Applied Mathematical Sciences, Vol. 6, 2012, no. 123, 6137 - 6146
On the Two Immiscible Fluids in Contact
with Each Other on a Solid Boundary
Hishyar Kh. Abdullah
Dept. of Mathematics, University of Sharjah
Sharjah, P.O. Box 27272, UAE
[email protected]
Abstract. In this paper class of two immiscible ‡uids in contact with each
other on a solid boundary, have been studied: The asymptotic solution near
contact line is …ned for two cases. First we assume that the external gas has
no e¤ect, then the case of external gas is assumed process viscosity comparable
to that of the ‡uid.
Mathematics Subject Classi…cation: 34C55, 76D05, 76D07, 76D08,
76D10
Keywords: Rayleigh-Ritz method, Fourier series expansion, Inclined rivulet
1. Introduction
Immiccible ‡uids ‡owing over an undulating solid represent an everday physical phenomenon that has eluded accurate continuum description so far. At
the heart of this probblem is the boundary condition for the moving contact
line (MCL), de…ned as the intersection of the ‡uid-‡uid interface with the solid
wall.
The no-slip boundary condition, i.e., zero ‡uid velocity relative to the solid
at the ‡uid-solid interface, has been very successful in describing many macroscopic ‡ows. A problem of principle arises when the no-slip boundary condition is used to model the hydrodynamics of immiscible-‡uid displacement in
the vicinity of the moving contact line, where the interface separating two immiscible ‡uids intersects the solid wall. Decades ago it was already known that
the moving contact line is incompatible with the no-slip boundary condition,
since the latter would imply in…nite dissipation due to a non-integrable singularity in the stress near.In [8] Qian staded the contact-line motion in immiscible
two-phase ‡ow, from molecular dynamics (MD) simulations to continuum hydrodynamics calculations.
6138
H. Kh. Abdullah
Multiphase ‡ows form when two or more partially or not miscible ‡uids are
brought in contact and subjected to a pressure gradient. The resulting ‡ows
display a rich phase behavior, e.g. as suspended droplets, bubbles, slugs or thin
…lms. The ‡ow behavior is dependent on the relative ‡ow rates of the ‡uid
phases involved, the resulting interaction between gravitational, interfacial,
inertial and viscous forces and the wetting behavior of the channel walls.The
alternating succession of immiscible ‡uid segments will play a particularly important role.Multiphase microchemical systems generally take advantage of
the large interfacial areas, rapid mixing and reduced mass transfer limitations
to achieve improved performance relative to conventional bench-scale systems
(see [14].[15], [16].[17]) .Multiphase ‡ows provide several mechanisms for enhancing and extending the performance of single-phase micro‡uidic systems.
The long di¤usion times and broad dispersion bands associated with singlephase pressure-driven ‡ow can be reduced by adding a second, immiscible,
‡uid stream. . Examples of gas–liquid reactions that were performed in microreactors include direct ‡uorination( [18],[19]).
Dynamic contact line phenomena involve two immiscible viscous ‡uids in
contact with a solid surface; as one of the ‡uids displaces the other, the contact
line moves along the solid surface. Applications range from droplet spreading
in various applications to coating ‡ows. Such ‡ows have been studied both
theoretically and experimentally (see [2], [6]). Simulating such ‡ows is complicated by the mathematical paradox of a contact line moving along a no-slip
solid surface. Analytical solutions of the Navier–Stokes equations lead to a
stress singularity at the contact line (see [4],[6]). This non-physical divergent
stress stems from the fact that continuum ‡uid mechanics breaks down at
molecular distances from the contact line; to circumvent the stress singularity,
the molecular interactions between the ‡uids and the solid would have to be
modeled.The standard model for two-phase ‡ow consists of the incompressible Navier–Stokes equations, combined with a mechanism to transport the
interface with the local ‡uid velocity. This model cannot be combined with
conventional no-slip boundary conditions at solids because this would lead to
a singularity in the stresses around the contact line (see [10]). The singularity
in the model can be avoided if the contact line is allowed to slip (see [11]).The
Navier condition is su¢ cient for ‡ows that are driven by external forces. For
capillary-driven ‡ows, on the other hand, the contact angle between the interface and the solid must also be taken into account. In numerical algorithms
based on the level set method [12] or the volume of ‡uid method [13].
