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Unsteady motion of a sphere in an elastico-viscous fluid
a32&7225/93 ss*oo+ 0.00 Copyright@ 1992PergamonPress Ltd
Int. J. Engng Sci. Vot. 31, No. 1, pp. 19-261993 Printedin Great Britain.All rightsreserved
UNSTEADY MOTION OF A SPHERE IN AN ELASTICO-VISCOUS FLUID HAROLD
BASSOON
Department of Mathematics, The University of the West Indies, St Augustine, Trinidad, W.I.
HAN SHIFANG Research Laboratory of Non-Newtonian Fluid Mechanics, Chengdu Branch, Academia Sinica, 610015 Chengdu, People’s Republic of China Ah&act--This paper examines the unsteady slow motion of a sphere in a short memory elastico-viscous Walters fluid. With the aid of a semi-direct variational approach, the flow field is determined and spcciai cases of constant acceleration and exponentially decreasing acceleration are then investigated. To examine the effect of elasticity, streamlines have been plotted and compared with their classical counterparts.
1. INTRODUCTION
In recent times a tremendous amount of work has been done on elastico-viscous problems. However, in comparison to steady tlows not much has been done on unsteady flows. Most of the theoretical studies in this area have been confined to oscillatory flows of small amplitude (l-31. In the particular case of the unsteady motion of a sphere, attempts have been made [4-71 to lift the above-mentioned restriction but limited results were obtained. Lai [4] obtained an exact solution for the drag on a sphere moving in an arbitrary manner along a rectilinear path in an otherwise still Maxwell fluid of infinite extent. Lai and Fan [S] examined the same problem for an Oldroyd fluid. The method of solution in both cases is similar to that used by Landau and Lifshitz [8] and is based on the fact that the drag on an accelerating body can be obtained by integrating the drag on an oscillating body over all possible frequencies. In both cases attention was focused exclusively on the drag, and the stream-function and consequently the velocity field, was not determined. In the work by Thomas and Walters [6] the Laplace technique was utilized to examine the linearized problem of a sphere failing in an elastico-viscous fluid. They were only able to determine the velocity of the falling sphere for small and large time t. King and Waters f7] later successfully obtained an exact solution valid for all t. However, in these works [6,7] attention was paid only to the velocity of the falling sphere. The flow field was not determined as was also the case in all the above-mentioned works. The objective of this work is to determine the flow field due to the unsteady slow motion of a sphere in a short memory elastico-viscous Walters fluid. The main ma~ematic~ tool used to solve this problem is the semi-direct variational approach of Kantovorich 191. This is a recent technique that has been successfully applied to unsteady flow problems [lo, 111. The special cases of constant acceleration and exponentially decreasing acceleration of the sphere are then examined.
2.
STATEMENT
AND
For the particular class of shop-memos rheological equation of state is given by [12]
SOLUTION
OF
elastico-viscous
Ti = -p&j + cj 6 rij = 2qneij - 2Ko6t
19
PROBLEM
fluids mentioned
above,
the (2.1)
eij,
(2.2)
20
H. RAMKISSOON
and H. SHIFANG
where Z$ is the stress tensor, p an arbitrary isotropic pressure, gii the metric tensor, eij the rate of strain tensor, m - sN(s) ds K. = 770= N(s) h, (2.3) I0 I0 with N(s) being the distribution function of relaxation times and 6/c% the convected differentiation of a tensor quantity. In the case of an unsteady incompressible creeping flow in the absence of external forces, the momentum equation takes the form
39, tji.i-P.i=
in any orthogonal
curvilinear co-ordinate
(2.4)
Pt
system. The continuity equation is of course given by qi,; =
0
(2.5)
with q being the velocity vector. We shall now consider the unsteady motion of a sphere of radius a, moving in an arbitrary manner along a rectilinear path in an otherwise still short-memory elastic0 viscous fluid of infinite extent under the above conditions. Working in spherical polar co-ordinates (r, 8, @) and assuming axially symmetric flow, we introduce the stream function I/J(~, 8, t) via the usual relation 4r = --
1
w
4e = --
r* sin 19de ’
1
w
(2.6)
r sin 19dr
where q = (q,, qe, 0). With the aid of (2.2), (2.4) and (2.6) satisfied by the stream function ly is given by
one can show that the equation
(2.7) where r+, P
a2 D*=%+
z ’ and
Assuming that the sphere is moving with velocity V(r), we need to solve (2.7) subject to the following conditions: +!JI_ = -i
(9
w
(ii)
dr
Vu* sin* 0
= Vu sin* 8
(24
IEa
!j*0
as
r--*m
lylt=o=o
(iv)
We note that in the case of steady flow, the solution of (2.7), (2.8) is given by &=~Vu*sin*e(~-z), To solve our general boundary-value of the form
r
a
problem (2.7), (2.8), it is reasonable t&(r, 8,
t) =f(r, t)sin* 8.
