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Pulsating poiseuille flow of a non-newtonian fluid
hematics
2%
Computers
PULSATING POISEUILLE FLOW OF A NON-hEWTONIAN
Simulation XXVI (1984) 276-288 North-Holland
FLUID
K.R. RAJAGOPAL Depurtment of Mechakd
Engirteering, Unicw.si!v oj‘ Pittsburgh, Attshurgh,
PA I Xtil,
U.S.A.
E. SCIWBBA Depurtment of Mechanical
I.
Engineering
The Cutholic Universi!v (,f America,
U.S.A.
analysis will provide some further insight
Introduction There are few non-linear problems in
mechanics which are amenable to a detailed numerical study.
Wushtngton. DC N&4,
A simple and interesting
non-linear problem arises in the case of
into the contradictory results.
In this
paper, we study the pulsating Poiseuille flow of a fiuid of third grade. The stress T in an incompressible
pulsating Poiseuille flow of a non-
homogeneous flui; of third grade is related
The problem under
to the fluid motion in the following manner
Newtonian fluid.
question provides the opportunity for evaluating the usefulness and applicability of several standard procedures in numerical analysis with respect to non-linear
(cf. Truesdell and No11 [5]) * T= -pl + uA, + a,A, + 01 A' wML 2-l + "_A3 + 82 @,A,
+ 221,A,l
problems.
(1)
The pulsating Poiseuille flow of nonNewtonian fluids has been studied by several authors (cf. Bird, Armstrong and Hassager [I], pp 373-378).
The study
carried out at small amplitudes by Barnes, Townsend and Walters [2] indicated that the increase in the volumetric flow rate, i.e.,
where p is the nressure, 11 the viscosity, CX, and a2 material ;!otluliusually referred to as the norrial stress moduli.
The kine-
matical tensors cl, A,,and A, are defined recursively through 7cf. Ri;;in and Ericksen [6]) hl = (grad v) + (grad v)IS
(3,
the flow enhancement, increases with both the frequency and amplitude of the pressure gradient.
However, the recent work of
A = $ ,n
$,J+ n=
!n_, (grad 1) + (qrad I)TAn_l* 2 and 3
(2)2
Sundstrom and Kaufman [3] (using the Ellis
In the above equations v, denotes the veloc-
model) at larger amplitudes and that of
ity and d the material time derivative. a The thermodynamics of fluids modeled
Phan-thien and Dudek [4] (using the powerlaw model) indicate contrary results. Thus, in addition to providing an interesting numerical solution of a non-linear problem, it is hoped that the present
0378-4754/84/$3.00
@TJ 1984,
by (1) has been the object of a detailed study by Fosdick and Rajagopal [7].
They
show that if the motions of the fluid are to be consistent with thermodynamics in
IMACS/Elsevier Science Publishers B.V. (North-Holland)
K.R. Rajagopal,E Sciubba / PoisedIe flow
the sense that all motions of the fluid meet the Clausius-Duhem inequality and the assumption that the specific Helmholtt free energy of the fluid be a minimum in equilibriurn,then u > 0, al
>
0, a1 + a2 < q
We shall assume that
,
$1 = 8* = 0, 63 i 0.
(3)
In this study we shall not restrict the material moduli by equation (3) but allow them all possible values, a situation which has relevance when (1) is used as a model
p (PO + Q, cos (tit)
g=_
(6
On defining a modified pressure i through h p = p - (2a, + a2) (+_)" rty + 2B2 &
in the sense of an approximation.
(A ) 9 ayat
ay
In the case of the steady plane
.
it follows that
Poiseuille flow of an incompressible fluid cf third grade, it is well known that the
2
$+
A l?-----
3Y
solution reduces to determining a quadra-
3
7
ay2,t
+ 6 ($2 + i;,) h
ture (cf. Coleman, Markovitr and No11 CD]). However, the quadrature is sufficiently complicated to warrant careful scrutiny (cf. Rajagopal, Sciubba and von Kertcek [9])
Thus,
and has been solved both by means of a finite difference method and by means of a perturbatton technique. Here, we shall
-)
restrict ourselves to obtaining a numerical solution for the problem in question. II.
