In the current study, we report the results of a detailed and systematic numerical investigation of developing pipe flow of inelastic non-Newtonian fluids obeying the power-law model. We are able to demonstrate that a judicious choice of the Reynolds number allows the development length at high Reynolds number to collapse onto a single curve (i.e., independent of the power-law index n). Moreover, at low Reynolds numbers, we show that the development length is, in contrast to existing results in the literature, a function of power-law index. Using a simple modification to the recently proposed correlation for Newtonian fluid flows (Durst, F. et al., 2005, “The Development Lengths of Laminar Pipe and Channel Flows,” J. Fluids Eng., 127, pp. 1154–1160) to account for this low Re behavior, we propose a unified correlation for XDD, which is valid in the range 0.4<n<1.5 and 0<Re<1000.

1.
Darbyshire
,
A. G.
, and
Mullin
,
T.
, 1995, “
Transition to Turbulence in Constant-Mass-Flux Pipe Flow
,”
J. Fluid Mech.
0022-1120,
289
, pp.
83
114
.
2.
Siegel
,
R.
, 1953, “
The Effect of Heating on Boundary Layer Transition for Liquid Flow in a Tube
,” Sc.D. thesis, Massachusetts Institute of Technology, Cambridge.
3.
Collins
,
M.
, and
Schowalter
,
W. R.
, 1963, “
Behaviour of Non-Newtonian Fluids in the Inlet Region of a Channel
,”
AIChE J.
0001-1541,
9
, pp.
98
102
.
4.
Chen
,
R. Y.
, 1973, “
Flow in the Entrance Region at Low Reynolds Numbers
,”
J. Fluids Eng.
0098-2202,
95
, pp.
153
158
.
5.
Hornbeck
,
R. W.
, 1964, “
Laminar Flow in the Entrance Region of a Pipe
,”
Appl. Sci. Res., Sect. A
0365-7132,
13
, pp.
224
236
.
6.
Friedmann
,
M.
,
Gillis
,
J.
, and
Liron
,
N.
, 1968, “
Laminar Flow in a Pipe at Low and Moderate Reynolds Numbers
,”
Appl. Sci. Res.
0003-6994,
19
(
6
), pp.
426
433
.
7.
Atkinson
,
B.
,
Brocklebank
,
M. P.
,
Card
,
C. C. H.
, and
Smith
,
J. M.
, 1969, “
Low Reynolds Number Developing Flows
,”
AIChE J.
0001-1541,
15
, pp.
548
553
.
8.
Nikuradse
,
J.
, 1950,
Applied Hydro and Aerodynamics
,
McGraw-Hill
,
New York
, p.
27
.
9.
McComas
,
S. T.
, and
Eckert
,
E. R. G.
, 1965, “
Laminar Pressure Drop Associated With the Continuum Entrance Region and for Slip Flow in a Circular Tube
,”
ASME J. Appl. Mech.
0021-8936,
32
, pp.
765
770
.
10.
Durst
,
F.
,
Ray
,
S.
,
Unsal
,
B.
, and
Bayoumi
,
O. A.
, 2005, “
The Development Lengths of Laminar Pipe and Channel Flows
,”
J. Fluids Eng.
0098-2202,
127
, pp.
1154
1160
.
11.
Mashelkar
,
R. A.
, 1975, “
Hydrodynamic Entrance-Region Flow of Pseudoplastic Fluids
,”
Proc. Inst. Mech. Eng.
0020-3483,
177
, pp.
683
689
.
12.
Soto
,
R. J.
, and
Shah
,
V. L.
, 1976, “
Entrance Flow of a Yield-Power Law Fluid
,”
Appl. Sci. Res.
0003-6994,
32
, pp.
73
85
.
13.
Matros
,
Z.
, and
Nowak
,
Z.
, 1983, “
Laminar Entry Length Problem for Power Law Fluids
,”
Acta Mech.
0001-5970,
48
, pp.
81
90
.
14.
Mehrota
,
A. K.
, and
Patience
,
G. S.
, 1990, “
Unified Entry Length for Newtonian and Power Law Fluids in Laminar Pipe Flow
,”
Can. J. Chem. Eng.
0008-4034,
68
, pp.
529
533
.
15.
Ookawara
,
S.
,
Ogawa
,
K.
,
Dombrowski
,
N.
,
Amooie-Foumeny
,
E.
, and
Riza
,
A.