(2.9) to assume a solution
(2.10)
Unsteady motion of a sphere in an elastico-viscous fluid
21
Substitution into (2.7) gives (2.11) where prime denotes differentiation
with respect to r. We now put
$va*(;- $) +I)@, t).
f(r, t) =
(2.12)
Substituting this into (2.11), we get wiu-~~f~+~W’-5~)-g(,r~-~,)-~~=0. L(w)+-K&)(
(2.13)
With the aid of (2.10) and (2.12), the conditions (2.8), assuming V(0) = 0, are now transformed to the homogeneous form (i) (ii)
(iii)
l&o=o.
(iv)
(2.14)
What we have achieved thus far is to transform the given boundary-value problem (2.7), (2.8) to a boundary-value problem (2.13), (2.14), with homogeneous boundary conditions. In this latter form the Kantovorich semi-direct variational technique is applicable. In this method we approximate v(r, t) by
This method differs from functions is required. homogeneous boundary that the variational form
the Ritz method in that only partial specification of the approximating The spatial functions gk(r) must be specified and must satisfy conditions (2.14) while hk(t) are to be determined. It can be shown of the problem (2.13)-(2.15) is given by [lo, 111 t(&)g,
6hk dr dt = 0.
(2.16)
The form of the spatial functions is, of course, not unique. We take them as g*(r)=~,
(2.17)
k=1,2,...
They satisfy all the boundary conditions given in (2.14). First approximation
To get the first approximation to the solution of the problem (2.13), (2.14), we put n = 1 in (2.15) and (2.17). That is, we take
and Mr, 0 = h,(t) If we now substitute into (2.16), that is
(r - a)’ 7.
22
H. RAMKISSOON
and H. SHIFANG
use (2.13) and integrate with respect to r from a to m, we get dt=O
(2.19)
where
1%
ff, =
63as
q =
16~ +z2 But t is arbitrary, hence
ah,
dt-qh,+cu,z=O.
dV
(2.20)
The solution of this equation subject to the initial condition h,(O) = 0 is given by f h,(t) = -cu,V - 0201eplf
The first approximation
I0
VeTalrdt.
(2.21)
to the original problem (2.7), (2.8) is therefore q(r, 8, t) = sin’ e[ a Va’(f - :)
h,(t)]
+ 9
(2.22)
where h,(t) is given by (2.20). Here we have utilized (2.18), (2.12) and (2.10). Second approximation
This is obtained by taking n = 2 in (2.16) and making use of (2.15) and (2.17). We thus get q,=h,(t)~+h,(t)~
Iff
L(I+I$
rs - ‘I’ r3
I
L(~+!J~)-
rs
(2.23)
’
6h 1 dr dt = 0
(2.24)
6h2 dr dt = 0.
(2.25)
Using (2.13) and repeating the above analysis, we obtain the following pair of simultaneous ordinary differential equations 4
-a4-~Ka2
16
~+~a2~h,+~;h2=~a7~,
105
grjh,=;a’$.
(2.26)
(2.27)
Applying the Laplace transform technique, it can be shown that the solution of the above system of equations (2.26), (2.27), subject to the initial conditions h,(O) = h2(0) = 0, takes the form h,(t) =A
[C,D,V
+ El/’ V(r)eel(‘-‘)dt 0
1
hz(f)=
-&z F,D, 1
V + G,
0
V(z)eel(‘-‘)
1 f %(1--t) VW dt1
+ E,(’ V(r)ee2(‘-r)dr
dr + G2 I0
(2.28a)
(2.28b)
Unsteady
motion of a sphere in an elastico-viscous
fluid
23
where 4 16 A 1 =za4-za2K,
1 82 AZ=ig-a2--GK,
AJ=;a2q,
B2 -_ - 3465 13 a2__ 1287 188 K’
B1=iga4-Ga2K, 1 82
C1 = 10B2 - A2,
A,=-$ 188 B4=- 1287 ’
B3=-&a2q,
C2 = 10B4 - A4
D, =A1B2-A2B,,
4 = A3B2 + AI B4 - A2B3 - A4B1
D3 = A3B4 - A4B3,
fi = lOBl -AI,
~@IC~--GD~)-C~D~
E,=
E
81-02 G
2
=e,(D&-44)-W3
G
62-91 =e,(D,F,-4fi)-44
’
e,-e2
F2= 10B3 - A3
=e,P,c,-CID,)-GD3
'
1
(2.29)
2
e2 - 65
and f3,, t12 are given by
z =css2+gs+ Thus, the second approximation
e,)(s - e,).
