An Analysis of Two Fluid Layers Enclosed Between Two Non-Porous
Surfaces
ASAD SALEM
Weisberg Department of Mechanical Engineering
Marshall University
1 John Marshall Drive., Huntington, WV 25755
UNITED STATES OF AMERICA
Abstract: The stability of a two-phase interface is a crucial occurrence that involves the design of many
engineering applications. It correlates the spatial and droplet size-distributions of many fluid spraying
applications and has a great effect on the estimations of the critical heat flux of systems that involves
phase change or evaporation. However, the existing hydrodynamic models are only able to predict the
stability of a plane fluid sheet, surrounded by an infinite pool of liquid. The case of a thin sheet of
liquid surrounding a vapor sheet and enclosed between two walls has not been studied yet. The
present paper solves this problem using a linearized stability analysis. Velocity potentials satisfying
these conditions are introduced and a complete analysis is presented.
Key-Words: Fluid Flow Stability, Nonporous Cavity, Sinusoidal Wave, Dilatational Wave.
Received: May 28, 2021. Revised: November 25, 2021. Accepted: December 19, 2021. Published: January 11, 2022.
1 Introduction
There are numerous theoretical and experimental
studies of fluid flow instability over the past few
decades since the work of Shea and Hagerty [1]
because of its prevalence in fluid and heat transfer
analysis and stability. The stability of fluid flow in
between non-porous medium channel finds many
important applications in geothermal and
geophysical engineering such as underground
disposal of nuclear wastes, spreading of chemical
pollutants in water-saturated soil and many other
applications. Shea and Hagerty studied the
stability of a liquid sheet surrounded by air. They
used a potential flow model for their analysis.
They assumed a potential function for the different
flow regimes and employed a sinusoidal waveform
for the initial disturbance. Further, the wave
formation at the film surface considerably
improves mass and heat transfer rates and play a
vital role in the process equipment, such as falling
film in absorption columns condensers, and
evaporators, [2,3]. The stability of fluid flow
between tow parallel walls of unknown surface
condition was studied by Chamkha [4]. The study
analysed the flow characteristics and thermal and
electrical properties of the fluid flow. It showed
that if any velocity profile is unstable for a
particular value of Reynold’s number, it will be
unstable at a lower value of the Reynold’s number
for the two-dimensional disturbances. The linear
flow stability of a contaminated fluid with a
monolayer flow down a slippery non-porous
inclined channel was studied by Bhat and Samanta
[5]. Bhat and Samanta used Orr-Sommerfeld
equation for analysis with Chebyshev spectral
collection method to obtain numerical solutions.
Salem 6] analysed the flow of two plane non-
Darcy fluid flow in narrow rectangular cavity
using Keller-box solution.
The present work attempts to explore the stability
of two-phase flow between layers non-porous
surfaces. The stability of vapor layers contained by
a liquid pool is an important phenomenon that
affects the design of many engineering devices.
This situation can arise in many practical
applications, for example, the flow between
parallel plates with internal heat generation such as
steam generators, or if boiling occurs in the narrow
gaps of heat exchangers. It is desirable to study the
parameters influencing critical heat flux and the
vapor removal mechanism from the heating
surface. However, the existing hydrodynamic
models are only able to predict the stability of a
plane fluid sheet surrounded by an infinite pool of
liquid. Many hydrodynamic models have been
developed to explain this phenomenon. The
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Asad Salem
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criterion for stable film boiling on a horizontal
surface facing upward can be developed from
Taylor instability. The stability of an interface of
waveform between two fluids of different densities
depends on the balance of surface tension energy
and the sum of the kinetic and potential energies of
the wave. This process is called Taylor's instability
wavelength. In this process, the length of the wave
grows fast and thus predominates during the
collapse of an infinite plane horizontal interface.
When two immiscible fluids flow relative to each
other along with the interface of separation, there is
a maximum relative velocity above which a small
disturbance of the interface will amplify and grow
and thereby distort the flow. This phenomenon is
known as the Helmholtz instability.
This paper solves a similar problem that includes the
addition of a solid boundary surrounding the outer
fluid. The instabilities are observed at the interface
of the fluid sheet as the vapor layer starts building at
the surface. The objective of this work is to find an
expression for the growth rate of these instabilities.