2 -_a2u 3Y2
=
.m
p
(p.
+ Q, ~0s
at>.
Equations c;fMotion
The appropriate boundary and initial
For the problem under considerationwe
conditions are
seek a velocity field of the form v = u (y,t) i,
(4)
u (I?,:) = u (-h,t) = 0, Wt
(IO)1
u (Y,O) = F (~1, tfy
UN2
ihere u is tie velocity in the x-coordinate
where F(y) is the solution to
directions and i is the unit vector in the
p F"
x-coordinatedilrection. It follows from (l), (2), (4) and the balance of linear momentum, when 8 = 0, that 2 3l 2 a2u u+ul+ + 6 (B2 + B3) (3) 7 ay dt 3Y 3Y
!!I! -og=ax,
(5),
(9)
+ 6 (f32+ B3) (F')' F" = -PPo
l
(N
We introduce the dimensiaqless velocity function f(y) = Q4 "6
12)
where Up is the centerline velocity for the steady Poireuille flow of the same fluid. Also, let us define a new coordinate through
278
K. R. Rajagopal, E. Sciubba / Poiseuille flow
y = CL,
(13) Let B = B2 + B3.
with L = (V/W)'.
ducing the time scales
Intro-
PLT k+l m - fi@[l = w; + AT f 'j u + 6 ('v fi)* Ilk+'
.
'Im
= l/w, Tm = b. T" = (B/l-#,
'f
Tn
B = a,/ll
(14)
and defining a dimensionless time T = t/'rf, we can rewrite (9) as
(Tfl *_
- ->
f”
= f*#[l + 6 (>_
+ q
:r-; f _ Pirm .
(19)
l
f*)*l
Tm (15)
Finally, f5+l = wk+'
.
(20)
The above procedure, being entirely explicit, is very efficient in terms of CPU-time (cf. [lo]).
Also, since numeri-
cal instability might be encountered in this kind of non-linear equation, this technique has the significant advantage
Equatic?r (15) jr equivalent to tr,e system PLTm ;Cw = -__ - f"[l+
-41
6 (h Tm
;:
w=-_ 'n f I'+f 'f
(164
.
(16)2
Poiseuille
e (for
the limiting cases 8 3 0 and 8 + 1, the above procedure is equivalent to the corrector time marching methods,
We now proceed to solve (16)1 2 numerically.
(quasi-linearized) stability region, via the tuning of the parameter
Euler explicit and the simple predictor--
Numerical Solution
III.
of allowing for some calibration of the
respectively).
Assuming the steady &ate
flow problem for the same value
No specific numerical
experimentation has been carried on to
of H as the initial guess (cf. [9]),
search for an optimal value of f3, the
equation (16), 5s first solved for w K+e: I PLTm wk+t: = wk
results presented here being obtained for
* . 1
~+olA_;&--
J
J
-fi# il + 6
-' v (% f;)*]lk,
where 0 < 8 < 1.
0=
(17)
In the above equation,
derivatives:
discretized time counter, and with subscript Next,
Tf
f”
+
J
f*k+e = J
w
j
k+e .
fj+l
_
2fj
+
fj-l ,
x2 fj = fj+12iCfj-l ,
we determ!ne f.kte from (l6)2: 3
-'n
Also, a standard
employed for detemlining the spatial
fi* =
we indicate with the superscript k the j, the discretized space marker.
.75 (cf. [ll]).
centered--space discretization has been
)i
IV.
(‘18)
k+e Then, wit1 is computed from f. by using J
Results and Discussions Detailed numerical solutions for the
velocity profile were obtained for several values of B/E_ and al/pand typical velocity profiles at various instants of time for
K. R. Rajagopal,
E. Sciubba / Poiseuille flow
279
specific values of the parameters are
- wall stress
provided in Figure 1.