1
1
to our solution is
q(r, 8, t) = sin2 8 [fVa2(f-:)
+yh,(t)+vh2(t)]
(2.30)
where h,(t) and h2(f) are given in (2.28).
3. SPECIAL
CASES
(a) Constant acceleration Here we take 0
t
w =I,;,t> 0
(i) The first approximation 1/)(r, 8,
t) =
is given by sin2 8 [’4 eta z(f_!J
(ii) The second approximation q(r, 8, t) = V sin’
e
+PZ.$!~(l
-em,‘)].
(3.1)
is given by [1( 4 cfa f-~)+q%(t)+~h,(r)]
(3.2)
where hl(r)=~[CID1l+~{esl~1
h2(t)=
fI,t-l}+${e@1
--&$ [&DIt+${eel’1 1
e,t - 1) 2
e,t -
1) + 4
Here V(f) = V,(l - e-‘3,
{eezr - e2t - 111 2
@) Exponentially decreasing acceleration r>O.
1
H. RAMKISSOON
24
-20
-15
-10
and H. SHIFANG
-5
for constant
Fig. l(a). Streamlines
acceleration at t = 0.1. (b) acceleration at I = 0.1.
(K
=
0) Streamlines
for constant
20 (a) I 15
10
5 0
-5
-10
-15
-20
-15
-10
-5
Fig. 2(a). Streamlines
0
5
for constant
IO
15
20
-20
-15
-10
-5
0
5
acceleration at t =0.5. (b). (K = 0) Streamlines acceleration at t = 0.5.
10
15
20
for constant
20
15
10
5
0
-5
-10
-15
-20 -20
-15
-10
-5
Fig. 3(a). Streamlines
0
0
for constant
acceleration at f = 1. (b). acceleration at t = 1.
(K
=O) Streamlines
for constant
25
Unsteady motion of a sphere in an elastico-viscous fluid (b)
l* : -20
-15
-10
-5
0
5
10
15
20
Fig. 4(a). Streamlines for exponentially decreasing acceleration at I = 0.1. (b). (K = 0) Streamlines for exponentially decreasing acceleration at t = 0.1.
20 I
(a)
15
10
5
0
-5
-10
-15
Fig. 5(a). Streamlines for exponentially decreasing acceleration at I = 0.5. (b). (K = 0) Streamlines for exponentially decreasing acceleration at t = 0.5.
T (b)
20 I
(a)
15
10
5
0
-5
-10
-15
-20
-15
-10
-5
0
5
10
15
2"
-20
-15
-10
-5
0
5
10
15
Fig. 6(a). Streamlines for exponentially decreasing acceleration at I = 1.0. (b). (K = 0) Streamlines for exponentially decreasing acceleration at t = 1.
20
H. RAMKISSOON and H. SHIFANG
26
(i) The first approximation
is given by
r#(r, 8, t) = V, sin* 8 $ (1 - ,+)a*( [
f - 5) + 9
h,(t)]
where (3.3) (ii) The second approximation r/~(r, 8,t)=V0sin20
[
is given by
~(l-ePcr)u2
(
(r - a)’
4-:
>
+r3
h,(t) + qf
h*(t)]
(3.4)
where
hW=$[
clQ(l_e-“?+El
1
+E,
1
ee2* 1 7--+--2 e2
1
1$_$+&_& e2 + c II ’
eWcr
e2 + c
1
ee*:
--c*
h2(r)=-~[F,D,(1-e~c~)+G,(~-~+-e------
1 1 e%’ 1 e-=r e&t +G2I 7--+--2 e2 e,+c e,+c
, e,+c
1
I eW
e,+c I
II.