Figures are plotted for the growth rate versus
frequency for different sheet velocities. The so1ut
ion used a linearized stability model where squares
of the velocities and their products are ignored.
Velocity potentials satisfying these conditions are
introduced and a complete analysis is presented.
2 Problem Formulation
The idealized system to be treated is shown in
Figure 1. The inner fluid will be considered the
vapor phase and the other fluid will be the liquid
phase (this can be reversed easily). The width of the
inner fluid sheet is “2 a” and the plates are a
distance “b” from the centerline of the inner fluid.
The density of the vapor phase is V and that for the
liquid phase is and . We assume the vapor
phase to be moving vertically with velocity
through the surrounding liquid phaseand ,
which moves with velocity . Following Rayleigh
(1954) the following assumptions are made:
1. The velocity of the sheet is great enough so
that the effect of gravity will be ignored.
2. Both fluids are inviscid.
3. The surface tension between the vapor and
liquid phase is .
4. The surface at (at the interface) is
disturbed by a small disturbance of amplitude and
frequency but out of arbitrary phase to each other.
5. Flow is in steady-state conditions
6. Nonlinear terms such as the squares and
products of the variables are ignored through the
derivation of equations (a linearized theory is used).
h1
h2
L
2
L
1
v’ v’
X
Y
-b b
-a a
V
L
Fig. 1: Idealized fluid-sheet System
Based on the above assumptions, an incompressible
inviscid flow between two parallel planes is
assumed: and , where is the
velocity potential function.
3 Problem Solution
Let us perturb the interface by a small wave of
amplitude . The equations of the disturbed
surfaces (the interface) are assumed to be
ℎ (1a)
and
ℎ (1b)
where phase angle is introduced to treat both
symmetrical and anti-symmetrical waves at the
same time.
The boundary conditions are as follow:
1. The velocity of the fluid normal to the
boundary is zero, i.e.
ℎ (2a)
and
ℎ (2b)
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2. the y-direction velocity at the wall is zero,
i.e.
(3a)
and
(3b)
the velocity potentials that satisfy the Laplace
equation ( ) as well as the boundary
conditions as presented above, are given as follow,
for the region ,
ℎℎ
(4a)
where
ℎ (4b)
and
ℎ (4c)
and the potential function for the surrounding fluid
in the region
ℎ
ℎ ′ (5a)
where ′
(5b)
the velocity potential for the surrounding fluid in the
region
ℎ
ℎ ′ (6a)
where ′
(6b)
The velocity potential functions presented above
satifies all the required conditions, the Laplace
equation and the boundary conditions at the solid
and the liquid-vapor interfaces.
3.1 Pressure and Surface-Tension Forces
The stability of the sheet depends on the growth rate
of the disturbance at the liquid-vapor interface. The
relationship between the surface tension and the
pressure force acting at the surface of the sheet will
determine the velocity of the vapor that will trigger
the collapse of the sheet as in Helmholtz instability.
The non-steady form of Bernoull’s equation is given
as
(7a)
Then
(7b)
Solving for the excess pressure by subsituting the
velocity potentials into the equation above for the
region will yield
ℎ
′ℎ
(8)
(8a)
or
ℎ
′ℎ (8b)
and
ℎ
′ℎ
(9)
(9a)
or
ℎ
′ℎ (9b)
Now, equation (9a) will yield which
corresponds to a dilatation wave solution while
equation (8a) will yield which corresponds
to a sinusoidal wave solution.
3.2 Analysis of the Sinusoidal and Di-
latational Wave
3.2.1 Sinusoidal Wave
For a sinuous wave , let ′ and
.
Equation (8) with rearrangement gives
ℎℎ
ℎℎ
(10)
Solving equation (10) yields,
ℎ
ℎℎℎℎ
ℎℎ
(11)
For the particular case of interest, that of a fluid
being sprayed into gas, the ratio is quite small and
can be neglected in comparison with ℎ
over most of the unstable regions.
