**8. References**

	- [17] Finalyson, B. A. (1968). The Galerkin Method applied to convective instability problems. *Journal of Fluid Mechanics*, Vol. 33, No. 1, 201 - 208.

[37] Martínez-Mardones, J., Tiemann, R., Walgraef, D. & Zeller, W.(1996). Amplitude equations and pattern selection in viscoelastic convection. *Physical Review E*, Vol. 54,

Viscoelastic Natural Convection 31

[38] Martínez-Mardones, J., Tiemann, R., Walgraef, D.(1999). Convective and absolute instabilities in viscoelastic fluid convection. *Physica A*, Vol. 268, No. 1, 14 - 23. [39] Newell, A. C. & Whitehead, J. A. (1969). Finite bandwidth, finite amplitude convection.

[40] Nield, D. A. (1964). Surface tension and buoyancy effects in cellular convection. *Journal*

[41] Oldroyd, J. G. (1950). On the formulation of rheological equations of state. *Proceedings*

[42] Ortiz-Pérez A. S. & Dávalos-Orozco, L. A. (2011). Convection in a horizontal fluid layer under an inclined temperature gradient. *Physics of Fluids*, Vol. 23, No. 084107, 1 - 11. [43] Park, H. M. & Lee, H. S. (1995). Nonlinear hydrodynamic stability of viscoelastic fluids heated from below. *Journal of Non-Newtonian Fluid Mechanics*, Vol. 60, No. 1, 1 - 26. [44] Park, H. M. & Lee, H. S. (1996). Hopf bifurcation of viscoelastic fluids heated from

[45] Pearson, J. R. A. (1958). On convection cells induced by surface tension. *Journal of Fluid*

[46] Pérez- Reyes, I. & Dávalos-Orozco, L. A. (2011). Effect of thermal conductivity and thickness of the walls in the convection of a viscoelastic Maxwell fluid layer. *International*

[47] Petrie, C. J. S. (1979). *Elongational Flows*, Pitman Publishing Limited, ISBN 0-273-08406-2

[48] Pérez- Reyes, I. & Dávalos-Orozco, L. A. (2012). Vertical vorticity in the non- linear long

[49] Pismen, L. M. (1986). Inertial effects in long-scale thermal convection. *Physics Letters A*,

[50] Proctor, M. R. E. (1981). Planform selection by finite-amplitude thermal convection between poorly conducting slabs. *Journal of Fluid Mechanics*, Vol. 113, 469 - 485. [51] Prosperetti, A. (2011). A simple analytic approximation to the Rayleigh-Bénard stability

[52] Rajagopal, K. R., Ruzicka, M. and Srinivasa, A. R. (1996) On the Oberbeck-Boussinesq approximation. *Mathematical Models and Methods in Applied Sciences*, Vol. 6, No. 8, 1157 -

[53] Lord Rayleigh (1916). On convective currents in a horizontal layer of fluid when the higher temperature is on the under side. *Philosophical Magazine Series 6*, Vol. 32, No. 192,

[54] Reid, W. H. & Harris, D. L. (1958). Some further results on the Bénard problem. *Physics*

[55] Rosenblat, S. (1986). Thermal convection in a viscoelastic liquid. *Journal of*

[56] Segel, L. A. (1969). Distant side-walls cause slow amplitude modulation of cellular

[57] Segel, L. A. & Stuart, J. T. (1962). On the question of the preferred mode in cellular

*Journal of Fluid Mechanics*, Vol. 38, No. 2, 279 - 303.

*of the Royal Society of London A*, Vol. 200, No. 1063, 523 - 541.

*Journal of Heat and Mass Transfer*, Vol. 54, No. 23, 5020 - 5029.

threshold. *Physics of Fluids*, Vol. 23, No. 124101, 1 - 8.

