Yamaguchi H. Engineering Fluid Mechanics (Fluid Mechanics and Its Applications)
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6 Newtonian Flow
,
fluctuating component
time average
Subscripts
e
entrance value
m
model
p
prototype
x
value based on axial position
w
wall value
0
initial value or base value
f
value at infinity
21,
inner and outer respectively
Bibliography
Recent development of Newtonian viscous fluid flow is well documented
together with fundamental concepts in fluid mechanics, including the flow
instability, in
1.
F.M. White, Viscous Fluid Flow (3rd Edition), McGraw Hill Interna-
tional, New York, 2006.
2.
F.S. Sherman, Viscous Flow, McGraw Hill Publishing Company, New
York, 1990.
The most classic text book in boundary layer theory is
3.
H. Schlichting, Boundary-Layer Theory (7th Edition 1979), McGraw-
Hill book company Inc., New York, 1955.
4.
L. Rosenhead, Laminar Boundary Layer, Oxford University, Oxford,
Press, 1963.
5.
D. Meksyn, New Methods in Laminar Boundary Theory, Pergamon,
London, 1961.
Basic treatment on incompressible viscous flows with details on dimen-
sional analysis is found in
6.
R.L. Panton, Incompressible Flow (2nd Edition), John Willey & Sons,
Inc., Hoboken, NJ, 1996.
Extensive literature review for developing flows are found in
7.
A.J. Word-Smith, Internal Fluid Flow, Clarendon Press, England,
1980.
394
Bibliography
Experimental techniques on Newtonian fluid flow and visualization meth-
ods are documented in
8.
W.J. Yang, Handbook of Flow Visualization, Hemisphere Publishing
Corp., New York, 1989.
Flow phenomena, specific data, correlations and approximations referred
in the text are presented in
9.
J. Boussinesq, Thiorie Analytique de la chaleur, 2, 172, Gauthier-
Villars, Paris, 1903.
10.
P.L. Silveston, Warmedurchgang in waagerechten flüssigkeitsschichten,
Part I Forsch. Ing. Wes. 24,29–32 and 59–69, 1958.
11.
H.S. Hele-Shaw, Investigation of the nature of surface resistance of
water and of stream motion under certain experimental conditions,
Frans. Inst. Naval Arch., 11, 1898.
12.
G.G. Stokes, Mathematical and physical papers I, 75, Trans. Camb.
Soc., 8, 1845.
13.
C.W. Oseen, On the Stokes’ problem and on the fundamental problems
of hydrodynamics (German), Ark. F. Mat. Astr. og Fys., 6 (29), 1910.
14.
J. Boussinesq, Comptes Reudus, 9, 1891.
15.
Prandtl-Tietjens, Hydro und Aeromechanics, Bd.II, 1931.
16.
R.K. Shah and A.L. London, Laminar Flow Forced Convection in
Ducts, Academic Press, New York, 1978.
17.
J. Nikuradse, Strömungsgesetze in rauhen Rohen, Forsch. Arb. Ing.-
Wes., (361), 1933.
21.
H. Blasius, Grenzschichten in flüssikeiten mit kleiner reibung,
1908.
22.
T. von Kàrmàn, Über den mechanismus des widerstandes, den ein
bewegter Körger in einer flüssigkeit erzeugt, Nachr. Ges. Wiss.
Göttingen Math. –Phys. Kl.II, 1921.
Z. Angew. Math. Phys., 56 [NACA Technical Memo.1256 in English],
395
18. L. Prandtl, Über flussigkeitsbewegung bei sehr kleiner reibung.
Verhundlungen des III. Internationalen Mathematiker-Kongrees,
Heidelberg, 1904.
19. L. Prandtl, Über flussigkeitsbewegung bei sehr kleiner reibung,
Proc.Third Int. Math. Congr. Heidelberg (NACA Technical Memo.452
in English), 1908.
20. L. Prandtl, Über die ausgebildete turblenz, Z. Angew. Math. Mech., 5,
1925.
6 Newtonian Flow
25. D.E. Coles and E.A. Hirst (editors), Proceedings on Computational
Turbulent Boundary Layers, 2, Stanford University Press, Stanford,
CA, 1968.
26.
E.R. van Driest, On turbulent flow near a wall, J. Aeronaut. Sci., 23,
1956.
28.
H. Tennekes and J.L. Lumley, A First Course in Turbulence,
M.I.T.Press, Cambridge, MA, 1972.
29.
G.S. Ambrok, Approximate solutions of equations for the thermal
boundary layer with variations in the boundary layer structure, Soviet
Phys. -Tech. Phys., 2 (9), 1957.
