Convective Heat Transfer of Viscoelastic Flow in a Curved Duct

In this paper, fully developed flow and heat transfer of viscoelastic materials in curved ducts with square cross section under constant heat flux have been investigated. Here, staggered mesh is used as computational grids and flow and heat transfer parameters have been allocated in this mesh with marker and cell method. Numerical solution of governing equations has being performed with FTCS finite difference method. Furthermore, Criminale-Eriksen- Filbey (CEF) constitutive equation has being used as viscoelastic model. CEF constitutive equation is a suitable model for studying steady shear flow of viscoelastic materials which is able to model both effects of the first and second normal stress differences. Here, it is shown that the first and second normal stresses differences have noticeable and inverse effect on secondary flows intensity and mean Nusselt number which is the main novelty of current research.

Authors:

• M. Norouzi
• M. H. Kayhani
• M. R. H. Nobari
• M. Karimi Demneh

Keywords:

• Viscoelastic
• fluid flow
• heat convection
• CEF model
• curved duct
• square cross section.

References:

[1] W. R., Dean, "Note on the motion of a fluid in a curved pipe," Phil.
Mag., Vol. 4, pp. 208-233, May. 1927.
[2] Dean, W. R., "The streamline motion of a fluid in a curved pipe," Phil.
Mag., Vol. 5, pp. 673-693, Sep. 1928.
[3] R. H. Thomas, and K., Walters, "On the flow of an elastico-viscous
liquid in a curved pipe under a pressure gradient," J. Fluid Mechanic,
Vol. 16, pp. 228-242, Feb. 1963.
[4] A. M., Robertson, and S. J., Muller, "Flow of Oldroyd-b fluids in curved
pipes of circular and annular cross-section," Int. J. Non-Linenr
Mechanics, Vol. 31, No.1, pp. l-20, Aug. 1996.
[5] V. B., Sarin, "Flow of an elastico-viscous liquid in a curved pipe of
slowly varying curvature," Int J. Biomed Comput, Vol. 32, pp. 135-149,
Sep. 1993.
[6] V. B., Sarin, "The steady laminar flow of an elastico-viscous liquid in a
curved pipe of varying elliptic cross section," Mathl. Comput.
Modelling, Vol. 26, No. 3, pp. 109-121, Sep. 1997.
[7] W. Jitchote, and A. M., Robertson, "Flow of second order fluids in
curved pipes," J. Non-Newtonian Fluid Mech., Vol. 90, pp. 91-116, July.
2000.
[8] P. J., Bowen, A. R., Davies, and K., Walters, "On viscoelastic effects in
swirling flows," J. Non-Newtonian Fluid Mech., Vol. 38, pp. 113-126,
June. 1991.
[9] H. G. Sharma, and A., Prakash, "Flow of a second order fluid in a
curved pipe," Indian Journal of Pure and Applied Mathematics, Vol. 8,
pp. 546-557, May. 1977.
[10] Y., Iemoto, M., Nagata, and F., Yamamoto, "Steady laminar flow of a
power-law fluid in a curved pipe of circular cross-section with varying
curvature," J. Non-Newtonian Fluid Mech., Vol. 19, pp. 161-183, May.
1985.
[11] Y., Iemoto, M., Nagata, and F., Yamamoto, "Steady laminar flow of
viscoelastic fluid in a curved pipe of circular cross-section with varying
curvature," J. Non-Newtonian Fluid Mech., Vol. 22, pp. 101-114, Apr.
1986.
[12] N. Phan-Thien, and R., Zheng, "Viscoelastic flow in a curved duct: a
similarity solution for the Oldroyd-b fluid," Journal of Applied
Mathematics and Physics, Vol. 41, pp. 766-781, Nov. 1990.
[13] Y., Fan, R. I., Tanner, and N., Phan-Thien, "Fully developed viscous and
viscoelastic flows in curved pipes," J. Fluid Mech., Vol. 440, pp. 327-
357, Feb. 2001.
[14] M. K., Zhang, X. R., Shen, J. F., Ma, and B. Z., Zhang, Flow of
Oldroyd-B fluid in rotating curved square ducts, Journal of
Hydrodynamics, Vol. 19, No. 1, pp. 36-41, Apr. 2007.
[15] L., Helin, L., Thais, and G., Mompean, "Numerical simulation of
viscoelastic Dean vortices in a curved duct," J. Non-Newtonian Fluid
Mech., Vol. 156, pp. 84-94, July. 2008.
[16] M., Boutabaa, L., Helin, G., Mompean, and L., Thais, "Numerical study
of Dean vortices in developing newtonian and viscoelastic flows through
a curved duct of square cross-section," J. Non-Newtonian Fluid Mech.,
Vol. 337, pp, 84-94, Nov. 2009.
[17] M., Zhang, X., Shen, J., Ma, and B., Zhang, "Theoretical analysis of
convective heat transfer of Oldroyd-B fluids in a curved pipe,"
International Journal of Heat and Mass Transfer, Vol. 40, pp. 1-11,
July. 2007.
[18] X. R., Shen, M. K., Zhang, J. F., Ma, and B., Zhang, "Flow and heat
transfer of Oldroyd-B fluids in a rotating curved pipe," Journal of
Hydrodynamics, Vol. 20, pp. 39-46, Jan. 2008.
[19] H. Y. Tsang, and D. F., James, "Reduction of secondary motion in
curved tubes by polymer additives," J. Rheol., Vol. 24, pp. 589-601, Oct.
1980.
[20] S., Yanase, N., Goto, and K., Yamamoto, "Dual solutions of the flow
through a curved tube," Fluid Dyn. Res., Vol. 5, pp. 191-201, May.
1989.
[21] W. M. Jones and O. H., Davies, "The flow of dilute aqueous solutions of
macromolecules in various geometries: III. Curved pipes and porous
materials," J. Phys. D: Appl. Phys., Vol. 9, pp. 753-770, Sep. 1976.
[22] R. B. Bird, R. C., Armstrong, and O., Hassager, Dynamics of Polymer
Liquids, Second Edition, Canada, John Wiley & Sons, Vol. 2, 1987, ch.
6.
[23] W. M. Kays and M. E. Crawford, Convective Heat and Mass Transfer,
New York, McGraw-Hill, Inc., Third Edition, 1993, ch. 8.
[24] R. B., Bird, and J. M., Wiest, "Constitutive equations for polymeric
liquids," Annual Review of Fluid Mechanics, Vol. 27, pp. 169-193, Apr.
1995.
[25] K. A., Hoffmann, S. T., Chiang, Computational Fluid Dynamics for
Engineers, First Edition, EES, Texas, 1989, ch. 8.
[26] A. j., Chorin, "A numerical method for solving incompressible viscous
flow problems,-- J. Comput., Vol. 2, pp. 12-26, May. 1967.
[27] P., Twonsend, K., Walters, and W. M., Waterhouse, ÔÇÿÔÇÿSecondary flows
in pipes of square cross-section and the measurement of the second
normal stress difference,-- J. Non-Newtonian Fluid Mech., Vol. 1, pp.
107-123, Sep. 1976.
[28] B. M., Bara, Experimental investigation of developing and fully
developed flow in a curved duct of square cross section, PhD Thesis,
University of Alberta, 1991, ch. 10.
[29] R. K., Shah and A. L. London, Advanced in Heat Transfer, New York,
Academic Press, 1978, ch. 7.