Theoretical Approaches for Heat Transfer Problems

Chapter 19

Abstract

1. Introduction

381

phenomena aimed to be controlled.

Feedback Control of Rayleigh

Convection in Viscoelastic

Ildebrando Pérez-Reyes, Cornelio Alvarez-Herrera

Control of Rayleigh convection in a viscoelastic Maxwell fluid is addressed here by considering a feedback from shadowgraphic visualizations. Here, a theoretical approach is made to the problem of the onset of convective motion through a source term in the lower thermal boundary condition. A numerical Galerkin technique is then used to study the linear hydrodynamic stability. Small relaxation times are considered for Prandtl numbers 1 and 10. Interesting results for the Rayleigh, the wavenumber, and the frequency of oscillations are presented along with discussion on the physical mechanism. In short, the linear hydrodynamic stability analysis

The idea of control in physics and engineering, and other areas of applications, is

not new. Several examples and interesting advances can be found in chemical processes where multivariable problems may appear in the way to obtain desired product quality or to guarantee safety of operators [1, 2]. At the same time, technology has been developed to meet the needs of the industry such as proportional integral derivative controllers, sensors, etc. On the other hand, new technology and lab developments needing control of variables may use different hardware. Onset of thermal convection and thermoconvective pattern formation are examples of

Control of Rayleigh convection is a subject that has called the attention of researchers because of its connection to applications in electronics [3] and genetics [4], for example. In these cases understanding and modulation of convective motions are important so that the problem of Rayleigh convection may be coupled to a feedback control scheme. It is straight that heat transfer along the fluid layer should be monitored and controlled in order to modulate the convective motions. The present manuscript is devoted to give an approach to the control of the Rayleigh convection phenomena in a viscoelastic fluid layer by a feedback scheme based on shadowgraphic visualization. This is a theoretical approach to the problem which actually introduces a modification in the lower thermal boundary conditions

to modulate the heat transfer so that convection may be changed [5]. Since

Maxwell Fluids

and Jonathan Rodríguez-Campos

states that suppression of convection may be favored.

Keywords: viscoelastic fluid, feedback control, Rayleigh convection
