The developing steady flow of Oldroyd-B and Phan-Thien-Tanner (PTT) fluids through a two-dimensional rectangular channel is investigated computationally by means of a finite volume technique incorporating uniform collocated grids. A second-order central difference scheme is employed to handle convective terms in the momentum equation, while viscoelastic stresses are approximated by a third-order accurate quadratic upstream interpolation for convective kinematics (QUICK) scheme. Momentum interpolation method (MIM) is used to evaluate both cell face velocities and coefficients appearing in the stress equations. Coupled mass and momentum conservation equations are then solved through an iterative semi-implicit method for pressure-linked equation (SIMPLE) algorithm. The entry length over which flow becomes fully developed is determined by considering gradients of velocity, normal and shear stress components, and pressure in the axial direction. The effects of the mesh refinement, inlet boundary conditions, constitutive equation parameters, and Reynolds number on the entry length are presented.

References

References
1.
Lécuyer
,
H. A.
,
Mmbaga
,
J. P.
,
Hayes
,
R. E.
,
Bertrand
,
F. H.
, and
Tanguy
,
P. A.
, 2009, “
Modelling of Forward Roll Coating Flows With a Deformable Roll: Application to Non-Newtonian Industrial Coating Formulations
,”
Comput. Chem. Eng.
,
33
, pp.
1427
1437
.
2.
Mu
,
Y.
,
Zhao
,
G.
,
Zhang
,
C.
,
Chen
,
A.
, and
Li
,
H.
, 2010, “
Three-Dimensional Simulation of Planar Contraction Viscoelastic Flow by Penalty Finite Element Method
,”
Int. J. Numer. Methods Fluids
,
63
, pp.
811
827
.
3.
Yapici
,
K.
,
Karasozen
,
B.
, and
Uludağ
,
Y.
, 2009, “
Finite Volume Simulation of Viscoelastic Laminar Flow in a Lid Driven Cavity
,”
J. Non-Newtonian Fluid Mech.
,
164
, pp.
51
65
.
4.
Na
,
Y.
, and
Yoo
,
J. Y.
, 1991, “
A Finite Volume Technique to Simulate the Flow of a Viscoelastic Fluid
,”
Comput. Mech.
,
8
, pp.
43
55
.
5.
Missirlis
,
K. A.
,
Assimacopoulos
,
D.
, and
Mitsoulis
,
E.
, 1998, “
A Finite Volume Approach in the Simulation of Viscoelastic Expansion Flows
,”
J. Non-Newtonian Fluid Mech.
,
78
, pp.
91
118
.
6.
Al Moatssime
,
H.
,
Esselaoi
,
D.
,
Hakim
,
A.
, and
Raghay
,
S.
, 2001, “
Finite Volume Multigrid Method of Planar Contraction Flow of a Viscoelastic Fluid
,”
Int. J. Numer. Methods Fluids
,
36
, pp.
885
902
.
7.
Alves
,
M. A.
,
Oliveira
,
P. J.
, and
Pinho
,
F. T.
, 2003, “
Benchmark Solutions for the Flow of Oldroyd-B and PTT Fluids in Planar Contractions
,”
J. Non-Newtonian Fluid Mech.
,
110
, pp.
45
75
.
8.
Durst
,
F.
,
Ray
,
S.
,
Unsal
,
B.
, and
Bayoumi
,
O.
, 2005, “
The Development Lengths of Laminar Pipe and Channel Flows
,”
ASME Trans. J. Fluids Eng.
,
127
, pp.
1154
1160
.
9.
Poole
,
R. J.
, and
Ridley
,
B. S.
, 2007, “
Development-Length Requirements for Fully Developed Laminar Pipe Flow of Inelastic Non-Newtonian Liquids
,”
ASME Trans. J. Fluids Eng.
,
129
, pp.
1281
1287
.
10.
Liang
,
J. Z.
, 1998, “
Determination of the Entry Region Length of Viscoelastic Fluid Flow in Channel
,”
Chem. Eng. Sci.
,
53
, pp.
3185
3187
.
11.
Phan-Thien
,
N.
, and
Tanner
,
R. I.
, 1977, “
A New Constitutive Equation Derived From Network Theory
,”
J. Non-Newtonian Fluid Mech.
,
2
, pp.
353
365
.
12.
