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Two-phase fluid flow and hysteresis in a periodic capillary tube
Two-Phase Fluid Flow and Hysteresis in a Periodic Capillary Tube S A M U E L L E V I N E , J A M E S L O W N D E S , AND P A U L R E E D
Department of Mathematics, University of Manchester, Manchester M13 9PL, England Received November 11, 1977; accepted September 20, 1979 The wetting process in a vertical capillary tube with nonuniform bore is examined. The model chosen consists of a sequence of conical sections alternately convergingdownwards and upwards. From the Navier-Stokes equation an expression is derived for the rate of fluid rise. This expression can be regarded as the generalization of Rideal's equation to a nonuniform capillary tube. We solve for both the static and the dynamicproperties of our model. Solutionfor the statics reveals that there is a band of alternate metastable and unstable equilibrium heights, the lowest of these being the true stable equilibrium position. However, solution for the dynamics shows that the lowest equilibrium height is not necessarily where the fluid comes to rest as acceleration and inertia terms may carry the meniscus past its stable equilibrium position to a higher metastable position. This tendency to overshoot is clearly illustrated in our plots of volume flux against height of rise. These plots also show that the volume flux is an oscillating function of height of rise. These oscillations are the socalled Haines jumps which arise in flow through any nonuniform capillary tube. level. The lowest position may be reached in imbibition and the highest in drainage, i.e., hysteresis is observed. An understanding of capillary hysteresis depends on the fact that the curvature C of the meniscus varies with the position that it occupies in the pore space. The equilibrium positions of the meniscus are given by the equation ogh = y C , where p is the density of the liquid, g the acceleration of gravity, h the height of the meniscus, and y the surface tension. For a model of a porous medium based on beds of spheres the determination of the equilibrium states is a formidable task, which remains unsolved. A simpler although less realistic model is an assemblage of cylindrical capillaries which may manifest capillary hysteresis if the capillary tubes have nonuniform bores. K u s a k o v and N e k r a s o v (4) have studied capillary hysteresis in a single capillary tube having a sinusoidal variation in the radius of its circular cross section. They obtain a range of heights in which the meniscus has alternate positions of stable and unstable
1. INTRODUCTION Two distinct types of hysteresis are associated with the wetting/dewetting or imbibition/drainage of porous media. The type most studied by physical and colloid chemists is contact angle hysteresis, which is usually attributed to mechanical and chemical heterogeneities on the solid surface (1). For zero contact angle between the wetting fluid and the solid surface, contact angle hysteresis is absent but there still remains capillary hysteresis (2-4) or hydrostatic hysteresis (5), which has received more attention from soil scientists. Because of the nonuniform cross section of the " b o r e " or space between the particles, the fluid-fluid (liquid-vapor) interface (the meniscus) expands and contracts with the advance of this interface through the porous medium. As a consequence, in addition to one stable equilibrium position for the meniscus, there is a multiplicity of quasiequilibrium positions representing metastable states above the true equilibrium
253 0021-9797/80/090253-11502.00/0 Journal of Colloid and Interface Science, Vol. 77, No. 1, September 1980
Copyright © 1980 by Academic Press, Inc. All rights of reproduction in any form reserved.
254
LEVINE,
LOWNDES,
equilibrium. No attempt is made to study the dynamics of the wetting process in the periodic capillary tube. These authors also determine the equilibrium positions of the meniscus in a vertical right circular single cone. For a cone converging downwards one equilibrium position of the meniscus is found. If the cone converges upwards, the meniscus has two equilibrium positions, the higher being unstable and the lower stable. Most authors (5-8) working in the field of hysteresis in porous media treat the problem from an essentially thermodynamic point of view and a key feature in such approaches is the condition for the stability of an interracial configuration. In the present paper two related topics, relevant to twophase flow in porous media, are discussed by a different approach. In Section 2 our main concern is the dynamics of the wetting process in a nonuniform bore capillary. We are considering the problem of unsteady twophase flow in a vertical capillary tube consisting of a sequence of conical sections alternately converging upwards and downwards. Here, the flow pattern is far more complicated than two-phase flow in uniform channels or tubes. Consequently, we have made certain simplifying approximations about the velocity profile in each conical section. In order to check the accuracy of our approximation we compare in Section 4 our results for the flow rate with direct computer simulations and find the agreement to be good. We therefore believe that our results display the essential features of twophase flow through a sequence of conical sections, in particular, so-called Haines jumps are clearly visible. Making use of the Navier-Stokes equations for an incompressible fluid and applying a method used previously by the authors (9) for a cylindrical capillary tube, a general equation governing the volume flux along the periodic corrugated tubes is set up. In a cone of infinite length, the streamlines in slow steady-state flow of an incompressible liquid are straight lines radiating from Journal of Colloid and Interface Science, Vol. 77, No. 1, September 1980
AND REED
the apex of the cone. Each conical section is regarded as part of such a single infinite cone with the simple steady-state radial flow and the same volume flux as the whole column. For this assumed flow pattern in a tube of identical conical sections, the component of fluid velocity parallel to the axis of the tube is continuous at a junction between two sections. A discontinuity in the velocity component perpendicular to the axis exists at a junction but this becomes less significant the smaller the cone angle. The departure of the streamlines from the assumed radial form and the small effect this has on the flow rate is discussed in Section 4. The volume flux is calculated assuming the meniscus is spherical with zero contact angle, except when the meniscus is passing through a junction. In order to obtain the curvature of the meniscus as it passes through the junction between two cones a numerical interpolation technique is used which matches the curvatures of the meniscus just before and just after it has passed through a junction. Our results are shown in Figs. 2 and 3 where volume flux is plotted against height of rise. The plots show a rapid oscillation of volume flux which is indicative of Haines jumps. Because the acceleration and inertial terms have been retained in our expression governing the rate of rise of the liquid, there is a tendency for the meniscus to overshoot its equilibrium position. This tendency to overshoot is particularly pronounced in our model because of the rapid change in curvature of the meniscus as it passes between cones. This tendency to overshoot can result in the meniscus coming to rest not at its true equilibrium position but at one of its higher metastable states. One of our plots show this effect. On Figs. 2 and 3 plots of the volume flux through a cylindrical capillary obtained from the Rideal equation are shown for comparison. In Section 6 the equation p g h = y C is solved graphically. As in the work of
255
CAPILLARY FLOW
n {nrz ~/
/o-, d' t
'\ '7 n-2 {n-2
/
~+{I tl'+{ ~'
\
r|
\,
"
3 {3
I
TY
{i r=
F]G. 1. (a) Model of two adjacent conical sections. (b) Model of a sequence of conical sections alternately converging downwards and upwards.
Kusakov and Nekrasov (4) a range of alternating stable and unstable equilibrium heights for the meniscus is found. 2. FLOW THROUGH A PAIR OF ADJACENT CONICAL SECTIONS
It is convenient to consider a single cone of infinite length with its axis in the upward vertical direction. We introduce cylindrical polar coordinates (r,O,z) with origin at the vertex (base) of the cone, the axis of the cone being coincident with the z axis (Fig. la). The angle al specifies the surface of the cone. The incompressible fluid filling the cone has an axially symmetric velocity with components (v,O,u) in the above coordinates, v, u, and the pressurep are functions of r, z, and the time t. The equation of continuity for the incompressible fluid is 0 (rv) +
0-7
0
(ru) = O.
[2.1]
Let /x and p be the viscosity and density, respectively, of the fluid, u = Ix/p the kinematic viscosity, and g the acceleration of gravity. Then the Navier-Stokes equation for the velocity component u reads Ou
Ou
Ou
- - + b t - - + V
Ot
Oz
+ u
10p =
Or
p Oz
02/,/
10/g +---+ Or2 r Or
02b/
Oz2
]
-g.
