**Author details**

26 Will-be-set-by-IN-TECH

They conclude that Oldroyd fluid viscoelastic convection is characterized both by a Hopf

Analytical and numerical methods were used by Martínez-Mardones et al. [36] to calculate the nonlinear critical parameters which lead to stationary convection as well as traveling and standing waves. By means of coupled Landau amplitude equations Martínez-Mardones et al. [37] investigated the pattern selection in terms of the viscoelastic parameters. They fix *Pr* and *E* and show that increasing *L* stationary convection changes into standing waves by means of a subcritical bifurcation. The convective and absolute instabilities for the three model time derivatives of the Oldroyd fluid were investigated by Martínez- Mardones et al. [38]. If the group velocity is zero at say *k* = *k*<sup>0</sup> and the real part of *σ*, *s*, in Equation 39 is positive, it is said that the instability is absolute. In this case, the perturbations grow with time at a fixed point in space. If the perturbations are carried away from the initial point and at that point the perturbation decays with time, the instability is called convective. By means of coupled complex Ginzburg-Landau equations (Landau equations with second derivatives in space and complex coefficients) they investigated problems for which oscillatory convection appears first. Besides, they investigated the effect the group velocity has on oscillatory convection. It is found that the conductive state of the fluid layer is absolutely unstable if *L* > 0 or *E* > *EC* and therefore, when 0 < *E* < *EC*, the state is convectively unstable. They also show that there is no traveling wave phenomena when passing from stationary convection to standing waves.

In this chapter many phenomena have been discussed in order to show the variety of problems which can be found in natural convection of Newtonian and viscoelastic fluids. One of the goals was to show that the different boundary conditions may give results which differ considerably from each other. Sometimes, the results are qualitatively the same and this is taken as an advantage to solve "simpler" problems as those corresponding to the linear and nonlinear equations with free-free boundary conditions. A change in the setting of the problem may produce large complications, as in the case of the free-free boundary conditions, but with one of them being deformable. In this case a new parameter appears, the Galileo number *G*, which complicates not only the number of numerical calculations, but also the physical interpretation of the results, as explained above. As have been shown, the introduction of viscoelasticity complicates even more the physics of convection. Depending on the boundary conditions, there can be stationary and oscillatory cells in linear convection. Nonlinear convection can be stationary but for other magnitudes of the parameters, traveling and standing waves may appear as the stable fluid motion. The problem is to find the conditions and magnitudes of the viscoelastic parameters when a particular convection phenomenon occurs. This is the thrilling part of viscoelastic convection. It is the hope of the present author that this review may motivate a number of readers to work in this rich area

The author would like to thank Joaquín Morales, Cain González, Raúl Reyes, Ma. Teresa

bifurcation (for very large value of *L*) and a subcritical bifurcation.

**7. Conclusions**

of research.

**Acknowledgements**

Vázquez and Oralia Jiménez for technical support.

L. A. Dávalos-Orozco *Instituto de Investigaciones en Materiales, Departamento de Polímeros, Universidad Nacional Autónoma de México, Ciudad Universitaria, Circuito Exterior S/N, Delegación Coyoacán, 04510 México D. F., México*
