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The generation of a draining liquid film by the vertical withdrawal of a flat plate from an elastico-viscous liquid
Journal of Non-Newtonian Fluid Mechanics, 26 (1987) 161-174 Elsevier Science Publishers B.V., Amsterdam - Printed in The Netherlands
161
THE GENERATION OF A DRAINING LIQUID FILM BY THE VERTICAL WITHDRAWAL OF A FLAT PLATE FROM AN ELASTICO-VISCOUS LIQUID
N.D. WATERS Department of Applied Mathematics Liverpool, L.69 3BX (Great Britain)
and Theoretical
Physics,
The University
of Liverpool,
and A.M. KEELEY British Aerospace, Military Aircraft Division, Warton Aerodrome, PR4 1AX (Great Britain)
Warton, Luncs,
(Received April 12,1987)
The analysis of Waters and Keeley [l] for the start-up of the drainage of an elastico-viscous liquid from a vertical surface is extended to include the effect of the vertical surface moving vertically with an arbitrary velocity. This is used to examine the effect of elasticity on the interesting drainage problems associated with the sudden withdrawal of a vertical plate from an elastico-viscous liquid.
1. Introduction We (Waters and Keeley [l]) have recently considered the development of the profile of a fihn of elastico-viscous liquid draining from rest down a vertical surface. In a similar fashion to the Newtonian solution of Annapuma and Ramanaiah [2] shall now extend that work by allowing the plate to move with a prescribed velocity given by a known function of time. In particular we shall investigate such industrially important flow situations as the withdrawal of the plate at a constant speed from the liquid and the stopping of the plate after it has been partially removed. These flows are encountered in many industrial processes such as dip-coating and electro0377-0257/87/$03.50
0 1987 Elsevier Science Publishers B.V.
162 plating. As in Ref. 1 the liquid is assumed to have a constant dynamic viscosity. In reality the liquid could well have a shear-dependent viscosity which would alter the flow characteristics. However, it is interesting to consider the effect that elasticity, in isolation, has on the formation of the draining film. Also, we would expect elasticity to be the dominant characteristic of the liquid during the initial formation of the film and when the plate experiences sudden changes in velocity. In future work we hope to consider more realistic models which will include both the effects of elasticity and a shear-dependent viscosity. However, it is unlikely that the neat analytical methods used in this paper will still be applicable and we may well have to attempt a numerical solution. 2. Theory Here, the elastico-viscous liquid will be characterized with the same Oldroyd [3] liquid ‘B’ model as in Waters and Keeley [l]. The equations of state are given by Pik
=
plik +
-Pgik
+ Pz’k 9
(1)
x,7aPtik =
Wk +
$e(l)ik A,
_
at
where pik is the stress tensor, pi;, the deviatoric or extra-stress tensor, e$’ the rate-of-strain tensor, gik is the metric tensor of the fixed coordinate system, p an isotropic pressure, p the viscosity, X, the relaxation time, A, the retardation time and a/& is the contravariant convected derivative defined by Oldroyd [4] as 8Aij _ aAij I Ur CIA” ,UjAi, &.i Akj &
at
---
axr
ax’
--
axk
*
Since this model is capable of predicting elastic behaviour but not a shear-dependent viscosity, we assume, as in [l], that elasticity will be the dominant mode of behaviour in the transitional stages of the flow and the liquid will be assumed to have a constant viscosity throughout the flow. We choose Cartesian axes x and y such that x points vertically upwards a along the plate and y is perpendicular to the plate. As in the start-up of drainage [l] the liquid is allowed to fall under gravity for t 2 0. On making the assumptions that the thickness of the film is small compared to its length the equations of motion reduce to
* Note that this is the opposite direction to that used in [l].
163 where p is the liquid density, the velocity field u = (u( x, y, t), u( x, y, t), 0) and u is the subject to the boundary conditions 4%
t > 0,
0, t) =f(t),
&.4/ay=Oon
y=h,
(4)
t>o.
(5)
initial condition is
The
+,
Ogygh.
Y, 0) = 0,
(6) Since there is no natural length scale in this problem we shall not non-dimensionalize in the manner used by Waters and Keeley [l] but adopt the length and time scales ( y2/g)l13 and ( y/g2)lj3 used by Annapurna and Ramanaiah [2] where v = p/p is the kinematic viscosity. Thus the new non-dimensional parameters (capital letters) are
s, = hl(g*/Vy3,
s* = A2(g*/vy3,
u=
v=
u/(vgy3,
u/(vgy3.
Taking the Laplace transform of (3) and using (6) and (2) we obtain -
!$_Q2&$,
(8)
with u( X, Y, y) satisfying
Clu/aY = 0 on Y =
H,
T>O.
00)
F(y)
is the Laplace transform of F(t) (=f( sional velocity of the plate. Q is given by Q*=:
t)/(
vg)l13),
the non-dimen-
YO + SlY>
1+s*y
(11)
-
Equation (8) has solution
G(x, v) =
($ +F(y)ico~oseh(&-;) - $.