We consider a vertical rivulet of liquid with density, and viscosity ( 1 , 1 )
in a second ‡uid with density, and viscosity ( 2 , 2 ), where 1 > 2 , the latter
being at rest at a large distance from the rivulet. In practical case of water
rivulet in air, we should expect the small ratio of the viscosity’s to ensure that
the motion of the air has little e¤ect on the ‡ow in the rivulet. But there are
many cases of ‡uid ‡ow in which, as some parameters ( her the viscosity ratio)
On two immiscible ‡uids
6139
tends to zero. The solution of the limiting form of the governing equations is
di¤erent from the limiting form of the solution of the full equations.
The objective of this paper is to answer the question: “ Dose the outer ‡uid
has an important e¤ect on the rivulet?”, and how dose it e¤ect on the stability
of the interface between the two ‡ows? We will obtain the explicit solution of
the conventional governing equations of ‡uid near contact line.
2. Flow Description and Governing Equations.
A vertical rivulet of liquid with density, and viscosity ( 1 , 1 ) in a second
‡uid with density, and viscosity ( 2 , 2 ) where 1 > 2 , the latter being
at rest a large distance from the rivulet which is ‡ow steadily in the negative
direction of the Z axis. Thus, if U1 ,V1 ; W and U2 ,V2 ; W2 are the velocity
components of the ‡uids then,U1 = U2 = V1 = V2 = 0; W1 = W1 (X; Y ), and
W2 = W2 (X; Y ):
The partisan of the solid boundary between AB (see …g-1-) is in contact
with the …rst ‡uid, the partisan above A and lowerB being in contact with the
second ‡uid. The interface between the two ‡uids is characterized by a uniform
surface tension , and by a contact angle , between the solid boundary and
the tangent line to S at A, and B. We adopt a plane polar coordinate system
(r, ). The speci…cation of the coordinate system is completed by taking the
ray = 0, and = to be the tangent lines to S atB and A respectively. The
Navier - Stokes equations inside and outside the rivulet becomes:
.
@P1
(2.1)
0 = 1g
+ 1 r2 w1 (r; )
@Z
(2.2)
(2.3)
0=
2g
@P2
+
@Z
2r
2
w2 (r; )
@P1
@P2
=
=0
@
@
@P1
@P2
=
=0
@r
@r
whereP1 ; P2 the pressures , are both independent of r; . The outer boundary
w2 =) 0 , as r =) 1 , then gives
(2.4)
@P2
= 2g
@Z
the pressure di¤erence P across the rivulet is independent of Z, and this
is because of the normal stress condition from the Laplace’s condition
(2.5)
(2.6)
( P )s =
R
H. Kh. Abdullah
6140
where R is the local radius of curvature of S, is the surface tension of the
liquid.
Since the interface S is a circular cylinder, and the pressure di¤erence P
across it is independent of Z, we must have
@P1
@P2
=
= 2g
@Z
@Z
Using (2.5), and (2.6) in (2.1), and (2.2) we get.
(2.7)
r2 w1 (r; ) =
(2.8)
where k = (
1
2
1
k;
)g
r2 w2 (r; ) = 0:
The boundary conditions are:
a- The slip condition on AB
(2.9)
w1 (r; 0) = w2 (r; 0) , for all values of r
(2.10)
w1 (r; ) = w2 (r; ); for all values of r
b- The shear stress condition
(2.11)
w1 (r; ) = w2 (r; );
(2.12)
1(
@w1
)s =
@r
2(
@w2
)s
@r
3. Local Theory Near a Contact Line
Now we will try to answer the question: Does the outer ‡uid has an important e¤ect on the rivulet? To answer this question , …rst we assume that
the ambient gas has no signi…cant e¤ect. The equation of motion with the
boundary conditions from (2.8).(2.10), and (2.11) will be
(3.1)
r2 w1 (r; ) =
k1 ; where k1 =
1g
,
1
and
(3.2)
w1 (r; ) = 0; at
= 0, for all values of r ,
On two immiscible ‡uids
6141
@w1
= 0; at = , for all values of r:
@
Using separation of variable to homogeneous form of (3.1) (i.e. to r2 w1 (r; ) =
0 ) by seeking the solution of the form w1 (r; ) = R(r) ( ) , we get
(3.3)
00
r2 R00 rR0
+
=
=
R
R
where eigenvalues. To …nd the solution w1 for (3.1) we have two cases:
case 1: If = 2 , let w1 (r; ) = r2 2 ( ) be the solution of (3.1) with the
boundary conditions (3,2), by substituting, we get
(3.4)
00
(3.5)
( )+4 ( )=
k1 ;
and
(3.6)
(0) =
0
( ) = 0:
The general solution of (3.1) will be
(3.7)
provided that
k1
(cos 2 + tan 2
4
and 6= 34 :
w1 (r; ) = r2
6=
4
sin 2
1);
H. Kh. Abdullah
6142
case 2: if
6= 2; let
(3.8)
X
( ) be the solution of (3.1) then,
r
w1 (r; ) =
1
X
bn r
( 12 +n) 2
n=1
Now the solution of (3.1) will be
I- for > 4 , then 2 < 2, and n = 0,
w1 = b0 r( 2 ) sin
II- for
>
3
4
w1 = b0 r( 2 ) sin
then
2
3
2
0
1
sin(( + ) :
2
2
=
, then
2
k1
(cos 2 + tan 2 sin 2
2
4
< 2, and n = 1, 1 = 23 , then
+ r2
3
+b1 r( 2 ) sin
3
2
+r2
1)
k1
(cos 2 + tan 2
4
sin 2
1)
III- for < 4 , 2 > 2 , n = 0 , 0 = 2 then the solution will be as in (I)
above.
