- Email: [email protected]
On the use of complex variables in the analysis of flows of an elastic fluid
Journal of Non-Newtonian Elsevier Science Publishers
221
Fluid Mechanics, 15 (1984) 227-238 B.V., Amsterdam - Printed in The Netherlands
ON THE USE OF COMPLEX VARIABLES FLOWS OF AN ELASTIC FLUID
IN THE ANALYSIS
OF
C.J. COLEMAN Department of Theoretical Mechanics, NC7 2RD (Ct. Britain) (Received
August
University of Nottingham,
12, 1983; in revised form November
University Park, Nottingham,
23, 1983)
Summary We investigate the use of complex variables in the study of elastic fluids and show how these techniques can simplify the analysis of plane flows. In particular, we investigate the flow of an elastic fluid over a circular cylinder.
1. Introduction Complex variable methods have proved extremely useful in studying the plane creeping flows of incompressible Newtonian fluids [l]. The usual approach is to describe the flow by means of a stream function \cIin terms of which the velocity components ui are given by u=-A!Landu=?!k.
1
ax2
For creeping
2
ax,
-
flows, the stream function
will satisfy the biharmonic
equation
v”#=O or
in terms of the complex variable z (= xi + ix2). integrate (1.1) to yield the general solution
for which +i and +2 are analytic 0377-0257/84/$03.00
functions
It is a simple
of z.
0 1984 Elsevier Science Publishers
B.V.
matter
to
228 Alternatively, it has been shown described by the complex equation
[2] that plane
creeping
flows may be
EL, (1.2) az2 ’ where x is a complex potential. The real part of x consists of the Airy stress function $I in terms of which the stresses uii are given by a’+
a2+ (Jll=y'o2‘22 ax*
a2+
=2'"12=-ax,
axlax
7
and the imaginary part of x consists of the stream function # multiplied by twice the viscosity. On integrating (1.2) twice with respect to 2, it is seen that (1.2) will have the general solution x = @,(z) + +2w for which +r and G2 are analytic functions of z. In what follows, we investigate modifications to eqn. (1.2) that enable the description of flows in which both inertial and non-Newtonian effects are present. We then apply the modified equations to the study of some flows with low Reynolds number. The complex-variable approach greatly simplifies the analysis since the perturbed form of (1.2) is far easier to integrate than its real-variable counterparts. In Section 2 we consider the modification of (1.2) for Newtonian flows in which inertial effects are important and use the resulting equation to investigate these effects in a low Reynolds number Hamel flow. In Section 3 we consider the modification of (1.2) for a corotational Maxwell fluid and use the resulting equation to investigate the effect of elasticity on low Reynolds number flow over a cylinder. This problem has been considered by several authors [3,4] who have solved the governing equations by applying an Oseen type approximation. Although this approximation is justifiable for the momentum equations, it does not appear so justifiable for the constitutive equations. Consequently, the present analysis should be of interest since it avoids the use of an Oseen approximation. Although the algebra becomes prohibitive as the analysis proceeds, the complex approach makes the solution process so systematic that a computer program has been written to carry it out. 2. Newtonian flows For plane flows, we can introduce which the stresses
3% -s+po 2
u 11 -
i
w 2 yjy2 i
7
a modified
Airy stress function
+ for
(2.1)
229
w 2
a2+
=-+po u22 ax: i z i 012=
a2+
-~-
(2.2)
’
1
w
a4
will automatically poujui,j = aij,j.
satisfy the steady state equations
In the case of a Newtonian uij = -pa,,
(2.3)
P”ax,%zy9
axlax
q( ui,j
+
+
of motion
fluid we shall have
z+),
from which 022 -
011 =
2r)b2,2
-
%,1)
and 012 =
77G41.2 +
u2.1).