(12)
Following the method used by Waters and King [S] and making use of the convolution theorem, the inverse of U is found to be
u(X, Y, T) = +(y_
2H)
+ 2H2
5 sin(z’H) n=l
x G,(K T) + ~=G;(K T- t)F(5)
de],
(13)
164 where GN( H, T) is the same as that defined by Waters and King [5] and Waters and Keeley [l], and is given by
with (Ye = 1+ N2S2/H2, & = {(a;
(15A)
- 4N2SJH2)
,
(15B)
yN = (Ye - 2N2Sl/H2
(15C)
and N = (2n - 1)7r/2. We shah now allow H to vary slowly with X. On using the condition V( X, 0, T) = 0 and the continuity equation, along with the condition at the free surface Y = H( X, T) V(X, H, T) = g
+ U(X,
H, T)g,
(16)
Tao,
we obtain X(H,
T)-X,(H)=
-H2T+2e
+
=G,(HX)dX
L-!!n=l H4 aH
iTiKG:(Hx -‘t)F(t)
dt
4x]},
07)
where X = X,(H) is the inverse of the function H(X) which describes the shape of the free surface initially. Let us now consider specific cases. 2.1. Drainage (withdrawal)
after
the plate
is impulsive
lifted
with constant
speed
U,
The plate is impulsively lifted in its own plane in the X direction with a constant non-dimensional speed U, for T 2 0 and is at rest for T-K 0. The liquid which has a free surface shape given by X,(H) at T = 0 is allowed to drain freely for T > 0. We can write u(X,
0, T) = U,x(T),
where S’(T) .8(T)
08)
is the Heavyside function defined as
= 1
for Tao,
= 0
for Tc
09) 0.
165 Using eqn. (18) in (13) we get
u(x,
yyH)
Y, T)=;(Y-2H)+2H2F n=l
x { G,(H, T) + U,Gk(H, T)},
(20)
and after much algebra eqn. (19) gives
X(H, T) - x,(H) =
-HZT+
x
i
F - H’(S,
- S,) + U,T-
T) + TB,(H,
A,(H,
UoH2 - H4 E N+ n=l
T) - F[&(H,
T) + TD,(H,
xexp(--a,TP&),
T)] (21)
where A,(%
T) = K3(B:(4
+ 3~
+ (P;(4Y,
+ ~YN) cosh&T/2S,)
+ 2% + 3%Y,
+ 3%)
- 2%&)
xsinh(P,T/2S,)), %(H,
T) = (S~~~)-‘((~~,Y,+P~(~,-~YN)) +&v(2%YN
C&C
T) = P,‘(3P,:
- Y‘z- Pi)
cosh(&T/2S,)
suW%T/2Sr)),
cosh(P,T/2S,)
+ (3YNPI: + 2% - Y;) suW,T/2S,)), D,(H,
T)=(S,P,:)-~((Y~-P~-P~(Y~-~,))~~~~(~~T/~S,) +&(a,Y,-
Pi> sirW%vT/2SA).
(22)
On putting U, = 0 eqn. (21) reduces to the free-drainage solution obtained in [l]. The only differences are due to the different non-dimensional scales and X being measured in the opposite direction. If we put S, = S, in eqns. (21) and (22) we obtain the Newtonian result derived by Annapuma and Ramanaiah [2] X(K
T) - X,(H)
= -HZT+
?$
_ UoH2 + L&T- 2 E N-6(H2(5H2
+ 2N’T)
n=l
-N2Uo(3H2 + 2N’T))
exp( -N2T/H2).
(23)
166
Fig. 1. Expected initial profile for a plate partially immersed in a bath of liquid.
The fihn profile for a liquid film produced by raising the plate with a constant speed U, is given by eqn. (21). We have not specified the initial profile X,(H) but it could be chosen to be the inverse of any function H(X) which represents the shape of the initial profile. If we were to consider the interesting case of the removal of the plate from a large bath of liquid then the initial profile would probably be of the form shown in Fig. 1 and we would expect to have a region of length 6 (say) immediately above the surface of the bath in which iflH/l+X is not small and where our theory is not valid. However, for a distance greater than 8 away from the surface of the bath the theory will hold, and the length of the thin fihn in which the theory holds will increase with time. In eqn. (21) the transient component for the withdrawal of a plate consists of two parts, that associated with the start-up of the drainage (the terms involving A, and BN) and that directly caused by the impulsive start up of the plate (the terms involving C, and 4). We note that since X( H, 0) = X,,(H), on putting T = 0 in eqn. (21) the transient terms, given by the summation, take the value -H2(&-,S2)-U,H2
I
at T=O.
