2. About viscoelastic fluids

Viscoelastic fluids are a type of non-Newtonian fluid formed by a viscous component and an elastic one. For short, viscoelastic fluids are the blend of a solvent and some polymer. Examples of these are paints, DNA suspensions, some biological fluids and others from the chemical industry. A number of features make the viscoelastic fluids very interesting and of industrial importance: polymers are almost everywhere. Take for example the case of paints whose annual production generates several USD millions. Proper understanding of viscoelasticity is key for industrial applications.

Polymeric suspensions show viscoelastic behavior but its stress - deformation relationship is not easily represented by a single model. In rheology these models are called constitutive equations and for which the books of Bird et al. [3] and Macosko [4] are two very good classic readings on this subject.

#### 2.1. Some models

On the other hand, not all viscoelastic fluids are paints and some of these may have biological nature or even may be intended for some optical application or the like. Also, viscoelastic fluids may be so different to each other that a reasonable number of constitutive equations have been developed to model their behavior. At the same time, some applications involving viscoelastic fluids may require physical mechanisms for proper working, like natural convection. Such is the case of deoxyribonucleic acid (DNA) replication and the fabrication of corrugated surfaces, for

The present contribution is devoted to exposed the link between viscoelasticity and Rayleigh and/or Marangoni convection and how these fit in lab or industrial applications. In fact, this is also a different point of view for polymer rheology applications. In this case some examples involving viscoelastic fluids and hydrodynamic stability can be addressed to show another connection of rheology with fluid mechanics. These type of applications are only a few years

The author has been working in similar theoretical problems and is trying to make experimental contributions that would finally improve applications like these. Hopefully, this proposal could show the link between rheology and other subjects in new multidisciplinary research topics and developments. The aim of this contribution is to call the attention of researchers and graduate students to interdisciplinary subjects were important technological advances are awaiting to be discovered. Hydrodynamic stability and rheology are both matured topics with

The chapter is organized as follows. In Section 2, general comments on viscoelasticity are given so that a connection with thermal hydrodynamic stability in Section 3 can be made. In Section 3 a summary of the physics of thermal convection in Newtonian and viscoelastic fluids is presented too. A brief presentation of the methods used to study the thermal hydrodynamics in viscoelastic fluids is given in Section 4. Next in Section 5 interesting application examples that use thermoconvection are given. Finally, a short discussion on the role played by thermal

Viscoelastic fluids are a type of non-Newtonian fluid formed by a viscous component and an elastic one. For short, viscoelastic fluids are the blend of a solvent and some polymer. Examples of these are paints, DNA suspensions, some biological fluids and others from the chemical industry. A number of features make the viscoelastic fluids very interesting and of industrial importance: polymers are almost everywhere. Take for example the case of paints whose annual production generates several USD millions. Proper understanding of viscoelasticity is

Polymeric suspensions show viscoelastic behavior but its stress - deformation relationship is not easily represented by a single model. In rheology these models are called constitutive equations and for which the books of Bird et al. [3] and Macosko [4] are two very good classic

example. Viscoelasticity is then at the same level of importance than natural convection.

old and remain relevant.

30 Polymer Rheology

interesting theoretical and experimental findings.

convection in the example applications is given.

2. About viscoelastic fluids

key for industrial applications.

readings on this subject.

Modeling viscoelasticity is a complicated matter since viscoelastic fluids may show, for example, linear and non-linear behavior. On the other hand, some features of the polymeric component may have too much importance to be explicitly introduced in these models. Such is the case for thixotropic behavior in viscoelastic micellar solutions and liquid crystalline dispersions [5]. Some of the constitutive equations of common usage and interest in the academic and industrial community are


The readers may find more details on the viscoelastic fluids in the papers of Larson [6, 7] and the textbook of Phan-Thien [8]. For thermal convective instabilities problems the constitutive equations for the viscoelastic Maxwell [9, 10], Jeffreys [11, 12] and second order [13, 14] fluids has some popularity. Several investigations has used these models to study the hydrodynamics of viscoelastic fluids heated from below. Also, these results are the foundation of a number of applications [15–17] like those presented in this contribution.

As a common and widely constitutive equation consider the case of Jeffreys viscoelastic fluids represented by the following expression

$$\left(\frac{\partial}{\partial t} + \lambda\_1\right)\tau = -\eta\_0 \left(\frac{\partial}{\partial t} + \lambda\_2\right)\dot{\mathbf{y}} \tag{1}$$

where τ is the stress tensor, γ\_ is the rate of strain tensor, η<sup>0</sup> is the zero shear rate viscosity, λ<sup>1</sup> is the stress relaxation time and λ<sup>2</sup> is the stress retardation time. An important feature of Eq. (1) is that it reduces to the model for viscoelastic Maxwell fluids when λ<sup>2</sup> ¼ 0, and to the constitutive equation for Newtonian fluids when λ<sup>1</sup> ¼ λ<sup>2</sup> ¼ 0. Several investigations dealing with Rayleigh and Marangoni convection has to do with this type of constitutive equations because of the previously mentioned feature.

## 3. Thermoconvection hydrodynamics

Thermal convection is of interest for the applications described below but mainly Rayleigh and Marangoni convection. A particular feature of these is the secondary flow generated solely by a thermal gradient. Also, Rayleigh and Marangoni convection form regular patterns which is related to the heat transfer across the fluid.

The problem of thermal convection in incompressible fluids is not new and several geometrical configurations have been considered. Also, the orientation of the thermal gradient and nature of the thermal source has been subjected to different arrangements. This has been driven mainly for potential industrial and/or lab applications [18, 19]. the following cases are of interest for the present contribution: Rayleigh and Marangoni convection in horizontal fluid layers and in vertical cylinders. Pattern formation and heat transfer are key for the proper understanding of the technological developments described below.