From (2.1) to (2.3) we obtain
which, together s42.2
u1.1)
-
the relation
with the relation
-w*.2
+
U2,l)
allows us to rewrite relations
=
-
-~i a21mx 29 a22 ’
(2.7)
(2.4) and (2.5) in the form (2.8)
Equation (2.8) generalises (1.2) to include inertial effects. Choosing a length scale L and a velocity scale U, we may introduce the dimensionless complex coordinate [ (= z/L) and the dimensionless potential 2 (= x/27$JL). Under this resealing, eqn. (2.8) becomes a2z
;‘R
aS”
*
a1wt2
i af
()
i
=’
(2-9)
where R = p. LU/TJ is the Reynolds number. In order to illustrate equation (2.9), we investigate the flow of a liquid between two converging plates (a Hamel flow) [l]. With the coordinate conventions of Fig. 1, the flow will be symmetric about the x, axis which is equivalent to the condition
g(s, f)
=m.
(2.10)
230
Fig. 1. Hamel flow.
We shall assume the Reynolds number to be small and expand 2 in powers of R; then
~=~,,+R~,+R2f,+... with &, satisfying a220 - 0 as”
(2.11)
and g1 satisfying 2 a221+, -iat2
awLI 2
af
=o I
(2.12) ’
Integrating (2.11) twice with respect to f yields $0 = s;p,(S) +42(S), where I#B~ and $2 are analytic functions of 1. We choose +i and +2 so that go will have the form Jqo = -KC/[
- C log 5,
(2.13)
where K and C are constants. Condition (2.10) will be satisfied if K and C are taken to be real. Noting the relations z+=URe(%-%)
and u2= UIm( F+$),
the no-slip condition (ui = u2 = 0) on the plates will yield a pair of homoge-
231 neous linear equations for C and K. These equations will be satisfied when C = 2K cos 2a. Constant K will be determined by the rate of flow between the plates. On substituting (2.13) into (2.12), (2.14) which, on integrating twice with respect to c, yields
(2.15) The last two terms of (2.15) represent eigensolutions that have been chosen to enable the boundary conditions to be satisfied. By choosing y and 6 to be real, we ensure that (2.10) is satisfied. From the no-slip condition on the plates, 6 = ~K’COS 4a - $(2K2 + C’) + +KC cos 2cu + aKC sin 2a + 2y cos 2a, (2.16) and from the condition that the perturbations
should not alter the flow rate,
K2sin 4~y- 6(2K2 + C2)a + 9KC sin 2a - 6a cos 2a +24y sin 2a - 24&x= 0.
(2.17)
Equations (2.16) and (2.17) determine the constants y and 8. 3. The corotational Maxwell model The constitutive equation for a corotational rij + Gij = qDij,
Maxwell model is given by (3 -1)
where Dij = ui,j + u~.~,
and rij represents the extra stress. For a steady plane flow, equation (3.1) implies that
(L-qM)+A
i
u~~+~~~-i(~~,,-u~,~))L=O, 1
x2
(3.2)
232 where L=+(rZ2-7i1)-+i7i2 and M=$(u,,,-
ui,i> - ti(ui,, + t+,r).
Noting relations (2.6), (2.7), 2 a21mx
u2.1 - u*,* = -rj ~azaz and
we may rewrite (3.2) in the form
i!k+f2-(iG.!gq+~(~~-~~-2~) a22
x i
a*Rex+ az2
p. a ImX 4172 ( az
2 =*
(3.3)
)I*
Equation (3.3) generalises (2.8) to include the corotational Maxwell fluid. In order to illustrate eqn. (3.3), we use it to investigate low Reynolds number flow over a circular cylinder (Fig. 2). The region of interest is split into inner and outer zones (see Appendix). The inner zone is the region close to the cylinder where inertial effects may be neglected, and so
&
a22
ahx ~-_--_ 17 i az
+G
a aZ
aImx
aZ
a az
2a21mx azaz
a2Rex -= 1 a22
o (3.4)
will describe the flow. However, far from the cylinder, inertial effects will
Fig. 2. Flow past a cylinder.
233 become
important
a*x f ar*
whilst non-Newtonian
PO
a1mx
4712i
az
effects will diminish
and so
*=o
(3.5)
i
will describe the flow in this outer zone. Solutions to (3.4) and (3.5) represent the first terms in the inner and outer expansions of a singular perturbation scheme. We shall rescale x and z in the same way as in Section 2, with length scale L taken to be the cylinder radius and the velocity scale U taken to be the relative velocity of the far zone fluid and the cylinder. Equation (3.4) will now become
a*2 +ix
a al
aIm2 ~-----
-V
i
af
aIm2
al
a af
-= ’
where i (= 2XU/L) is a dimensionless elasticity parameter. i to be small and expand 2 in powers of x; then
We shall assume
2 = go + XRi + X*2* + . . .)
