A new approach for obtaining a normalized closed-form frequency domain analytical model for the non-Newtonian shear thinning effects on the pressure and shear stress transients in a pretransient turbulent flow of fluids in smooth circular lines is formulated. The Oldroyd-B model is utilized to analyze these shear thinning effects on these transients. The process of converting the analytical frequency domain model to the time domain using an inverse frequency algorithm commonly used in system identification is explained and demonstrated. The boundary conditions at the ends of the line are defined by the flow and pressure variables, which are in general functions of time or defined by causality relationships. Corresponding equations for the transient changes in the velocity profile and shear stress are also formulated. Two examples demonstrating the application versatility of the model and the sensitivity of the transients to the shear thinning parameters are included. For these specific examples, the sensitivity of the pressure and velocity transients is observed to be relatively low compared to the sensitivity of the wall shear stress. Insight into when the non-Newtonian complexities associated with shear thinning need to be included in a model for fluid transients considering the mode frequencies and/or the input frequencies is provided. The analytical model can easily be simplified for laminar flow and Newtonian fluids.

References

References
1.
Wahba
,
E. M.
,
2013
, “
Non-Newtonian Fluid Hammer in Elastic Circular Pipes: Shear-Thinning and Shear-Thickening Effects
,”
J. Non-Newtonian Fluid Mech.
,
198
, pp.
24
30
.
2.
Duan
,
F.
,
Kwek
,
D.
, and
Crivoi
,
A.
,
2011
, “
Viscosity Affected by Nanoparticle Aggregation in Al2O3-Water Nanofluids
,”
Nanoscale Res. Lett.
,
6
(
1
), p.
248
.
3.
AbuYousef
,
I. A.
,
Martinez
,
D. M.
,
Olson
,
J. A.
, and
Green
,
S.
,
2010
, “
Pumping Performance Increase Through the Addition of Turbulent Drag-Reducing Polymers to Pulp Fibre Suspensions
,”
ASME
Paper No. IMECE2010-37697
.
4.
Japper-Jaafar
,
A.
,
Escudier
,
M. P.
, and
Poole
,
R. J.
,
2010
, “
Laminar, Transitional and Turbulent Annular Flow of Drag-Reducing Polymer Solutions
,”
J. Non-Newtonian Fluid Mech.
,
165
(
19–20
), pp.
1357
1372
.
5.
Ptasinski
,
P. K.
,
Nieuwstadt
,
F. T. M.
,
Van Den Brule
,
G. H. A. A.
, and
Hulsen
,
M. A.
,
2001
, “
Experiments in Turbulent Pipe Flow With Polymer Additives at Maximum Drag Reduction
,”
J. Flow, Turbul. Combust.
,
66
, pp.
159
182
.
6.
Den Toonder
,
J. M. J.
,
Hulsen
,
M. A.
,
Kuiken
,
G. D. C.
, and
Nieuwstadt
,
F. T. M.
,
1996
, “
Drag Reduction by Polymer Additives in a Turbulent Pipe Flow: Numerical and Laboratory Experiments
,”
J. Fluid Mech.
,
337
, pp.
193
231
.
7.
Hullender
,
D. A.
,
Snyder
,
N.
, and
Gans
,
J.
,
2017
, “
Application of an Analytical Model for Simulating Hydraulic Systems Containing Internal Lines With Turbulent Flow
,”
ASME
Paper No. FPMC2017-4202.
8.
Jonášová
,
A.
, and
Vimmr
,
J.
,
2017
, “
Noninvasive Assessment of Carotid Artery Stenosis by the Principle of Multiscale Modelling of Non-Newtonian Blood Flow in Patient-Specific Models
,”
Appl. Math. Comput.
,
319
, pp.
598
616
.
9.
Mamun
,
K.
,
Rahman
,
M. M.
,
Akhter
,
M. N.
, and
Ali
,
M.
,
2016
, “
Physiological Non-Newtonian Blood Flow Through Single Stenosed Artery
,”
AIP Conf. Proc.
,
1754
(
1
), p.
040001
.
10.
Mei
,
C. C.
, and
Jing
,
H.
,
2016
, “
Pressure and Wall Shear Stress in Blood Hammer—Analytical Theory
,”
Math. Biosci.
,
280
, pp.
62
70
.
11.
Eshtehardi
,
P.
, and
Teng
,
Z.
,
2016
, “
Protective or Destructive: High Wall Shear Stress and
,”
Atheroscler.,” J. Atheroscler.
,
251
, pp.
501
503
.
12.
Vinoth
,
R.
,
Kumar
,
D.
,
Raaviraj
,
A.
, and
Shankar
,
V.