In order to get an insight into the form of the streamlines we take a = 4.0,
r] = 0.2,
K = 0.6,
c = 1.0
and plot $(r, 8, t) at different times. From Figs l-6 the streamlines seem to bunch closer to the sphere with increasing time. To examine the effect of elasticity, streamlines have been plotted for the same times but for K = 0. In other words these streamlines, depicted in Figs l(b)-6(b) are the streamlines for classical viscous fluids. Comparison of Figs l(b)-6(b) with the corresponding ones in Figs l(a)-6(a) show that the effect of elasticity is to push the streamlines slightly closer to the sphere and to create a slightly greater “bunching” around the sphere. Acknowledgemenf.r-One of the authors (H.R.) would like to acknowledge with thanks the Third World Academy of Sciences Fellowship granted under the South Fellowship Programme. The support granted by the Chinese Academy of Sciences and the facilities provided by the Research Laboratory of Non-Newtonian Fluid Mechanics at its Chengdu Branch are also acknowledged.
REFERENCES [I] K. WALTERS, Q. J. Me&. Appl. Marh. W, 444 (1960). [2] K. R. FRATER, /. Fluid Mech. 20, 369 (1964). [3] K. R. FRATER, Z. Angew. Marh. Phys. 18,798 (1967). [4] R. Y. S. LA1 and C. P. FAN, Int. J. Engng Sci. 12, 645 (1974). (51 R. Y. S. LA1 and C. P. FAN, Int. J. Engng Sci. 16,303(1978). (61R. H. THOMAS and K. WALTERS, Rheol. Acra 5, 23 (1966). (71 M. J. KING and N. D. WATERS, J. Phys. D: Appl. Phys. 5, 141 (1972). [8] L. D. LANDAU and E. M. LIFSCHITZ, Fluid Mechanics, p. 97. Pergamon Press, Oxford (1959). [9] L. V. KANTOVORICH and V. I. KRYLOV, Approximate Methods of Higher Analysis. Interscience New York (1958). [lo] R. S. R. GORLA and P. E. MADDEN, J. Non-Newronian Fluid Me& 16, 251 (1984). [ll] HAN SHIFANG, Continuum Mechanics ofNon-NewtonianFluids. Sichuan Academic Press, China (1988). [12] K. WALTERS, J. &f&z.1, 474 (1962). (Revision received 20 March 1992; accepred 26 March 1992)
Int. J. Engng Sci. Vot. 31, No. 1, pp. 19-261993 Printedin Great Britain.All rightsreserved
UNSTEADY MOTION OF A SPHERE IN AN ELASTICO-VISCOUS FLUID HAROLD
BASSOON
Department of Mathematics, The University of the West Indies, St Augustine, Trinidad, W.I.
HAN SHIFANG Research Laboratory of Non-Newtonian Fluid Mechanics, Chengdu Branch, Academia Sinica, 610015 Chengdu, People’s Republic of China Ah&act--This paper examines the unsteady slow motion of a sphere in a short memory elastico-viscous Walters fluid. With the aid of a semi-direct variational approach, the flow field is determined and spcciai cases of constant acceleration and exponentially decreasing acceleration are then investigated. To examine the effect of elasticity, streamlines have been plotted and compared with their classical counterparts.
1. INTRODUCTION
In recent times a tremendous amount of work has been done on elastico-viscous problems. However, in comparison to steady tlows not much has been done on unsteady flows. Most of the theoretical studies in this area have been confined to oscillatory flows of small amplitude (l-31. In the particular case of the unsteady motion of a sphere, attempts have been made [4-71 to lift the above-mentioned restriction but limited results were obtained. Lai [4] obtained an exact solution for the drag on a sphere moving in an arbitrary manner along a rectilinear path in an otherwise still Maxwell fluid of infinite extent. Lai and Fan [S] examined the same problem for an Oldroyd fluid. The method of solution in both cases is similar to that used by Landau and Lifshitz [8] and is based on the fact that the drag on an accelerating body can be obtained by integrating the drag on an oscillating body over all possible frequencies. In both cases attention was focused exclusively on the drag, and the stream-function and consequently the velocity field, was not determined. In the work by Thomas and Walters [6] the Laplace technique was utilized to examine the linearized problem of a sphere failing in an elastico-viscous fluid. They were only able to determine the velocity of the falling sphere for small and large time t. King and Waters f7] later successfully obtained an exact solution valid for all t. However, in these works [6,7] attention was paid only to the velocity of the falling sphere. The flow field was not determined as was also the case in all the above-mentioned works. The objective of this work is to determine the flow field due to the unsteady slow motion of a sphere in a short memory elastico-viscous Walters fluid. The main ma~ematic~ tool used to solve this problem is the semi-direct variational approach of Kantovorich 191. This is a recent technique that has been successfully applied to unsteady flow problems [lo, 111. The special cases of constant acceleration and exponentially decreasing acceleration of the sphere are then examined.