, 2000, “
Unified Entry Length Correlation for Newtonian, Power Law and Bingham Fluids in Laminar Pipe Flow at Low Reynolds Number
,”
J. Chem. Eng. Jpn.
0021-9592,
33
, pp.
675
678
.
16.
Gupta
,
R. C.
, 2001, “
On Developing Laminar Non-Newtonian Flow in Pipes and Channels
,”
Nonlinear Anal.: Real World Appl.
1468-1218,
2
, pp.
171
193
.
17.
Chebbi
,
R.
, 2002, “
Laminar Flow of Power-Law Fluids in the Entrance Region of a Pipe
,”
Chem. Eng. Sci.
0009-2509,
57
, pp.
4435
4463
.
18.
Escudier
,
M. P.
,
O’Leary
,
J.
, and
Poole
,
R. J.
, 2007, “Flow Produced in a Conical Container by a Rotating Endwall,” Int. J. Heat Fluid Flow (to be published).
19.
Fellouah
,
H.
,
Castelain
,
C.
,
El Moctar
,
A. O.
, and
Peerhossaini
,
H.
, 2006, “
A. Numerical Study of Dean Instability in Non-Newtonian Fluids
,”
ASME Trans. J. Fluids Eng.
0098-2202,
128
, pp.
34
41
.
20.
Huang
,
Z.
,
Olsen
,
J. A.
,
Kerekes
,
R. J.
, and
Green
,
S. I.
, 2006, “
Numerical Simulation of the Flow Around Rows of Cylinders
,”
Comput. Fluids
0045-7930,
35
, pp.
485
491
.
21.
Taha
,
T.
, and
Cui
,
Z. F.
, 2006, “
CFD Modeling of Slug Flow in Vertical Tubes
,”
Chem. Eng. Sci.
0009-2509,
61
, pp.
676
687
.
22.
Hu
,
L. Y.
,
Zhou
,
L. X.
,
Zhang
,
J.
, and
Shi
,
M. X.
, 2005, “
Studies on Strongly Swirling Flows in the Full Space of a Volute Cyclone Separator
,”
AIChE J.
0001-1541,
51
(
3
), pp.
740
749
.
23.
Patankar
,
S.
, 1980,
Numerical Heat Transfer and Fluid Flow
,
Hemisphere
,
Washington
.
24.
Celik
,
I. B.
, and
Li
,
J.
, 2005, “
Assessment of Numerical Uncertainty for the Calculations of Turbulent Flow Over a Backward-Facing Step
,”
Int. J. Numer. Methods Fluids
0271-2091,
49
(
9
), pp.
1015
1031
.
25.
Bird
,
R. B.
,
Armstrong
,
R. C.
, and
Hassager
,
O.
, 1987,
Fluid Mechanics
,
Dynamics of Polymeric Fluids
, Vol.
1
,
2nd ed.
,
Wiley-Interscience
,
New York
.
26.
Ferziger
,
J. H.
, and
Peric
,
M.
, 2001,
Computational Methods for Fluid Dynamics
,
Springer
,
New York
.
27.
Chhabra
,
R. P.
, and
Richardson
,
J. F.
, 1999,
Non-Newtonian Flow in the Process Industries: Fundamentals and Engineering Applications
,
Butterworth-Heinemann
,
Oxford
.
28.
Escudier
,
M. P.
,
Poole
,
R. J.
,
Presti
,
F.
,
Dales
,
C.
,
Nouar
,
C.
,
Desaubry
,
C.
,
Graham
,
L.
, and
Pullum
,
L.
, 2005, “
Observations of Asymmetrical Flow Behaviour in Transitional Pipe Flow of Yield-Stress and Other Shear-Thinning Liquids
,”
J. Non-Newtonian Fluid Mech.
0377-0257,
127
, pp.
143
155
.
29.
Poole
,
R. J.
, and
Escudier
,
M. P.
, 2004, “
Turbulent Flow of Viscoelastic Liquids Through an Axisymmetric Sudden Expansion
,”
J. Non-Newtonian Fluid Mech.
0377-0257,
117
, pp.
25
46
.
30.
Metzner
,
A. B.
, and
Reed
,
J. C.
, 1955, “
Flow of Non-Newtonian Fluids—Correlation of the Laminar, Transition, and Turbulent-Flow Regions
,”
AIChE J.
0001-1541,
1
, pp.
434
440
.
31.
Bird
,
R. B.
, 1956, “
Correlation of Friction Factors in Non-Newtonian Flow
,”
AIChE J.
0001-1541,
2
, pp.
428
429
.
You do not currently have access to this content.