4. Methods

a thermal gradient. Also, Rayleigh and Marangoni convection form regular patterns which is

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

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

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

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)

related to the heat transfer across the fluid.

32 Polymer Rheology

3.1. Convection in Newtonian fluids

areas.

Basic state (b) Beginning of convection.

understanding of the technological developments described below.

cooler surface regions. Next, the convective motions take place.

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 critical conditions that defined the onset of convection and later the formed patterns. In Section 5 the previously mentioned critical conditions make sense through the brief explanation of the physical mechanisms of each application.

#### 4.1. Theoretical approach

The theoretical approach uses the common mathematical techniques of hydrodynamic stability for linear and non-linear problems. In either, Rayleigh or Marangoni convection these techniques are used since both are eigenvalue problems. In linear Rayleigh convection the analysis in made to find critical values of the Rayleigh number (Ra), and those of the Marangoni number (Ma) in linear Marangoni convection. As the mathematical procedure for different geometrical and heat source orientation, for example, are different only that for the convection in a fluid layer heated from below shall be presented.

Consider a horizontal Maxwell viscoelastic fluid layer heated from below and bounded by two horizontal solid walls which are very good thermal conductors. The physical arrangement is very similar to that shown in Figure 1 with a Maxwell viscoelastic fluid instead of a Newtonian fluid. If the thermal convection in this system is to be studied then the momentum, the continuity, the heat conduction and a constitutive equations should be considered. These are,

$$(1+\text{F}i\omega)\left[\frac{\text{i}\omega}{\text{Pr}}\left(\frac{d^2}{dz^2}-k^2\right)\mathcal{W}-\text{R}k^2\theta\right]=\left(\frac{d^2}{dz^2}-k^2\right)^2\mathcal{W}\tag{2}$$

$$
\left[i\omega - \left(\frac{d^2}{dz^2} - k^2\right)W\right]\theta = W\tag{3}
$$

time. Certain polymeric suspensions may fit the Maxwell viscoelastic fluid model. With the working fluid relaxation time F the theoretical methodology may help to find the

Table 2. List of critical values for the onset of convection of viscoelastic Maxwell fluids in horizontal fluid layers heated from below. These data correspond to perfect thermal conducting horizontal walls. These are shown here as

Applications of Viscoelastic Fluids Involving Hydrodynamic Stability and Heat Transfer

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

35

Ra k ω Pr F 226.7151 7.26 76.2593 10 0.1 0.04623 3.44 1.9625 10 100

Also, experiments in thermal hydrodynamics of convection in fluids are mainly based on visual techniques like Schlieren and shadowgraph. Some authors have also used particle image velocimetry to study the flow field of convective motions. Here, the shadowgraph techniques is considered because the evolution of the convective patterns is key. Besides, the temperature difference and the geometrical dimensions are sufficient for a discussion on the

The experimental setup considered is sketched in Figure 2. The shadowgraph technique is very suitable for this type of investigations because it outputs important results at very low costs and time. It is based in the fact that fluid density changes also modify how the light is reflected by it. Then, an optical arrangement is built in ordered to detect light reflexion

The aim of the experimental tests is to help find the real critical conditions at which a given convective pattern is formed. This is of paramount importance for improvement of existent

Figure 2. Experimental setup for the shadowgraph technique applied on the convection in a viscoelastic Maxwell fluid

corresponding critical conditions for the onset of convection.

physics of this phenomena.

representative values [9].

and potential applications.

layer heated from below.

variations.

subjected to the following boundary conditions

$$dW = \frac{dW}{dz} = \theta = 0 \text{ at } z = 0, 1\tag{4}$$

The system of differential Eqs. (2-4) is an eigenvalue problem for the Rayleigh number Ra. In Eqs. (2)–(4) Pr is the Prandtl number, ω is the frequency of oscillation, W is the vertical fluid velocity, θ is the temperature and k is the wavenumber. This eigenvalue problem can be analytically approached with the Galerkin method without fully solving for W and θ.

For short, as inputs, the Galerkin method needs approximated W and θ which are obtained as functions satisfying the corresponding boundary conditions. Then as the approximated functions are used to calculate the residual and find an analytical expression or numerical value of the Rayleigh number. This is a brief explanation of the solution process and further details can be found in Refs. [9, 10], for example. Then, the Rayleigh number Ra, the wavenumber k and the frequency of oscillation ω are obtained as outputs of the Galerkin method, for fixed values of Pr and F, as shown in the following Table 2

#### 4.2. Experimental techniques

As the working fluids are viscoelastic, these should be characterized. In the case of Maxwell viscoelastic fluids a rheological study is necessary in order to find the corresponding relaxation


critical conditions that defined the onset of convection and later the formed patterns. In Section 5 the previously mentioned critical conditions make sense through the brief explanation of the