[2.2]
On multiplying [2.2] by r and using [2.1], it is readily verified that 0
--(ru) + ot
_
0
(ru 2) +
0
r _O p + v I°_(r p Oz LOr \
(ruv) o") Or )
OZu ] + r~ - gr. 0z 2
[2.31
Journal of Coaoid and Interface Science, Vol. 77, No. 1, September 1980
256
LEVINE, LOWNDES, AND REED
T h e no-slip condition at the cone s u r f a c e yields u =0 at r = z t a n o q . [2.4] N o t i n g that u = u(r,z,t) it follows f r o m [2.4] that at given t, tan a l ( Ou ) \ Or /r=z
tan
as
+
= o.
[2.5]
--
z tan as
rudr
r2 = (~ + 11) tan a i > r~,
I
2 (P )" = r~
022
2 r2
(P )'O+ll
f? I?
rp(r,~o,t)dr,
rp(r, rl + ll, t)dr.
[2.10]
W e n o w integrate [2.8] with r e s p e c t to z f r o m z = ~q to z = ~/ + 11. N o t i n g that
02/1
r
[2.9]
[2.5]
d e p e n d s only on t and in particular is indep e n d e n t o f z. F r o m [2.5] and the t w o relations dq/dz = O, d2q/dz 2 = 0, it is readily verified that
Jo
[2.8]
and the c o r r e s p o n d i n g m e a n p r e s s u r e s on t h e s e c r o s s sections are
0
z tan al
1 "~ gz 2 tan 2 al.
~1
rl = ~ tan ~1,
F o r i n c o m p r e s s i b l e flow, the v o l u m e t r i c flow rate or v o l u m e flux along the c o n e
f
tan
W e n o w c h o o s e a section of the c o n e of length li along the axis f r o m z = ~q to z = + 11. T h e radii o f the h o r i z o n t a l circular c r o s s sections of the b o t t o m and top of this section are r e s p e c t i v e l y
r=z tan as
q = 21r
\ Or ]r=z
+ vz tan a l sec 2
dr
d
=-ztan
2
~i(~OZ/r=ztanas 0u I
as
rpdr = ~o
Op
r Oz dr
[2.7]
Integrating [2.3] with r e s p e c t to r a c r o s s a h o r i z o n t a l section of the c o n e f r o m the axis r = 0 to the b o u n d a r y r = z t a n a l and m a k i n g use o f [2.4], [2.5], and the p r o p e r t y that q is i n d e p e n d e n t o f z, w e obtain 1 dq + 0 I ~tan -- --2~r dt Oz Jo
(ztana~
fztanus
~z )o
+ z tan 2 a l p(z tan oq,z,t), the first t e r m on the right-hand side o f [2.8] contributes
l~n+~'ffta'~lOP dz r dr p ~ Oz
-~
=
ru2dr
1
1
[r~(p)~+l~ - r~(p)~] + - 2p p 'o+ll
1 I f tan cq
--
19
x tan ~ 0/1
Op dr r
I
zp(z tan a l , z , t ) d z .
[2.11]
a~q
OZ
I n t e g r a t i o n o f [2.8] yields t h e r e f o r e r (z tan al
pll dq + 2~rP[Jo dt ×
[,1
lgt='O+/1
ru~dr
= zr[r~(p)n - r~(P)~+ts] + 2rr tan 2 a l /Jz=.
zp(z tan oq,z,t)dz + 27rpv tan a l se& al
a~ Journalof Colloidand InterfaceScience,VoL 77, No. 1, September 1980
z a~
dz - pgV1, \ Or ]r=z tan al
[2.12]
257
CAPILLARY FLOW
where
cone to the pressure of the fluid in the volume V~. Using [2.5],
V~ = - - tan 2 oq[(/1 + ~/)3 _ ~ ] , 3
[2.13] 2zrpv fan a~ sec 2 as z
which is the volume o f the conical section. It is convenient to integrate by parts the second term on the right-hand side o f [2.12] to obtain
I
where Ou/On is the rate of change of the velocity u normal to the conical boundary. Since u = u~ cos oq, where us is the component of the fluid velocity parallel to the boundary, the third term on the right-hand side of [2.12] is the frictional force in the z direction on the fluid in V1 due to its viscosity. The last term in [2.12] is the force due to gravity. We now imagine a second conical section in inverted position having radii of the top and bottom cross sections equal to rl and r2, respectively, and resting on the conical section considered above (Fig. la). This upper section has angle a2 and length 12 along the vertical axis, so that it is situated between z = ~7 +l~ and z = ~7 +11 + I s . Consider new axes r, 0, z' with origin at the vertex of the cone to which the upper section belongs, and the z' axis pointing downwards. Let the upper section lie between z' = ~0' and z' = "O' + 12. If the fluid flow downwards in the upper section is described by velocity components u', v', volume flux q', and pressure p ' , then an equation similar to [2.12] is obtained provided g is replaced by - g and the appropriate quantities are labeled with a prime. For upward flow q', u', and p ' become - q , - u , and p, respectively, in this equation which, when added to [2.13], yields for the pair of conical sections the exact relation
zp(z tan a~,z,t)dz
a~ = 7r[r~p(r2, ~ + l~, t) - r~p(rt,~,t)] - zr tan 2 ax X (~+h z 2 jrt
Op(z tan
Otl,Z,t)
dz.
tan al
=
rt+lx
2~r tan 2 a~
dz r=z
[2.14]
OZ
Equation [2.12], an exact relation, is the statement of momentum balance in the z direction. Since q depends only on t, the first term in [2.12] may be written as z tan aa 0/,/ pll Oq = 27rpl~ r dr, Ot ~o Ot
which is rate of momentum increase in the volume V~. The second term on the left-hand side o f [2.12] is the rate of momentum loss by convection across the top and bottom boundaries o f the conical section. The first term on the right-hand side is the force on the volume V~ in the z direction due to the pressure difference between these two boundaries. Noting that we can write 27r tan 2 oq zpdz = 27rrp sin oq ds, where ds is an element of length along a generator of the cone, the second term on the right-hand side of [2.12] is the reaction in the z direction by the boundary wall of the
dq i i i tan otl ]z=o+/1 [ f0,' tarl a2 ]z'=~'+/z ru~dr j~=n - 27tO ru~dr j~,=~, [ p(ll + 12) at + 27tO )+h = 7rr~[(p)~ - ( P ) , + t l + j + 27r tan 2 a~
zp(z tan al,Z,t)dz - 2zr tan 2 a2 art Journal o f Colloid and Interface Science,
[2.151
Vol. 77, No. 1, September 1980
258
LEVINE, LOWNDES, AND REED
z ' p ( z ' tan az,z~,t)dz'
+ 27rpv tan a l " s e c 2 al
z
Jn'
~rl
dz \ Or
+ 27rpv tan a s ' s e c 2 a2 n'+t2 z' ( 0u ) '
where V2 is the volume of the upper section. The relation [2.15] is an exact integrated form of the N a v i e r - S t o k e s equation for the velocity c o m p o n e n t u. The velocity component v in the radial direction, which occurs in Eqs. [2.1] and [2.2], is eliminated when [2.3] is integrated with respect to r. To make further progress expressions for the velocity c o m p o n e n t u and the pressure p are required. As a first approximation, we shall assume steady-state creeping flow for an infinite cone. F o r the lower conical section, this means (10, 11)
dz'-pg(Vl+Vz), r = z ' t a n a2
~
Then, using {2.91
If2
z=n
'1 = 2i (;q
[2.21]
where E(Ol) = 1//4 -- 2/3 COS e a
+ V2 cos 4 a - 1/1z cos s a
[2.22]
and
u = qK(o~l). ( re + ze)~/2
(0÷)
zdz r=z tan C~l
~rl
g COS 20Q ] (r e + z S ) S/z~ ,
,)
r~
× Ke(al) tan s al"e(al),
tan c~1 sec e al _
r=z tan aa
[2.16] where
P =Po-
pgz-
~t.tqK(al)
X(a) = 2K(c~) sin 3 c~ cos 2 a.
