6. Results and discussion

Very interesting results were found. Figures 5a–5c and 6a–6c present the main findings on the linear hydrodynamic stability of the Maxwell viscoelastic fluid layer. Weak viscoelastic fluids, with F = 0.1, are investigated for understanding of the role played by the controller gain γ.

For the case of Pr = 1, the critical Rayleigh, the wave number, and the frequency of oscillation show unexpected behavior since at small values of γ the magnitude of

Curve of criticality for the Rayleigh convection in a Newtonian fluid corresponding to Rc against γ (a) and to kc against γ. This curve of criticality extends the results of [10]. At γ = 100, Rc = 3976.59, and kc = 3.97.

#### Figure 5.

Curve of criticality for the Rayleigh convection in a Maxwell viscoelastic fluid with Pr = 1 and F = 0.1. These curves correspond to Rc, kc, and ω<sup>c</sup> against γ.

Rc, kc, and ω<sup>c</sup> decreases. However, at certain values of γ, the same parameters start growing monotonically. On the other hand, for the case of Pr = 10, only the critical Rayleigh number behaves as for Pr = 1. The critical wavenumber and frequency of oscillation decrease monotonically with γ for Pr = 10.

The results on the hydrodynamics are unexpected and can be attributed to a coupling of the viscoelastic property F and to the nonzero heat flux bottom boundary condition. For the two values of the Prandtl number investigated, the fluid layer always stabilizes after certain critical value of γ. From the comparison with the curves for the Newtonian case, it can be said that fluid viscoelasticity triggers stronger nonlinear behavior of Rc, kc, and ω<sup>c</sup> (Figures 4–6).

The physical interpretation of the present results is as follows. Increasing γ means that temperature at the bottom is increased too. As the heat flux is increased, viscoelasticity helps to destabilize the system. At the same time, there must be a limit for the effect of small γ since the Rayleigh number R depends on the temperature difference which cannot be indefinitely increased. If temperature is increased with no limit, along with larger values of γ, the thermal energy should be released or converted into fluid motions, for example. Then, the oscillations in the fluid would help to diffuse the heat very quickly, while the layer becomes more stable with γ. This behavior is found in both systems (Figures 5 and 6).

Feedback Control of Rayleigh Convection in Viscoelastic Maxwell Fluids DOI: http://dx.doi.org/10.5772/intechopen.84915

#### Figure 6.

Curve of criticality for the Rayleigh convection in a Maxwell viscoelastic fluid with Pr=10 and F=0.1. These curves correspond to Rc, kc, and ω<sup>c</sup> against γ.

#### 7. Conclusions

In the present work, the effect of controller gain in the linear hydrodynamic stability of a viscoelastic Maxwell fluid was studied.

The main conclusion of this work is that convection in the fluid layer can be controlled, or at least it can be suppressed. This is a direct conclusion since the curves of criticality state that the hydrodynamic stability of the fluid layer is increased with γ. The coupling of the stability parameters gives unexpected behaviors at small γ, but to the best knowledge of the authors, it could happen experimentally.

#### Acknowledgements

I. Pérez-Reyes and J. Rodríguez-Campos would like to thank the financial support from CONACyT through the project Ciencia Básica-255839. Numerical computations were performed using Maple™.