2.
STATEMENT
AND
For the particular class of shop-memos rheological equation of state is given by [12]
SOLUTION
OF
elastico-viscous
Ti = -p&j + cj 6 rij = 2qneij - 2Ko6t
19
PROBLEM
fluids mentioned
above,
the (2.1)
eij,
(2.2)
20
H. RAMKISSOON
and H. SHIFANG
where Z$ is the stress tensor, p an arbitrary isotropic pressure, gii the metric tensor, eij the rate of strain tensor, m - sN(s) ds K. = 770= N(s) h, (2.3) I0 I0 with N(s) being the distribution function of relaxation times and 6/c% the convected differentiation of a tensor quantity. In the case of an unsteady incompressible creeping flow in the absence of external forces, the momentum equation takes the form
39, tji.i-P.i=
in any orthogonal
curvilinear co-ordinate
(2.4)
Pt
system. The continuity equation is of course given by qi,; =
0
(2.5)
with q being the velocity vector. We shall now consider the unsteady motion of a sphere of radius a, moving in an arbitrary manner along a rectilinear path in an otherwise still short-memory elastic0 viscous fluid of infinite extent under the above conditions. Working in spherical polar co-ordinates (r, 8, @) and assuming axially symmetric flow, we introduce the stream function I/J(~, 8, t) via the usual relation 4r = --
1
w
4e = --
r* sin 19de ’
1
w
(2.6)
r sin 19dr
where q = (q,, qe, 0). With the aid of (2.2), (2.4) and (2.6) satisfied by the stream function ly is given by
one can show that the equation
(2.7) where r+, P
a2 D*=%+
z ’ and
Assuming that the sphere is moving with velocity V(r), we need to solve (2.7) subject to the following conditions: +!JI_ = -i
(9
w
(ii)
dr
Vu* sin* 0
= Vu sin* 8
(24
IEa
!j*0
as
r--*m
lylt=o=o
(iv)
We note that in the case of steady flow, the solution of (2.7), (2.8) is given by &=~Vu*sin*e(~-z), To solve our general boundary-value of the form
r
a
problem (2.7), (2.8), it is reasonable t&(r, 8,
t) =f(r, t)sin* 8.
(2.9) to assume a solution
(2.10)
Unsteady motion of a sphere in an elastico-viscous fluid
21
Substitution into (2.7) gives (2.11) where prime denotes differentiation
with respect to r. We now put
$va*(;- $) +I)@, t).
f(r, t) =
(2.12)
Substituting this into (2.11), we get wiu-~~f~+~W’-5~)-g(,r~-~,)-~~=0. L(w)+-K&)(
(2.13)
With the aid of (2.10) and (2.12), the conditions (2.8), assuming V(0) = 0, are now transformed to the homogeneous form (i) (ii)
(iii)
l&o=o.
(iv)
(2.14)
What we have achieved thus far is to transform the given boundary-value problem (2.7), (2.8) to a boundary-value problem (2.13), (2.14), with homogeneous boundary conditions. In this latter form the Kantovorich semi-direct variational technique is applicable. In this method we approximate v(r, t) by
This method differs from functions is required. homogeneous boundary that the variational form
the Ritz method in that only partial specification of the approximating The spatial functions gk(r) must be specified and must satisfy conditions (2.14) while hk(t) are to be determined. It can be shown of the problem (2.13)-(2.15) is given by [lo, 111 t(&)g,
6hk dr dt = 0.
(2.16)
The form of the spatial functions is, of course, not unique. We take them as g*(r)=~,
(2.17)
k=1,2,...
They satisfy all the boundary conditions given in (2.14). First approximation
To get the first approximation to the solution of the problem (2.13), (2.14), we put n = 1 in (2.15) and (2.17). That is, we take
and Mr, 0 = h,(t) If we now substitute into (2.16), that is
(r - a)’ 7.
