# Turbulent Drag Reduction with Polymers in Rotating Disk Flow

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

_{s}and f

_{a}, are the friction factors, Δp

_{s}and Δp

_{a}are the pressure gradients for the pure solvent medium and solution with an additive, respectively. Among various types of additives such as solid particles, polymers [7,8], and surfactants [9,10,11], linear flexible polymers—those are the ones without branches—are known to be the most effective DR agents. Even carbon nanotubes by themselves do not provide DR in a turbulent flow, it was recently reported that carbon nanotubes can enhance the drag-reducing characteristics of polymer additives [12]. The polymer or surfactant additives were found to strongly influence the turbulence behavior, which has broadened the range of possibilities for turbulence manipulation in engineering applications [13]. Some examples of engineering applications of polymer or surfactant additives include crude oil pipeline transportation [14], slurry transport [15], and biomedical applications [16].

## 2. Mechanism of DR

_{L}(t) + w(L/2)N

_{L/2}(t) + c

= w

_{1}e

^{−at}+ 2w

_{2}(1 − e

^{−at}) + c

_{L}(t) and N

_{L/2}(t) are the number of polymers with lengths L and L/2 at time t, respectively, and w

_{1}, w

_{2}, and c are the fitting parameters. The rate constant a, with units of reciprocal time (1/s), contains information on the dynamics of mechanical degradation of the polymer chain.

^{−ht})]

## 3. Rotating Disk Flow

_{s}is the measured torque of the solvent alone and T

_{p}is the measured torque of the polymer solution at a constant angular velocity of the disk at a fixed rotation speed. The pipe flow and rotating disk apparatus (RDA) system flow are shown schematically in Figure 1.

**Figure 1.**Schematic diagram for the methods of added polymer to solution with (

**a**) pipeline turbulent flow (Reprinted from [3]) and (

**b**) rotating disk apparatus (RDA) turbulent flow.

**Figure 2.**Schematic diagram of experimental set-up of flowing soap films (Reprinted from [52]).

## 4. Polymer Concentration Effect

**Figure 3.**Drag reduction (DR) efficiency as faction of poly(ethylene oxide) (PEO), and polyacrylamide (PAAM) concentration (Reprinted from [54]).

**Figure 5.**DR of commercial guar gum, purified guar gum, and grafted guar gum in a pipe flow (Reprinted from [57]).

**Figure 6.**DR as a function of time at 2800 rpm for three different PAAM (Reprinted from [54]).

**Figure 7.**Drag reduction versus time with virgin guar gum and different concentrations (Reprinted from [62]).

## 5. Polymer Chain Conformation Effect

**Figure 9.**Coil-globule transition of DNA by SPD (Reprinted from [66]).

**Figure 10.**The percent drag reduction versus time for 1.35 wppm λ-DNA in buffer solution at 1157 rpm (N

_{Re}= 5.9 ×10

^{5}) and 25 °C with and without spermidine (SPD). The inset shows the magnification of initial change of DR by SPD injection (Reprinted from [66]).

**Figure 11.**DR for 1.35 and 2.70 wppm λ-DNA in buffer solution compared with PEO (Mw = 5 × 10

^{6}) at 1980 rpm. Inset shows the same data at early times (Reprinted from [35]).

**Figure 12.**Comparison of λ-DNA percent drag reduction (1.35, 2.03, and 2.70 wppm) with PAAM (Mw) 18 × 10

^{6}g/mol) on long-term scale (1 h) at 1980 rpm (N

_{Re}) 1 × 10

^{6}) and 25 °C. The inset represents the initial changes in the drag reducing efficiency for λ-DNA and PAAM at 1.35 wppm (Reprinted from [61]).

**Figure 13.**Difference between measured %DR of 1.35 wppm λ-DNA at Re = 7 × 10

^{5}in buffer solution and distilled water (Reprinted from [35]).

## 6. Molecular Weight and Polydispersity Effect

**Figure 14.**Drag reduction behavior of PEO polymers in 0.514 M NaCl tested by rotating disk: solvent (*); WSR-N-12K, M

_{W}= 1.7 × 10

^{6}(○), WSR-N-60K, M

_{W}= 4.0 × 10

^{6}(□); WSR-301, M

_{W}= 5.3 × 10

^{6}(□). (

**a**) Friction factor versus Reynolds number for 3 ppm PEO solutions; (

**b**) Friction factor versus Reynolds number at constant polymer volume fraction, [η]C = 0.007; (

**c**) Drag reduction efficiency measured at Re = 520,000 versus PEO polymer volume fraction (Reprinted from [68]).

_{De}, which depends on polymer molecular weight and relaxation time as follows [71]:

_{A}is Avogadro’s number. A single curve for polymers with different molecular weights was obtained when DR (reduced for polymer volume fraction DR(%) [η]C) was plotted against N

_{De}. Here, C is the polymer concentration.

## 7. Reynolds Number Effect

_{av}/η, where D is the diameter of a pipe or disk, u

_{av}is the average flow velocity, ρ is the fluid mass density, and η is the fluid viscosity. Meanwhile, the formation of what is known as the maximum DR asymptote has been revealed. Usually, maximum DR can be achieved in a dilute polymer solution by even minute amounts of suitable polymers [75]. As a maximum DR asymptote, the power-law form equation was reported from correlating the DR and the Reynolds number.

^{3}for a laminar flow of polymer solutions in a pipe flow, the magnitude of the DR effect is either zero or negative, i.e., polymer additives lead to an increase in the effective viscosity and, naturally, do not act as agents to reduce the resistance to the flow. No DR was observed at Reynolds numbers below 3 × 10

^{5}using a rotating disk flow system [77]. When Re > (3–5) × 10

^{3}, the DR effect comes into play and, as the Re increases, the magnitude of the DR effect increases until reaching a maximum.

**Figure 15.**Conceptual model for the streamwise variation of the turbulent structure (Reprinted from [40]).

## 8. Temperature Effect

**Figure 16.**Drag reduction of 50 wppm PAAM and PEO at three different temperatures (Reprinted from [80]).

## 9. Conclusion

## Acknowledgments

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**MDPI and ACS Style**

Hong, C.H.; Jang, C.H.; Choi, H.J.
Turbulent Drag Reduction with Polymers in Rotating Disk Flow. *Polymers* **2015**, *7*, 1279-1298.
https://doi.org/10.3390/polym7071279

**AMA Style**

Hong CH, Jang CH, Choi HJ.
Turbulent Drag Reduction with Polymers in Rotating Disk Flow. *Polymers*. 2015; 7(7):1279-1298.
https://doi.org/10.3390/polym7071279

**Chicago/Turabian Style**

Hong, Cheng Hai, Chun Hag Jang, and Hyoung Jin Choi.
2015. "Turbulent Drag Reduction with Polymers in Rotating Disk Flow" *Polymers* 7, no. 7: 1279-1298.
https://doi.org/10.3390/polym7071279