(3.7)
with z. satisfying
a220 -
0
(3.8)
af*
and z1 satisfying --a
aims,
ac
al
ahz, al
--a
at
’
(3.9)
For first order perturbations, the approximate constitutive equations will be those of a second order fluid. Consequently, since we only have velocity boundary conditions. Tanner’s result [5] implies that Img,=O
(3.10)
and so x2 will satisfy a*ji*
at*
+
--a al
Hmji,
i i
ai
ah%, al
--a af
Solutions to eqn. (3.8) represent zone flow and this approximation potential ii0 is given by
2a*Imno alaf
I
a% _. ai*
(3.11) *
the Newtonian approximation to the near has been studied in great detail [6]. The
(3.12)
iio=C{(5-f)ln5-5-1/S}, where constant
C is determined
by matching
(3.12) to a solution
that is valid
234 in the far zone [6]. C will have the value -OS( A.- 0.87A3 + . . .), where A = {ln(3.705/R)}-‘. Substitution of (3.12) into (3.9) yields the equation
(3.13) On integrating (3.13) twice with respect to c, and choosing the integration “constants” for no slip on the cylinder (a Im x/a{ = 0 when St= l), we obtain (3.14) Substitution
of (3.13) into (3.11) yields an equation
2, = C3(316[5f4 + 6015f2(ln
for g2 from which
+ 6015f2(1n 1)2 + 12035f21n S In s + 3015f21n 5 i)2 + 30c5f21n i + 3515c2 - 316c4f5
- 60c4f3(1n [)’ - 120{4f31n l In f - 39014f31n { - 60c4j3(ln
f)’ - 390{4f31n f - 31514i3
- 12014f(1n S)’ - 24014i In 5 In f - 120343(1n !)2 - 4514i - 3013f4(ln
1)2 - 60c3f41n [ In f + 12013f41n S
- 30c3f4(ln
Q2 + 12013c41n f + 40113f4
+ 36013i21n { + 36013f21n f + 1013j2 + 1813 + 3012f5(ln
[)2 + 6012f51n [ In f + 3012f5(1n f)’
+ 28712c5 + 180[‘f31n -6012f3
5 + 180{2f31n f
- 24O{‘f In { - 24012f In f + 90[‘?
- 601f4(ln
{)2 - 120{r41n 5 In f - 1805f41n I-
- 180{pln
f - 3395c4 - 1205f’ln
I-
the 0( A4) term
is an eigensolution
f)’
1205c21n f
+ 1605f2 - 361+ 60f3 - 120f}/(4805434) where
603f4(ln
+ 0( A4), that
behaves
(3.15) as a Newtonian
235
Stokes flow and whose order is determined by matching with a solution valid in the outer zone. In the above perturbation solution we have utilised two expansion parameters. However, it will be noticed that the ith power of parameter i always appears multiplied by a term having the order of Ati+i). This suggests that our solution is valid for a considerable range of x and that the expansion in terms of x was unnecessary. Indeed, we may repeat the analysis without the assumption of an expansion in ?i and obtain an asymptotic solution in powers of A. In addition to the normal assumption that both inner and outer solutions be expressible in terms of asymptotic sequences as R --) 0, we need only assume that the Newtonian solution be the limit as i + 0 of the non-Newtonian solution. Up to terms in O(bp), the new solution is identical with the one given above. A quantity of particular interest is the drag coefficient. Consider the surface generated by translating a curve joining points A and B of the x,x,-plane in a direction perpendicular to this plane. We wish to calculate the force F acting on a unit width of this surface 4 = ,“( ej2dx, - u;idx,). / In terms of x, F is given by 2ia
F, +iF2=
[
al1 A
(3.16)
B
Using (3.16), we find the drag coefficient has the Newtonian value - 8mC/R in the present degree of approximation. The first non-Newtonian correction to the drag coefficient will be a term in 0( x2A4/R) and such a correction would appear to be more in line with the experimental results [7] than the 0( x2A/R) correction suggested in reference [4]. The above calculations are straightforward but extremely lengthy. Consequently, they were performed on a computer using a program written in the ALTRAN symbolic computation language. Such an approach not only reduces the amount of effort needed to carry out the analysis but also reduces the possibility of algebraic error. 4. Extensions The main purpose of the present paper has been to demonstrate that the equations describing plane flows of certain fluids may be reduced to a single complex equation and that such a reduction can lead to simplifications in the analysis of these flows. The first step in the reduction is to eliminate the momentum and continuity equations by use of an Airy stress function and a
236
