Table Of ContentNON-NEWTONIAN
FLUID
MECHANICS
G. BOHME
Hochschule der Bundeswehr Hamburg
Hamburg, F.R.G.
1987
NORTH-HOLLAND-AMSTERDAM NEW YORK OXFORD · TOKYO
©ELSEVIER SCIENCE PUBLISHERS B.V., 1987
All rights reserved. No part of this publication may be reproduced,
stored in a retrieval system, or transmitted, in any form or by any means,
electronic, mechanical, photocopying, recording or otherwise, without the prior
permission of the copyright owner.
ISBN: 0 444 70186 9
Translation of:
Stromungsmechanik nicht-newtonscher Fluide
©Teubner, Stuttgart, 1981
Translated by J .C. Harvey
Publishers:
ELSEVIER SCIENCE PUBLISHERS B.V.
P.O. Box 1991
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The Netherlands
Sole distributors for the U.S.A. and Canada:
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Library of Congress Cataloging-in-Publication Data
Bohme, G. (Gert), 19^2-
Non-Newtonian fluid mechanics.
(North-Holland series in applied mathematics and
mechanics ; v. 31)
Translation of: Stromungsmechanik nicht-newtonscher
Fluide.
Bibliography: p.
Includes index.
1. Non-Newtonian fluids. 2. Fluid mechanics.
I. Title. II. Series.
QA929.5.B61H3 198T 532 86-32937
ISBN 0-M*l*-70l86-9 (U.S.)
PRINTED IN THE NETHERLANDS
ν
PREFACE
This book is a translation of the German textbook 1Stromungsmechanik
nicht-Newtonscher Fluide' published by B.G. Teubner, Stuttgart, 1981.
Because of its friendly reception by readers I was encouraged to bring
out an English edition in order to reach a wider range of readers.
Dr. J.C. Harvey of Yelverton, Devon, England has contributed to this
project not only by translating the text, but also by producing the
camera-ready copy, for which I am very grateful.
The book has its origin in lectures which I have several times given
to engineering students after their intermediate diploma examination,
and to co-workers in Hamburg who are interested in this subject. The
book is intended for use in technical universities, and as a help to
practising engineers who are involved with flow problems of non-Newtonian
fluids.
The treatment of the subject is based throughout on continuum mechan
ics model concepts and methods. Because in non-Newtonian fluids the
material properties operating depend critically on the kinematics of
the flow, special attention is paid to the deriving and explanation
of the adequate constitutive equations used. Thus I avoid as much
as possible formal arguments, but instead use obvious arguments and
I give detailed comment on the Theological law concerned, before a flow
is analysed thoroughly, and the study of the chosen Theological process
is brought up to definite results.
In order to ensure that the book can be read without reference to
other sources it is necessary initially to consider some general prin
ciples of continuum mechanics. After this I begin with the study of
simple motions, namely steady and unsteady shear flows, and I then pro
ceed by degrees to kinematically more complex motions. Thus at the
start I deal with the topic in greater detail and later I treat the
topics rather more briefly. Problems of various degrees of difficulty
at the end of each chapter invite active participation by the reader.
Numerous stimulating topics from the literature are considered in
the book. I have however not allowed myself to present the problems
dealt with in a more or less disconnected sequence, but have always
been concerned to work out what is essential from a didactic viewpoint,
vi Preface
to omit unnecessary ancillary details, and to bring the individual parts
together into a unified whole. Hence many times merely the formulation
of a problem and the result of a contribution were useful for my purpose.
The solution method has always been harmonised with the methods desc
ribed in the book. Therefore I have refrained from quoting the many
journal articles which refer to the material being described and which
I have found useful. It is impossible to give a complete reference
list.
'Everything' cannot be included in a textbook, i.e. the author has
to make a careful choice of his material. Thus I have considered only
laminar flows throughout, and I have in the actual applications mostly
assumed the fluid to be incompressible. This means no significant
limitation when dealing with the highly viscous substances such as occur
in plastics technology and processing techniques. It is rather differ
ent in the case of the strict limitation to one-phase flows. Multi
phase flows with participation of a non-Newtonian fluid could not be
considered for lack of space. Thermal effects had to be treated relat
ively briefly. The reader who has read the book thoroughly should
however immediately be able to apply himself to a study of these and
other topics in non-Newtonian fluid mechanics.