The variation with time of the transient terms for a Newtonian liquid is shown in Fig. 2 and that for an elastico-viscous liquid, with S, = 0.12 and
167
05
-05
1.0
TIME
I
-1.0
Fig. 2. The transient terms occuming in eqn. (23), the Newtonian sional time T for H = 0.3 (a), 0.6 (b) and 0.9 (c), respectively.
case, versus non-dimen-
S, = 0.02, is shown in Fig. 3. It can be seen that near the wall (H = 0.3) the presence of elasticity causes the transient terms to persist longer than their Newtonian equivalent. However, further away from the wall (H = 0.9) the presence of elasticity causes the transient terms to fade more quickly. For larger values of S, and S, the elastic transient terms persist longer than the equivalent Newtonian terms for all H. The amplitude of the oscillations increases for increasing values of the elastic ratio of the liquid S,/S, for
Fig. 3. The transient terms occurring in equation (21) versus non-dimensional S, = 0.12, S, = 0.02, U, = 1.0 for H = 0.3 (a), 0.6 (b) and 0.9 (c), respectively.
time T when
168 fixed values of the elastic difference (S, - S,). Whereas increasing (S, - S,) for fixed S,/S, increases the persistence of the oscillations; these oscillations can continue for a much longer time than the Newtonian equivalent, especially in the region near the wall. When the transient terms have tended to zero the film profile, for a Newtonian liquid, is given by X(H,
T) -X,,(H)
= -H’T+
F
-
UoH2 + U,T.
This equation contains both the drainage inertial correction 2H4/3 and the extra terms associated with the impulsively started moving plate - &Hz + U,T, found by Annapurna and Ramanaiah [2]. For large T eqn. (24) yields the well known expression for the constant film thickness at high capillary number H=fi.
(25)
For an elastico-viscous liquid the long-time solution is given by X(H,
T) -X,(H)
= -H’T+
y
- H2(S,
- S2) + U,T-
UoH2.
(26)
We observe the same extra term H2( S1 - S,) as obtained by Waters and Keeley [l] in start-up of the drainage from a stationary plate. In Fig. 4 we show for a fixed distance, e.g. H = 0.3, away from the wall the distance the surface of the film at X = X,(0.3) at T = 0 has moved as a
Fig. 4. X- X0 versus T when H= 0.3, U, =l.O, S,/S, and 10.0 (c), respectively.
= 0.125 for S, - S, = 0.1 (a), 1.0 (b)
169
Y
Fig. 5. The development of non-dimensional fihn profiles for an elastico-viscous liquid with S, = 0.12, S, = 0.02 (continuous curves) compared to the Newtonian equivalent (discontinuous curves). X - X0 versus H when Us = 1.0, T = 0.2 (a), 0.4 (b) and 0.6 (c), respectively.
function of time for U, = 1.0 and different values of S, - S,. It can be seen that for the elastic liquids the distance upstream oscillates before tending to the quasi-steady solution (26). In Fig. 5 we illustrate the fihn profile generation, for an elastic liquid and a Newtonian liquid; a surface wave can be seen to propagate across the film for the elastic liquid. The elastic liquid produces a thinner film than the
Fig. 6. The effect of the elastic correction on the withdrawal profile. X- X0 versus H when U, = 1.0, T = 5.0for (a) S, = S, (Newtonian case), (b) S, - S, = 0.1, (c) S1 - S, = 1.0 and (d) S, - S, = 10.0, respectively.
170 Newtonian liquid. This thinning of the film is due to the same H2(S, - S,) term that produced the dramatic effects in the start-up of the drainage from a stationary plate. Figure 6 shows the fihn profile at T = 5.0 for varying values of (S, - S,) and the large correction that elasticity makes to the film thickness at this stage of the film development. 2.2. Drainage after the plate is impulsively brought to rest (post withdrawal) If the plate moves with a constant non-dimensional speed U, for a non-dimensional time T, and then stopped then the plate velocity function will be given by F(T) = U,@‘(T)
-x(T-
G)),
where &‘(T) and &‘( T - T,) are Heavyside functions. For 0 < T d T, U( X,Y, T) and X( H, T) are given by eqns. (20) and (21) respectively but for T > T, we find that (13) becomes U(X, Y, T)=+(Y-2H)IZH’f
sn(F’H) n=l
x {Giv(H, T) + U,Gh(K T) - U,Gk(H, T- T,)}
(27)
and eqn. (17) gives us X(K
7’) - X,(H)
= -H2T+
x
?$
A,(H,
-HZ@,
- S,) + U,T, - H4 E IV+ ?l=l
T) + TB,(H,
x exp( - a,T/2S,)
T) - F[,(H,
+ y[CN(H,
T-
T) + TD,(H,
T)]
T,)
+(T- T,)D,(H, T- G)] exd-+dT-- T&W)
I
-
(28)
The corresponding Newtonian result for T > T,, obtained by putting S, = S,, is given by X(H, T) - X,(H) = -H’T+
F
+ U,T, - 2 f
A+(
H2(5H2
+ 2N2T)
n=l
-N2Uo(3H2
+ 2N2T))
+N2U0(3H2
+ 2N2(T-
exp( -iV2T/H2) T,,))exp( -N*(T-
T,)/H’).