stream function, the real and imaginary parts of a complex potential. The constitutive equations then provide a set of three equations that determine the stream, stress and pressure functions. For fluids with a trace free extra stress, a further reduction is possible. The pressure may be eliminated to provide two equations in terms of the stress and stream functions, and these equations combined to form a single complex equation. Up to now we have concentrated on the corotational Maxwell fluid, but a further example is given by the corotational Oldroyd model B fluid T;~
+
X,fij = n(Dij + XIDij),
for which plane flows are described by
We may extend the calculation of Section 3 to such fluids, but the conclusions concerning drag are unaltered. Further examples are given by the group of fluids with constitutive equation of the form ‘ij
=
Ilf(I1)tui,j
+
uj,i)7
where f is a function of the second strain rate invariant II. Plane flows of such fluids are described by
The complex variable methods of the present paper may be used to extend the boundary integral methods of reference [l] to flows in which both inertial and non-Newtonian effects are important. However, this work is still in progress and so will be discussed in a future report. Acknowledgements
The author wishes to acknowledge the financial support of the Nuffield Foundation and the hospitality of the Mechanical Engineering Department of Sydney University where much of this work was undertaken. He would also like to thank referees for helpful comments.
237 References 1 2 3 4 5 6 7
W.E. Langlois, Slow Viscous Flow, Macmillan, New York, 1964. C.J. Coleman, Q. J. Mech. appl. Math., XXXIV (1981) 453-464. J.S. Ultman and M.M. Denn, Chem. Eng. J., 2 (1971) 81. B. Mena and B. Caswell, Chem. Eng. J., 8 (1974) 125. RI. Tanner, Phys. Fluids, 9 (1966) 12461247. M. Van Dyke, Perturbation Methods in Fluid Mechanics, Parabolic Press, Stanford, 1975. J.M. Broadbent and B. Mena, Chem. Eng. J., 8 (1974) 11.
Appendix The governing equations for the inner and outer solutions On transforming (2.9), we obtain
(3.3) into
dimensionless
Stokes
variables,
as for eqn.
(Al) For small Reynolds number, an inner or Stokes’ solution from (Al) by use of standard perturbation techniques. solution will be invalid in regions far removed from the regions, we must consider (Al) in terms of the outer or (= [R), that is
may be obtained However, such a cylinder. In such Oseen variable 6
642) The first term in the outer solution will be a uniform stream and so 2 will have the behaviour 0(1/R) when I = O(1). Consequently, on ignoring terms of O(1) in (A2), we obtain the equation
for the leading terms of the outer solution. We assume that the inner solution is bounded as R + 0 and so will have behaviour no stronger than
238 O(1) for S = O(1). Consequently, obtain the equation
on ignoring terms of O(R)
in (Al),
we
for the leading terms of the inner solution. In between the inner and outer zones there is the possibility of a zone in which both inertial and non-Newtonian effects will be important. Such a zone occurs when { = 0( R-'12),but requires that 2 behave as O( R-') for both effects to be important. However, an O( R-') behaviour in the intermediate zone would imply an unbounded velocity field as R + 0,which is physically unacceptable. In order to maintain a bounded velocity field, k needs to have behaviour no worse than O( R-*12)in this zone. Consequently, we find the intermediate zone to be a region in which both inertial and non-Newtonian effects are of no importance. Both inner and outer solutions will be valid in this zone and approximate to a solution of the equation that governs a Newtonian Stokes’ flow. Equation (A4) may be solved by ‘the methods shown in Section 3. However, the solution will contain eigensolutions whose coefficients must be determined by matching to a solution that is valid in the outer zone. Correct to terms in O(l), the outer solution will take the same form as the Newtonian outer solution and as a consequence the matching in the first three orders of parameter A will be virtually identical to that for the Newtonian case [6 and references therein].
