058:0160
Jianming Yang
Fall 2012
Chapters 3&4
97
12.3
Two Immicsible Liquid Films Flowing down an Inclined Wall
Consider two immiscible liquid films flowing down an inclined wal under the influence
of gravity. Assume that the flow is fully established and the thicknesses of the films are
and , respectively, and that the atmospheric pressure is and the viscosity of air is
much smaller than those of the liquids. Solve for the velocity distributions in each of the
liquids.
058:0160
Jianming Yang
Chapters 3&4
98
Fall 2012
For both liquids a and b
;
;
IC (Initial condition): steady flow:
BC: Fully developed 1D flow, i.e.,
( )
( )
No-slip at the wall,
Kinematic Condition at the interface:
Continuous velocity at the interface,
: ( )
Dynamic Condition at the interface: (continuous stresses)
Continuous shear stress at the interface,
:
Neglect shear stress at the air/liquid interface,
(
)
:
Continuous normal stress and neglect viscous normal stress at air/liquid interface
:
:
058:0160
Jianming Yang
Chapters 3&4
99
Fall 2012
Continuity equation:
for both a and b
x-momentum:
(
Integrate for both a and b with
BC:
(
|
)
|
)
for both a and b
058:0160
Jianming Yang
Chapters 3&4
100
Fall 2012
|
|
(
(
)
)
(
(
)
(
)
[
(
)
(
{
(
)]
)
[
(
)]}
(
[
)
[
)
(
)
]
[
{[
(
(
)] ( )
)]
( ) }
]
058:0160
Jianming Yang
Chapters 3&4
101
Fall 2012
[
[
( )
( )
z-momentum:
y-momentum:
for both a and b
( )
BC:
:
(
( )
)
( )
(
)
]
(
)
]
058:0160
Jianming Yang
Chapters 3&4
102
Fall 2012
( )
:
(
( )
(
(
)
)
(
)
( )
(
( )
(
Also,
)
)
)
(no flow)
( )
Single layer of liquid film:
{[
(
{( )
)] ( )
( ) }
( ) }