*Non-Newtonian Fluid Mechanics*, Vol. 21, No. 2, 201 - 223.

convection. *Journal of Fluid Mechanics*, Vol. 38, No. 1, 203 - 224.

thermal convection. *Journal of Fluid Mechanics*, Vol. 13, No. 2, 289 - 306.

below. *Journal of Non-Newtonian Fluid Mechanics*, Vol. 66, No. 1, 1 - 34.

wavelength instability of a viscoelastic fluid layer. To be submitted.

*of Fluid Mechanics*, Vol. 19, No. 3, 341 - 352.

*Mechanics*, Vol. 4, No. 5, 489 - 500.

Vol. 116, No. 5, 241 - 244.

*of Fluids*, Vol. 1, No. 2, 102 - 110.

, London.

1167.

529 - 546.

No. 2, 1478 - 1488.


[37] Martínez-Mardones, J., Tiemann, R., Walgraef, D. & Zeller, W.(1996). Amplitude equations and pattern selection in viscoelastic convection. *Physical Review E*, Vol. 54, No. 2, 1478 - 1488.

28 Will-be-set-by-IN-TECH

[17] Finalyson, B. A. (1968). The Galerkin Method applied to convective instability problems.

[18] Finlayson, B. A. (1972). *The Method of Weighted Residuals and Variational Principles*,

[19] Gershuni, G. Z. & Zhukhovitskii, E. M. (1976). *Convective Stability of Incompressible Fluids*,

[20] Giesekus, H. (1963). Die simultane Translations- und Rotationsbewegun einer Kugel in

[21] Green III, T. (1968). Oscillating convection in an elasticoviscous liquid. *Physics of Fluids*,

[22] Hoyle, R. B. (1998). Universal instabilities of rolls, squares and hexagones, In: *Time-Dependent Nonlinear Convection*, Tyvand, P. A., (Ed.), 51 - 82, Computer Mechanics

[23] Hoyle, R. B. (2006). *Pattern Formation, An Introduction to Methods*, Cambridge University

[24] Huilgol, R. R. (1973). On the solution of the Bénard problem with boundaries of finite

[25] Hurle, D. T. J., Jakeman, E. & Pike E. R. (1967). On the solution of the Bénard problem with boundaries of finite conductivity. *Proceeding of the Royal Society of London A*, Vol.

[26] Jakeman, E (1968). Convective instability in fluids of high thermal diffusivity. *Physics of*

[27] Jeffreys, H. (1926). The stability of a layer of fluid heated from below. *Philosophical*

[28] Kapitaniak, T. (2000). *Chaos for Engineers*, Springer-Verlag, ISBN 3-540-66574-9, Berlin. [29] Khayat, R. E. (1995). Fluid elasticity and the transition to chaos in thermal convection.

[30] Kolkka, R. W. & Ierley, G. R. (1987). On the convected linear instability of a viscoelastic Oldroyd B fluid heated from below. *Journal of Non-Newtonian Fluid Mechanics*, Vol. 25,

[31] Landau, L. D. & Lifshitz, E. M. (1987). *Fluid Mechanics*, Pergamon Press, ISBN

[32] Lorenz, E. N. (1963). Deterministic non- periodic flows. *Journal Atmospheric Science*, Vol.

[33] Malkus, W. V. R. & Veronis, G. (1958). Finite amplitude cellular convection. *Journal of*

[34] Markovitz, H. & Coleman, B. D. (1964). Incompressible Second-Order Fluids, In: *Advances in Applied Mechanics Volume 8*, Dryden, H. L. & von Kármán, T., (Ed.), 69 -

[35] Martínez-Mardones, J. & Pérez-García, C. (1990). Linear instability in viscoelastic fluid

[36] Martínez-Mardones, J., Tiemann, R., Zeller, W. & Pérez-García, C. (1994). Amplitude equation in polymeric fluid convection. *International Journal of Bifurcation and Chaos*, Vol.

convection. *Journal of Physics: Condensed Matter*, Vol. 2, No. 5, 1281 - 1290.

conductivity. *SIAM Journal of Applied Mathematics*, Vol. 24, No. 2, 226 - 233.