221
Fluid Mechanics, 15 (1984) 227-238 B.V., Amsterdam - Printed in The Netherlands
ON THE USE OF COMPLEX VARIABLES FLOWS OF AN ELASTIC FLUID
IN THE ANALYSIS
OF
C.J. COLEMAN Department of Theoretical Mechanics, NC7 2RD (Ct. Britain) (Received
August
University of Nottingham,
12, 1983; in revised form November
University Park, Nottingham,
23, 1983)
Summary We investigate the use of complex variables in the study of elastic fluids and show how these techniques can simplify the analysis of plane flows. In particular, we investigate the flow of an elastic fluid over a circular cylinder.
1. Introduction Complex variable methods have proved extremely useful in studying the plane creeping flows of incompressible Newtonian fluids [l]. The usual approach is to describe the flow by means of a stream function \cIin terms of which the velocity components ui are given by u=-A!Landu=?!k.
1
ax2
For creeping
2
ax,
-
flows, the stream function
will satisfy the biharmonic
equation
v”#=O or
in terms of the complex variable z (= xi + ix2). integrate (1.1) to yield the general solution
for which +i and +2 are analytic 0377-0257/84/$03.00
functions
It is a simple
of z.
0 1984 Elsevier Science Publishers
B.V.
matter
to
228 Alternatively, it has been shown described by the complex equation
[2] that plane
creeping
flows may be
EL, (1.2) az2 ’ where x is a complex potential. The real part of x consists of the Airy stress function $I in terms of which the stresses uii are given by a’+
a2+ (Jll=y'o2‘22 ax*
a2+
=2'"12=-ax,
axlax
7
and the imaginary part of x consists of the stream function # multiplied by twice the viscosity. On integrating (1.2) twice with respect to 2, it is seen that (1.2) will have the general solution x = @,(z) + +2w for which +r and G2 are analytic functions of z. In what follows, we investigate modifications to eqn. (1.2) that enable the description of flows in which both inertial and non-Newtonian effects are present. We then apply the modified equations to the study of some flows with low Reynolds number. The complex-variable approach greatly simplifies the analysis since the perturbed form of (1.2) is far easier to integrate than its real-variable counterparts. In Section 2 we consider the modification of (1.2) for Newtonian flows in which inertial effects are important and use the resulting equation to investigate these effects in a low Reynolds number Hamel flow. In Section 3 we consider the modification of (1.2) for a corotational Maxwell fluid and use the resulting equation to investigate the effect of elasticity on low Reynolds number flow over a cylinder. This problem has been considered by several authors [3,4] who have solved the governing equations by applying an Oseen type approximation. Although this approximation is justifiable for the momentum equations, it does not appear so justifiable for the constitutive equations. Consequently, the present analysis should be of interest since it avoids the use of an Oseen approximation. Although the algebra becomes prohibitive as the analysis proceeds, the complex approach makes the solution process so systematic that a computer program has been written to carry it out. 2. Newtonian flows For plane flows, we can introduce which the stresses
3% -s+po 2
u 11 -
i
w 2 yjy2 i
7
a modified
Airy stress function
+ for
(2.1)
229
w 2
a2+
=-+po u22 ax: i z i 012=
a2+
-~-
(2.2)
’
1
w
a4
will automatically poujui,j = aij,j.
satisfy the steady state equations
In the case of a Newtonian uij = -pa,,
(2.3)
P”ax,%zy9
axlax
q( ui,j
+
+
of motion
fluid we shall have
z+),
from which 022 -
011 =
2r)b2,2
-
%,1)
and 012 =
77G41.2 +
u2.1).
From (2.1) to (2.3) we obtain
which, together s42.2
u1.1)
-
the relation
with the relation
-w*.2
+
U2,l)
allows us to rewrite relations
=
-
-~i a21mx 29 a22 ’
(2.7)
(2.4) and (2.5) in the form (2.8)
Equation (2.8) generalises (1.2) to include inertial effects. Choosing a length scale L and a velocity scale U, we may introduce the dimensionless complex coordinate [ (= z/L) and the dimensionless potential 2 (= x/27$JL). Under this resealing, eqn. (2.8) becomes a2z
;‘R
aS”
*
a1wt2
i af
()
i
=’
(2-9)
where R = p. LU/TJ is the Reynolds number. In order to illustrate equation (2.9), we investigate the flow of a liquid between two converging plates (a Hamel flow) [l]. With the coordinate conventions of Fig. 1, the flow will be symmetric about the x, axis which is equivalent to the condition
g(s, f)
=m.
