Turbulent flow viscoelastic

The transition to turbulent flow begins at Re R in the range of 2,000 to 2,500 (Metzuer and Reed, AIChE J., 1, 434 [1955]). For Bingham plastic materials, K and n must be evaluated for the condition in question in order to determine Re R and establish whether the flow is laminar. An alternative method for Bingham plastics is by Hanks (Hanks, AIChE J., 9, 306 [1963] 14, 691 [1968] Hanks and Pratt, Soc. Petrol. Engrs. J., 7, 342 [1967] and Govier and Aziz, pp. 213-215). The transition from laminar to turbulent flow is influenced by viscoelastic properties (Metzuer and Park, J. Fluid Mech., 20, 291 [1964]) with the critical value of Re R increased to beyond 10,000 for some materials. 

This classification of material behavior is summarized in Table 3-1 (in which the subscripts have been omitted for simplicity). Since we are concerned with fluids, we will concentrate primarily on the flow behavior of Newtonian and non-Newtonian fluids. However, we will also illustrate some of the unique characteristics of viscoelastic fluids, such as the ability of solutions of certain high polymers to flow through pipes in turbulent flow with much less energy expenditure than the solvent alone. 

Because of the interaction of the two complicated and not well-understood fields, turbulent flow and non-Newtonian fluids, understanding of DR mechanism(s) is still quite limited. Cates and coworkers (for example, Refs. " ) and a number of other investigators have done theoretical studies of the dynamics of self-assemblies of worm-like micelles. Because these so-called living polymers are subject to reversible scission and recombination, their relaxation behavior differs from reptating polymer chains. An additional form of stress relaxation is provided by continuous breaking and repair of the micellar chains. Thus, stress relaxation in micellar networks occurs through a combination of reptation and breaking. For rapid scission kinetics, linear viscoelastic (Maxwell) behavior is predicted and is observed for some surfactant systems at low frequencies. In many cationic surfactant systems, however, the observed behavior in Cole-Cole plots does not fit the Maxwell model. 

Finally, we cannot overlook the development of computational tools for the solution of problems in fluid mechanics and transport processes. Methods of increasing sophistication have been developed that now enable quantitative solutions of some of the most complicated and vexing problems at least over limited parameter regimes, including direct numerical simulation of turbulent flows so-called free-boundary problems that typically involve large interface or boundary deformations induced by flow and methods to solve flow problems for complex fluids, which are typically characterized by viscoelastic constitutive equations and complicated flow behavior. 

While the stress tensor component tfor purely viscous fluids can be determined from the instantaneous values of the rate of deformation tensor 4, the past history of deformation together with the current value of 4, may become an important factor in determining t, for viscoelastic fluids. Constitutive equations to describe stress relaxation and normal stress phenomena are also needed. Unusual effects exhibited by viscoelastic fluids include rod climbing (Weis-senberg effect), die swell, recoil, tubeless siphon, drag, and heat transfer reduction in turbulent flow. 

In dealing with viscoelastic fluids, especially under turbulent flow conditions, it is necessary to introduce a dimensionless number to take account of the fluid elasticity [29-33], Either the Deborah or the Weissenberg number, both of which have been used in fluid mechanical studies, satisfies this requirement. These dimensionless groups are defined as follows ... 

TURBULENT FLOW OF VISCOELASTIC FLUIDS IN CIRCULAR TUBES... 

When an aqueous solution of a high-molecular-weight polymer is used in a practical engineering system, the solvent is generally predetermined by the system. However, the importance of the solvent on the pressure drop and heat transfer behavior with these viscoelastic fluids has often been overlooked. Since the heat transfer performance in turbulent flow is critically dependent on the viscous and elastic nature of the polymer solution, it is important to understand the solvent effects on the rheological properties of a viscoelastic fluid. 

The fully established friction factor for turbulent flow of a viscoelastic fluid in a rectangular channel is dependent on the aspect ratio, the Reynolds number, and the Weissenberg number. As in the case of the circular tube, at small values of Ws, the friction factor decreases from the newtonian value. It continues to decrease with increasing values of Ws, ultimately reaching a lower asymptotic limit. This limiting friction factor may be calculated from the following equation ... 

The behavior of a viscoelastic fluid in turbulent flow in the hydrodynamic entrance region of a rectangular channel can be estimated by assuming that the circular tube results are applicable provided that the hydraulic diameter replaces the tube diameter. 

Studies of the heat transfer behavior of viscoelastic aqueous polymer solutions have been carried out for turbulent flow in a rectangular channel having an aspect ratio of 0.5. These experimental results obtained with aqueous polyacrylamide solutions are shown in Fig. 10.37, where the minimum asymptotic values of the dimensionless heat transfer coefficient, jH, are compared with the values reported by Cho and Hartnett for turbulent pipe flow. The turbulent pipe flow results are correlated by... 

An exception to the generally observed drag reduction in turbulent channel flow of aqueous polymer solutions occurs in the case of aqueous solutions of polyacrylic acid (Carbopol, from B.F. Goodrich Co.). Rheological measurements taken on an oscillatory viscometer clearly demonstrate that such solutions are viscoelastic. This is also supported by the laminar flow behavior shown in Fig. 10.20. Nevertheless, the pressure drop and heat transfer behavior of neutralized aqueous Carbopol solutions in turbulent pipe flow reveals little reduction in either of these quantities. Rather, these solutions behave like clay slurries and they have been often identified as purely viscous nonnewtonian fluids. The measured dimensionless friction factors for the turbulent channel flow of aqueous Carbopol solutions are in agreement with the values found for clay slurries and may be correlated by Eq. 10.65 or 10.66. The turbulent flow heat transfer behavior of Carbopol solutions is also found to be in good agreement with the results found for clay slurries and may be calculated from Eq. 10.67 or 10.68. 

W. A. Meyer, A Correlation of the Friction Characteristics for TUrbulent Flow of Dilute Viscoelastic non-Newtonian Fluids in Pipes, AIChE J. (12) 522,1966. 

This equation was derived by following procedure similar to that given by Hannah et al. (29) but using the Virk (37) equation. It is claimed that the equation is valid for turbulent flows of viscoelastic fluids and is applicable to most of the available field data. 

It is known that a viscoelastic fluid, e.g., a solution with a trace amount of highly deformable polymers, can lead to elastic flow instability at Reynolds number well below the transition number (Re 2,000) for turbulence flow. Such chaotic flow behavior has been referred to as elastic turbulence by Tordella [2]. Indeed, the proper characterization of viscoelastic flows requires an additional nondimensional parameter, namely, the Deborah number, De, which is the ratio of elastic to viscous forces. Viscoelastic fluids, which are non-Newtonian fluids, have a complex internal microstructure which can lead to counterintuitive flow and stress responses. The properties of these complex fluids can be varied through the length scales and timescales of the associated flows [3]. Typically the elastic stress, by shear and/or elongational strains, experienced by these fluids will not immediately become zero with the cessation of fluid motion and driving forces, but will decay with a characteristic time due to its elasticity. 

A linear theory of viscoelasticity is sufficient for obtaining a significant drag reduction effect qualitative arguments which assume that either or 14 depend on the kinematics of turbulent flows appear unnecessary (see, esp., Lumley (8)). 

Ruckenstein, E On the Mechanism of Drag Reduction in Turbulent Flow of Viscoelastic Liquids. Chem. Eng. Sci. 26 (1971) 1075. 

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