The theoretical approach uses the common mathematical techniques of hydrodynamic stability for linear and non-linear problems. In either, Rayleigh or Marangoni convection these techniques are used since both are eigenvalue problems. In linear Rayleigh convection the analysis in made to find critical values of the Rayleigh number (Ra), and those of the Marangoni number (Ma) in linear Marangoni convection. As the mathematical procedure for different geometrical and heat source orientation, for example, are different only that for the

Consider a horizontal Maxwell viscoelastic fluid layer heated from below and bounded by two horizontal solid walls which are very good thermal conductors. The physical arrangement is very similar to that shown in Figure 1 with a Maxwell viscoelastic fluid instead of a Newtonian fluid. If the thermal convection in this system is to be studied then the momentum, the continuity, the heat conduction and a constitutive equations should be considered. These are,

<sup>W</sup> � Rk<sup>2</sup>

θ

W

<sup>¼</sup> <sup>d</sup><sup>2</sup> dz<sup>2</sup> � <sup>k</sup> 2 <sup>2</sup>

W (2)

θ ¼ W (3)

dz <sup>¼</sup> <sup>θ</sup> <sup>¼</sup> 0 at <sup>z</sup> <sup>¼</sup> <sup>0</sup>, <sup>1</sup> (4)

physical mechanisms of each application.

convection in a fluid layer heated from below shall be presented.

iω Pr

d2 dz<sup>2</sup> � <sup>k</sup> 2 

<sup>W</sup> <sup>¼</sup> dW

<sup>i</sup><sup>ω</sup> � <sup>d</sup><sup>2</sup>

dz<sup>2</sup> � <sup>k</sup> 2 

The system of differential Eqs. (2-4) is an eigenvalue problem for the Rayleigh number Ra. In Eqs. (2)–(4) Pr is the Prandtl number, ω is the frequency of oscillation, W is the vertical fluid velocity, θ is the temperature and k is the wavenumber. This eigenvalue problem can be

For short, as inputs, the Galerkin method needs approximated W and θ which are obtained as functions satisfying the corresponding boundary conditions. Then as the approximated functions are used to calculate the residual and find an analytical expression or numerical value of the Rayleigh number. This is a brief explanation of the solution process and further details can be found in Refs. [9, 10], for example. Then, the Rayleigh number Ra, the wavenumber k and the frequency of oscillation ω are obtained as outputs of the Galerkin method, for fixed values

As the working fluids are viscoelastic, these should be characterized. In the case of Maxwell viscoelastic fluids a rheological study is necessary in order to find the corresponding relaxation

analytically approached with the Galerkin method without fully solving for W and θ.

ð Þ 1 þ Fiω

subjected to the following boundary conditions

of Pr and F, as shown in the following Table 2

4.2. Experimental techniques

4.1. Theoretical approach

34 Polymer Rheology

Table 2. List of critical values for the onset of convection of viscoelastic Maxwell fluids in horizontal fluid layers heated from below. These data correspond to perfect thermal conducting horizontal walls. These are shown here as representative values [9].

time. Certain polymeric suspensions may fit the Maxwell viscoelastic fluid model. With the working fluid relaxation time F the theoretical methodology may help to find the corresponding critical conditions for the onset of convection.

Also, experiments in thermal hydrodynamics of convection in fluids are mainly based on visual techniques like Schlieren and shadowgraph. Some authors have also used particle image velocimetry to study the flow field of convective motions. Here, the shadowgraph techniques is considered because the evolution of the convective patterns is key. Besides, the temperature difference and the geometrical dimensions are sufficient for a discussion on the physics of this phenomena.

The experimental setup considered is sketched in Figure 2. The shadowgraph technique is very suitable for this type of investigations because it outputs important results at very low costs and time. It is based in the fact that fluid density changes also modify how the light is reflected by it. Then, an optical arrangement is built in ordered to detect light reflexion variations.

The aim of the experimental tests is to help find the real critical conditions at which a given convective pattern is formed. This is of paramount importance for improvement of existent and potential applications.

Figure 2. Experimental setup for the shadowgraph technique applied on the convection in a viscoelastic Maxwell fluid layer heated from below.

The theoretical results for Ra, k and ω are linked to the experimental observations as follows. The critical value of the Rayleigh number predicts the critical temperature difference at which the convective motions set and the critical value of the wavenumber indicates the number of formed convective cells. Then the theoretical results help for tuning the lab experiments and to control the formation of patterns in the fluid.

At this point both, theory and experiments are connected for the formation of convective patterns in viscoelastic fluids. As explain in the next section, this background is the foundation that makes it possible.