( re + zZ) 3/z
3ze ] ( r 2 -4- Z2) 5/2
'
[2.17]
[2.24]
Also, rewriting the first t e r m on the righthand side of [2.141 as 7r(r~ - r~)p(re, ~7 + 11, t)
where
+ rrr~[p(r2, .1 + 11, t) - p ( r , * 1 , t ) l ,
K(a) 3
1
27r (1 + 2 cos a)(1 - cos oOz
[2.18]
'o+ll
We do not require the equation for the velocity c o m p o n e n t v if [2.16]-[2.18] are assumed a n d also if L a p l a c e ' s equation for the meniscus is used. On the conical b o u n d a r y [2.17] reduces to P =Po-
pgz
f(oq)
- , 3zrz 3 tan s al
[2.19]
where f(a) =
3 sin s a ( l - 3 cos 2 a) (1 + 2 cos a)(1 - cos a) 2
substitution of [2.19] into [2.14] yields
[2.20]
Journal of Colloid and Interface Science, Vol. 77, No. 1, September 1980
27r tan s a~
zp(z tan a l , z , t ) d z J~
= 7r(r~ - r~)p(r2, '1 + 11, t) - t z q f ( a ) x
( 2 3rl
1
ld I
+-~ +ogVI, r2 3 r~ ]
[2.25]
where V~ is the volume of fluid situated in the lower conical section b e t w e e n the conical boundary and the central circular cylinder of radius rl and length l~ (Fig. la). V~ = Va - V~ where V1 is given by [2.13] and V~ = ~'l~ s tan 2 al. Equation [2.12] is n o w
CAPILLARY FLOW
259
replaced by the approximate form
dq +
pla ~
(
1\ ]K2(a,) tan 2 al.E(Ogl) /
7rpq 2 r2
2 × p(r2, *1 + ll, t) - /zqf(al) ~ r I
1 +.
r2
=
7r[r2(p), - r~(p)n+h] + ~r(r 2 - r 2)
(,
3r~
- 27rpz/ . r,
1) X(al) -
rz
pgV~.
[2.26]
The corresponding equation for flow through the upper conical section is obtained simply by replacingq, g, al, and 11by - q , - g , a2, and 12, respectively. The equation [2.15] becomes
dq
0(ll + 12)--~t + zrpq 2 = 7rr2[(p ),
(1
1 )[K2(al) tans
r2
- (P)n+h+l~]
-
0/l'E(al)
/xq[f(%) +f(azl]
K2(0/~ ) tan2
--
2 3rl
O/2 "E(0/2)1
1 + r~] r2 3r~/
[2.27]
1 )[X(0/O + X(az)] - pg(V~ + V~) 1"2
sincez = r/ + / l a n d z ' = "O' + / 2 a r e t h e s a m e p o i n t s a n d l i k e w i s e z = 9 + 11 Jr 12andz' = r/. 3. FLOW THROUGH A VERTICAL COLUMN OF ALTERNATING CONICAL SECTIONS We now consider a vertical column of n conical sections alternately converging downwards and upwards and numbered i = 1, . . . , n from bottom to top (Fig. lb). Odd i signifies a cone converging downwards and even i one converging upwards. The conical angle and vertical lengths of section i are a~ and lz, respectively. These angles and lengths are so chosen that the radii of the c o m m o n horizontal circular cross sections, where two adjacent sections join, alternately equal rl and r2 where r2 > rl and r~ is the radius of the base of the column. Otherwise at and l~ (i = 1. . . . . n) are arbitrary. It is now convenient to measure vertical distances from the base of the column, i.e., from the bot-
PY dq + rrpq dt
q = 7rr2(y)f4(y).
[3.1]
The momentum equation [2.27] for a pair of sections is readily generalized to describe the flow through the column. The form obtained depends on whether n is even or odd. Let rl/2 equal r2 for n even and rl for n odd. Then [2.27] is generalized to
~ (-1)i+iK2(a0"tan ~ ai'e(0/0
z(
- zcpq
tom of conical section l, which is converging downward. The vertical axis of the column intersects the meniscus at its lowest point which is situated in the nth conical section at a distance y above the base of the column (Fig. lb). If the circular cross section at this point has radius r(y) and 5(y) is the mean of the vertical component of the fluid velocity on this cross section, then the volume flux through the column is
1 7~y)
-
-
J )K2(a,).tan2 0/,.e(a,) = Tr[r~(p)y=o - rZ(y)(p)u] rl/2 / + zr(r2(y) - r~)p(r(y),y,t) - zrr~pgy - 27r/zqP~(y)
[3.21
Journal o f Colloid and Interface Science, Vol. 77, No. 1, September 1980
260
LEVINE, LOWNDES, AND REED
where =
r2
1 (2
1
27r 3rl
rz
E X(a~) ÷ (-1)" - ~=~ r(y) +
r~ )"-1
Z f(ozi) + 3r~/ ~=~
(-1)"
rlt2
[.
1
X(C~.)
1
r21
+-2~r Lra/2 3r~/2
r(y)
r~ ]f(ot.). 3r3(y)
[3.31
The sums from i = 1 to i = n - 1 are absent in [3.2] and [3.3] i f n = 1. 4. EXAMINATIONOF APPROXIMATIONS The only approximation made so far in deriving Eq. [3.2] has been to assume that each conical section can be regarded as part of a single infinite cone. We therefore assumed a simple radial flow pattern in each conical section which is the solution for the steady-state Navier-Stokes equations for creeping flow through an infinite cone. This assumption necessarily introduces nonphysical singularities at the junctions between conical sections but more importantly ignores the effect of fluid inertia. Now, for a tube consisting of identical conical sections the actual two-phase flow can, except in the initial stages of the flow, be identified with single-phase flow through the given tube extended axially (by indefinitely adding extra conical sections) to infinity. To estimate the errors introduced by our assumed form of the velocity profile a finite element technique was used to solve the steady-state N a v i e r - S t o k e s equations for single-phase flow through the infinitely long tube subjected to a specified pressure drop in each pair of adjacent conical sections. The results for the volume flux were then compared with the corresponding volume flux with the same pressure drop in each pair of adjacent conical sections but where the flow through each conical section is radial. This was done for a spread of typical pressure drops and several cone angles and it was found that the results for the volume flux agreed to within acceptable errors. We therefore conclude that the use of the simple radial flow pattern in each conical section leads to a quite acceptable Journal of Colloid and Interface Science, Vol. 77, No. 1, September 1980
approximation to the volume flux of singlephase flow, and hence to two-phase flow, through a vertical capillary tube consisting of a sequence Of identical conical sections alternately converging downwards and upwards. The details of these calculations will be published elsewhere. 5. CONDITIONS AT ENTRY AND AT THE MENISCUS To complete Eq. [3.2] expressions for (P)~=0 and (p)u must be inserted. In a previous paper (9) a theory was developed for the evaluation of (P)u=0 and the results of (9) can be directly applied here to yield 2gq (P)~=o = Pa --
rrra~
370 dq 367rrl dt
70 qZ, 67r2r4
--
[5.1]
wherepa is atmospheric pressure. To evaluate (p)~ we assume that the pressure just below the meniscus is independent of r and that the meniscus is spherical. With these assumptions and applying Laplace's equation we obtain (p)y = p(r(y),y,t) = Pa - ~
2y
R(y)
[5.2]