Hamburg, October 1986 G. Bohme
χ
LIST OF THE MOST IMPORTANT SYMBOLS
Symbol Dimension1 Meaning
-2
a LT Acceleration vector
2
A L Area
A T~n nth Rivlin-Ericksen tensor (n = integer)
η
b L Width, semi-axis of ellipse
Br - Brinkman number
c L^"^"1 Specific heat
c LT 1 Wave speed
c L Semi-axis of ellipse
C - Relative right-Cauchy-Green tensor
fc
d L Diameter
D Τ 1 Strain rate tensor
2 -2
e L Τ Specific internal energy
e - Unit vector
Ε - Unit tensor
f - Normalised flow function
-2 -2
f ML Τ Force per unit volume
F MLT"2 Force
F - Relative deformation gradient
-2
g LT Acceleration caused by gravity
-1 -2
G ML Τ Linear viseoelastic influence function
h L Height
-1 -2
Gi*i=G,'i+iG" ML Τ ICnovmaprlieaxn tssh eoafr amo sdyumlmuest rical tensor
r 2 3 -2 -2
k ML Τ Pressure drop per unit length
Κ - Dimensionless pressure parameter
I L Length
L τ"1 Velocity gradient tensor
2 -2
Μ ML Τ Torque
Μ - Mean molecular weight
-1 -2
Ν,Η Normal stress functions
χ 2 ML Τ
-1 -2
Ρ ML Τ Pressure
1 Mass [M], length [L], time [T] and temperature [Θ] are basic quantities
in the International System of units.
List of the most important symbols xi
Ρ Laplace transform variable
2 -3 Power
Ρ ML Τ3 Heat flux density vector
q ΜΤ"3 Dimensionless discharge
Q Cylindrical coordinate, radius
r L Spherical coordinate
R L Reynolds number
Re Time delay
s Τ Dimensionless velocity parameter
S Stress tensor
-1 -2
S ML Τ Sommerfeld number
So
Stokes number
St
Time
t Τ
-1 -2 Extra-stress tensor
ML Τ
Τ
-1 Cartesian components of velocity
u,v,w LT Mean velocity
ΰ LT"1 Constant reference velocity
U LT"1 Velocity vector
Volume
LT'1
Volume flux
V L3
3 -1 Vorticity tensor
V L^T Weissenberg number
W τ"1 Cartesian coordinates
We Second order material coefficients
x,y,z L Angle
ot,a ML Second flow function
1 2
Third order material coefficients
3
β ML"1 Angle of shear
Shear rate
e,3,3 ML_1T
1 2 3
Ύ Boundary layer displacement thickness
-1
Dimensionless displacement thickness
γ Τ Laplace operator
δ* L Elongation rate
Δ Shear viscosity
Δ L-2 Lower Newtonian limiting viscosity, zero-shear
έ Τ"1 viscosity
η ML~1T"1
n ML"1!"1
0
xii List of the most important symbols
-1 -1
Upper Newtonian limiting viscosity
ML Τ
Reference value for viscosity, in general =
η*
ML"1!'1
η*=η'-ίη" Complex viscosity
-1 -1
η - Pumping efficiency
ΗΕ'ΗΕΒ'ΗΕΡ MMLL" Τ1? "1 Elongational viscosities
θ - Dimensionless temperature difference
α
Spherical coordinate, angle
Θ Θ Absolute temperature
λ MLT"3e~ Thermal conductivity
λ Τ Relaxation time, material specific time scale
VV2 ML"1 Normal stress coefficients
π - Dimensionless power parameter
Ρ ML"3 Density
σ - Normalised shear stress = τ/τ
-1 -2
σ. . ML Τ Total stresses, elements of S
-1 -2
τ ML Τ Shear stress
-1 -2
τ Stress reference value
* ML Τ
Extra-stresses, elements of Τ
τ. .
-1 -2
φ 1J Cylindrical coordinate, angle
Φ ML- 1Τ -3 Dissipation function
ML Τ
Φ - Reduced discharge in a pipe flow
ψ Λ"1 Stream function for plane flows
ψ LV1 Stream function for rotational symmetric flows
ω Τ"1 Angular velocity
-1 -3
2 Potential of the extra-stresses of generalised
ML ΤJ
Newtonian fluids
1
/ PRINCIPLES OF CONTINUUM MECHANICS
1.1 Basic concepts
Fluid mechanics is a field theory in the broad sense. It describes
the observed phenomena by considering the material as a continuum.
The true atomic structure of the material is not considered. This
modelling assumes that the molecular dimensions are negligible when
compared with the global dimensions of the flow field. Applications
for which this assumption is not valid, for example in the case of
the dynamics of highly rarefied gases, are not considered here.