,
2016
, “
Non-Newtonian and Newtonian Blood Flow in Human Aorta: A Transient Analysis
,”
Biomedical Research, Dept. Electronics and Communication Eng, Manipal Inst. of Tech.
, Karnataka, India.
13.
Tazraei
,
P.
,
Riasi
,
A.
, and
Takabi
,
B.
,
2015
, “
The Influence of the Non-Newtonian Properties of Blood on Blood-Hammer Through the Posterior Cerebral Artery
,”
Math. Biosci.
,
264
, pp.
119
127
.
14.
Bird
,
R. B.
,
Armstrong
,
R. C.
, and
Hassager
,
O.
,
1987
,
Dynamics of Polymeric Liquids
, Vol.
1
,
2nd ed.
,
Wiley
,
New York
.
15.
Riaz
,
M. B.
,
Imran
,
M. A.
, and
Shabbir
,
K.
,
2016
, “
Analytic Solutions of Oldroyd-B Fluid With Fractional Derivatives in a Circular Duct That Applies a Constant Couple
,”
Alexandria Eng. J.
,
55
(
4
), pp.
3267
3275
.
16.
Sajid
,
M.
,
Zaman
,
A.
,
Ali
,
N.
, and
Siddiqui
,
M.
,
2015
, “
Pulsatile Flow of Blood in a Vessel Using an Oldroyd-B Fluid
,”
Int. J. Nonlinear Sci. Numer. Simul.
,
16
(
5
), pp.
197
206
.
17.
Casanellas
,
L.
, and
Ortin
,
J.
,
2011
, “
Laminar Oscillatory Flow of Maxwell and Oldroyd-B Fluids: Theoretical Analysis
,”
J. Non-Newtonian Fluid Mech.
,
166
(
23–24
), pp.
1315
1326
.
18.
Tazraei
,
P.
, and
Riasi
,
A.
,
2016
, “
Quasi-Two-Dimensional Numerical Analysis of Fast Transient Flow Considering Non-Newtonian Effects
,”
ASME J. Fluids Eng.
,
138
(
1
), p.
011203
.
19.
Weinerowska-Bords
,
K.
,
2015
, “
Alternative Approach to Convolution Term of Viscoelasticity in Equations of Unsteady Pipe Flow
,”
ASME J. Fluids Eng.
,
137
(
5
), p.
054501
.
20.
Bizhani
,
M.
, and
Kuru
,
E.
,
2015
, “
Modeling Turbulent Flow of Non-Newtonian Fluids Using Generalized Newtonian Models
,”
ASME
Paper No. OMAE2015-41427
.
21.
Roy
,
A. S.
,
Back
,
L. H.
, and
Banerjee
,
R. K.
,
2005
, “
Guidewire Flow Obstruction Effect on Pressure Drop-Flow Relationship in Moderate Coronary Artery Stenosis
,”
J. Biomech.
,
39
(
5
), pp.
853
864
.
22.
Meniconi
,
S.
,
Duan
,
H. F.
,
Brunone
,
B.
,
Ghidaoui
,
M. S.
,
Lee
,
P. J.
, and
Ferrante
,
M.
,
2014
, “
Further Developments in Rapidly Decelerating Turbulent Pipe Flow Modeling
,”
ASCE J. Hydraul. Eng.
,
140
(
7
), p.
04014028
.
23.
Pezzinga
,
G.
,
Brunone
,
B.
,
Cannizzaro
,
D.
,
Ferrante
,
M.
,
Meniconi
,
S.
, and
Berni
,
A.
,
2014
, “
Two-Dimensional Features of Viscoelastic Models of Pipe Transients
,”
ASCE J. Hydraul. Eng.
,
140
(
8
), p.
04014036
.
24.
Azouz
,
I.
, and
Shirazi
,
S. A.
,
1998
, “
Evaluation of Several Turbulence Models for Turbulent Flow in Concentric and Eccentric Annuli
,”
ASME J. Energy Resour. Technol.
,
120
(
4
), pp.
268
275
.
25.
Anderson
,
A.
, and
Bergant
,
A.
,
2008
, “
Issues in ‘Benchmarking’ Fluid Transients Software Models
,”
BHR Group, Tenth International Conference on Pressure Surges
, Edinburgh, UK, May 14–16, pp.
519
537
.
26.
Adamkowski
,
A.
, and
Lewandowski
,
M.
,
2006
, “
Experimental Examination of Unsteady Friction Models for Transient Pipe Flow Simulation
,”
ASME J. Fluids Eng.
,
128
(
6
), pp.
1351
1363
.
27.
Pinho
,
F. T.
, and
Whitelaw
,
J. H.
,
1990
, “
Flow of Non-Newtonian Fluids in a Pipe
,”
J. Non-Newtonian Fluid
,
34
(
2
), pp.