R(y) is the radius of curvature of the meniscus and ,/is the surface tension. The meniscus is assumed spherical and to make zero contact angle with the walls of the capillary while it is entirely contained in a single coni-
CAPILLARY FLOW
261
I
~o~ S
I
I
1~
MS
MS
Fro. 2. Plot of volume flux q against height of rise y for a capillary made up of conical sections. Values of y (integers) in units of section length ! and all other distances in centimeters, r~ = 0.03, r2 = 0.06, cone angle a = 11.1°, and length l = 0.153. The broken curve is a comparison curve showing q for a cylindrical capillary of radius 0.046. In both plots the oscillation of the meniscus about its equilibrium height has been suppressed. cal s e c t i o n . Its c u r v a t u r e is thus g i v e n b y
R(y) =
r(y) cos an 1 + ( - 1 ) n sin a ,
[5.3]
T o o b t a i n a n e s t i m a t e of R ( y ) as the m e n i s cus p a s s e s b e t w e e n c o n e s a n u m e r i c a l interp o l a t i o n t e c h n i q u e is u s e d w h i c h m a t c h e s the r a d i u s o f c u r v a t u r e j u s t b e f o r e a n d j u s t after p a s s i n g t h r o u g h the i n t e r s e c t i o n o f t w o c o n e s . Plots of 2y/R(y) are s h o w n in Figs. 4 a n d 5. U s i n g [5.1] a n d [5.2] in [3.2] we o b t a i n
dq (y 37 ) p--~ +-~rl + ¢rpqZ(r?2 - r~ 2)
a c o m m o n c o n i c a l s e c t i o n a n g l e (a,, = a for all n). T y p i c a l v a l u e s of the r e l e v a n t p h y s i cal q u a n t i t i e s c h o s e n are p = 1 g/cm 3, /z = 0.01 p, a n d 3' = 32 d y n / c m . 6. DISCUSSION F i g u r e s 2 a n d 3 s h o w plots o f the v o l u m e flux q as a f u n c t i o n of height of rise, for v e r t i c a l capillaries o f i d e n t i c a l c o n i c a l sec-
\
\
\
n--1
× ~ (-1)i+lKZ(ai) t a n z ai e(ai) i=l -
7rpqZ(r(y) -2 - r~)K2(an) t a n 2 a. e(an)
7 2lzq + _ _ pq2 + 2zrpxtPn(y ), + 6rrr~ rl =
-~)-
pgy ~rr~. [5.4]
E q u a t i o n [5.4] d e s c r i b e s the rate of rise o f fluid t h r o u g h a s e q u e n c e of c o n i c a l s e c t i o n s a n d is the m a i n r e s u l t o f this p a p e r . E q u a t i o n [5.4] will n o w b e a p p l i e d to the case of
I, 5
V
.~
MS
FIG. 3. Plot of volume flux q against height of rise y for a capillary made up of conical sections. Values of y (integers) in units of section length l and all other distances in centimeters, rl = 0.04, r2 = 0.06, cone angle a = 5°, and length l = 0.229. The broken curve shows q for a cylindrical capillary of radius 0.050. In both plots oscillation about the equilibrium height has been suppressed.
Journal of Colloid and Interface Science, Vol. 77, N o . 1, S e p t e m b e r 1980
262
LEVINE, LOWNDES, AND REED
:000k
I\/ \/ \/ %_. ~ I
\/'
~
o~
,
utt
t
]~
,
~
,
~
i
tt t
t
i,o
I
'~z
ttt
'
1~
Y/z F]~. 4. Plot of pgy and 2"y/R(y) against y / / f o r a = 11.1 °, rl = 0.03, r~ = 0.06, and l = 0.153. T h e stable, unstable, and metastable positions are m a r k e d S, U, and MS, respectively.
tions, obtained by integrating [5.4] with both acceleration and inertia terms retained. The plots clearly show the oscillating volume flux which is indicative of Haines jumps. However, the retention of the acceleration terms has tended to smear out this effect and the wetting process is seen to be dominated by the rapid acceleration during the initial stages of imbibition. Because of the acceleration terms, there is a tendency for the meniscus to overshoot its equilibrium position before returning to this position. This effect can clearly be seen in Figs. 2 and 3. The equilibrium positions for the meniscus are given by the solutions to the equation 27 [6.11 ogY R(y) -
,
200° I
~
I000
0~ 0
l 2
I t+
S
MS
Sl
l ~is 6
I
I
I
B
FIG. 5. Plot of pgy and 2y/R(y) against y/l for a = 5°,r~ = 0.04, r2 = 0.06, a n d / = 0.229. T h e stable, unstable, and metastable positions are m a r k e d S, U, and MS, respectively.
Journal of Colloid and Interface Science, Vol. 77, No. I, September 1980
which is obtained from [5.4] by putting q = 0. Graphical solutions of Eq. [6.1] are shown in Figs. 4 and 5 where both pgy and 27/R(y) are plotted. The solution of Eq. [6.1] shows a band of alternating metastable and unstable equilibrium positions; these are denoted on Figs. 4 and 5 by MS and U, respectively. The stable equilibrium position is marked with an S. This result is similar to the result of Kusakov and Nekrasov (4) for a sinusoidal capillary. The stable and metastable equilibrium positions are marked on Figs. 2 and 3 by S and MS, respectively. It is seen in Fig. 3 that the fluid has not come to rest at its lowest stable position but at its first metastable position. This shows that the dynamical processes can effect the equilibrium properties in a model which exhibits capillary hysteresis. Also plotted on Figs. 2 and 3 is the solution to the Rideal equation for a cylindrical capillary obtained by taking the limit a ~ 0 in Eq. [5.4]. The radius of the cylindrical capillary is chosen such that the volume per conical section length of the cylindrical capillary and the alternating capillary are the same. The fluid in the cylindrical capillary is seen to accelerate more rapidly in the initial stages. This will always be true for any cylindrical capillary of radius less than r2 and is a consequence of the first conical section converging downwards.
CAPILLARY FLOW ACKNOWLEDGMENTS We are indebted to the Science Research Council of the United Kingdom and to Tioxide International Limited, for postdoctoral research assistantships to J.L. and P.R.
5. 6.
REFERENCES
7.
1. Johnson, R. E., and Dettre, R. H., in "Surface and Colloid Science" (E. Matijevic, Ed.), Vol. 2, p. 85. Wiley-Interscience, New York, 1969. 2. Haines, W. B.,J. Agric. Sci. 20, 97 (1930). 3. Miller, E. E., and Miller, R. D., J. Appl. Phys. 27, 324 (1956). 4. (a) Kusakov, M. M., and Nekrasov, D. N., in "Research in Surface Forces" (B. V. Deryaguin, Ed.), Vol. 2, p. 193. Consultants Bureau,
8. 9.
10.
11.
263
New York, 1966; (b) Dokl. Akad. Nauk SSSR 119, 107 (1958); (c) Zh. Fiz. Khim. 34, 1602 (1960). Melrose, J. C.,J. Soc. Petrol. Eng. 5, 259 (1965). Everett, D. H., and Haynes, J. M., Z. Phys. Chem. (Frankfurt am Main) 82, 36 (1972); 97, 301 (1975). Everett, D. H., J. Colloid Interface Sci. 52, 189 (1975). Morrow, N. R., Ind. Eng. Chem. 62, 32 (1970). Levine, S., Reed, P., Watson, E. J., and Neale, G., in "Colloid and Interface Science" (M. Kerker, Ed.), Vol. III, p. 403. Academic Press, New York, 1976. Happel, J., and Brenner, H., " L o w Reynolds Number Hydrodynamics," Prentice-Hall, Englewood, N. J., 1965. p. 138. Ackerberg, R. C., J. Fluid Mech. 21, 47 (1965).