(29)
171
Fig. 7. The transient terms associated with the stopping of the plate after a time T’ when U, = 1.0, S, = 0.12, S, = 0.02 for H = 0.3 (a), 0.6 (b) and 0.9 (c), respectively.
The transient terms associated with the stopping of the plate at a time T = To (To being large enough so that the withdrawal and drainage transient
terms have already tended to zero) shown in Fig. 7, are equal to, but opposite in sign to, the transient terms associated with the impulsively starting of the plate at T = 0 given in equation (21), but with T replaced by T-T,,andare -2&H’
E W4(CN(H,
T-T,)
+ TD,(H,
T-
To))
?l==l
which reduce to UoH2 at T = To. As these transient terms fade away they cancel out the - UoH2 obtained from the start-up of the withdrawal giving us the long-time solution X(H,
T) -X,(H)
= -H2T-
F
- H2( S1 - S,) + U,T.
The corresponding Newtonian long-time result is X(H,
T) -X,(H)
= -H’T-
F
+ U,T.
(31)
The post-withdrawal drainage of an elastico-viscous liquid is again influenced by the additional term H2( S, - S,) which causes an elastic correction to the long-time solution. The “stopping” transient terms are less noticeable because the fihn thickness has already diminished and they
172 11 0
x-x,
90
10.0
11.0
12.0
TIME
Fig. 8. X - X,, versus T for H = 0.3, the plate initially moving at speed Z&= 1.0, is stopped at To = 10.0 for (a) S, = S,, (b) S, = 0.12, S, = 0.02 and (c) S, = 1.2, S, = 0.2, respectively.
subsequently fade away quicker those associated with the withdrawal. In Fig. 8 we show, for a fixed distance H = 0.3 away from the wall, the position of the free surface initially at X = X,(0.3) as a function of time T. The transient terms associated with the drainage and the withdrawal have already faded to zero by the time the plate is stopped. We observe a
Fig. 9. Stopping the plate shortly after the initiation of the withdrawal. X- X0 versus T when I!&=l.O, To = 0.5, H = 0.3 for (a) S, = S,, (b) S, = 0.12, S, = 0.02 and (c) S, =1.2, S, = 0.2, respectively.
173
Fig. 10. Development of the profile after the plate has been stopped shortly after the initiation of the withdrawal. X - X,, versus H when U, = 1.0, To = 0.5, S, = 0.12, S, = 0.2 for T = 0.5 (a), 1.0 (b) and 1.5 (c), respectively.
dramatic elastic overshoot for the non-Newtonian liquids before they tend to the final quasi-steady state. The film profile for a plate that is suddenly stopped after it has been withdrawn for a large time is only slightly perturbed by the stopping of the plate, this is due to the thickness of the film causing the transient terms to fade very rapidly. If the plate is stopped before the withdrawal and drainage transient terms have decayed then we have an interaction between the starting and stopping transient terms which can combine to cause a larger or smaller elastic overshoot, see Figs. 9 and 10. Figure 9 shows the variation with time T of X( H, T) - X,,(H) for H = 0.3 and Fig. 10 shows the film profile for fixed values of T, both of these graphs are for U, = 1 and To = 0.5. In Fig. 10 we can see the presence of two distinct waves propagating across the film, one is due to the start-up of the withdrawal (the one furthest from the wall) and the other is due to the stopping of the plate. 3. Conclusions The transient terms associated with the sudden start-up and stopping of the plate can be considerably larger than those associated with the commencement of drainage and can cause the fihn profile to be visibly perturbed by the presence of a comparatively large surface wave.
174
The long-time solution for the withdrawal is given by eqn. (26) and the long-time post-withdrawal drainage solution is given by eqn. (31); both contain the “elastic correction” I%‘( S, - S,) to the Newtonian results obtained by Armapurna and Ramanaiah [2]. Waters and Keeley [l] found that this term is present in the drainage from a plate which is stationary. We have shown that it is still present in the drainage from a plate which suddenly starts moving with a constant speed, and persists for all time even when the plate is suddenly stopped. This correction ensures that at any given point on the plate the film thickness is less for an elastico-viscous liquid than for a Newtonian with the same viscosity. Thus the presence of elasticity makes a permanent contribution to the quasi-steady flow and causes the f&n to drain faster than the corresponding Newtonian liquid. Acknowledgement
The authors discussions.
would like to thank Dr. G.K. Retie
for some useful
References 1 2 3 4 5
N.D. Waters and A.M. Keeley, J. Non-Newtonian Fluid Mech., 22 (1987) 325. N. Annapurna and G. Ramanaiah, AIChE J., 22 (5) (1976) 940. J.G. Oldroyd, Proc. Roy. Sot., A200 (1950) 523. J.G. Oldroyd, F’roc. Roy. Sot., A245 (1958) 278. N.D. Waters and M.J. King, Rheol. Acta, 9 (1970) 345.