(2.10)
230
Fig. 1. Hamel flow.
We shall assume the Reynolds number to be small and expand 2 in powers of R; then
~=~,,+R~,+R2f,+... with &, satisfying a220 - 0 as”
(2.11)
and g1 satisfying 2 a221+, -iat2
awLI 2
af
=o I
(2.12) ’
Integrating (2.11) twice with respect to f yields $0 = s;p,(S) +42(S), where I#B~ and $2 are analytic functions of 1. We choose +i and +2 so that go will have the form Jqo = -KC/[
- C log 5,
(2.13)
where K and C are constants. Condition (2.10) will be satisfied if K and C are taken to be real. Noting the relations z+=URe(%-%)
and u2= UIm( F+$),
the no-slip condition (ui = u2 = 0) on the plates will yield a pair of homoge-
231 neous linear equations for C and K. These equations will be satisfied when C = 2K cos 2a. Constant K will be determined by the rate of flow between the plates. On substituting (2.13) into (2.12), (2.14) which, on integrating twice with respect to c, yields
(2.15) The last two terms of (2.15) represent eigensolutions that have been chosen to enable the boundary conditions to be satisfied. By choosing y and 6 to be real, we ensure that (2.10) is satisfied. From the no-slip condition on the plates, 6 = ~K’COS 4a - $(2K2 + C’) + +KC cos 2cu + aKC sin 2a + 2y cos 2a, (2.16) and from the condition that the perturbations
should not alter the flow rate,
K2sin 4~y- 6(2K2 + C2)a + 9KC sin 2a - 6a cos 2a +24y sin 2a - 24&x= 0.
(2.17)
Equations (2.16) and (2.17) determine the constants y and 8. 3. The corotational Maxwell model The constitutive equation for a corotational rij + Gij = qDij,
Maxwell model is given by (3 -1)
where Dij = ui,j + u~.~,
and rij represents the extra stress. For a steady plane flow, equation (3.1) implies that
(L-qM)+A
i
u~~+~~~-i(~~,,-u~,~))L=O, 1
x2
(3.2)
232 where L=+(rZ2-7i1)-+i7i2 and M=$(u,,,-
ui,i> - ti(ui,, + t+,r).
Noting relations (2.6), (2.7), 2 a21mx
u2.1 - u*,* = -rj ~azaz and
we may rewrite (3.2) in the form
i!k+f2-(iG.!gq+~(~~-~~-2~) a22
x i
a*Rex+ az2
p. a ImX 4172 ( az
2 =*
(3.3)
)I*
Equation (3.3) generalises (2.8) to include the corotational Maxwell fluid. In order to illustrate eqn. (3.3), we use it to investigate low Reynolds number flow over a circular cylinder (Fig. 2). The region of interest is split into inner and outer zones (see Appendix). The inner zone is the region close to the cylinder where inertial effects may be neglected, and so
&
a22
ahx ~-_--_ 17 i az
+G
a aZ
aImx
aZ
a az
2a21mx azaz
a2Rex -= 1 a22
o (3.4)
will describe the flow. However, far from the cylinder, inertial effects will
Fig. 2. Flow past a cylinder.
233 become
important
a*x f ar*
whilst non-Newtonian
PO
a1mx
4712i
az
effects will diminish
and so
*=o
(3.5)
i
will describe the flow in this outer zone. Solutions to (3.4) and (3.5) represent the first terms in the inner and outer expansions of a singular perturbation scheme. We shall rescale x and z in the same way as in Section 2, with length scale L taken to be the cylinder radius and the velocity scale U taken to be the relative velocity of the far zone fluid and the cylinder. Equation (3.4) will now become
a*2 +ix
a al
aIm2 ~-----
-V
i
af
aIm2
al
a af
-= ’
where i (= 2XU/L) is a dimensionless elasticity parameter. i to be small and expand 2 in powers of x; then
We shall assume
2 = go + XRi + X*2* + . . .)