The points from which the continuum is assembled are called material
points. It is expedient in the representation of a material point
to start out from a small particle of finite extent, an element of
fluid which one may think of as being coloured. The smaller the spot of
colour, i.e. the smaller the particle of fluid is chosen, the better
is its approximation to a material point. The path of the coloured
spot for any motion of the fluid which can be detected with the eye there
fore approximates to the path travelled by a material point.
We understand by the term field a function of space and time. It
may thus be assumed that a characterising condition for this point can
be assigned to each point in a space filled by a liquid or a gas. There
occur in continuum mechanics scalar functions of state like pressure
and temperature, vector functions of state like velocity and acceleration,
and second order tensor fields, particularly the stress tensor.
Because the processes which concern us here occur in the three dimens
ional space of the observation, we require three space coordinates for
their description. In general we take as a basis a fixed reference
system composed of Cartesian coordinates. It may however be more app
ropriate for certain applications to use other forms of spatial coord
inates, especially cylindrical or spherical coordinates. We shall decide
on this as the case may be in the treatment of actual examples. It
is of course advantageous for the derivation of general statements to
start from the Cartesian coordinates x, y, z.
We shall generally designate the Cartesian coordinates of a vector
η by subscripts x, y, ζ and enclose them in a column matrix which repres
ents the vector n, e.g.
2 1 Principles of continuum mechanics
η =
We make an exception for the position vector r, whose components
we shall designate x, y, z, and for the frequently occurring velocity vec
tor v. In order to simplify the writing we shall not designate the Cart
esian velocity components by indices, but by various letters, u,v,w:
A direction is assigned to each material point by the vector field
of the flow velocity, which may possibly vary with time. The integral
curves which make up this direction field at a certain time are called
stream lines. The following applies for advance along a stream line
dx:dy:dz = u:v:w (t constant). The stream lines consequently satisfy
the following system of the ordinary differential equations ( σ is the
curve parameter for the stream lines; the time t remains constant):
dx / *\ dy / .\ dz
— = u(x,y,z,t); ^ = v(x,y,z,t); — = w (x, y, z, t) (1.1)
Simultaneous integration of this system of equations yields a stream line
for time t. The three constants of integration which arise can be
considered as the space coordinates of a point through which the stream
line will pass. Different stream lines are distinguished by various
values of the constant of integration. A surface made up of stream
lines only is called a stream surface.
One understands by the word path line the path traversed by a material
point. One can determine the path lines for a given velocity field
by integrating the set of equations
dx , Ax dy , x dz
— = u (x, y, z, t); -jL = ν (χ, y, ζ, t); — = w (x, y, z, t) (1.2)
Note that time plays the part of the parameter of the function. The
constants of integration can be identified with the coordinates of that
point which defines the position of the fluid particle singled out at
a certain time (say t = t).
n
1.1 Basic concepts 3
A flow is called steady when it has a time independent velocity field
(3v/3t =0); otherwise it is described as unsteady. In the case of
steady flows it is evident that according to equation (1.1) there is
at all times the same stream line picture, and at the same time each
stream line is simultaneously a path line. The following example shows
that this can also occur with unsteady flows. However in general the
stream line structure of an unsteady flow varies with time, and the
stream lines and path lines no longer coincide.
For an example we now consider an unsteady flow with the velocity
field u = a(t)x; - v = -a(t)y; w = 0 (j )
3
Integrating equation (1.1) we obtain the parameter representation
for the stream line passing through the point χ = x^, y = y^, ζ = z^
x = xe»w«'; y = yoe-»<*><'; z = z
0 0 ( M)
This is a planar curve (in the plane ζ = ZQ)« The curve parameterσ can
obviously be easily eliminated. Hence the time dependent factor a(t)
also does not apply. The result
xy = xoy (L 5)
0
in all cases no longer contains time as a parameter, so that the stream
lines according to equation (1.5) are rectangular hyperbolas, unchanging
with time. The same holds for the path lines. One then obtains directly
from equation (1.2) the representation for the path line which passes
through the point χ = XQ, y = y^, ζ = z at time t = t^:
Q
t t
/ a(r)dr -/ a(r)dr
x = xoet0 ; y = ye t0 ; z = zo (1.6)
0
Here also the function parameter (that is to say t) can be easily
eliminated. Hence the relationship (1.5) again follows. Stream lines
and path lines therefore coincide and form a system of rectangular hyper
bolas. The flow field under discussion has in the origin of the coord
inates a stagnation point, where all the velocity components vanish.
This therefore describes the events in the vicinity of a free stagnation
point and hence is described as a plane stagnation point flow.
The expression (1.5) could also have been derived thus. Because
w = 0 the stream and path lines are two-dimensional curves. The follow
ing applies for the two groups of curves