161
THE GENERATION OF A DRAINING LIQUID FILM BY THE VERTICAL WITHDRAWAL OF A FLAT PLATE FROM AN ELASTICO-VISCOUS LIQUID
N.D. WATERS Department of Applied Mathematics Liverpool, L.69 3BX (Great Britain)
and Theoretical
Physics,
The University
of Liverpool,
and A.M. KEELEY British Aerospace, Military Aircraft Division, Warton Aerodrome, PR4 1AX (Great Britain)
Warton, Luncs,
(Received April 12,1987)
The analysis of Waters and Keeley [l] for the start-up of the drainage of an elastico-viscous liquid from a vertical surface is extended to include the effect of the vertical surface moving vertically with an arbitrary velocity. This is used to examine the effect of elasticity on the interesting drainage problems associated with the sudden withdrawal of a vertical plate from an elastico-viscous liquid.
1. Introduction We (Waters and Keeley [l]) have recently considered the development of the profile of a fihn of elastico-viscous liquid draining from rest down a vertical surface. In a similar fashion to the Newtonian solution of Annapuma and Ramanaiah [2] shall now extend that work by allowing the plate to move with a prescribed velocity given by a known function of time. In particular we shall investigate such industrially important flow situations as the withdrawal of the plate at a constant speed from the liquid and the stopping of the plate after it has been partially removed. These flows are encountered in many industrial processes such as dip-coating and electro0377-0257/87/$03.50
0 1987 Elsevier Science Publishers B.V.
162 plating. As in Ref. 1 the liquid is assumed to have a constant dynamic viscosity. In reality the liquid could well have a shear-dependent viscosity which would alter the flow characteristics. However, it is interesting to consider the effect that elasticity, in isolation, has on the formation of the draining film. Also, we would expect elasticity to be the dominant characteristic of the liquid during the initial formation of the film and when the plate experiences sudden changes in velocity. In future work we hope to consider more realistic models which will include both the effects of elasticity and a shear-dependent viscosity. However, it is unlikely that the neat analytical methods used in this paper will still be applicable and we may well have to attempt a numerical solution. 2. Theory Here, the elastico-viscous liquid will be characterized with the same Oldroyd [3] liquid ‘B’ model as in Waters and Keeley [l]. The equations of state are given by Pik
=
plik +
-Pgik
+ Pz’k 9
(1)
x,7aPtik =
Wk +
$e(l)ik A,
_
at
where pik is the stress tensor, pi;, the deviatoric or extra-stress tensor, e$’ the rate-of-strain tensor, gik is the metric tensor of the fixed coordinate system, p an isotropic pressure, p the viscosity, X, the relaxation time, A, the retardation time and a/& is the contravariant convected derivative defined by Oldroyd [4] as 8Aij _ aAij I Ur CIA” ,UjAi, &.i Akj &
at
---
axr
ax’
--
axk
*
Since this model is capable of predicting elastic behaviour but not a shear-dependent viscosity, we assume, as in [l], that elasticity will be the dominant mode of behaviour in the transitional stages of the flow and the liquid will be assumed to have a constant viscosity throughout the flow. We choose Cartesian axes x and y such that x points vertically upwards a along the plate and y is perpendicular to the plate. As in the start-up of drainage [l] the liquid is allowed to fall under gravity for t 2 0. On making the assumptions that the thickness of the film is small compared to its length the equations of motion reduce to
* Note that this is the opposite direction to that used in [l].
163 where p is the liquid density, the velocity field u = (u( x, y, t), u( x, y, t), 0) and u is the subject to the boundary conditions 4%
t > 0,
0, t) =f(t),
&.4/ay=Oon
y=h,
(4)
t>o.
(5)
initial condition is
The
+,
Ogygh.
Y, 0) = 0,
(6) Since there is no natural length scale in this problem we shall not non-dimensionalize in the manner used by Waters and Keeley [l] but adopt the length and time scales ( y2/g)l13 and ( y/g2)lj3 used by Annapurna and Ramanaiah [2] where v = p/p is the kinematic viscosity. Thus the new non-dimensional parameters (capital letters) are
s, = hl(g*/Vy3,
s* = A2(g*/vy3,
u=
v=
u/(vgy3,
u/(vgy3.
Taking the Laplace transform of (3) and using (6) and (2) we obtain -
!$_Q2&$,
(8)
with u( X, Y, y) satisfying
Clu/aY = 0 on Y =
H,
T>O.
00)
F(y)
is the Laplace transform of F(t) (=f( sional velocity of the plate. Q is given by Q*=:
t)/(
vg)l13),
the non-dimen-
YO + SlY>
1+s*y
(11)
-
Equation (8) has solution
G(x, v) =
($ +F(y)ico~oseh(&-;) - $.