After a brief transient (of the same
The principal conclusions which could be
order of magnitude as the one followed by
drawn from an analysis of the results are
the centerline velocity) ~~~~~ reaches a
the following:
steady state, fig. 2.
- influence of the ratio al/p
delay between centerline- and wall velocity
The numerical value of this coefficient
oscillations,
Due to the phase
the wall stress has a
does not affect the flow significantly.
larger phase lag than the centerline
This fact can be proven with an order-of-
velocity, for all frequencies (fig. 3).
magnitude preliminary analysis of the
This phase lag decreases steadily with
constitutive equations (Rajagopal L Sciubba,
increasing p/p (fig. 8) .
1981, unpublished) and is of general validity for all flows of this type.
Figs. 5 show the dependence of -rwall on In
the nuaerical simulation, al/u was varied between +l and -1, with no significant effect on thefluiddynamic details of the flow.
All results presented here
correspond to al/u = -0.01 except where
the frequency o and the added viscosity 8.
Notice that the actual wall stress
is related to the dimensionless T wall
depicted in fig. 5 by: 'actual ='wall
($--) B
otherwise indicated.
with L defined as in Section II and U6
- velocity profiles
the centerline velocity of the steady
The centerline velocity follows the pulsat-
Poiseuille flow of the same fluid.
In
ing pressure with a phase lag whict is
other words, the apparent decrsease in the
inversely proportional to the frequency of
wall stress -for a given frequency of the
the oscillation (fig. 3).
forcing pressure oscillation- with an
Steady state
is achieved after few (typically, ten)
increasing B is due to the fact that
complete cycles (fig. 2):
the scaling factor k =
after that,
L
increases
the phase lag is steady and the velocity profile displays a "viscous layer
drastically with increasing P.
behaviour" near the wall where, for certain
scaled expression for 'T is:
The non-
values of 6, flow reversal is attained (fig. 1).
The maximum amplitude of the
velocity oscillation (equal to twice the centerline velocity value given in fig. 4) decreases asymptotically with increasing frequency, as expected, and decreases
TxY
y=o
=b 2.a ay
1
g
(Cl + a3 ,$“,
y-o
and, for 6 > 0 and for a fixed frequency 0, is always increasirg with P/p. As mentioned earlier, the effect of
also drastically with increasing B/p
the frequency and amplitude of the pressure
(fig. 4b).
gradient on the mean volumetric flow rate
This result, too, had to be
expected, since fi is, viscosity" coefficient.
asically, an "added
has been a matter of some controversy.
K. R. Rajagopal, E. Sciubba / PoiwuilJe flow
280
The flow enhancement parameter J is defined
173
R. L. Fosdick and K. R. Rajagopal,
PI
B. D. Coleman, H. Markovitz and W.
tg3
K. R. Rajagopal, E. Sciubba and C. von Kerzcek, Proceedings of the
through (cf. [4])
whm-e Q, is the volumetric flow rate for the steady Poiseuille flow of a fluid with the same material parameters 0, 0~~and 8. The parameter J has been computed using the following procedure.
reduced using an extrapolation similar to
that of Aitken% (cf. [12]). The asymptote
of the reduced series has been
taken to be Qmean.
It is found that 3
seems to be small but positive for all valLes of B/11 (see Figure 6), the dependance of 3 from w being essentially in agreement with the results shown in [4]
(Figure
7). REFERENCES
R, B. Bird, R, C. Armstrong and 0. Hassager, Dynamics of polymeric Liquids, Vol. 1, Fluid Mechanics, John Wiley and Sons, New York (1977). Ii.A. Barnes, P. Townsend and K. Walters, Rheologica Acta, -10, 517 (1971).