W.P. Jones and B.E. Lunder, The prediction of laminarization with a
two-equation model of turbulence, Int. J. Heat Mass Transfer, 15, 1972.
33.
F. Szymanski, Quelques solutions exactes des equations de
l’hydrodynamique de fluide visqueux dans le cas d’un tuke cylindrique,
J., Math., pure Appl., Ser. 9, 11, 1932.
34.
T.C. Papanastasiou, G.C. Georgiou and A.N. Alexandrou, Viscous
Fluid Flow, CRC Press LLC, Boca Raton, FL, 2000.
35.
K. Hiemez, Die Grenzschicht an einem in den gleich förmigen
Flüssigkeittsstrom eingretauchen geraden Kreiszylinder, Dingler’s
Polytech. J., 326, 1911.
36.
E. Achenback, Influence of surface roughness on the cross-flow
around a circular cylinder, J. Fluid Mech., 46, 1971, and Experiments
on the flow past spheres at very high Reynolds number, J. Fluid Mech.,
54, 1972.
27.
S.J. Kline, M.V. Morkovin, G. Sovran and D.J. Cockrell, Computation
of Turbulent Boundary Layers, AFOSR-IFP Stanford conference,
Proceedings of the 1968 Conference, 1, Stanford University Press,
Stanford, CA, 1968.
31.
B.A. Kader, Temperature and concentration profile in fully turbulent
boundary layers, Int. J. Heat Mass transfer, 24(9), 1981.
32.
R.B. Bird, W.E. Stewart and E.N. Lightfoot, Transport Phenomena
(2nd Edition) John Wiley & Sons, Inc., Hoboken, NJ, 2002.
30.
23.
H.W. Liepmann, Investigation on curved boundary layer stability and
transition on curved boundaries, NACA Wartime Report W107 (ACR
3H30) or [NACA Technical Memo.1196], 1943.
24.
D.C. Wilcox, Turbulence Modeling for CFD (2nd Edition), DCW In-
dustries, La Cañada, CA, 1998.
37.
M. Couette, Etudes sur le frottement des liquides, Ann. Chim. Phys.,
ser. 6, 21, 433–510.
38.
G.I. Taylor, Stability of a Viscous Liquid Contained between Two
Rotating Cylinders, Philosophical Transactions of the Royal Society of
London. Series A, Containing Papers of a Mathematical or Physical
Character, 223, 289–343,1923.
396
The vorticity vortex dynamics is well documented with details in
J.Z. Wu, H.Y. Ma and M.D. Zhou, Vorticity and Vortex Dynamics,
Springer, New York, 2006.
Bibliography
39. O. Reynolds, On the Theory of Lubrication and Its Application to
Mr. Beauchamp Tower’s Experiments, Including an Experimental
Determination of the Viscosity of Olive Oil, Philosophical
Transactions of the Royal Society of London, 177, 157–234, 1886.
40.
O. Reynolds, On the dynamical theory of incompressible viscous
fluids and the determination criterion. Philosophical Transactions of
the Royal Society of London, 14, 123–164, 1895.
41.
C.F. Colebrook, Turbulent flow in pipes with particular reference to
the transition region between the smooth and rough pipe laws.
42.
W.P. Jones and B.E. Launder, The Prediction of Laminarization with a
Two-equation Model of Turbulence, Int. J. Heat Mass Transfer, 15,
301–314, 1972.
43.
397
J. Institution Civil Engineers, 1939.
7. Non-Newtonian Fluid and Flow
In a general sense, fluids that exhibit characters not predicted by the New-
tonian constitutive equation given in Eq. (6.4) are non-Newtonian. The ex-
ceptions to the Newtonian fluids are not of rare occurrence, and in fact
many common fluids are non-Newtonian. Some examples are: paints, solu-
tions of various polymers and molten plastics; food products such as apple
sauce, ketchup and other mammalian whole foods; synovial fluid found in
joints, blood and other organic fluids; many solid-liquid and liquid-liquid
suspensions such as fibers in a liquid paper pulp, coal slurries, emulsions
of water in oil or oil in water, and so on. The so-called non-Newtonian flu-
ids, as mentioned above, are often found in many fields of engineering
fluid mechanics as well as in bio-medical fields, and exhibit interesting,
useful and even exciting characteristics differed from those found in New-
tonian fluids.
Many difficulties are encountered to predict non-Newtonian flow due
to the reason that the theoretical predictions are usually based upon the use
of idealizing rheological fluid properties with associated constitutive equa-
tions that are often difficult to verify under conditions of complex flows.