(12)
Following the method used by Waters and King [S] and making use of the convolution theorem, the inverse of U is found to be
u(X, Y, T) = +(y_
2H)
+ 2H2
5 sin(z’H) n=l
x G,(K T) + ~=G;(K T- t)F(5)
de],
(13)
164 where GN( H, T) is the same as that defined by Waters and King [5] and Waters and Keeley [l], and is given by
with (Ye = 1+ N2S2/H2, & = {(a;
(15A)
- 4N2SJH2)
,
(15B)
yN = (Ye - 2N2Sl/H2
(15C)
and N = (2n - 1)7r/2. We shah now allow H to vary slowly with X. On using the condition V( X, 0, T) = 0 and the continuity equation, along with the condition at the free surface Y = H( X, T) V(X, H, T) = g
+ U(X,
H, T)g,
(16)
Tao,
we obtain X(H,
T)-X,(H)=
-H2T+2e
+
=G,(HX)dX
L-!!n=l H4 aH
iTiKG:(Hx -‘t)F(t)
dt
4x]},
07)
where X = X,(H) is the inverse of the function H(X) which describes the shape of the free surface initially. Let us now consider specific cases. 2.1. Drainage (withdrawal)
after
the plate
is impulsive
lifted
with constant
speed
U,
The plate is impulsively lifted in its own plane in the X direction with a constant non-dimensional speed U, for T 2 0 and is at rest for T-K 0. The liquid which has a free surface shape given by X,(H) at T = 0 is allowed to drain freely for T > 0. We can write u(X,
0, T) = U,x(T),
where S’(T) .8(T)
08)
is the Heavyside function defined as
= 1
for Tao,
= 0
for Tc
09) 0.
165 Using eqn. (18) in (13) we get
u(x,
yyH)
Y, T)=;(Y-2H)+2H2F n=l
x { G,(H, T) + U,Gk(H, T)},
(20)
and after much algebra eqn. (19) gives
X(H, T) - x,(H) =
-HZT+
x
i
F - H’(S,
- S,) + U,T-
T) + TB,(H,
A,(H,
UoH2 - H4 E N+ n=l
T) - F[&(H,
T) + TD,(H,
xexp(--a,TP&),
T)] (21)
where A,(%
T) = K3(B:(4
+ 3~
+ (P;(4Y,
+ ~YN) cosh&T/2S,)
+ 2% + 3%Y,
+ 3%)
- 2%&)
xsinh(P,T/2S,)), %(H,
T) = (S~~~)-‘((~~,Y,+P~(~,-~YN)) +&v(2%YN
C&C
T) = P,‘(3P,:
- Y‘z- Pi)
cosh(&T/2S,)
suW%T/2Sr)),
cosh(P,T/2S,)
+ (3YNPI: + 2% - Y;) suW,T/2S,)), D,(H,
T)=(S,P,:)-~((Y~-P~-P~(Y~-~,))~~~~(~~T/~S,) +&(a,Y,-
Pi> sirW%vT/2SA).
(22)
On putting U, = 0 eqn. (21) reduces to the free-drainage solution obtained in [l]. The only differences are due to the different non-dimensional scales and X being measured in the opposite direction. If we put S, = S, in eqns. (21) and (22) we obtain the Newtonian result derived by Annapuma and Ramanaiah [2] X(K
T) - X,(H)
= -HZT+
?$
_ UoH2 + L&T- 2 E N-6(H2(5H2
+ 2N’T)
n=l
-N2Uo(3H2 + 2N’T))
exp( -N2T/H2).
(23)
166
Fig. 1. Expected initial profile for a plate partially immersed in a bath of liquid.
The fihn profile for a liquid film produced by raising the plate with a constant speed U, is given by eqn. (21). We have not specified the initial profile X,(H) but it could be chosen to be the inverse of any function H(X) which represents the shape of the initial profile. If we were to consider the interesting case of the removal of the plate from a large bath of liquid then the initial profile would probably be of the form shown in Fig. 1 and we would expect to have a region of length 6 (say) immediately above the surface of the bath in which iflH/l+X is not small and where our theory is not valid. However, for a distance greater than 8 away from the surface of the bath the theory will hold, and the length of the thin fihn in which the theory holds will increase with time. In eqn. (21) the transient component for the withdrawal of a plate consists of two parts, that associated with the start-up of the drainage (the terms involving A, and BN) and that directly caused by the impulsive start up of the plate (the terms involving C, and 4). We note that since X( H, 0) = X,,(H), on putting T = 0 in eqn. (21) the transient terms, given by the summation, take the value -H2(&-,S2)-U,H2
I
at T=O.