Journal of Colloid and Interface Science, Vol.77, No. 1, September1980
Department of Mathematics, University of Manchester, Manchester M13 9PL, England Received November 11, 1977; accepted September 20, 1979 The wetting process in a vertical capillary tube with nonuniform bore is examined. The model chosen consists of a sequence of conical sections alternately convergingdownwards and upwards. From the Navier-Stokes equation an expression is derived for the rate of fluid rise. This expression can be regarded as the generalization of Rideal's equation to a nonuniform capillary tube. We solve for both the static and the dynamicproperties of our model. Solutionfor the statics reveals that there is a band of alternate metastable and unstable equilibrium heights, the lowest of these being the true stable equilibrium position. However, solution for the dynamics shows that the lowest equilibrium height is not necessarily where the fluid comes to rest as acceleration and inertia terms may carry the meniscus past its stable equilibrium position to a higher metastable position. This tendency to overshoot is clearly illustrated in our plots of volume flux against height of rise. These plots also show that the volume flux is an oscillating function of height of rise. These oscillations are the socalled Haines jumps which arise in flow through any nonuniform capillary tube. level. The lowest position may be reached in imbibition and the highest in drainage, i.e., hysteresis is observed. An understanding of capillary hysteresis depends on the fact that the curvature C of the meniscus varies with the position that it occupies in the pore space. The equilibrium positions of the meniscus are given by the equation ogh = y C , where p is the density of the liquid, g the acceleration of gravity, h the height of the meniscus, and y the surface tension. For a model of a porous medium based on beds of spheres the determination of the equilibrium states is a formidable task, which remains unsolved. A simpler although less realistic model is an assemblage of cylindrical capillaries which may manifest capillary hysteresis if the capillary tubes have nonuniform bores. K u s a k o v and N e k r a s o v (4) have studied capillary hysteresis in a single capillary tube having a sinusoidal variation in the radius of its circular cross section. They obtain a range of heights in which the meniscus has alternate positions of stable and unstable
1. INTRODUCTION Two distinct types of hysteresis are associated with the wetting/dewetting or imbibition/drainage of porous media. The type most studied by physical and colloid chemists is contact angle hysteresis, which is usually attributed to mechanical and chemical heterogeneities on the solid surface (1). For zero contact angle between the wetting fluid and the solid surface, contact angle hysteresis is absent but there still remains capillary hysteresis (2-4) or hydrostatic hysteresis (5), which has received more attention from soil scientists. Because of the nonuniform cross section of the " b o r e " or space between the particles, the fluid-fluid (liquid-vapor) interface (the meniscus) expands and contracts with the advance of this interface through the porous medium. As a consequence, in addition to one stable equilibrium position for the meniscus, there is a multiplicity of quasiequilibrium positions representing metastable states above the true equilibrium
253 0021-9797/80/090253-11502.00/0 Journal of Colloid and Interface Science, Vol. 77, No. 1, September 1980
Copyright © 1980 by Academic Press, Inc. All rights of reproduction in any form reserved.
254
LEVINE,
LOWNDES,
equilibrium. No attempt is made to study the dynamics of the wetting process in the periodic capillary tube. These authors also determine the equilibrium positions of the meniscus in a vertical right circular single cone. For a cone converging downwards one equilibrium position of the meniscus is found. If the cone converges upwards, the meniscus has two equilibrium positions, the higher being unstable and the lower stable. Most authors (5-8) working in the field of hysteresis in porous media treat the problem from an essentially thermodynamic point of view and a key feature in such approaches is the condition for the stability of an interracial configuration. In the present paper two related topics, relevant to twophase flow in porous media, are discussed by a different approach. In Section 2 our main concern is the dynamics of the wetting process in a nonuniform bore capillary. We are considering the problem of unsteady twophase flow in a vertical capillary tube consisting of a sequence of conical sections alternately converging upwards and downwards. Here, the flow pattern is far more complicated than two-phase flow in uniform channels or tubes. Consequently, we have made certain simplifying approximations about the velocity profile in each conical section. In order to check the accuracy of our approximation we compare in Section 4 our results for the flow rate with direct computer simulations and find the agreement to be good. We therefore believe that our results display the essential features of twophase flow through a sequence of conical sections, in particular, so-called Haines jumps are clearly visible. Making use of the Navier-Stokes equations for an incompressible fluid and applying a method used previously by the authors (9) for a cylindrical capillary tube, a general equation governing the volume flux along the periodic corrugated tubes is set up. In a cone of infinite length, the streamlines in slow steady-state flow of an incompressible liquid are straight lines radiating from Journal of Colloid and Interface Science, Vol. 77, No. 1, September 1980
AND REED
the apex of the cone. Each conical section is regarded as part of such a single infinite cone with the simple steady-state radial flow and the same volume flux as the whole column. For this assumed flow pattern in a tube of identical conical sections, the component of fluid velocity parallel to the axis of the tube is continuous at a junction between two sections. A discontinuity in the velocity component perpendicular to the axis exists at a junction but this becomes less significant the smaller the cone angle. The departure of the streamlines from the assumed radial form and the small effect this has on the flow rate is discussed in Section 4. The volume flux is calculated assuming the meniscus is spherical with zero contact angle, except when the meniscus is passing through a junction. In order to obtain the curvature of the meniscus as it passes through the junction between two cones a numerical interpolation technique is used which matches the curvatures of the meniscus just before and just after it has passed through a junction. Our results are shown in Figs. 2 and 3 where volume flux is plotted against height of rise. The plots show a rapid oscillation of volume flux which is indicative of Haines jumps. Because the acceleration and inertial terms have been retained in our expression governing the rate of rise of the liquid, there is a tendency for the meniscus to overshoot its equilibrium position. This tendency to overshoot is particularly pronounced in our model because of the rapid change in curvature of the meniscus as it passes between cones. This tendency to overshoot can result in the meniscus coming to rest not at its true equilibrium position but at one of its higher metastable states. One of our plots show this effect. On Figs. 2 and 3 plots of the volume flux through a cylindrical capillary obtained from the Rideal equation are shown for comparison. In Section 6 the equation p g h = y C is solved graphically. As in the work of
255
CAPILLARY FLOW
n {nrz ~/
/o-, d' t
'\ '7 n-2 {n-2
/
~+{I tl'+{ ~'
\
r|
\,
"
3 {3
I
TY
{i r=
F]G. 1. (a) Model of two adjacent conical sections. (b) Model of a sequence of conical sections alternately converging downwards and upwards.
Kusakov and Nekrasov (4) a range of alternating stable and unstable equilibrium heights for the meniscus is found. 2. FLOW THROUGH A PAIR OF ADJACENT CONICAL SECTIONS
It is convenient to consider a single cone of infinite length with its axis in the upward vertical direction. We introduce cylindrical polar coordinates (r,O,z) with origin at the vertex (base) of the cone, the axis of the cone being coincident with the z axis (Fig. la). The angle al specifies the surface of the cone. The incompressible fluid filling the cone has an axially symmetric velocity with components (v,O,u) in the above coordinates, v, u, and the pressurep are functions of r, z, and the time t. The equation of continuity for the incompressible fluid is 0 (rv) +
0-7
0
(ru) = O.
[2.1]
Let /x and p be the viscosity and density, respectively, of the fluid, u = Ix/p the kinematic viscosity, and g the acceleration of gravity. Then the Navier-Stokes equation for the velocity component u reads Ou
Ou
Ou
- - + b t - - + V
Ot
Oz
+ u
10p =
Or
p Oz
02/,/
10/g +---+ Or2 r Or
02b/
Oz2
]
-g.