? W. Sundstrorn and A. Kaufman, Ind. Eng, Chem. Process Des. Dev., -16, 320 (1977). Y
h’. Phan-thien and J, Dudek, Journal of Non-Newtonian Fluid Mechanics, -11, 147 (1982). C. Truesdell and W. Noll, The Nonlinear field theories of mechanics, !0ndbuch der Physik, III/3, SpringerVlzrlag,Berlin-Heidelberg-New York (1965). I?,S. Rivlin and J. L. Ericksen, %I1 Rat. Mech. Analysis, 4, 323 (1954).
Roll, Viscometry of simple fluids, Springer Tracts in Natural Philosophy, Springer-Verlag,BerlinHeidelberg (1965).
Xth IMACS World Congress, Montreal (1982).
The average volumetric
flow rate Q,, was computed each 100 time steps and the series thus generated was
Proc. Roy. Sot. London, Ser A. 339, 351 (1980).
II101 R. Vichnevetsky,Computer Methods in P.D.E., Prentice-Hall,New York (1982). Cl11
R. Vichnevetsky, Stability charts in
the numerical approximationof P.D.E., Math. and Computer in Simulation, Xx1, 170 (1979).
II121 D. Shanks, Phys, Rev., Vol. 91
n. 2 (1954).
U. R. Rajagopai,
0.4
0.2
0.
E. Sciubba /
dImensionless l-
281
Poiseuille flow
0.6
velocity
0.8
1.0
u/W ?
fluid
nevronian
In
-6
2 d,
d
0.
0.2
ii-
Fig. 1.
plr
=
0.4
0.001
0.6
0.b
--
12
---
25
_m__._--.
50
--3
95
1.0
282
K. R. Rcybgopal, E. Schbba / Poiseuille flow
t/At --
12
_--
25
e-.m--w,
50
v_-
9s .
1
0.
0.2
0.4
0.6
0:a
0.
0.2
0.4
0.6
0.8
iv-
B/pi
- 0.623
d/pp Fig.
= -1.
l(continued).
I
K.R. Rajagopul, E. Scwbba / Poiseuiile fl w
0.
0.2
0.4
0.6
283
0.8
- 0.625
l+
-10.001
t//It
0.
0.2
vi-
p/k- 3.125
Fig. l(continued).
0.4
0.6
--
12
---
25
*--._.I..
50
-.-
95
K. A. Rajagopal, E. Sciubba / Poiseuille flow
!
1 @e
.
!-
dimensionless Fig.
2 - Pressure Initial
gradieltt, condition:
vchcity steady
time and Wall
Poiseuille
8tre8lr
08CillatiOna
flow.
Bjv = 0.01
‘4s
K. R. Rajagopal, E. S&bba
/ Poiseuile flow
285
w
d
Fig.
3-
I
I
I
I
I
I
6.0
4.0
2.0
0.
I
I
I
8.0
and wall stress phase lag (with respect sure oscillation) as function of frequency
Velocity-
Id.
to the forcing
0
pres-
vu I I
;
0. Fig.
I
1
1.0 ta- Maximumcenterline coefficient
I
I
2.0 vetocity
ratio
I
3.0
I
U/U@ as function
4.0
i
I
5.0
I
of the added viscosity
0
l
d
Fig.
4b- Mm&mm ccaterllne
velocity
ratio
u/V
8
ius function of frequency
Fig. 5a- Maximum wall stress as function of frequency
K.R.
0. Fig.
2.0
1.0 5b- Maximumwall
Rajagopa4 E. ScMba / Poiseuiirre flow
287
3.0
atream as function
4.0
of the added viscosity
5.0 coefficient
0
N’ 1
0. Fig.
1
I
I
1.0 6- Flow enhancement factor
1
1
2.0 J as fuction
1
D
3.0 of
1
4.0 the
added
viscosity
r
B/v coefficient
1
5.0
K. R. Rajagopal,E. Sciubh / Poisedle f?mv
lo3
Jx
0
ed
d
I
0.
Pig.
I
1
1
7- Flow enhancement
factor
.