(3.7)
with z. satisfying
a220 -
0
(3.8)
af*
and z1 satisfying --a
aims,
ac
al
ahz, al
--a
at
’
(3.9)
For first order perturbations, the approximate constitutive equations will be those of a second order fluid. Consequently, since we only have velocity boundary conditions. Tanner’s result [5] implies that Img,=O
(3.10)
and so x2 will satisfy a*ji*
at*
+
--a al
Hmji,
i i
ai
ah%, al
--a af
Solutions to eqn. (3.8) represent zone flow and this approximation potential ii0 is given by
2a*Imno alaf
I
a% _. ai*
(3.11) *
the Newtonian approximation to the near has been studied in great detail [6]. The
(3.12)
iio=C{(5-f)ln5-5-1/S}, where constant
C is determined
by matching
(3.12) to a solution
that is valid
234 in the far zone [6]. C will have the value -OS( A.- 0.87A3 + . . .), where A = {ln(3.705/R)}-‘. Substitution of (3.12) into (3.9) yields the equation
(3.13) On integrating (3.13) twice with respect to c, and choosing the integration “constants” for no slip on the cylinder (a Im x/a{ = 0 when St= l), we obtain (3.14) Substitution
of (3.13) into (3.11) yields an equation
2, = C3(316[5f4 + 6015f2(ln
for g2 from which
+ 6015f2(1n 1)2 + 12035f21n S In s + 3015f21n 5 i)2 + 30c5f21n i + 3515c2 - 316c4f5
- 60c4f3(1n [)’ - 120{4f31n l In f - 39014f31n { - 60c4j3(ln
f)’ - 390{4f31n f - 31514i3
- 12014f(1n S)’ - 24014i In 5 In f - 120343(1n !)2 - 4514i - 3013f4(ln
1)2 - 60c3f41n [ In f + 12013f41n S
- 30c3f4(ln
Q2 + 12013c41n f + 40113f4
+ 36013i21n { + 36013f21n f + 1013j2 + 1813 + 3012f5(ln
[)2 + 6012f51n [ In f + 3012f5(1n f)’
+ 28712c5 + 180[‘f31n -6012f3
5 + 180{2f31n f
- 24O{‘f In { - 24012f In f + 90[‘?
- 601f4(ln
{)2 - 120{r41n 5 In f - 1805f41n I-
- 180{pln
f - 3395c4 - 1205f’ln
I-
the 0( A4) term
is an eigensolution
f)’
1205c21n f
+ 1605f2 - 361+ 60f3 - 120f}/(4805434) where
603f4(ln
+ 0( A4), that
behaves
(3.15) as a Newtonian
235
Stokes flow and whose order is determined by matching with a solution valid in the outer zone. In the above perturbation solution we have utilised two expansion parameters. However, it will be noticed that the ith power of parameter i always appears multiplied by a term having the order of Ati+i). This suggests that our solution is valid for a considerable range of x and that the expansion in terms of x was unnecessary. Indeed, we may repeat the analysis without the assumption of an expansion in ?i and obtain an asymptotic solution in powers of A. In addition to the normal assumption that both inner and outer solutions be expressible in terms of asymptotic sequences as R --) 0, we need only assume that the Newtonian solution be the limit as i + 0 of the non-Newtonian solution. Up to terms in O(bp), the new solution is identical with the one given above. A quantity of particular interest is the drag coefficient. Consider the surface generated by translating a curve joining points A and B of the x,x,-plane in a direction perpendicular to this plane. We wish to calculate the force F acting on a unit width of this surface 4 = ,“( ej2dx, - u;idx,). / In terms of x, F is given by 2ia
F, +iF2=
[
al1 A
(3.16)
B
Using (3.16), we find the drag coefficient has the Newtonian value - 8mC/R in the present degree of approximation. The first non-Newtonian correction to the drag coefficient will be a term in 0( x2A4/R) and such a correction would appear to be more in line with the experimental results [7] than the 0( x2A/R) correction suggested in reference [4]. The above calculations are straightforward but extremely lengthy. Consequently, they were performed on a computer using a program written in the ALTRAN symbolic computation language. Such an approach not only reduces the amount of effort needed to carry out the analysis but also reduces the possibility of algebraic error. 4. Extensions The main purpose of the present paper has been to demonstrate that the equations describing plane flows of certain fluids may be reduced to a single complex equation and that such a reduction can lead to simplifications in the analysis of these flows. The first step in the reduction is to eliminate the momentum and continuity equations by use of an Airy stress function and a
236
