1983).
M. Sheikholeslami, M. Gorji-Bandpy, D.D. Ganji, Soheil Soleimani, Adv. Powder Technol. 24, 980 (2013).
Mohsen Sheikholeslami, Davood Domiri Ganji, Mohammad Mehdi Rashidi, J. Taiwan Inst. Chem. Eng. (2014) http://dx.doi.org/10.1016/j.jtice.2014.09.026.
M. Sheikholeslami, M. Gorji-Bandpy, D.D. Ganji, Soheil Soleimani, Trans. Mech. Eng. 38, 217 (2014).
M. Sheikholeslami, M. Gorji-Bandpy, D.D. Ganji, P. Rana, Soheil Soleimani, Comput. Fluids 94, 147 (2014).
M. Sheikholeslami, M. Gorji-Bandpy, D.D. Ganji, Soheil Soleimani, J. Mol. Liq. 194, 179 (2014).
M. Sheikholeslami, M. Gorji-Bandpy, D.D. Ganji, Soheil Soleimani, J. Mol. Liq. 193, 174 (2014).
M. Sheikholeslami, M. Gorji-Bandpy, Soheil Soleimani, Int. Commun. Heat Mass Transfer 47, 73 (2013).
M. Sheikholeslami, M. Gorji Bandpy, R. Ellahi, Mohsan Hassan, Soheil Soleimani, J. Magn. Magn. Mater. 349, 188 (2014).
M. Sheikholeslami, M. Gorji-Bandpy, D.D. Ganji, Soheil Soleimani, J. Taiwan Inst. Chem. Eng. 45, 40 (2014).
M. Sheikholeslami, I. Hashim, Soheil Soleimani, Math. Prob. Eng. 2013, 831725 (2013).
M. Sheikholeslami, M. Gorji-Bandpy, D.D. Ganji, Soheil Soleimani, Adv. Powder Technol. 24, 980 (2013).
M. Sheikholeslami, M. Gorji-Bandpy, D.D. Ganji, Soheil Soleimani, Neural Comput. Appl. 24, 873 (2014).
M. Sheikholeslami, M. Gorji-Bandpy, D.D. Ganji, Soheil Soleimani, S.M. Seyyedi, Int. Commun. Heat Mass Transfer 39, 1435 (2012).
Soheil Soleimani, M. Sheikholeslami, D.D. Ganji, M. Gorji-Bandpay, Int. Commun. Heat Mass Transfer 39, 565 (2012).
Mohsen Sheikholeslami, Davood Domiri Ganji, Energy 75, 400 (2014).
Xidong Zhang, Hulin Huang, Int. Commun. Heat Mass Transfer 51, 31 (2014).
Tswen-Chyuan Jue, Int. Commun. Heat Mass Transfer 33, 846 (2006).
R. Azizian, E. Doroodchi, T. McKrell, J. Buongiorno, L.W. Hu, B. Moghtaderi, Int. J. Heat Mass Transfer 68, 94 (2014).
Mohsen Sheikholeslami, Mofid Gorji-Bandpy, Powder Technol. 256, 490 (2014).
Fatih Selimefendigil, Hakan F. Oztop, Int. J. Heat Mass Transfer 71, 142 (2014).
C.E. Nanjundappa, I.S. Shivakumara, M. Ravisha, Int. Commun. Heat Mass Transfer 37, 1246 (2010).
Mohsen Sheikholeslami Kandelousi, Phys. Lett. A 378, 3331 (2014).
Mohsen Sheikholeslami, Davood Domiri Ganji, Physica A 417, 273 (2014).
Mohsen Sheikholeslami, Mofid Gorji-Bandpy, Kuppalapalle Vajravelu, Int. J. Heat Mass Transfer 80, 16 (2015).
M. Sheikholeslami, J Braz. Soc. Mech. Sci. Eng. (2014) DOI:10.1007/s40430-014-0242-z.
M. Sheikholeslami, M. Gorji-Bandpy, S.M. Seyyedi, D.D. Ganji, Houman B. Rokni, Soheil Soleimani, Powder Technol. 247, 87 (2013).
W.A. Khan, O.D. Makinde, Int. J. Therm. Sci. 81, 118 (2014).
W.A. Khan, O.D. Makinde, Z.H. Khan, Int. J. Heat Mass Transfer 74, 285 (2014).
Winifred Nduku Mutuku, Oluwole Daniel Makinde, Comput. Fluids 95, 88 (2014).
O.D. Makinde, A. Aziz, Int. J. Therm. Sci. 50, 1326 (2011).