I
4.0
2.0
J ae function
1
8.0
610
I
1
w
10.0
of frequency
+? , rad
Fig. 8- Phase Lag for wall
stress
as
function
of
the added
viscosity
coefficient
2%
Computers
PULSATING POISEUILLE FLOW OF A NON-hEWTONIAN
Simulation XXVI (1984) 276-288 North-Holland
FLUID
K.R. RAJAGOPAL Depurtment of Mechakd
Engirteering, Unicw.si!v oj‘ Pittsburgh, Attshurgh,
PA I Xtil,
U.S.A.
E. SCIWBBA Depurtment of Mechanical
I.
Engineering
The Cutholic Universi!v (,f America,
U.S.A.
analysis will provide some further insight
Introduction There are few non-linear problems in
mechanics which are amenable to a detailed numerical study.
Wushtngton. DC N&4,
A simple and interesting
non-linear problem arises in the case of
into the contradictory results.
In this
paper, we study the pulsating Poiseuille flow of a fiuid of third grade. The stress T in an incompressible
pulsating Poiseuille flow of a non-
homogeneous flui; of third grade is related
The problem under
to the fluid motion in the following manner
Newtonian fluid.
question provides the opportunity for evaluating the usefulness and applicability of several standard procedures in numerical analysis with respect to non-linear
(cf. Truesdell and No11 [5]) * T= -pl + uA, + a,A, + 01 A' wML 2-l + "_A3 + 82 @,A,
+ 221,A,l
problems.
(1)
The pulsating Poiseuille flow of nonNewtonian fluids has been studied by several authors (cf. Bird, Armstrong and Hassager [I], pp 373-378).
The study
carried out at small amplitudes by Barnes, Townsend and Walters [2] indicated that the increase in the volumetric flow rate, i.e.,
where p is the nressure, 11 the viscosity, CX, and a2 material ;!otluliusually referred to as the norrial stress moduli.
The kine-
matical tensors cl, A,,and A, are defined recursively through 7cf. Ri;;in and Ericksen [6]) hl = (grad v) + (grad v)IS
(3,
the flow enhancement, increases with both the frequency and amplitude of the pressure gradient.
However, the recent work of
A = $ ,n
$,J+ n=
!n_, (grad 1) + (qrad I)TAn_l* 2 and 3
(2)2
Sundstrom and Kaufman [3] (using the Ellis
In the above equations v, denotes the veloc-
model) at larger amplitudes and that of
ity and d the material time derivative. a The thermodynamics of fluids modeled
Phan-thien and Dudek [4] (using the powerlaw model) indicate contrary results. Thus, in addition to providing an interesting numerical solution of a non-linear problem, it is hoped that the present
0378-4754/84/$3.00
@TJ 1984,
by (1) has been the object of a detailed study by Fosdick and Rajagopal [7].
They
show that if the motions of the fluid are to be consistent with thermodynamics in
IMACS/Elsevier Science Publishers B.V. (North-Holland)
K.R. Rajagopal,E Sciubba / PoisedIe flow
the sense that all motions of the fluid meet the Clausius-Duhem inequality and the assumption that the specific Helmholtt free energy of the fluid be a minimum in equilibriurn,then u > 0, al
>
0, a1 + a2 < q
We shall assume that
,
$1 = 8* = 0, 63 i 0.
(3)
In this study we shall not restrict the material moduli by equation (3) but allow them all possible values, a situation which has relevance when (1) is used as a model
p (PO + Q, cos (tit)
g=_
(6
On defining a modified pressure i through h p = p - (2a, + a2) (+_)" rty + 2B2 &
in the sense of an approximation.
(A ) 9 ayat
ay
In the case of the steady plane
.
it follows that
Poiseuille flow of an incompressible fluid cf third grade, it is well known that the
2
$+
A l?-----
3Y
solution reduces to determining a quadra-
3
7
ay2,t
+ 6 ($2 + i;,) h
ture (cf. Coleman, Markovitr and No11 CD]). However, the quadrature is sufficiently complicated to warrant careful scrutiny (cf. Rajagopal, Sciubba and von Kertcek [9])
Thus,
and has been solved both by means of a finite difference method and by means of a perturbatton technique. Here, we shall
-)
restrict ourselves to obtaining a numerical solution for the problem in question. II.