[2.2]
On multiplying [2.2] by r and using [2.1], it is readily verified that 0
--(ru) + ot
_
0
(ru 2) +
0
r _O p + v I°_(r p Oz LOr \
(ruv) o") Or )
OZu ] + r~ - gr. 0z 2
[2.31
Journal of Coaoid and Interface Science, Vol. 77, No. 1, September 1980
256
LEVINE, LOWNDES, AND REED
T h e no-slip condition at the cone s u r f a c e yields u =0 at r = z t a n o q . [2.4] N o t i n g that u = u(r,z,t) it follows f r o m [2.4] that at given t, tan a l ( Ou ) \ Or /r=z
tan
as
+
= o.
[2.5]
--
z tan as
rudr
r2 = (~ + 11) tan a i > r~,
I
2 (P )" = r~
022
2 r2
(P )'O+ll
f? I?
rp(r,~o,t)dr,
rp(r, rl + ll, t)dr.
[2.10]
W e n o w integrate [2.8] with r e s p e c t to z f r o m z = ~q to z = ~/ + 11. N o t i n g that
02/1
r
[2.9]
[2.5]
d e p e n d s only on t and in particular is indep e n d e n t o f z. F r o m [2.5] and the t w o relations dq/dz = O, d2q/dz 2 = 0, it is readily verified that
Jo
[2.8]
and the c o r r e s p o n d i n g m e a n p r e s s u r e s on t h e s e c r o s s sections are
0
z tan al
1 "~ gz 2 tan 2 al.
~1
rl = ~ tan ~1,
F o r i n c o m p r e s s i b l e flow, the v o l u m e t r i c flow rate or v o l u m e flux along the c o n e
f
tan
W e n o w c h o o s e a section of the c o n e of length li along the axis f r o m z = ~q to z = + 11. T h e radii o f the h o r i z o n t a l circular c r o s s sections of the b o t t o m and top of this section are r e s p e c t i v e l y
r=z tan as
q = 21r
\ Or ]r=z
+ vz tan a l sec 2
dr
d
=-ztan
2
~i(~OZ/r=ztanas 0u I
as
rpdr = ~o
Op
r Oz dr
[2.7]
Integrating [2.3] with r e s p e c t to r a c r o s s a h o r i z o n t a l section of the c o n e f r o m the axis r = 0 to the b o u n d a r y r = z t a n a l and m a k i n g use o f [2.4], [2.5], and the p r o p e r t y that q is i n d e p e n d e n t o f z, w e obtain 1 dq + 0 I ~tan -- --2~r dt Oz Jo
(ztana~
fztanus
~z )o
+ z tan 2 a l p(z tan oq,z,t), the first t e r m on the right-hand side o f [2.8] contributes
l~n+~'ffta'~lOP dz r dr p ~ Oz
-~
=
ru2dr
1
1
[r~(p)~+l~ - r~(p)~] + - 2p p 'o+ll
1 I f tan cq
--
19
x tan ~ 0/1
Op dr r
I
zp(z tan a l , z , t ) d z .
[2.11]
a~q
OZ
I n t e g r a t i o n o f [2.8] yields t h e r e f o r e r (z tan al
pll dq + 2~rP[Jo dt ×
[,1
lgt='O+/1
ru~dr
= zr[r~(p)n - r~(P)~+ts] + 2rr tan 2 a l /Jz=.
zp(z tan oq,z,t)dz + 27rpv tan a l se& al
a~ Journalof Colloidand InterfaceScience,VoL 77, No. 1, September 1980
z a~
dz - pgV1, \ Or ]r=z tan al
[2.12]
257
CAPILLARY FLOW
where
cone to the pressure of the fluid in the volume V~. Using [2.5],
V~ = - - tan 2 oq[(/1 + ~/)3 _ ~ ] , 3
[2.13] 2zrpv fan a~ sec 2 as z
which is the volume o f the conical section. It is convenient to integrate by parts the second term on the right-hand side o f [2.12] to obtain
I
where Ou/On is the rate of change of the velocity u normal to the conical boundary. Since u = u~ cos oq, where us is the component of the fluid velocity parallel to the boundary, the third term on the right-hand side of [2.12] is the frictional force in the z direction on the fluid in V1 due to its viscosity. The last term in [2.12] is the force due to gravity. We now imagine a second conical section in inverted position having radii of the top and bottom cross sections equal to rl and r2, respectively, and resting on the conical section considered above (Fig. la). This upper section has angle a2 and length 12 along the vertical axis, so that it is situated between z = ~7 +l~ and z = ~7 +11 + I s . Consider new axes r, 0, z' with origin at the vertex of the cone to which the upper section belongs, and the z' axis pointing downwards. Let the upper section lie between z' = ~0' and z' = "O' + 12. If the fluid flow downwards in the upper section is described by velocity components u', v', volume flux q', and pressure p ' , then an equation similar to [2.12] is obtained provided g is replaced by - g and the appropriate quantities are labeled with a prime. For upward flow q', u', and p ' become - q , - u , and p, respectively, in this equation which, when added to [2.13], yields for the pair of conical sections the exact relation
zp(z tan a~,z,t)dz
a~ = 7r[r~p(r2, ~ + l~, t) - r~p(rt,~,t)] - zr tan 2 ax X (~+h z 2 jrt
Op(z tan
Otl,Z,t)
dz.
tan al
=
rt+lx
2~r tan 2 a~
dz r=z
[2.14]
OZ
Equation [2.12], an exact relation, is the statement of momentum balance in the z direction. Since q depends only on t, the first term in [2.12] may be written as z tan aa 0/,/ pll Oq = 27rpl~ r dr, Ot ~o Ot
which is rate of momentum increase in the volume V~. The second term on the left-hand side o f [2.12] is the rate of momentum loss by convection across the top and bottom boundaries o f the conical section. The first term on the right-hand side is the force on the volume V~ in the z direction due to the pressure difference between these two boundaries. Noting that we can write 27r tan 2 oq zpdz = 27rrp sin oq ds, where ds is an element of length along a generator of the cone, the second term on the right-hand side of [2.12] is the reaction in the z direction by the boundary wall of the
dq i i i tan otl ]z=o+/1 [ f0,' tarl a2 ]z'=~'+/z ru~dr j~=n - 27tO ru~dr j~,=~, [ p(ll + 12) at + 27tO )+h = 7rr~[(p)~ - ( P ) , + t l + j + 27r tan 2 a~
zp(z tan al,Z,t)dz - 2zr tan 2 a2 art Journal o f Colloid and Interface Science,
[2.151
Vol. 77, No. 1, September 1980
258
LEVINE, LOWNDES, AND REED
z ' p ( z ' tan az,z~,t)dz'
+ 27rpv tan a l " s e c 2 al
z
Jn'
~rl
dz \ Or
+ 27rpv tan a s ' s e c 2 a2 n'+t2 z' ( 0u ) '
where V2 is the volume of the upper section. The relation [2.15] is an exact integrated form of the N a v i e r - S t o k e s equation for the velocity c o m p o n e n t u. The velocity component v in the radial direction, which occurs in Eqs. [2.1] and [2.2], is eliminated when [2.3] is integrated with respect to r. To make further progress expressions for the velocity c o m p o n e n t u and the pressure p are required. As a first approximation, we shall assume steady-state creeping flow for an infinite cone. F o r the lower conical section, this means (10, 11)
dz'-pg(Vl+Vz), r = z ' t a n a2
~
Then, using {2.91
If2
z=n
'1 = 2i (;q
[2.21]
where E(Ol) = 1//4 -- 2/3 COS e a
+ V2 cos 4 a - 1/1z cos s a
[2.22]
and
u = qK(o~l). ( re + ze)~/2
(0÷)
zdz r=z tan C~l
~rl
g COS 20Q ] (r e + z S ) S/z~ ,
,)
r~
× Ke(al) tan s al"e(al),
tan c~1 sec e al _
r=z tan aa
[2.16] where
P =Po-
pgz-
~t.tqK(al)
X(a) = 2K(c~) sin 3 c~ cos 2 a.
