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SPECIAL SECTION ON THE FLUID MECHANICS AND RHEOLOGY OF NONLINEAR MATERIALS AT THE MACRO, MICRO AND NANO SCALE

# A Numerical Study of Dean Instability in Non-Newtonian Fluids

[+] Author and Article Information
H. Fellouah, C. Castelain, A. Ould El Moctar

Thermofluids, Complex Flows and Energy, Laboratoire de Thermocinétique, UMR CNRS 6607, Ecole Polytechnique de I’Université de Nantes, La Chantrerie, BP 50609, F-44306, Nantes, France

H. Peerhossaini

Thermofluids, Complex Flows and Energy, Laboratoire de Thermocinétique, UMR CNRS 6607, Ecole Polytechnique de I’Université de Nantes, La Chantrerie, BP 50609, F-44306, Nantes, Francehassan.peerhossaini@univ-nantes.fr

J. Fluids Eng 128(1), 34-41 (May 17, 2005) (8 pages) doi:10.1115/1.2136926 History: Received August 02, 2004; Revised May 17, 2005

## Abstract

We present a numerical study of Dean instability for non-Newtonian fluids in a laminar $180deg$ curved-channel flow of rectangular cross section. A methodology based on the Papanastasiou model (Papanastasiou, T. C., 1987, J. Rheol., 31(5), pp. 385–404) was developed to take into account the Bingham-type rheological behavior. After validation of the numerical methodology, simulations were carried out (using FLUENT CFD code) for Newtonian and non-Newtonian fluids in curved channels of square or rectangular cross section and for a large aspect and curvature ratios. A criterion based on the axial velocity gradient was defined to detect the instability threshold. This criterion was used to optimize the grid geometry. The effects of curvature and aspect ratio on the Dean instability are studied for all fluids, Newtonian and non-Newtonian. In particular, we show that the critical value of the Dean number decreases with increasing curvature ratio. The variation of the critical Dean number with aspect ratio is less regular. The results are compared to those for Newtonian fluids to emphasize the effect of the power-law index and the Bingham number. The onset of Dean instability is delayed with increasing power-law index. The same delay is observed in Bingham fluids when the Bingham number is increased.

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## Figures

Figure 1

Coordinate system

Figure 2

Dimensionless velocity profiles for Newtonian fluid (n=1) and pseudoplastic fluid (n=0.5)

Figure 3

Dimensionless velocity profiles for Bingham fluids (Reg=14)

Figure 4

Relative differences between analytical and calculated velocity profile for different regularization models

Figure 5

Dimensionless velocity profiles for Bingham fluids

Figure 6

Contour plots of helicity in square duct section: (a) flow without instability and (b) flow with instability

Figure 7

Axial velocity gradient along line AA of Fig. 6

Figure 8

Effect of curvature ratio in Dean instability

Figure 9

Effect of aspect ratio on Dean instability

Figure 10

Effect of aspect ratio on axial velocity profile: Dn=55 and Rc∕Dh=10

Figure 11

Vortex organization for two aspect ratios at Dn=300: (a)b∕a=8 and (b)b∕a=0.5

Figure 12

Effect of power-law index on the axial velocity profiles for a generalized Dean number of 120 in curved square channel with Rc∕Dh=10: (a) vertical midsurfaces (y=a∕2) and (b) horizontal midsurfaces (x=0)

Figure 13

Effect of power-law index on the onset of instability

Figure 14

Effect of Bingham number on the appearance of instability, Rc∕Dh=10

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