The variation with time of the transient terms for a Newtonian liquid is shown in Fig. 2 and that for an elastico-viscous liquid, with S, = 0.12 and
167
05
-05
1.0
TIME
I
-1.0
Fig. 2. The transient terms occuming in eqn. (23), the Newtonian sional time T for H = 0.3 (a), 0.6 (b) and 0.9 (c), respectively.
case, versus non-dimen-
S, = 0.02, is shown in Fig. 3. It can be seen that near the wall (H = 0.3) the presence of elasticity causes the transient terms to persist longer than their Newtonian equivalent. However, further away from the wall (H = 0.9) the presence of elasticity causes the transient terms to fade more quickly. For larger values of S, and S, the elastic transient terms persist longer than the equivalent Newtonian terms for all H. The amplitude of the oscillations increases for increasing values of the elastic ratio of the liquid S,/S, for
Fig. 3. The transient terms occurring in equation (21) versus non-dimensional S, = 0.12, S, = 0.02, U, = 1.0 for H = 0.3 (a), 0.6 (b) and 0.9 (c), respectively.
time T when
168 fixed values of the elastic difference (S, - S,). Whereas increasing (S, - S,) for fixed S,/S, increases the persistence of the oscillations; these oscillations can continue for a much longer time than the Newtonian equivalent, especially in the region near the wall. When the transient terms have tended to zero the film profile, for a Newtonian liquid, is given by X(H,
T) -X,,(H)
= -H’T+
F
-
UoH2 + U,T.
This equation contains both the drainage inertial correction 2H4/3 and the extra terms associated with the impulsively started moving plate - &Hz + U,T, found by Annapurna and Ramanaiah [2]. For large T eqn. (24) yields the well known expression for the constant film thickness at high capillary number H=fi.
(25)
For an elastico-viscous liquid the long-time solution is given by X(H,
T) -X,(H)
= -H’T+
y
- H2(S,
- S2) + U,T-
UoH2.
(26)
We observe the same extra term H2( S1 - S,) as obtained by Waters and Keeley [l] in start-up of the drainage from a stationary plate. In Fig. 4 we show for a fixed distance, e.g. H = 0.3, away from the wall the distance the surface of the film at X = X,(0.3) at T = 0 has moved as a
Fig. 4. X- X0 versus T when H= 0.3, U, =l.O, S,/S, and 10.0 (c), respectively.
= 0.125 for S, - S, = 0.1 (a), 1.0 (b)
169
Y
Fig. 5. The development of non-dimensional fihn profiles for an elastico-viscous liquid with S, = 0.12, S, = 0.02 (continuous curves) compared to the Newtonian equivalent (discontinuous curves). X - X0 versus H when Us = 1.0, T = 0.2 (a), 0.4 (b) and 0.6 (c), respectively.
function of time for U, = 1.0 and different values of S, - S,. It can be seen that for the elastic liquids the distance upstream oscillates before tending to the quasi-steady solution (26). In Fig. 5 we illustrate the fihn profile generation, for an elastic liquid and a Newtonian liquid; a surface wave can be seen to propagate across the film for the elastic liquid. The elastic liquid produces a thinner film than the
Fig. 6. The effect of the elastic correction on the withdrawal profile. X- X0 versus H when U, = 1.0, T = 5.0for (a) S, = S, (Newtonian case), (b) S, - S, = 0.1, (c) S1 - S, = 1.0 and (d) S, - S, = 10.0, respectively.
170 Newtonian liquid. This thinning of the film is due to the same H2(S, - S,) term that produced the dramatic effects in the start-up of the drainage from a stationary plate. Figure 6 shows the fihn profile at T = 5.0 for varying values of (S, - S,) and the large correction that elasticity makes to the film thickness at this stage of the film development. 2.2. Drainage after the plate is impulsively brought to rest (post withdrawal) If the plate moves with a constant non-dimensional speed U, for a non-dimensional time T, and then stopped then the plate velocity function will be given by F(T) = U,@‘(T)
-x(T-
G)),
where &‘(T) and &‘( T - T,) are Heavyside functions. For 0 < T d T, U( X,Y, T) and X( H, T) are given by eqns. (20) and (21) respectively but for T > T, we find that (13) becomes U(X, Y, T)=+(Y-2H)IZH’f
sn(F’H) n=l
x {Giv(H, T) + U,Gh(K T) - U,Gk(H, T- T,)}
(27)
and eqn. (17) gives us X(K
7’) - X,(H)
= -H2T+
x
?$
A,(H,
-HZ@,
- S,) + U,T, - H4 E IV+ ?l=l
T) + TB,(H,
x exp( - a,T/2S,)
T) - F[,(H,
+ y[CN(H,
T-
T) + TD,(H,
T)]
T,)
+(T- T,)D,(H, T- G)] exd-+dT-- T&W)
I
-
(28)
The corresponding Newtonian result for T > T,, obtained by putting S, = S,, is given by X(H, T) - X,(H) = -H’T+
F
+ U,T, - 2 f
A+(
H2(5H2
+ 2N2T)
n=l
-N2Uo(3H2
+ 2N2T))
+N2U0(3H2
+ 2N2(T-
exp( -iV2T/H2) T,,))exp( -N*(T-
T,)/H’).