( re + zZ) 3/z
3ze ] ( r 2 -4- Z2) 5/2
'
[2.17]
[2.24]
Also, rewriting the first t e r m on the righthand side of [2.141 as 7r(r~ - r~)p(re, ~7 + 11, t)
where
+ rrr~[p(r2, .1 + 11, t) - p ( r , * 1 , t ) l ,
K(a) 3
1
27r (1 + 2 cos a)(1 - cos oOz
[2.18]
'o+ll
We do not require the equation for the velocity c o m p o n e n t v if [2.16]-[2.18] are assumed a n d also if L a p l a c e ' s equation for the meniscus is used. On the conical b o u n d a r y [2.17] reduces to P =Po-
pgz
f(oq)
- , 3zrz 3 tan s al
[2.19]
where f(a) =
3 sin s a ( l - 3 cos 2 a) (1 + 2 cos a)(1 - cos a) 2
substitution of [2.19] into [2.14] yields
[2.20]
Journal of Colloid and Interface Science, Vol. 77, No. 1, September 1980
27r tan s a~
zp(z tan a l , z , t ) d z J~
= 7r(r~ - r~)p(r2, '1 + 11, t) - t z q f ( a ) x
( 2 3rl
1
ld I
+-~ +ogVI, r2 3 r~ ]
[2.25]
where V~ is the volume of fluid situated in the lower conical section b e t w e e n the conical boundary and the central circular cylinder of radius rl and length l~ (Fig. la). V~ = Va - V~ where V1 is given by [2.13] and V~ = ~'l~ s tan 2 al. Equation [2.12] is n o w
CAPILLARY FLOW
259
replaced by the approximate form
dq +
pla ~
(
1\ ]K2(a,) tan 2 al.E(Ogl) /
7rpq 2 r2
2 × p(r2, *1 + ll, t) - /zqf(al) ~ r I
1 +.
r2
=
7r[r2(p), - r~(p)n+h] + ~r(r 2 - r 2)
(,
3r~
- 27rpz/ . r,
1) X(al) -
rz
pgV~.
[2.26]
The corresponding equation for flow through the upper conical section is obtained simply by replacingq, g, al, and 11by - q , - g , a2, and 12, respectively. The equation [2.15] becomes
dq
0(ll + 12)--~t + zrpq 2 = 7rr2[(p ),
(1
1 )[K2(al) tans
r2
- (P)n+h+l~]
-
0/l'E(al)
/xq[f(%) +f(azl]
K2(0/~ ) tan2
--
2 3rl
O/2 "E(0/2)1
1 + r~] r2 3r~/
[2.27]
1 )[X(0/O + X(az)] - pg(V~ + V~) 1"2
sincez = r/ + / l a n d z ' = "O' + / 2 a r e t h e s a m e p o i n t s a n d l i k e w i s e z = 9 + 11 Jr 12andz' = r/. 3. FLOW THROUGH A VERTICAL COLUMN OF ALTERNATING CONICAL SECTIONS We now consider a vertical column of n conical sections alternately converging downwards and upwards and numbered i = 1, . . . , n from bottom to top (Fig. lb). Odd i signifies a cone converging downwards and even i one converging upwards. The conical angle and vertical lengths of section i are a~ and lz, respectively. These angles and lengths are so chosen that the radii of the c o m m o n horizontal circular cross sections, where two adjacent sections join, alternately equal rl and r2 where r2 > rl and r~ is the radius of the base of the column. Otherwise at and l~ (i = 1. . . . . n) are arbitrary. It is now convenient to measure vertical distances from the base of the column, i.e., from the bot-
PY dq + rrpq dt
q = 7rr2(y)f4(y).
[3.1]
The momentum equation [2.27] for a pair of sections is readily generalized to describe the flow through the column. The form obtained depends on whether n is even or odd. Let rl/2 equal r2 for n even and rl for n odd. Then [2.27] is generalized to
~ (-1)i+iK2(a0"tan ~ ai'e(0/0
z(
- zcpq
tom of conical section l, which is converging downward. The vertical axis of the column intersects the meniscus at its lowest point which is situated in the nth conical section at a distance y above the base of the column (Fig. lb). If the circular cross section at this point has radius r(y) and 5(y) is the mean of the vertical component of the fluid velocity on this cross section, then the volume flux through the column is
1 7~y)
-
-
J )K2(a,).tan2 0/,.e(a,) = Tr[r~(p)y=o - rZ(y)(p)u] rl/2 / + zr(r2(y) - r~)p(r(y),y,t) - zrr~pgy - 27r/zqP~(y)
[3.21
Journal o f Colloid and Interface Science, Vol. 77, No. 1, September 1980
260
LEVINE, LOWNDES, AND REED
where =
r2
1 (2
1
27r 3rl
rz
E X(a~) ÷ (-1)" - ~=~ r(y) +
r~ )"-1
Z f(ozi) + 3r~/ ~=~
(-1)"
rlt2
[.
1
X(C~.)
1
r21
+-2~r Lra/2 3r~/2
r(y)
r~ ]f(ot.). 3r3(y)
[3.31
The sums from i = 1 to i = n - 1 are absent in [3.2] and [3.3] i f n = 1. 4. EXAMINATIONOF APPROXIMATIONS The only approximation made so far in deriving Eq. [3.2] has been to assume that each conical section can be regarded as part of a single infinite cone. We therefore assumed a simple radial flow pattern in each conical section which is the solution for the steady-state Navier-Stokes equations for creeping flow through an infinite cone. This assumption necessarily introduces nonphysical singularities at the junctions between conical sections but more importantly ignores the effect of fluid inertia. Now, for a tube consisting of identical conical sections the actual two-phase flow can, except in the initial stages of the flow, be identified with single-phase flow through the given tube extended axially (by indefinitely adding extra conical sections) to infinity. To estimate the errors introduced by our assumed form of the velocity profile a finite element technique was used to solve the steady-state N a v i e r - S t o k e s equations for single-phase flow through the infinitely long tube subjected to a specified pressure drop in each pair of adjacent conical sections. The results for the volume flux were then compared with the corresponding volume flux with the same pressure drop in each pair of adjacent conical sections but where the flow through each conical section is radial. This was done for a spread of typical pressure drops and several cone angles and it was found that the results for the volume flux agreed to within acceptable errors. We therefore conclude that the use of the simple radial flow pattern in each conical section leads to a quite acceptable Journal of Colloid and Interface Science, Vol. 77, No. 1, September 1980
approximation to the volume flux of singlephase flow, and hence to two-phase flow, through a vertical capillary tube consisting of a sequence Of identical conical sections alternately converging downwards and upwards. The details of these calculations will be published elsewhere. 5. CONDITIONS AT ENTRY AND AT THE MENISCUS To complete Eq. [3.2] expressions for (P)~=0 and (p)u must be inserted. In a previous paper (9) a theory was developed for the evaluation of (P)u=0 and the results of (9) can be directly applied here to yield 2gq (P)~=o = Pa --
rrra~
370 dq 367rrl dt
70 qZ, 67r2r4
--
[5.1]
wherepa is atmospheric pressure. To evaluate (p)~ we assume that the pressure just below the meniscus is independent of r and that the meniscus is spherical. With these assumptions and applying Laplace's equation we obtain (p)y = p(r(y),y,t) = Pa - ~
2y
R(y)
[5.2]
R(y) is the radius of curvature of the meniscus and ,/is the surface tension. The meniscus is assumed spherical and to make zero contact angle with the walls of the capillary while it is entirely contained in a single coni-