IV- for < 4 , 2 > 2 , n = 1 , 1 = 23 then the solution will be as in (II)
above.
Now for = 4 , or
= 34 , let the solution of (3.1) be of the form
w1 = r2 ln r f ( ) + r2 g( ) substituting in (3.1) we will get the following
di¤erential equations with boundary conditions
(3.9)
f 00 ( ) + 4f ( ) = 0;
With the boundary conditions
(3.10)
f 0 ( ) + 4f ( ) = 0
and
(3.11)
g 0 ( ) + 4g( ) = 4f (0)
k1
With the boundary conditions
(3.12)
g(0) = g 0 ( ) = 0:
The solution of (3.9) is f (0) = A sin 2 , where A is constant and the solution
of (3.11) is
(3.13)
g( ) = B sin 2 +
k1
[(
4
) cos 2
]:
Where B is arbitrary constant.
Then, from (3.12) and (3.13) we get A = 4k1 ;thus f (0) =
the solution w1 (r; ) of (3.1) will take the form
k1
4
sin 2 , then
On two immiscible ‡uids
w1 (r; ) =
(3.14)
For
=
4
k1 2
k1 2
r ln r(sin 2 ) +
r [(
4
4
1
X
1
1
+
bn r( 2 +n) sin
+n
2
n=1
; n = 0 and
0
=
4
6143
] + Br2 sin 2
) cos 2
:
; the solution will be of the form
(3.15)
k1 2
k1 2
r (sin 2 ) +
r [(
) cos 2
] + Br2 sin 2 + b0 r2 sin 2 :
4
4
= 34 ; n = 1 and 1 = 23 ; the solution will be of the form
w1 (r; ) =
For
w1 (r; ) =
(3.16)
k1 2
k1 2
r (sin 2 ) +
r [(
4
4
+b1 r2 sin 2
] + Br2 sin 2 + b0 r2 sin 2
) cos 2
4. Two Immiscible Fluids in Contact With Each Other on a
Vertical Plane
For the equations (2.6) and (2.9), let the solution be of the form w1 (r; ) =
r 1 ( ); and w2 (r; ) = r 2 ( ); where is the eigenvalues to be found and is
of two cases
I- if = 2 then the solution will be
(4.1)
w1 (r; ) =
k1 2
r (sin 2
4
sin 2
cos 2
1);
and
(4.2)
II- if
(4.3)
w2 (r; ) =
k1 2
r 1
8
1 2
sin 2 (1
cos 2 ) sin 2 :
2 1
6= 2;using the boundary condition (2.9) we get
tan(
)=
c tan 2 ;
where
(4.4)
w1 (r; ) = Br sin
1
cos2
c
where B is arbitrary constant and c =
ferent cases:
(4.5)
w2 (r; ) = Br (sin2
+
:
) (sin
2 1
1 2
tan
cos 2 ) ;
the parameter c has three dif-
6144
H. Kh. Abdullah
i - if c = 0 then = 2 :
ii - if c =) 0 then =
:
iii - if c =) 1 then = 2( ) :
These cases are shown in …g-2-.
5. Conclusion
from …g-2- we conclude that the dominate solution will be in the interval
< < 32 , and the point ( 2 ; 1) is a node point , ( 3 ; 23 ) is a saddle point . It
means that the the system reaches stability at ( 2 ; 1). For the range < 3 the
system is unstable and this range of is the range of most previous work.
Now as 2 =) 0; then equation (4.1) will be exactly the same form as (3.7),
1
and as 2 =) 0; the de…nition or r in the two formulas become identical. these
1
results therefore con…rm the negligible e¤ect of the ambient atmosphere.
1
2
Acknowledgement 1. I would like to extend my thanks to the University of
Sharjah for its support.
On two immiscible ‡uids
6145
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Received: July, 2012