Keter Publishing House Jerusalem Ltd., ISBN 0-7065-1562-5, Jerusalem.

einer elastoviscosen flussigkeit. *Rheologica Acta*, Vol. 03, No. 1, 59 - 71.

*Journal of Fluid Mechanics*, Vol. 33, No. 1, 201 - 208.

Publications, ISBN 1-85312-521-0, Southampton.

Press, ISBN 978-0-521-81750-9, Cambridge.

*Magazine Series 7*, Vol. 2, No. 10, 833 - 844.

*Physical Review E*, Vol. 51, No. 1, 380 - 399.

*Fluid Mechanics*, Vol. 4, No. 3, 225 - 260.

102, Academic Press, ISBN 978-0120020089, London.

Vol. 11, No. 7, 1410 - 1412.

296, No. 1447, 469 - 475.

No. 2, 209 - 237.

20, No. 2, 130 - 141.

4, No. 5, 1347 - 1351.

0-08-033933-6 , New York.

*Fluids*, Vol. 11, No. 1, 10 - 14.

Academic Press, ISBN 978-0-122-57050-6, New York.

	- [58] Siddheshwar, P. G. & Sri Krishna C. V. (2002). Unsteady non-linear convection in a second- order fluid. *International Journal of Non-Linear Mechanics*, Vol. 37, No. 2, 321 - 330.

**Turbulent Flow of Viscoelastic Fluid**

Viscoelastic liquids with very small amounts of polymer/surfactant additives can, as well known since B.A. Toms' observation in 1948, provide substantial reductions in frictional drag of wall-bounded turbulence relative to the corresponding Newtonian fluid flow. Friction reductions of up to 80% compared to the pure water flow can be occasionally achieved with smooth channel/pipe flow of viscoelastic surfactant solution [11, 54]. This friction-reducing effect, referred to as turbulent drag reduction (DR) or Toms effect, has been identified as an efficient technology for a large variety of applications, e.g. oil pipelines [25] and heating/cooling systems for buildings [43], because of major benefits in reducing energy

**Chapter 2**

It has been known that long, high-molecular-weight, flexible polymers or rod-like micelle networks of surfactant are particularly efficient turbulence suppressor, so that those solutions lead to different turbulent states both qualitatively and quantitatively, resulting in dramatic DRs. One of promising additives, which may allow their solutions to induce DR, is a cationic surfactant such as "cetyltrimethyl ammonium chloride (CTAC)" under appropriate conditions of surfactant chemical structure, concentration, counter-ion, and temperature to form micellar networks in the surfactant solution. Those resulting micro-structures give rise to viscoelasticity in the liquid solution. The properties and characteristics of the viscoelastic fluids measured even in simple shear or extensional flows are known to exhibit appreciably different from those of the pure solvent. From a phenomenological perspective, their turbulent flow is also peculiar as is characterized by extremely elongated streaky structures with less bursting events. Therefore, the viscoelastic turbulence has attracted much attention of researchers during past 60 years. Intensive analytical, experimental, and numerical works have been well documented and many comprehensive reviews are available dealing with this

Although the mechanism of DR is still imperfectly understood, but some physical insights have emerged. In particular, with the aid of recent advanced supercomputers, direct

and reproduction in any medium, provided the original work is properly cited.

©2012 Tsukahara and Kawaguchi, licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original

© 2012 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution,

**Through Complicated Geometry**

Takahiro Tsukahara and Yasuo Kawaguchi

http://dx.doi.org/10.5772/52049

topic: [cf., 18, 19, 26, 35, 51, , and others].

work is properly cited.

**1. Introduction**

consumption.

Additional information is available at the end of the chapter


#### 32 Viscoelasticity – From Theory to Biological Applications **Chapter 0 Chapter 2**