However, in recent years extensive efforts have been carried out to con-
struct more general constitutive equations and to calculate the flows of
practical configurations by using high-speed computers. Knowledge
gained from CFD efforts presents the greatest amount of insight into many
unexpected flow phenomena and explain the causes. The discovery, for ex-
ample, so-called Toms effect (1948), that high molecular weight additives
could lower the friction factor for flow of a polymer solution below that of
the Newtonian solvent, is recognized nowadays that the drag reduction is a
much wider spread phenomenon than originally thought. These problems
are studied extensively, although not being quoted in the text, in conjunc-
tion with turbulence structure and flow instability with an aid of CFD.
In this chapter we will begin to describe characteristics and a rheologi-
cal classification of non-Newtonian fluids. Then we will proceed to de-
scribe the standard flows, which are commonly used to characterize the
rheology of non-Newtonian fluids. Subjecting a fluid to a standard flow we
will then be able to define material functions, which are to be derived from
a constitutive equation. After this initial background, we will study some
399
7 Non-Newtonian Fluid and Flow
of the non-Newtonian constitutive equations with applications to basic en-
gineering flows.
7.1 Non-Newtonian Fluids and Generalized Newtonian
Fluid Flow
There exists a variety of substances for which the mechanical properties
differ from Newton’s viscous laws to a greater or lesser extent. In the pre-
sent section, we shall discuss the rheological characteristics of non-
Newtonian fluids that are chiefly based on a shear stress and their associ-
ated viscosity. Viscoelasticity is also introduced later in this section for fur-
ther interpretations into non-Newtonian fluids.
Some typical flows are studied in view of engineering applications in
connection with a discussion of rheological characteristics, considering an
interpretation of generalized Newtonian viscous laws.
7.1.1 Rheological Classifications
We shall deal with incompressible medium of non-Newtonian fluids as
commonly found in practice of engineering. The rheological characteriza-
tion is done by experimental measurements. To convey the measurements
of the subject, one approach specifies a deformation imposed on by a fluid
and then measures the stresses generated by the flowing fluid through a
specified channel geometry.
(i) Pure viscous non-Newtonian fluids
The most of the non-Newtonian fluids that we encounter practically
usually fall into this category. These fluids are also termed as time-
independent fluids (fully reversible without time-lag), and are subdivided
into several groups, as graphically shown by Fig. 7.1. A Newtonian fluid is
defined by Newton’s law of viscosity, with reference to Eq. (6.1) as fol-
lows
JKKW
y
u
yx
(7.1.1)
where
K
is the viscosity and
J
the shear rate. It is noted that Fig. 7.1(a) is
called the flow curve, which is the plot of
yx
W
and
J
, the shear stress ver-
sus the shear rate. Similarly, Fig. 7.1(b) is a plot of
K
and
J
, the shear
400
7.1 Non-Newtonian Fluids and Generalized Newtonian Fluid Flow
viscosity versus the shear rate.
K
is a function of
J
and is sometimes
called the apparent viscosity
a
K
, which is defined, from Eq. (7.1.1), as
J
W
K
yxa
(7.1.2)
In case of a Newtonian fluid,
K
=
a
K
=
0
K
=constant, as is indicated in Fig.
7.1(b).
Fig. 7.1 Pure viscous Non-Newtonian fluids
A. Pseudoplastic fluids (shear thinning fluids)
Pseudoplastic fluids, as the ones indicated on curve A in Fig. 7.1(a) and
(b), are characterized by a decreasing slop of a flow curve; equivalently,
401
7 Non-Newtonian Fluid and Flow
the apparent viscosity decreases with increasing shear rate. The pseudo-
plastic fluid often has a rheological behavior that at very low and very high
shear rates the apparent viscosity becomes constant; respectively, we have
0
K
(called zero-shear rate viscosity) and
f
K
, regions which are respectively
called the first Newtonian region and the second Newtonian region, where
the flow curves are linear. Pseudoplastic fluids are found in many real flu-
ids, such as polymer melts and solutions, starch paste, and glass melt.
There are a number of empirical relationships, among which the sim-
plest is the power law suggested by Ostwald (1925). It may be written as
JJW
1
n
xy
m
(7.1.3)
where m and n are constant for a particular fluid. The fluids described in
Eq. (7.1.3) are generally called power law fluids. In particular, when
n <1,
the fluid shows the shear thinning behavior and is characterized as a
pseudoplastic fluid. It appears that for a Newtonian fluid
n =1 and
0
K
m
.
Equation (7.1.3) is the most widely used in a less rigorous manner in many
practical applications. There are some difficulties that should be kept in
mind in applications of Eq. (7.1.3); one such instant occurs where the di-
mensions of
m depend upon n ; therefore, m is not a material property.