(29)
171
Fig. 7. The transient terms associated with the stopping of the plate after a time T’ when U, = 1.0, S, = 0.12, S, = 0.02 for H = 0.3 (a), 0.6 (b) and 0.9 (c), respectively.
The transient terms associated with the stopping of the plate at a time T = To (To being large enough so that the withdrawal and drainage transient
terms have already tended to zero) shown in Fig. 7, are equal to, but opposite in sign to, the transient terms associated with the impulsively starting of the plate at T = 0 given in equation (21), but with T replaced by T-T,,andare -2&H’
E W4(CN(H,
T-T,)
+ TD,(H,
T-
To))
?l==l
which reduce to UoH2 at T = To. As these transient terms fade away they cancel out the - UoH2 obtained from the start-up of the withdrawal giving us the long-time solution X(H,
T) -X,(H)
= -H2T-
F
- H2( S1 - S,) + U,T.
The corresponding Newtonian long-time result is X(H,
T) -X,(H)
= -H’T-
F
+ U,T.
(31)
The post-withdrawal drainage of an elastico-viscous liquid is again influenced by the additional term H2( S, - S,) which causes an elastic correction to the long-time solution. The “stopping” transient terms are less noticeable because the fihn thickness has already diminished and they
172 11 0
x-x,
90
10.0
11.0
12.0
TIME
Fig. 8. X - X,, versus T for H = 0.3, the plate initially moving at speed Z&= 1.0, is stopped at To = 10.0 for (a) S, = S,, (b) S, = 0.12, S, = 0.02 and (c) S, = 1.2, S, = 0.2, respectively.
subsequently fade away quicker those associated with the withdrawal. In Fig. 8 we show, for a fixed distance H = 0.3 away from the wall, the position of the free surface initially at X = X,(0.3) as a function of time T. The transient terms associated with the drainage and the withdrawal have already faded to zero by the time the plate is stopped. We observe a
Fig. 9. Stopping the plate shortly after the initiation of the withdrawal. X- X0 versus T when I!&=l.O, To = 0.5, H = 0.3 for (a) S, = S,, (b) S, = 0.12, S, = 0.02 and (c) S, =1.2, S, = 0.2, respectively.
173
Fig. 10. Development of the profile after the plate has been stopped shortly after the initiation of the withdrawal. X - X,, versus H when U, = 1.0, To = 0.5, S, = 0.12, S, = 0.2 for T = 0.5 (a), 1.0 (b) and 1.5 (c), respectively.
dramatic elastic overshoot for the non-Newtonian liquids before they tend to the final quasi-steady state. The film profile for a plate that is suddenly stopped after it has been withdrawn for a large time is only slightly perturbed by the stopping of the plate, this is due to the thickness of the film causing the transient terms to fade very rapidly. If the plate is stopped before the withdrawal and drainage transient terms have decayed then we have an interaction between the starting and stopping transient terms which can combine to cause a larger or smaller elastic overshoot, see Figs. 9 and 10. Figure 9 shows the variation with time T of X( H, T) - X,,(H) for H = 0.3 and Fig. 10 shows the film profile for fixed values of T, both of these graphs are for U, = 1 and To = 0.5. In Fig. 10 we can see the presence of two distinct waves propagating across the film, one is due to the start-up of the withdrawal (the one furthest from the wall) and the other is due to the stopping of the plate. 3. Conclusions The transient terms associated with the sudden start-up and stopping of the plate can be considerably larger than those associated with the commencement of drainage and can cause the fihn profile to be visibly perturbed by the presence of a comparatively large surface wave.
174
The long-time solution for the withdrawal is given by eqn. (26) and the long-time post-withdrawal drainage solution is given by eqn. (31); both contain the “elastic correction” I%‘( S, - S,) to the Newtonian results obtained by Armapurna and Ramanaiah [2]. Waters and Keeley [l] found that this term is present in the drainage from a plate which is stationary. We have shown that it is still present in the drainage from a plate which suddenly starts moving with a constant speed, and persists for all time even when the plate is suddenly stopped. This correction ensures that at any given point on the plate the film thickness is less for an elastico-viscous liquid than for a Newtonian with the same viscosity. Thus the presence of elasticity makes a permanent contribution to the quasi-steady flow and causes the f&n to drain faster than the corresponding Newtonian liquid. Acknowledgement
The authors discussions.
would like to thank Dr. G.K. Retie
for some useful
References 1 2 3 4 5
N.D. Waters and A.M. Keeley, J. Non-Newtonian Fluid Mech., 22 (1987) 325. N. Annapurna and G. Ramanaiah, AIChE J., 22 (5) (1976) 940. J.G. Oldroyd, Proc. Roy. Sot., A200 (1950) 523. J.G. Oldroyd, F’roc. Roy. Sot., A245 (1958) 278. N.D. Waters and M.J. King, Rheol. Acta, 9 (1970) 345.