stream function, the real and imaginary parts of a complex potential. The constitutive equations then provide a set of three equations that determine the stream, stress and pressure functions. For fluids with a trace free extra stress, a further reduction is possible. The pressure may be eliminated to provide two equations in terms of the stress and stream functions, and these equations combined to form a single complex equation. Up to now we have concentrated on the corotational Maxwell fluid, but a further example is given by the corotational Oldroyd model B fluid T;~
+
X,fij = n(Dij + XIDij),
for which plane flows are described by
We may extend the calculation of Section 3 to such fluids, but the conclusions concerning drag are unaltered. Further examples are given by the group of fluids with constitutive equation of the form ‘ij
=
Ilf(I1)tui,j
+
uj,i)7
where f is a function of the second strain rate invariant II. Plane flows of such fluids are described by
The complex variable methods of the present paper may be used to extend the boundary integral methods of reference [l] to flows in which both inertial and non-Newtonian effects are important. However, this work is still in progress and so will be discussed in a future report. Acknowledgements
The author wishes to acknowledge the financial support of the Nuffield Foundation and the hospitality of the Mechanical Engineering Department of Sydney University where much of this work was undertaken. He would also like to thank referees for helpful comments.
237 References 1 2 3 4 5 6 7
W.E. Langlois, Slow Viscous Flow, Macmillan, New York, 1964. C.J. Coleman, Q. J. Mech. appl. Math., XXXIV (1981) 453-464. J.S. Ultman and M.M. Denn, Chem. Eng. J., 2 (1971) 81. B. Mena and B. Caswell, Chem. Eng. J., 8 (1974) 125. RI. Tanner, Phys. Fluids, 9 (1966) 12461247. M. Van Dyke, Perturbation Methods in Fluid Mechanics, Parabolic Press, Stanford, 1975. J.M. Broadbent and B. Mena, Chem. Eng. J., 8 (1974) 11.
Appendix The governing equations for the inner and outer solutions On transforming (2.9), we obtain
(3.3) into
dimensionless
Stokes
variables,
as for eqn.
(Al) For small Reynolds number, an inner or Stokes’ solution from (Al) by use of standard perturbation techniques. solution will be invalid in regions far removed from the regions, we must consider (Al) in terms of the outer or (= [R), that is
may be obtained However, such a cylinder. In such Oseen variable 6
642) The first term in the outer solution will be a uniform stream and so 2 will have the behaviour 0(1/R) when I = O(1). Consequently, on ignoring terms of O(1) in (A2), we obtain the equation
for the leading terms of the outer solution. We assume that the inner solution is bounded as R + 0 and so will have behaviour no stronger than
238 O(1) for S = O(1). Consequently, obtain the equation
on ignoring terms of O(R)
in (Al),
we
for the leading terms of the inner solution. In between the inner and outer zones there is the possibility of a zone in which both inertial and non-Newtonian effects will be important. Such a zone occurs when { = 0( R-'12),but requires that 2 behave as O( R-') for both effects to be important. However, an O( R-') behaviour in the intermediate zone would imply an unbounded velocity field as R + 0,which is physically unacceptable. In order to maintain a bounded velocity field, k needs to have behaviour no worse than O( R-*12)in this zone. Consequently, we find the intermediate zone to be a region in which both inertial and non-Newtonian effects are of no importance. Both inner and outer solutions will be valid in this zone and approximate to a solution of the equation that governs a Newtonian Stokes’ flow. Equation (A4) may be solved by ‘the methods shown in Section 3. However, the solution will contain eigensolutions whose coefficients must be determined by matching to a solution that is valid in the outer zone. Correct to terms in O(l), the outer solution will take the same form as the Newtonian outer solution and as a consequence the matching in the first three orders of parameter A will be virtually identical to that for the Newtonian case [6 and references therein].