22
H. RAMKISSOON
and H. SHIFANG
use (2.13) and integrate with respect to r from a to m, we get dt=O
(2.19)
where
1%
ff, =
63as
q =
16~ +z2 But t is arbitrary, hence
ah,
dt-qh,+cu,z=O.
dV
(2.20)
The solution of this equation subject to the initial condition h,(O) = 0 is given by f h,(t) = -cu,V - 0201eplf
The first approximation
I0
VeTalrdt.
(2.21)
to the original problem (2.7), (2.8) is therefore q(r, 8, t) = sin’ e[ a Va’(f - :)
h,(t)]
+ 9
(2.22)
where h,(t) is given by (2.20). Here we have utilized (2.18), (2.12) and (2.10). Second approximation
This is obtained by taking n = 2 in (2.16) and making use of (2.15) and (2.17). We thus get q,=h,(t)~+h,(t)~
Iff
L(I+I$
rs - ‘I’ r3
I
L(~+!J~)-
rs
(2.23)
’
6h 1 dr dt = 0
(2.24)
6h2 dr dt = 0.
(2.25)
Using (2.13) and repeating the above analysis, we obtain the following pair of simultaneous ordinary differential equations 4
-a4-~Ka2
16
~+~a2~h,+~;h2=~a7~,
105
grjh,=;a’$.
(2.26)
(2.27)
Applying the Laplace transform technique, it can be shown that the solution of the above system of equations (2.26), (2.27), subject to the initial conditions h,(O) = h2(0) = 0, takes the form h,(t) =A
[C,D,V
+ El/’ V(r)eel(‘-‘)dt 0
1
hz(f)=
-&z F,D, 1
V + G,
0
V(z)eel(‘-‘)
1 f %(1--t) VW dt1
+ E,(’ V(r)ee2(‘-r)dr
dr + G2 I0
(2.28a)
(2.28b)
Unsteady
motion of a sphere in an elastico-viscous
fluid
23
where 4 16 A 1 =za4-za2K,
1 82 AZ=ig-a2--GK,
AJ=;a2q,
B2 -_ - 3465 13 a2__ 1287 188 K’
B1=iga4-Ga2K, 1 82
C1 = 10B2 - A2,
A,=-$ 188 B4=- 1287 ’
B3=-&a2q,
C2 = 10B4 - A4
D, =A1B2-A2B,,
4 = A3B2 + AI B4 - A2B3 - A4B1
D3 = A3B4 - A4B3,
fi = lOBl -AI,
~@IC~--GD~)-C~D~
E,=
E
81-02 G
2
=e,(D&-44)-W3
G
62-91 =e,(D,F,-4fi)-44
’
e,-e2
F2= 10B3 - A3
=e,P,c,-CID,)-GD3
'
1
(2.29)
2
e2 - 65
and f3,, t12 are given by
z =css2+gs+ Thus, the second approximation
e,)(s - e,).
1
1
to our solution is
q(r, 8, t) = sin2 8 [fVa2(f-:)
+yh,(t)+vh2(t)]
(2.30)
where h,(t) and h2(f) are given in (2.28).
3. SPECIAL
CASES
(a) Constant acceleration Here we take 0
t
w =I,;,t> 0
(i) The first approximation 1/)(r, 8,
t) =
is given by sin2 8 [’4 eta z(f_!J
(ii) The second approximation q(r, 8, t) = V sin’
e
+PZ.$!~(l
-em,‘)].
(3.1)
is given by [1( 4 cfa f-~)+q%(t)+~h,(r)]
(3.2)
where hl(r)=~[CID1l+~{esl~1
h2(t)=
fI,t-l}+${e@1
--&$ [&DIt+${eel’1 1
e,t - 1) 2
e,t -
1) + 4
Here V(f) = V,(l - e-‘3,
{eezr - e2t - 111 2
@) Exponentially decreasing acceleration r>O.
1
H. RAMKISSOON
24
-20
-15
-10
and H. SHIFANG
-5
for constant
Fig. l(a). Streamlines
acceleration at t = 0.1. (b) acceleration at I = 0.1.
(K
=
0) Streamlines
for constant
20 (a) I 15
10
5 0
-5
-10
-15
-20
-15
-10
-5
Fig. 2(a). Streamlines
0
5
for constant
IO
15
20
-20
-15
-10
-5
0
5
acceleration at t =0.5. (b). (K = 0) Streamlines acceleration at t = 0.5.