CAPILLARY FLOW
261
I
~o~ S
I
I
1~
MS
MS
Fro. 2. Plot of volume flux q against height of rise y for a capillary made up of conical sections. Values of y (integers) in units of section length ! and all other distances in centimeters, r~ = 0.03, r2 = 0.06, cone angle a = 11.1°, and length l = 0.153. The broken curve is a comparison curve showing q for a cylindrical capillary of radius 0.046. In both plots the oscillation of the meniscus about its equilibrium height has been suppressed. cal s e c t i o n . Its c u r v a t u r e is thus g i v e n b y
R(y) =
r(y) cos an 1 + ( - 1 ) n sin a ,
[5.3]
T o o b t a i n a n e s t i m a t e of R ( y ) as the m e n i s cus p a s s e s b e t w e e n c o n e s a n u m e r i c a l interp o l a t i o n t e c h n i q u e is u s e d w h i c h m a t c h e s the r a d i u s o f c u r v a t u r e j u s t b e f o r e a n d j u s t after p a s s i n g t h r o u g h the i n t e r s e c t i o n o f t w o c o n e s . Plots of 2y/R(y) are s h o w n in Figs. 4 a n d 5. U s i n g [5.1] a n d [5.2] in [3.2] we o b t a i n
dq (y 37 ) p--~ +-~rl + ¢rpqZ(r?2 - r~ 2)
a c o m m o n c o n i c a l s e c t i o n a n g l e (a,, = a for all n). T y p i c a l v a l u e s of the r e l e v a n t p h y s i cal q u a n t i t i e s c h o s e n are p = 1 g/cm 3, /z = 0.01 p, a n d 3' = 32 d y n / c m . 6. DISCUSSION F i g u r e s 2 a n d 3 s h o w plots o f the v o l u m e flux q as a f u n c t i o n of height of rise, for v e r t i c a l capillaries o f i d e n t i c a l c o n i c a l sec-
\
\
\
n--1
× ~ (-1)i+lKZ(ai) t a n z ai e(ai) i=l -
7rpqZ(r(y) -2 - r~)K2(an) t a n 2 a. e(an)
7 2lzq + _ _ pq2 + 2zrpxtPn(y ), + 6rrr~ rl =
-~)-
pgy ~rr~. [5.4]
E q u a t i o n [5.4] d e s c r i b e s the rate of rise o f fluid t h r o u g h a s e q u e n c e of c o n i c a l s e c t i o n s a n d is the m a i n r e s u l t o f this p a p e r . E q u a t i o n [5.4] will n o w b e a p p l i e d to the case of
I, 5
V
.~
MS
FIG. 3. Plot of volume flux q against height of rise y for a capillary made up of conical sections. Values of y (integers) in units of section length l and all other distances in centimeters, rl = 0.04, r2 = 0.06, cone angle a = 5°, and length l = 0.229. The broken curve shows q for a cylindrical capillary of radius 0.050. In both plots oscillation about the equilibrium height has been suppressed.
Journal of Colloid and Interface Science, Vol. 77, N o . 1, S e p t e m b e r 1980
262
LEVINE, LOWNDES, AND REED
:000k
I\/ \/ \/ %_. ~ I
\/'
~
o~
,
utt
t
]~
,
~
,
~
i
tt t
t
i,o
I
'~z
ttt
'
1~
Y/z F]~. 4. Plot of pgy and 2"y/R(y) against y / / f o r a = 11.1 °, rl = 0.03, r~ = 0.06, and l = 0.153. T h e stable, unstable, and metastable positions are m a r k e d S, U, and MS, respectively.
tions, obtained by integrating [5.4] with both acceleration and inertia terms retained. The plots clearly show the oscillating volume flux which is indicative of Haines jumps. However, the retention of the acceleration terms has tended to smear out this effect and the wetting process is seen to be dominated by the rapid acceleration during the initial stages of imbibition. Because of the acceleration terms, there is a tendency for the meniscus to overshoot its equilibrium position before returning to this position. This effect can clearly be seen in Figs. 2 and 3. The equilibrium positions for the meniscus are given by the solutions to the equation 27 [6.11 ogY R(y) -
,
200° I
~
I000
0~ 0
l 2
I t+
S
MS
Sl
l ~is 6
I
I
I
B
FIG. 5. Plot of pgy and 2y/R(y) against y/l for a = 5°,r~ = 0.04, r2 = 0.06, a n d / = 0.229. T h e stable, unstable, and metastable positions are m a r k e d S, U, and MS, respectively.
Journal of Colloid and Interface Science, Vol. 77, No. I, September 1980
which is obtained from [5.4] by putting q = 0. Graphical solutions of Eq. [6.1] are shown in Figs. 4 and 5 where both pgy and 27/R(y) are plotted. The solution of Eq. [6.1] shows a band of alternating metastable and unstable equilibrium positions; these are denoted on Figs. 4 and 5 by MS and U, respectively. The stable equilibrium position is marked with an S. This result is similar to the result of Kusakov and Nekrasov (4) for a sinusoidal capillary. The stable and metastable equilibrium positions are marked on Figs. 2 and 3 by S and MS, respectively. It is seen in Fig. 3 that the fluid has not come to rest at its lowest stable position but at its first metastable position. This shows that the dynamical processes can effect the equilibrium properties in a model which exhibits capillary hysteresis. Also plotted on Figs. 2 and 3 is the solution to the Rideal equation for a cylindrical capillary obtained by taking the limit a ~ 0 in Eq. [5.4]. The radius of the cylindrical capillary is chosen such that the volume per conical section length of the cylindrical capillary and the alternating capillary are the same. The fluid in the cylindrical capillary is seen to accelerate more rapidly in the initial stages. This will always be true for any cylindrical capillary of radius less than r2 and is a consequence of the first conical section converging downwards.
CAPILLARY FLOW ACKNOWLEDGMENTS We are indebted to the Science Research Council of the United Kingdom and to Tioxide International Limited, for postdoctoral research assistantships to J.L. and P.R.
5. 6.
REFERENCES
7.
1. Johnson, R. E., and Dettre, R. H., in "Surface and Colloid Science" (E. Matijevic, Ed.), Vol. 2, p. 85. Wiley-Interscience, New York, 1969. 2. Haines, W. B.,J. Agric. Sci. 20, 97 (1930). 3. Miller, E. E., and Miller, R. D., J. Appl. Phys. 27, 324 (1956). 4. (a) Kusakov, M. M., and Nekrasov, D. N., in "Research in Surface Forces" (B. V. Deryaguin, Ed.), Vol. 2, p. 193. Consultants Bureau,
8. 9.
10.
11.
263
New York, 1966; (b) Dokl. Akad. Nauk SSSR 119, 107 (1958); (c) Zh. Fiz. Khim. 34, 1602 (1960). Melrose, J. C.,J. Soc. Petrol. Eng. 5, 259 (1965). Everett, D. H., and Haynes, J. M., Z. Phys. Chem. (Frankfurt am Main) 82, 36 (1972); 97, 301 (1975). Everett, D. H., J. Colloid Interface Sci. 52, 189 (1975). Morrow, N. R., Ind. Eng. Chem. 62, 32 (1970). Levine, S., Reed, P., Watson, E. J., and Neale, G., in "Colloid and Interface Science" (M. Kerker, Ed.), Vol. III, p. 403. Academic Press, New York, 1976. Happel, J., and Brenner, H., " L o w Reynolds Number Hydrodynamics," Prentice-Hall, Englewood, N. J., 1965. p. 138. Ackerberg, R. C., J. Fluid Mech. 21, 47 (1965).
Journal of Colloid and Interface Science, Vol.77, No. 1, September1980