129
144
.
28.
Shemer
,
L.
,
Wygnanski
,
I.
, and
Kit
,
E.
,
1985
, “
Pulsating Flow in a Pipe
,”
J. Fluid Mech.
,
153
(
1
), pp.
313
337
.
29.
Shemer
,
L.
, and
Kit
,
E.
,
1984
, “
An Experimental Investigation of the Quasi-Steady Turbulent Pulsating Flow in a Pipe
,”
Phys. Fluids
,
27
(
1
), pp.
72
76
.
30.
Nouri
,
J. M.
,
Umur
,
H.
, and
Whitelaw
,
J. H.
,
1993
, “
Flow of Newtonian and Non-Newtonian Fluids in Concentric and Eccentric Annuli
,”
J. Fluid Mech.
,
253
(
1
), pp.
617
641
.
31.
Escudier
,
M. P.
,
Gouldson
,
L. W.
, and
Jones
,
D. M.
,
1994
, “
Flow of Shear-Thinning Fluids in a Concentric Annulus
,”
Exp. Fluids
,
18
(
4
), pp.
225
238
.
32.
Warholic
,
M. D.
,
Massah
,
H.
, and
Hanratty
,
T. J.
,
1999
, “
Influence of Drag-Reducing Polymers on Turbulence: Effects of Reynolds Number, Concentration and Mixing
,”
Exp. Fluids
,
27
(
5
), pp.
461
472
.
33.
Warholic
,
M. D.
,
Heist
,
D. K.
,
Katcher
,
M.
, and
Hanratty
,
T. J.
,
2001
, “
A Study With Particle-Image Velocimetry of the Influence of Drag-Reducing Polymers on the Structure of Turbulence
,”
Exp. Fluids
,
31
(
5
), pp.
474
483
.
34.
Martin
,
J. R.
, and
Shapella
,
B. D.
,
2003
, “
The Effect of Solvent Solubility Parameter on Turbulent Flow Drag Reduction in Polyisobutylene Solutions
,”
Exp. Fluids
,
34
(
5
), pp.
535
539
.
35.
He
,
S.
,
Ariyaratne
,
C.
, and
Vardy
,
A. E.
,
2007
, “
A Computational Study of Wall Friction and Turbulence Dynamics in Accelerating Pipe Flows
,”
Comput. Fluids
,
37
(
6
), pp.
674
689
.
36.
Ivannikov
,
V. G.
, and
Rozenberg
,
G. D.
,
1973
, “
Experimental Study of the Attenuation of Pressure Waves in Flows of Weak Polyacrylamide Solutions
,”
Inzhenerno-Fizicheskii Zh.
,
25
(
6
), pp.
1045
1049
.
37.
Szymkiewicz
,
R.
, and
Mitosek
,
M.
,
2014
, “
Alternative Convolution Approach to Friction in Unsteady Pipe Flow
,”
ASME J. Fluids Eng.
,
136
(
1
), p.
011202
.
38.
Woods
,
R. L.
,
Hsu
,
C. H.
,
Chung
,
C. H.
, and
Keyser
,
D. R.
,
1983
, “
Comparison of Theoretical and Experimental Fluid Line Responses With Source and Load Impedances
,”
Fluid Transmission Lines Dynamics
,
ASME Special Publication II
,
New York
.
39.
Stephens
,
M. L.
,
Lambert
,
A. M.
,
Simpson
,
A. R.
, and
Vítkovský
,
J. P.
,
2011
, “
Calibrating the Water Hammer Response of a Field Pipe Network by Using a Mechanical Damping Model
,”
ASCE J. Hydraul. Eng.
,
137
(
10
), pp.
1225
1237
.
40.
Bergant
,
A.
,
Simpson
,
A. R.
, and
Vítkovský
,
J.
,
2001
, “
Developments in Unsteady Pipe Flow Friction Modelling
,”
ASCE J. Hydraul. Res.
,
39
(
3
), pp.
249
257
.
41.
Chen
,
H. Y.
,
Liu
,
H. J.
,
Chen
,
J. H.
, and
Liu
,
S.
,
2012
, “
Numerical Simulation of Fluid Transients by Chebyshev Super Spectral Viscosity Method for Propellant Lines
,”
J. Propul. Technol.
,
33
(
5
), pp.
804
808
.
42.
Bizhani
,
M.
,
Corredor
,
F. F. R.
, and
Kuru
,
E.
,
2014
, “
An Experimental Study of Turbulent Non-Newtonian Flow in Concentric Annuli Using Particle Image Velocimetry Technique
,”
Flow Turbul. Combust.
,
94
(
3
), pp.