2 -_a2u 3Y2
=
.m
p
(p.
+ Q, ~0s
at>.
Equations c;fMotion
The appropriate boundary and initial
For the problem under considerationwe
conditions are
seek a velocity field of the form v = u (y,t) i,
(4)
u (I?,:) = u (-h,t) = 0, Wt
(IO)1
u (Y,O) = F (~1, tfy
UN2
ihere u is tie velocity in the x-coordinate
where F(y) is the solution to
directions and i is the unit vector in the
p F"
x-coordinatedilrection. It follows from (l), (2), (4) and the balance of linear momentum, when 8 = 0, that 2 3l 2 a2u u+ul+ + 6 (B2 + B3) (3) 7 ay dt 3Y 3Y
!!I! -og=ax,
(5),
(9)
+ 6 (f32+ B3) (F')' F" = -PPo
l
(N
We introduce the dimensiaqless velocity function f(y) = Q4 "6
12)
where Up is the centerline velocity for the steady Poireuille flow of the same fluid. Also, let us define a new coordinate through
278
K. R. Rajagopal, E. Sciubba / Poiseuille flow
y = CL,
(13) Let B = B2 + B3.
with L = (V/W)'.
ducing the time scales
Intro-
PLT k+l m - fi@[l = w; + AT f 'j u + 6 ('v fi)* Ilk+'
.
'Im
= l/w, Tm = b. T" = (B/l-#,
'f
Tn
B = a,/ll
(14)
and defining a dimensionless time T = t/'rf, we can rewrite (9) as
(Tfl *_
- ->
f”
= f*#[l + 6 (>_
+ q
:r-; f _ Pirm .
(19)
l
f*)*l
Tm (15)
Finally, f5+l = wk+'
.
(20)
The above procedure, being entirely explicit, is very efficient in terms of CPU-time (cf. [lo]).
Also, since numeri-
cal instability might be encountered in this kind of non-linear equation, this technique has the significant advantage
Equatic?r (15) jr equivalent to tr,e system PLTm ;Cw = -__ - f"[l+
-41
6 (h Tm
;:
w=-_ 'n f I'+f 'f
(164
.
(16)2
Poiseuille
e (for
the limiting cases 8 3 0 and 8 + 1, the above procedure is equivalent to the corrector time marching methods,
We now proceed to solve (16)1 2 numerically.
(quasi-linearized) stability region, via the tuning of the parameter
Euler explicit and the simple predictor--
Numerical Solution
III.
of allowing for some calibration of the
respectively).
Assuming the steady &ate
flow problem for the same value
No specific numerical
experimentation has been carried on to
of H as the initial guess (cf. [9]),
search for an optimal value of f3, the
equation (16), 5s first solved for w K+e: I PLTm wk+t: = wk
results presented here being obtained for
* . 1
~+olA_;&--
J
J
-fi# il + 6
-' v (% f;)*]lk,
where 0 < 8 < 1.
0=
(17)
In the above equation,
derivatives:
discretized time counter, and with subscript Next,
Tf
f”
+
J
f*k+e = J
w
j
k+e .
fj+l
_
2fj
+
fj-l ,
x2 fj = fj+12iCfj-l ,
we determ!ne f.kte from (l6)2: 3
-'n
Also, a standard
employed for detemlining the spatial
fi* =
we indicate with the superscript k the j, the discretized space marker.
.75 (cf. [ll]).
centered--space discretization has been
)i
IV.
(‘18)
k+e Then, wit1 is computed from f. by using J
Results and Discussions Detailed numerical solutions for the
velocity profile were obtained for several values of B/E_ and al/pand typical velocity profiles at various instants of time for
K. R. Rajagopal,
E. Sciubba / Poiseuille flow
279
specific values of the parameters are
- wall stress
provided in Figure 1.