Cruz
,
D. O. A.
,
Pinho
,
F. T.
, and
Oliveira
,
P. J.
, 2005, “
Analytical Solution for Fully Developed Laminar Flow of Some Viscoelastic Liquids With a Newtonian Solvent Contribution
,”
J. Non-Newtonian Fluid Mech.
,
132
, pp.
28
35
.
13.
Patankar
,
S. V.
, 1980,
Numerical Heat Transfer and Fluid Flow
,
Hemisphere
,
Washington, D.C
.
14.
Versteeg
,
H. K.
, and
Malalasekera
,
W.
, 1995,
An Introduction to Computational Fluid Dynamics: The Finite Volume Method
,
Prentice Hall
,
Englewood Cliffs, NJ
.
15.
Leonard
,
B. P.
, 1979, “
A Stable and Accurate Convective Modeling Procedure Based on Quadratic Interpolation
,”
Comput. Methods Appl. Mech. Eng.
,
19
, pp.
59
98
.
16.
Peric
,
M.
,
Kesser
,
R.
, and
Scheuerer
,
G.
, 1998, “
Comparison of Finite Volume Numerical Methods With Staggered and Collocated Grids
,”
Comput. Fluids
,
16
, pp.
389
403
.
17.
Majumdar
,
S.
, 1988, “
Role of Underrelaxation in Momentum Interpolation for Calculation of Flow With Nonstaggered Grids
,”
Numer. Heat Transfer
,
13
, pp.
125
132
.
18.
Yu
,
B.
,
Tau
,
W. Q.
,
Wei
,
J. J.
,
Kawaguchi
,
Y.
,
Tagawa
,
T.
, and
Ozoe
,
H.
, 2002, “
Discussion on Momentum Interpolation Method for Calculated Grids of Incompressible Flow
,”
Numer. Heat Transfer, Part B
,
42
, pp.
141
166
.
19.
Leonard
,
B. P.
, and
Drummond
,
J. E.
, 1995, “
Why You Should Not Use ‘Hybrid’, ‘Power-Law’ or Related Exponential Schemes for Convective Modeling: There Are Much Better Alternatives
,”
Int. J. Numer. Methods Fluids
,
20
, pp.
421
442
.
20.
Khosla
,
P. K.
, and
Rubin
,
S. G.
, 1974, “
A Diagonally Dominant Second Order Accurate Implicit Scheme
,”
Comput. Fluids
,
2
, pp.
207
209
.
21.
Hayase
,
T.
,
Humphrey
,
J. A. C.
, and
Greif
,
R. A.
, 1992, “
Consistently Formulated Quick Scheme for Fast and Stable Convergence Using Finite-Volume Iterative Calculation Procedures
,”
J. Comput. Phys.
,
98
, pp.
108
118
.
22.
Nacer
,
B.
,
David
,
L.
,
Pascal
,
B.
, and
Gérard
,
J.
, 2007, “
Contribution to the Improvement of the QUICK Scheme for the Resolution of the Convection-Diffusion Problems
,”
Heat Mass Transfer
,
43
, pp.
1075
1085
.
23.
Xue
,
S. C.
,
Phan-Thien
,
N.
, and
Tanner
,
R. I.
, 1995, “
Numerical Study of Secondary Flows of Viscoelastic Fluid in Straight Pipes by an Implicit Finite Volume Method
,”
J. Non-Newtonian Fluid Mech.
,
59
, pp.
191
213
.
24.
Gaidos
,
R. E.
, and
Darby
,
R.
, 1988, “
Numerical Simulation and Change in Type in the Developing Flow of a Nonlinear Viscoelastic Fluid
,”
J. Non-Newtonian Fluid Mech.
,
29
, pp.
59
79
.
25.
Deng
,
Q. H.
, and
Tang
,
G. F.
, 2002, “
Special Treatment of Pressure Correction Based on Continuity Conservation in a Pressure Based Algorithm
,”
Numer. Heat Transfer, Part B
,
42
, pp.
73
92
.
26.
Fiétier
,
N.
, and
Deville
,
M. O.
, 2003, “
Time-Dependent Algorithms for the Simulation of Viscoelastic Flows With Spectral Element Methods: Applications and Stability
,”
J. Comput. Phys.
,
186
, pp.
93
121
.
27.
van Os
,
R. G. M.
, and
Phillips
,
T. N.
, 2004, “
Spectral Element Methods for Transient Viscoelastic Flow Problems
,”
J. Comput. Phys.
,
201
, pp.
286
314
.
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