M. Sheikholeslami, M. Gorji-Bandpy, D.D. Ganji, Sci. Iran., Trans. B: Mech. Eng. 20, 1241 (2013).
M. Sheikholeslami, M. Gorji-Bandpy, G. Domairry, Appl. Math. Mech. 34, 1 (2013).
M. Sheikholeslami, D.D. Ganji, Powder Technol. 235, 873 (2013).
M. Sheikholeslami, M. Gorji-Bandpay, D.D. Ganji, Int. Commun. Heat Mass Transfer 39, 978 (2012).
Mohsen Sheikholeslami, Davood Domiri Ganji, M. Younus Javed, R. Ellahi, J. Magn. Magn. Mater. 374, 36 (2015).
Mohsen Sheikholeslami, Shirley Abelman, Davood Domiri Ganji, Int. J. Heat Mass Transfer 79, 212 (2014).
Mohsen Sheikholeslami, Davood Domiri Ganji, J Braz. Soc. Mech. Sci. Eng. (2014) DOI:10.1007/s40430-014-0228-x.
Wubshet Ibrahim, O.D. Makinde, Comput. Fluids 86, 433 (2013).
S.A. Shehzad, T. Hayat, M. Qasim, S. Asghar, Braz. J. Chem. Eng. 30, 187 (2013).
S.A. Shehzad, M. Qasim, A. Alsaedi, T. Hayat, M.S. Alhuthali, Eur. Phys. J. Plus 128, 7 (2013).
T. Hayat, S.A. Shehzad, S. Asghar, Walailak J. Sci. Technol. 10, 29 (2013).
M. Sheikholeslami, M. Gorji-Bandpay, D.D. Ganji, Arabian J. Sci. Eng. 39, 5007 (2014).
M. Sheikholeslami, D.D. Ganji, J. Appl. Fluid Mech. 7, 535 (2014).
Mohsen Sheikholeslami, Mofid Gorji Bandpy, R. Ellahi, A. Zeeshan, J. Magn. Magn. Mat. 369, 69 (2014).
M. Sheikholeslami, M. Gorji-Bandpy, D.D. Ganji, J. Taiwan Inst. Chem. Eng. 45, 1204 (2014).
M. Sheikholeslami, D.D. Ganji, Sci. Iran. B 21, 203 (2014).
T. Hayat, S.A. Shehzad, A. Alsaedi, M.S. Alhuthali, Appl. Math. Mech. 34, 489 (2013).
T. Hayat, M. Waqas, S.A. Shehzad, A. Alsaedi, J. Mech. 29, 403 (2013).
S.A. Shehzad, F.E. Alsaadi, S.J. Monaquel, T. Hayat, Eur. Phys. J. Plus 128, 56 (2013).
T. Hayat, S.A. Shehzad, M. Qasim, S. Asghar, A. Alsaedi, J. Thermophys. Heat Transfer 28, 155 (2014).
T. Hayat, S.A. Shehzad, M. Qasim, S. Asghar, Int. J. Numer. Methods Heat Fluid Flow 24, 342 (2014).
M. Sheikholeslami, D.D. Ganji, J. Mol. Liq. 194, 13 (2014).
M. Sheikholeslami, M. Gorji-Bandpy, D.D. Ganji, Powder Technol. 254, 82 (2014).
M. Sheikholeslami, D.D. Ganji, Powder Technol. 253, 789 (2014).
T. Hayat, S.A. Shehzad, M. Qasim, A. Alsaedi, Braz. J. Chem. Eng. 31, 109 (2014).
S.A. Shehzad, A. Alsaedi, T. Hayat, M.S. Alhuthali, J. Taiwan Inst. Chem. Eng. 45, 787 (2014).
S.A. Shehzad, T. Hayat, M.S. Alhuthali, S. Asghar, J. Cent. South Univ. 21, 1428 (2014).
T. Hayat, S.A. Shehzad, S. Al-Mezel, A. Alsaedi, J. Hydrol. Hydromech. 62, 117 (2014).
M. Sheikholeslami, M. Gorji-Bandpy, D.D. Ganji, Energy 60, 501 (2013).
M. Sheikholeslami, D.D. Ganji, M. Gorji-Bandpy, Soheil Soleimani, J. Taiwan Inst. Chem. Eng. 45, 795 (2014).
M. Sheikholeslami, D.D. Ganji, Houman B. Rokni, Int. J. Eng. Trans. C 26, 653 (2013).
V. Loukopoulos, E. Tzirtzilakis, Int. J. Eng. Sci. 42, 571 (2004).
H. Aminfar, M. Mohammadpourfard, F. Mohseni, J. Magn. Magn. Mater. 324, 830 (2012).
K. Khanafer, K. Vafai, M. Lightstone, Int. J. Heat Mass Transfer 46, 3639 (2003).
G. De Vahl Davis, Int. J. Numer. Methods Fluids 3, 249 (1962).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Kandelousi, M.S. Effect of spatially variable magnetic field on ferrofluid flow and heat transfer considering constant heat flux boundary condition. Eur. Phys. J. Plus 129, 248 (2014). https://doi.org/10.1140/epjp/i2014-14248-2
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1140/epjp/i2014-14248-2