After a brief transient (of the same
The principal conclusions which could be
order of magnitude as the one followed by
drawn from an analysis of the results are
the centerline velocity) ~~~~~ reaches a
the following:
steady state, fig. 2.
- influence of the ratio al/p
delay between centerline- and wall velocity
The numerical value of this coefficient
oscillations,
Due to the phase
the wall stress has a
does not affect the flow significantly.
larger phase lag than the centerline
This fact can be proven with an order-of-
velocity, for all frequencies (fig. 3).
magnitude preliminary analysis of the
This phase lag decreases steadily with
constitutive equations (Rajagopal L Sciubba,
increasing p/p (fig. 8) .
1981, unpublished) and is of general validity for all flows of this type.
Figs. 5 show the dependence of -rwall on In
the nuaerical simulation, al/u was varied between +l and -1, with no significant effect on thefluiddynamic details of the flow.
All results presented here
correspond to al/u = -0.01 except where
the frequency o and the added viscosity 8.
Notice that the actual wall stress
is related to the dimensionless T wall
depicted in fig. 5 by: 'actual ='wall
($--) B
otherwise indicated.
with L defined as in Section II and U6
- velocity profiles
the centerline velocity of the steady
The centerline velocity follows the pulsat-
Poiseuille flow of the same fluid.
In
ing pressure with a phase lag whict is
other words, the apparent decrsease in the
inversely proportional to the frequency of
wall stress -for a given frequency of the
the oscillation (fig. 3).
forcing pressure oscillation- with an
Steady state
is achieved after few (typically, ten)
increasing B is due to the fact that
complete cycles (fig. 2):
the scaling factor k =
after that,
L
increases
the phase lag is steady and the velocity profile displays a "viscous layer
drastically with increasing P.
behaviour" near the wall where, for certain
scaled expression for 'T is:
The non-
values of 6, flow reversal is attained (fig. 1).
The maximum amplitude of the
velocity oscillation (equal to twice the centerline velocity value given in fig. 4) decreases asymptotically with increasing frequency, as expected, and decreases
TxY
y=o
=b 2.a ay
1
g
(Cl + a3 ,$“,
y-o
and, for 6 > 0 and for a fixed frequency 0, is always increasirg with P/p. As mentioned earlier, the effect of
also drastically with increasing B/p
the frequency and amplitude of the pressure
(fig. 4b).
gradient on the mean volumetric flow rate
This result, too, had to be
expected, since fi is, viscosity" coefficient.
asically, an "added
has been a matter of some controversy.
K. R. Rajagopal, E. Sciubba / PoiwuilJe flow
280
The flow enhancement parameter J is defined
173
R. L. Fosdick and K. R. Rajagopal,
PI
B. D. Coleman, H. Markovitz and W.
tg3
K. R. Rajagopal, E. Sciubba and C. von Kerzcek, Proceedings of the
through (cf. [4])
whm-e Q, is the volumetric flow rate for the steady Poiseuille flow of a fluid with the same material parameters 0, 0~~and 8. The parameter J has been computed using the following procedure.
reduced using an extrapolation similar to
that of Aitken% (cf. [12]). The asymptote
of the reduced series has been
taken to be Qmean.
It is found that 3
seems to be small but positive for all valLes of B/11 (see Figure 6), the dependance of 3 from w being essentially in agreement with the results shown in [4]
(Figure
7). REFERENCES
R, B. Bird, R, C. Armstrong and 0. Hassager, Dynamics of polymeric Liquids, Vol. 1, Fluid Mechanics, John Wiley and Sons, New York (1977). Ii.A. Barnes, P. Townsend and K. Walters, Rheologica Acta, -10, 517 (1971).
? W. Sundstrorn and A. Kaufman, Ind. Eng, Chem. Process Des. Dev., -16, 320 (1977). Y
h’. Phan-thien and J, Dudek, Journal of Non-Newtonian Fluid Mechanics, -11, 147 (1982). C. Truesdell and W. Noll, The Nonlinear field theories of mechanics, !0ndbuch der Physik, III/3, SpringerVlzrlag,Berlin-Heidelberg-New York (1965). I?,S. Rivlin and J. L. Ericksen, %I1 Rat. Mech. Analysis, 4, 323 (1954).
