Abstract.
The two-dimensional differential transform method (DTM) is applied to analyse the transient natural convection of cold water in a vertical channel. The cold water gives rise to a density variation with temperature that may not be linearized. The vertical channel is composed of doubly infinite parallel plates, one of which has a constant prescribed temperature and the other of which is insulated. Considering the temperature-dependent viscosity and thermal conductivity of the water, approximate analytical (series) solutions for the temperature and flow velocity are derived. The transformed functions included in the solutions are obtained through a simple recursive procedure. Numerical computation is performed for the entire range of water temperature conditions around the temperature at the density extremum point, i.e. \( 4 {}^{\circ} C\) . Numerical results illustrate the effects of the temperature-dependent properties on the transient temperature and flow velocity profiles, volumetric flow rate, and skin friction. The DTM is a powerful tool for solving nonlinear transient problems as well as steady problems.
Similar content being viewed by others
References
Y. Joshi, B. Gebhart, Int. J. Heat Mass Transfer 27, 1573 (1984)
D.R. Moore, N.O. Weiss, J. Fluid Mech. 61, 553 (1973)
C.F. Kettleborough, Int. J. Heat Mass Transfer 15, 883 (1972)
H.M. Joshi, Int. Commun. Heat Mass Transfer 15, 227 (1988)
M.A. Al-Nimr, M.A.I. El-Shaarawi, Heat Mass Transfer 30, 241 (1995)
A.K. Singh, Defence Sci. J. 38, 35 (1988)
T. Paul, B.K. Jha, A.K. Singh, Heat Mass Transfer 32, 61 (1996)
A.K. Singh, H.R. Gholami, V.M. Soundalgekar, Heat Mass Transfer 31, 329 (1996)
T. Paul, B.K. Jha, A.K. Singh, Int. J. Appl. Mech. Eng. 6, 913 (2001)
B.K. Jha, A.K. Singh, H.S. Takhar, Int. J. Appl. Mech. Eng. 8, 497 (2003)
A.K. Singh, T. Paul, Int. J. Appl. Mech. Eng. 11, 143 (2006)
B.K. Jha, A.O. Ajibade, J. Process Mech. Eng. 224, 247 (2010)
B.K. Jha, Int. J. Appl. Mech. Eng. 6, 279 (2001)
A.O. Ajibade, B.K. Jha, Int. J. Heat Technol. 27, 85 (2009)
B.K. Jha, A.K. Samaila, A.O. Ajibade, Int. Commun. Heat Mass Transfer 38, 633 (2011)
B.K. Jha, A.O. Ajibade, Commun. Nonlinear Sci. Numer. Simulat. 17, 1576 (2012)
C. Mandal, S. Das, R.N. Jana, Int. J. Appl. Inf. Syst. 2, 49 (2012)
C. Bervillier, Appl. Math. Comput. 218, 10158 (2012)
R. Chiba, Int. J. Thermophys. 33, 363 (2012)
R. Chiba, Nonlinear Eng. 3, 215 (2014)
R. Chiba, Abst. Appl. Anal. 2014, 684293 (2014)
R. Chiba, Appl. Mech. Mater. 627, 145 (2014)
B.L. Kuo, Appl. Math. Comput. 165, 63 (2005)
H.A. Peker, G. Oturanc, arXiv:1212.1706 (2012)
C.K. Chen, H.Y. Lai, C.C. Liu, Int. Commun. Heat Mass Transfer 38, 285 (2011)
C.K. Chen, B.S. Chen, C.C. Liu, Int. J. Heat Mass Transfer 79, 750 (2014)
M.M. Rashidi, T. Hayat, M. Keimanesh, A.A. Hendi, Int. J. Numer. Methods Heat Fluid Flow 23, 436 (2013)
M. Hatami, D.D. Ganji, Case Studies Thermal Eng. 2, 14 (2014)
M. Hatami, J. Hatami, M. Jafaryar, G. Domairry, J. Brazil. Soc. Mech. Sci. Eng. 38, 589 (2016)
J.C. Umavathi, M. Shekar, Meccanica 51, 71 (2016)
M.J. Jang, C.L. Chen, Y.C. Liu, Appl. Math. Comput. 121, 261 (2001)
M.J. Jang, Y.L. Yeh, C.L. Chen, W.C. Yeh, Appl. Math. Comput. 216, 2339 (2010)
E. Fukumori, Doctoral Dissertation, State University of New York at Buffalo (1987)
E. Fukumori, A. Wake, Distributed Parameter Systems: Modelling and Simulation, edited by T. Futagami, S.G. Tzafestas, Y. Sunahara (Elsevier, North-Holland, 1989)
R.C. Weast, Handbook of Chemistry and Physics, 59th edition (CRC Press, West Palm Beach, 1978)
A. Kurnaz, G. Oturanc, M.E. Kiris, Int. J. Comput. Math. 82, 369 (2005)
Y. Khan, Z. Svoboda, Z. Smarda, Adv. Diff. Eq. 2012, 174 (2012)
F. Mirzaee, Appl. Math. Sci. 5, 3465 (2011)
N. Patil, A. Khambayat, Res. J. Math. Stat. Sci. 2, 4 (2014)
B. Kundu, K.S. Lee, Energy Conv. Manag. 110, 469 (2014)
I.Ç. Süngü, H. Demir, Appl. Math. 3, 246 (2012)
Z.M. Odibat, C. Bertelle, M.A. Aziz-Alaoui, G.H.E. Duchamp, Comput. Math. Appl. 59, 1462 (2010)
H. Saberi Nik, F. Soleymani, Alexandria Eng. J. 52, 543 (2013)
M.M. Rashidi, N. Laraqi, S.M. Sadri, Int. J. Thermal Sci. 49, 2405 (2010)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Chiba, R. Transient natural convection of cold water in a vertical channel. Eur. Phys. J. Plus 131, 135 (2016). https://doi.org/10.1140/epjp/i2016-16135-2
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1140/epjp/i2016-16135-2