10
15
20
for constant
20
15
10
5
0
-5
-10
-15
-20 -20
-15
-10
-5
Fig. 3(a). Streamlines
0
0
for constant
acceleration at f = 1. (b). acceleration at t = 1.
(K
=O) Streamlines
for constant
25
Unsteady motion of a sphere in an elastico-viscous fluid (b)
l* : -20
-15
-10
-5
0
5
10
15
20
Fig. 4(a). Streamlines for exponentially decreasing acceleration at I = 0.1. (b). (K = 0) Streamlines for exponentially decreasing acceleration at t = 0.1.
20 I
(a)
15
10
5
0
-5
-10
-15
Fig. 5(a). Streamlines for exponentially decreasing acceleration at I = 0.5. (b). (K = 0) Streamlines for exponentially decreasing acceleration at t = 0.5.
T (b)
20 I
(a)
15
10
5
0
-5
-10
-15
-20
-15
-10
-5
0
5
10
15
2"
-20
-15
-10
-5
0
5
10
15
Fig. 6(a). Streamlines for exponentially decreasing acceleration at I = 1.0. (b). (K = 0) Streamlines for exponentially decreasing acceleration at t = 1.
20
H. RAMKISSOON and H. SHIFANG
26
(i) The first approximation
is given by
r#(r, 8, t) = V, sin* 8 $ (1 - ,+)a*( [
f - 5) + 9
h,(t)]
where (3.3) (ii) The second approximation r/~(r, 8,t)=V0sin20
[
is given by
~(l-ePcr)u2
(
(r - a)’
4-:
>
+r3
h,(t) + qf
h*(t)]
(3.4)
where
hW=$[
clQ(l_e-“?+El
1
+E,
1
ee2* 1 7--+--2 e2
1
1$_$+&_& e2 + c II ’
eWcr
e2 + c
1
ee*:
--c*
h2(r)=-~[F,D,(1-e~c~)+G,(~-~+-e------
1 1 e%’ 1 e-=r e&t +G2I 7--+--2 e2 e,+c e,+c
, e,+c
1
I eW
e,+c I
II.
In order to get an insight into the form of the streamlines we take a = 4.0,
r] = 0.2,
K = 0.6,
c = 1.0
and plot $(r, 8, t) at different times. From Figs l-6 the streamlines seem to bunch closer to the sphere with increasing time. To examine the effect of elasticity, streamlines have been plotted for the same times but for K = 0. In other words these streamlines, depicted in Figs l(b)-6(b) are the streamlines for classical viscous fluids. Comparison of Figs l(b)-6(b) with the corresponding ones in Figs l(a)-6(a) show that the effect of elasticity is to push the streamlines slightly closer to the sphere and to create a slightly greater “bunching” around the sphere. Acknowledgemenf.r-One of the authors (H.R.) would like to acknowledge with thanks the Third World Academy of Sciences Fellowship granted under the South Fellowship Programme. The support granted by the Chinese Academy of Sciences and the facilities provided by the Research Laboratory of Non-Newtonian Fluid Mechanics at its Chengdu Branch are also acknowledged.
REFERENCES [I] K. WALTERS, Q. J. Me&. Appl. Marh. W, 444 (1960). [2] K. R. FRATER, /. Fluid Mech. 20, 369 (1964). [3] K. R. FRATER, Z. Angew. Marh. Phys. 18,798 (1967). [4] R. Y. S. LA1 and C. P. FAN, Int. J. Engng Sci. 12, 645 (1974). (51 R. Y. S. LA1 and C. P. FAN, Int. J. Engng Sci. 16,303(1978). (61R. H. THOMAS and K. WALTERS, Rheol. Acra 5, 23 (1966). (71 M. J. KING and N. D. WATERS, J. Phys. D: Appl. Phys. 5, 141 (1972). [8] L. D. LANDAU and E. M. LIFSCHITZ, Fluid Mechanics, p. 97. Pergamon Press, Oxford (1959). [9] L. V. KANTOVORICH and V. I. KRYLOV, Approximate Methods of Higher Analysis. Interscience New York (1958). [lo] R. S. R. GORLA and P. E. MADDEN, J. Non-Newronian Fluid Me& 16, 251 (1984). [ll] HAN SHIFANG, Continuum Mechanics ofNon-NewtonianFluids. Sichuan Academic Press, China (1988). [12] K. WALTERS, J. &f&z.1, 474 (1962). (Revision received 20 March 1992; accepred 26 March 1992)