Roll, Viscometry of simple fluids, Springer Tracts in Natural Philosophy, Springer-Verlag,BerlinHeidelberg (1965).
Xth IMACS World Congress, Montreal (1982).
The average volumetric
flow rate Q,, was computed each 100 time steps and the series thus generated was
Proc. Roy. Sot. London, Ser A. 339, 351 (1980).
II101 R. Vichnevetsky,Computer Methods in P.D.E., Prentice-Hall,New York (1982). Cl11
R. Vichnevetsky, Stability charts in
the numerical approximationof P.D.E., Math. and Computer in Simulation, Xx1, 170 (1979).
II121 D. Shanks, Phys, Rev., Vol. 91
n. 2 (1954).
U. R. Rajagopai,
0.4
0.2
0.
E. Sciubba /
dImensionless l-
281
Poiseuille flow
0.6
velocity
0.8
1.0
u/W ?
fluid
nevronian
In
-6
2 d,
d
0.
0.2
ii-
Fig. 1.
plr
=
0.4
0.001
0.6
0.b
--
12
---
25
_m__._--.
50
--3
95
1.0
282
K. R. Rcybgopal, E. Schbba / Poiseuille flow
t/At --
12
_--
25
e-.m--w,
50
v_-
9s .
1
0.
0.2
0.4
0.6
0:a
0.
0.2
0.4
0.6
0.8
iv-
B/pi
- 0.623
d/pp Fig.
= -1.
l(continued).
I
K.R. Rajagopul, E. Scwbba / Poiseuiile fl w
0.
0.2
0.4
0.6
283
0.8
- 0.625
l+
-10.001
t//It
0.
0.2
vi-
p/k- 3.125
Fig. l(continued).
0.4
0.6
--
12
---
25
*--._.I..
50
-.-
95
K. A. Rajagopal, E. Sciubba / Poiseuille flow
!
1 @e
.
!-
dimensionless Fig.
2 - Pressure Initial
gradieltt, condition:
vchcity steady
time and Wall
Poiseuille
8tre8lr
08CillatiOna
flow.
Bjv = 0.01
‘4s
K. R. Rajagopal, E. S&bba
/ Poiseuile flow
285
w
d
Fig.
3-
I
I
I
I
I
I
6.0
4.0
2.0
0.
I
I
I
8.0
and wall stress phase lag (with respect sure oscillation) as function of frequency
Velocity-
Id.
to the forcing
0
pres-
vu I I
;
0. Fig.
I
1
1.0 ta- Maximumcenterline coefficient
I
I
2.0 vetocity
ratio
I
3.0
I
U/U@ as function
4.0
i
I
5.0
I
of the added viscosity
0
l
d
Fig.
4b- Mm&mm ccaterllne
velocity
ratio
u/V
8
ius function of frequency
Fig. 5a- Maximum wall stress as function of frequency
K.R.
0. Fig.
2.0
1.0 5b- Maximumwall
Rajagopa4 E. ScMba / Poiseuiirre flow
287
3.0
atream as function
4.0
of the added viscosity
5.0 coefficient
0
N’ 1
0. Fig.
1
I
I
1.0 6- Flow enhancement factor
1
1
2.0 J as fuction
1
D
3.0 of
1
4.0 the
added
viscosity
r
B/v coefficient
1
5.0
K. R. Rajagopal,E. Sciubh / Poisedle f?mv
lo3
Jx
0
ed
d
I
0.
Pig.
I
1
1
7- Flow enhancement
factor
.
I
4.0
2.0
J ae function
1
8.0
610
I
1
w
10.0
of frequency
+? , rad
Fig. 8- Phase Lag for wall
stress
as
function
of
the added
viscosity
coefficient