527
554
.
43.
Knisely
,
C. W.
,
Nishihara
,
K.
, and
Iguchi
,
M.
,
2010
, “
Critical Reynolds Number in Constant-Acceleration Pipe Flow From an Initial Steady Laminar State
,”
ASME J. Fluids Eng.
,
132
(
9
), p.
091202
.
44.
Book
,
W. J.
, and
Watson
,
C.
,
2000
, “
Alternatives in the Generation of Time Domain Models of Fluid Lines Using Frequency Domain Techniques
,”
Math. Comput. Simul.
,
53
(
4–6
), pp.
353
365
.
45.
Brunone
,
B.
, and
Morelli
,
L.
,
1999
, “
Automatic Control Valve Induced Transients in an Operative Pipe System
,”
ASCE J. Hydraul. Eng.
,
125
(
5
), pp.
534
542
.
46.
Soumelidis
,
M. I.
,
Johnston
,
D. N.
, and
Edge
,
K. A.
,
2005
, “
A Comparative Study of Modeling Techniques for Laminar Flow Transients in Hydraulic Pipelines
,”
JFPS Int. Symp. Fluid Power
,
6
, pp.
100
105
.
47.
Kojima
,
E.
,
Yamazaki
,
T.
, and
Shinada
,
M.
,
2006
, “
Development of a New Simulation Technique Based on the Modal Approximation for Fluid Transients in Complex Pipeline Systems With Time-Variant Nonlinear Boundary Conditions
,”
ASME J. Fluids Eng.
,
129
(
6
), pp.
791
798
.
48.
Vítkovský
,
J. P.
,
Lee
,
P. J.
,
Zecchin
,
A. C.
,
Simpson
,
A. R.
, and
Lambert
,
M. F.
,
2011
, “
Head-and Flow-Based Formulations for Frequency Domain Analysis of Fluid Transients in Arbitrary Pipe Networks
,”
ASCE J. Hydraul. Eng.
,
137
(
5
), pp.
556
568
.
49.
Goodson
,
R. E.
, and
Leonard
,
R. G.
,
1971
, “
A Survey of Modeling Techniques for Fluid Line Transients
,”
ASME J. Basic Eng.
,
94
(
2
), pp.
474
482
.
50.
Hsue
,
C. Y.
, and
Hullender
,
D. A.
,
1983
, “
Modal Approximations for the Fluid Dynamics of Hydraulic and Pneumatic Transmission Lines
,”
Fluid Transmission Lines Dynamics
,
ASME Special Publication II
,
New York
.
51.
Hullender
,
D. A.
, and
Woods
,
R. L.
,
1983
, “
Time Domain Simulation of Fluid Transmission Lines Using Minimum Order State Variable Models
,”
Fluid Transmission Lines Dynamics
,
ASME Special Publication, II
,
New York
.
52.
Wongputorn
,
P.
,
2001
, “
Time Domain Simulation of Systems With Fluid Transmission Lines
,” M.S. thesis, The University of Texas at Arlington, Arlington, TX.
53.
Wongputorn
,
P.
,
Hullender
,
D.
,
Woods
,
R.
, and
King
,
J.
,
2003
, “
A Simplified Method for Formulating the Simulation Diagram for Systems Containing Lines With Fluid Transients
,”
ASME
Paper No. FEDSM2003-45246
.
54.
Wongputorn
,
P.
,
Hullender
,
D.
,
Woods
,
R.
, and
King
,
J.
,
2005
, “
Application of MATLAB Functions for Simulation of Systems With Lines With Fluid Transients
,”
ASME J. Fluids Eng.
,
127
(
1
), pp.
177
182
.
55.
Hullender
,
D. A.
,
2016
, “
Alternative Approach to Modeling Transients in Smooth Pipe With Low Turbulent Flow
,”
ASME J. Fluids Eng.
,
138
(
12
), p.
121202
.
56.
McGinty
,
S.
,
McKee
,
S.
, and
McDermott
,
R.
,
2009
, “
Analytic Solutions of Newtonian and Non-Newtonian Pipe Flows Subject to a General Time-Dependent Pressure Gradient
,”
J. Non-Newtonian Fluid Mech
,
162
(
1–3
), pp.
54
77
.
57.
Nursilo
,
W. S.
,
2000
, “
Fluid Transmission Line Dynamics
,” Ph.D. dissertation, The University of Texas at Arlington, Arlington, TX.
58.
Hullender
,
D. A.
,
2018
, “
Water Hammer Peak Pressures and Decay Rates of Transients in Smooth Lines With Turbulent Flow
,”
ASME J. Fluids Eng.
,
140
(
6
), p.
061204
.
You do not currently have access to this content.