Another difficulty is that for real fluids, as mentioned above,
n
may not be
constant over the entire range of
J
, with which the apparent viscosity
a
K
is written as
1
n
a
m
JK
(7.1.4)
However,
n
is effective in a limited range of
J
for
m
to be constant.
Many other empirical formulae have been proposed to overcome some
of the objections of the power law. One example is given where
Ree-Eyring;
JEW
i
m
i
ixy
C
¦
1
1
sinh
(7.1.5)
i
C and
i
E
are constants to be determined as specific molecular parameters.
Some rheological formulae for the apparent viscosity are
Carreau-Yasuda;
>
@
a
n
a
1
0
1
JO
KK
KK
(7.1.6)
where
0
K
is the zero-shear rate viscosity,
a
K
K
0
as 0o
J
, a is a con-
stant and
O
is the time constant.
402
7.1 Non-Newtonian Fluids and Generalized Newtonian Fluid Flow
Oldroyd;
¸
¸
¹
·
¨
¨
©
§
2
2
2
1
0
1
1
J
J
K
K
C
C
(7.1.7)
Here,
1
C and
2
C
are constants.
B. and C. Bingham-plastic and viscoplastic fluids
often characterized by the generalized Bingham model, which is written as
JJWW
1-n
yxy
m
(7.1.8)
where
y
W
is termed as the yield stress. It should be kept in mind that the
yield stress is a finite stress required to initiate flow and represents the
value of
xy
W
as 0o
J
. The Bingham plastics (fluids) behave like a straight
line in a flow curve after the yield stress, as indicated in curve B in Fig. 7.1,
with the setting
1 n
and
p
m
K
in Eq. (7.1.8). The formula of a Bing-
ham plastic (fluid) is thus expressed where
JKWW
pyxy
(7.1.9)
p
K
is called the plastic viscosity. Some real fluids behave like the Bing-
ham plastics, such as slurries, plastics, emulsions such as paint, and sus-
pensions of finely divided solids in a liquid. Some electro-rheological flu-
ids (ER fluid) have been found to be closely approximated to the behavior
of a Bingham fluid, in which the yield stress is a function of the strength of
electric field.
D. Dilatant fluids (shear thickening fluids)
Dilatant fluids, as indicated in curve D in Fig. 7.1(a) and (b), are simi-
lar to pseudoplastic fluids, but differ in that the slop of the flow curve de-
ceases with an increasing of the shear rate, and equivalently, the apparent
viscosity increases with increasing of the shear rate. These fluids are less
common than with pseudoplastic fluids. As with pseudoplastic fluids, they
are represented by the power law found in Eq. (7.1.3), where
1!n . The di-
latant fluids have been found to closely approximate the behavior of some
real fluids, such as starch in water and an appropriate mixture of sand and
water. The term dilatant is also used to describe volumetric dilatancy,
which is the tendency of suspensions to expand in a volume during flow-
ing situations. It must be noted that dilatant fluids in this text are rheologi-
cal dilatancy.
Viscoplastic fluids, as indicated on curve C in Fig. 7.1(a) and (b), are
403
7 Non-Newtonian Fluid and Flow
(ii) Thixotropic and rheopectic fluids
There are some fluids, which are more complex in structure than those
found in time independent (pure viscous) fluids, and to which the apparent
viscosity depends not only on the shear rate, but also on the time when the
shear has been applied or ceased. For example, in a certain colloidal sys-
tem, more particularly in a suspension, the apparent viscosity decreases
during the mixing process, but on cessation of the shearing the viscosity
recovers to its original value.
Fig. 7.2 Time dependent fluids
There are two general classes of such fluids: A. thixotropic fluids, in
which the shear stress decreases with time as the fluids are sheared; and B.
rheopectic fluids, in which the shear stress increases with time as the fluids
are sheared. Figure 7.2 shows a schematic of the thixotropic and rheopectic
fluid behavior in a symmetrical manner. Figure 7.2 is the simplest case of a
symmetrical thixotropy or rheopexy in which the structure of the fluid is
broken down (thixotropic fluid) or built up (rheopectic fluid) under in-
creasing duration of applying a constant shear rate. In case of the symmet-
rical thixotropy, on cessation of shearing the structure can build up at the
same rate as the break-down process. Therefore, it is symmetrical in this
sense; the opposite type is the symmetrical rheopexy. There are many cases
in which the structure does not recover completely upon the cessation of
shearing, often indicating hysteresis, and as a result, some factors other
than time, such as total strain, may be influential. Certain type of ink
shows thixotropy. Other examples include the beating and thickening of
egg whites, which shows rheopexy. It appears that many substances,
404