The unstable region consists of those waves which
render the term under the radical negative. The sheet
is unstable if
ℎ
(12)
From equation (11) the real circular frequency
and the upper bound of unstable
frequencies is therefore
ℎ
(13)
For frequencies below the growth rate is
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ℎℎ
ℎℎ
ℎℎ
(14)
3.2.2 Dilatational Wave
For a dailational wave , ′ , and
.
Now, frequency equation (8b) is reduced to
ℎℎ
ℎℎ
(15)
Assuming ℎ over most of the
unstable region
ℎ
ℎℎℎℎ
ℎℎ
(16)
This unstable region consists of those wave numbers
, which render the second term imaginary. The
relationship is the same as in equation (12). As in
the case of the sinuous wave, the upper bound of the
unstable frequencies is given by equation (13). For
the frequencies below the growth rate is is
ℎℎ
ℎℎ
ℎℎ
(17)
3.3 Representation of Equations
3.3.1 Logarithmic decrement For A
Sinusoidal Wave
The dominant wave in the unstable region has the
form
(18)
During one complete cycle, increases by
, and
increases by
. The ratio of the amplitude at the
end of the cycle , to the amplitude at the
beginning of the cycle is
(19)
The logarithmic decrement is defined as the
natural logarthim of the ratio of the amplitude in
successive cycles. Therefore
(20)
For the sinuous wave, using equations (13) and (14)
and defining the dimensionless parameters as
(21)
(22)
Equation (20) reduces to
ℎ
ℎℎ (23)
whare is Weber’s number.
3.3.2 Logarithmic Decrement For A Di-
latational Wave
Similarly, using equations (13), (19), (21), and (22),
the logarithmic decrement for dalational wave
reduces to
ℎ
ℎ (24)
The growth rate could be plotted as a function of
frequency for several different sheet velocities for
both sinusoidal and dilatational waves, and for
various Weber numbers in the range of interest.
4 Conclusion
As a result of the theoretical work done in this
paper, the following conclusions can be drawn. An
expression for the growth rates as well as the
logarithmic decrement for a sinusoidal and
dilatational wave were determined, which helped to
study the stability of plane fluid sheets with
sidewalls. There are only two kinds of waves that
are possible at any given frequency. Either the two
surfaces of the sheet oscillate in phase to produce
sinusoidal waves, or they oscillate out of phase as in
a dilatational wave. The frequency, velocity, and
wavelength are related in the same way for both
types of waves. This relation is helpful in
determining the velocity of the sheet. The equation
is given by
. It was, also, observed from the
reduced functions, that the growth rate for a
sinusoidal wave is always greater than that of a
dilatational wave for the same given frequency.
The equation is given by
ℎ.
References:
[1] "A Study of the Stability of Plane Fluid Sheets"
by Hagerty, W.W. and Shea, J.F. 1955, Journal of
Applied Mechanics, December 1955, pp 510-514.
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[2] "Gravity-driven flow of continuous thin liquid
film on non-porous substrates with topography”". by
P.H. Gaskell, P.K. Jimack, M. Seller, H. M.
Thompson, and M.C. T. Wilson, J. Fluid Mech.
(2004), Vol. 509, pp. 253-280.
[3] "Unsteady Flow Evolution in non-Porous
Chamber with Surface Mass Injection, Part 1: Free
Oscillation”, by S. Apte, and V. Yang, AIAA
Journal, Vol, 39, No. 8. August 2001.
[4] "Flow of Two-Immiscible Fluids in Porous and
Nonporous Channels”, by Ali J. Chamkha, Journal
of Fluids Engineering Vol, 122, March 2000.
[5] "Linear stability of a contaminated fluid flow
down a slippery inclined plane”, by F. A. Bhat and
A. Samanta, American Physical Society, 033108,
2018.
[6] " Analysis of Two Plane Fluid Layers in Narrow
Rectangular Cavity’, by Asad Salem, Proceedings of
the 5th Inter. Conf. on Fluid Mechanics and Heat
&Mass Transfer, Lisbon Portugal, Oct. 30-Nov. 1,
2014
[7] " Micropolar fluid flow between a non-porous
disk and a porous disk with slip: Keller-box
solution”, by A. Bhat and N. N. Katgi, Ain Shams
Engineering Journal, 11 (2020), 149-159.
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This article is published under the terms of the Creative
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