> motions are called convective cells. One important fact is that the problem of Rayleigh convection has been widely studied so that for certain cases the critical conditions at which convective motions set are well known as it is shown in Table 1. For the case convection of Newtonian fluids in horizontal fluid layers heated from below the critical conditions of perfect thermal conducting and thermal insulators are a common result. From these two cases, that of bounding perfect thermal conductors has been widely studied because it fits better with lab

Table 1. List of critical values for the onset of convection of Newtonian fluids in horizontal fluid layers heated from below. These data correspond to perfect thermal conducting or insulating bounding horizontal walls. For viscoelastic

Applications of Viscoelastic Fluids Involving Hydrodynamic Stability and Heat Transfer

http://dx.doi.org/10.5772/intechopen.76122

33

The Rayleigh number and the wavenumber are dimensionless parameters featuring the hydrodynamic stability of the fluid layer. As these two parameters achieve critical values thermal

Convection in viscoelastic fluids has been widely studied too [9–12, 14, 16]. The discussion in this section shall be restricted to viscoelastic Maxwell and Jeffreys fluids. Then, the physical mechanisms is completely changed due to the introduction of two parameters featuring visco-

Also, convection may be set starting as oscillatory motions. Fortunately, a number of investigators have been working on this subject and the hydrodynamics is well known for some cases. For viscoelastic Maxwell and Jeffreys fluids physics is known to some extent with both the relaxation and the retardation time featuring its behavior. Rayleigh and Marangoni convection in viscoelastic fluids appear in industrial applications and are of interest because the

Research on hydrodynamics of viscoelastic fluids involves two different approaches. Theoretical and experimental studies are to be linked in order to improve practical applications, which are explained later in this chapter. The aim in hydrodynamic stability studies is to find the

convection sets in. Otherwise the fluid motions eventually stop.

Rayleigh number Wavenumber Walls

Marangoni number Wavenumber Walls

1708.6 3.12 Conductors [9, 20] 720 0 Insulators [9, 21]

48 0 Insulators [22]

industry of polymers generates millions of USD per year.

and industrial applications.

fluids these critical numbers change.

elasticity as shown in Eq. (1) • stress relaxation time λ1, • stress retardation time λ2.

4. Methods

3.2. Convection in viscoelastic fluids

## 3.1. Convection in Newtonian fluids

For Newtonian fluids the physics behind these two classical problems of fluid mechanics is as follows. In horizontal fluid layers heated from below and cooled from above (see Figure 1a), near the bottom where the heating source is located the fluid changes its density by becoming lighter. At the same time the fluid near the top is heavy since because of the top cooling. This is an unstable arrangement of the fluid since portions of fluid with higher density tend to fall pushing portions of lower density fluid to the top (see Figure 1b). Next, the movement of the fluid occurs only if a critical temperature is achieved. This is called Rayleigh convection and investigation of the critical conditions at which the convective motions are set is key [20].

For the case of Marangoni convection the physical mechanism is quite different since the surface tension variations with temperature, at the surface, trigger fluid motions. This type of thermal convection occurs in very thin fluid layers or in low gravity conditions. Briefly, as the fluid layer is heated from below the energy is transferred by diffusion to the fluid surface. As the fluid surface tension depends on temperature, in hot surface spots the fluid moves away to cooler surface regions. Next, the convective motions take place.

Both, Rayleigh and Marangoni convection are connected and this has been demonstrated theoretical and experimentally. Most important is that the physical mechanisms has been studied and can be identified not only in the examples shown here but in other engineering areas.

Figure 1 only shows the beginning of the convective motion in the core of the fluid layer. As the process is reinforced, the motions become ordered in a periodical fashion. These convective

Figure 1. Schematics of the physical mechanism of thermal convection in a Newtonian fluid layer heated from below. (a) Basic state (b) Beginning of convection.


Table 1. List of critical values for the onset of convection of Newtonian fluids in horizontal fluid layers heated from below. These data correspond to perfect thermal conducting or insulating bounding horizontal walls. For viscoelastic fluids these critical numbers change.

motions are called convective cells. One important fact is that the problem of Rayleigh convection has been widely studied so that for certain cases the critical conditions at which convective motions set are well known as it is shown in Table 1. For the case convection of Newtonian fluids in horizontal fluid layers heated from below the critical conditions of perfect thermal conducting and thermal insulators are a common result. From these two cases, that of bounding perfect thermal conductors has been widely studied because it fits better with lab and industrial applications.

The Rayleigh number and the wavenumber are dimensionless parameters featuring the hydrodynamic stability of the fluid layer. As these two parameters achieve critical values thermal convection sets in. Otherwise the fluid motions eventually stop.

#### 3.2. Convection in viscoelastic fluids

Convection in viscoelastic fluids has been widely studied too [9–12, 14, 16]. The discussion in this section shall be restricted to viscoelastic Maxwell and Jeffreys fluids. Then, the physical mechanisms is completely changed due to the introduction of two parameters featuring viscoelasticity as shown in Eq. (1)


Also, convection may be set starting as oscillatory motions. Fortunately, a number of investigators have been working on this subject and the hydrodynamics is well known for some cases. For viscoelastic Maxwell and Jeffreys fluids physics is known to some extent with both the relaxation and the retardation time featuring its behavior. Rayleigh and Marangoni convection in viscoelastic fluids appear in industrial applications and are of interest because the industry of polymers generates millions of USD per year.
