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J. of Thermal Science and Engineering, The Heat Transfer Society of Japan
Vol. 12, No. 1, pp. 27-34, 2004
Heat transfer and two-dimensional deformations in locally heated
liquid film with co-current gas flow
Elizaveta Ya. GATAPOVA†, Igor V. MARCHUK† and Oleg A. KABOV‡,†
Abstract
A two-dimensional model of a steady laminar flow of liquid film and co-current gas flow in a plane
channel is considered. It is supposed that the height of a channel is much less than its width. There is a
local heat source on the bottom wall of the channel. An analytical solution for the temperature distribution problem in locally heated liquid film is obtained, when the velocity profile is linear. An analytical
solution of the linearized equation for thermocapillary film surface deformation is found. A liquid bump
caused by the thermocapillary effect in the region where thermal boundary layer reaches the film surface
is obtained. Damped perturbations of the free surface may exist before the bump turned out into a flow;
this is obtained according to the solution of the problem in an inclined channel. It depends on the forces
balance in the film. The defining criterion is found for this effect.
Key Words: Marangoni effect, Local Heating, Two-phase flow, Channel
Nomenclature
a
ag
B
Bi:
C
cp
g
H
h
h0
h1
h
L
lσ
P
p
pa
p0
Pe
q0
R
Re
Reg
: liquid temperature conductivity coefficient
: gas temperature conductivity coefficient
: channel width
Biot number
: analogue of capillary number
: specific heat capacity
: gravity acceleration
: channel height
: film thickness
: initial film thickness
: dimensionless relative thickening
: dimensionless film thickness
: heater length
: capillary length
: dimensionless pressure
: pressure
: pressure of ambient gas
: characteristic pressure
: Peclet number
: heat flux density
: curvature radius of the free surface
: film Reynolds number
: gas Reynolds number
s
:
Tl :
Tg :
Ta :
T0 :
Tw :
∆T :
T :
U, V :
ul, vl :
ug, vg:
u,v :
X,Y :
x,y :
α
:
Г
:
γ
:
θ
:
θa :
θ :
ε
:
ϕ :
ρl, :
ρg :
element of film surface
film temperature
gas temperature
constant temperature of ambient gas
initial temperature of the film
temperature of the top wall
characteristic temperature drop
temperature on the film surface
dimensionless liquid velocity components
liquid velocity components in x-, y-axis
gas velocity components in x-, y-axis
characteristic liquid velocity components
dimensionless Cartesian coordinates
Cartesian coordinates
heat-transfer coefficient
specific flow rate
dimensionless specific liquid flow rate
dimensionless temperature of the film
dimensionless temperature of ambient gas
dimensionless film surface temperature
ratio of linear film scales
inclination angle of the channel
liquid density
gas density
* Received: , , Editor:
† Institute of Thermophysics, Russian Academy of Sciences (prosp. Lavrentyev, 1, Novosibirsk, 630090, RUSSIA)
‡ Microgravity Research Center, Universite Libre de Bruxelles (Av. Roosevelt 50, B-1050 Brussels, BELGIUM)
Thermal Science & Engineering Vol. 12, No.1 (2004)
λ :
λg :
µl, :
µg, :
η(X) :
χ(X):
T
:
:
:
:
:
:
:
τ
τ0
Σ
σ
σ0
σT
liquid thermal conductivity coefficient
gas thermal conductivity coefficient
liquid dynamic viscosity
gas dynamic viscosity
finite function
Heaviside function
dimensionless tangential stress on interface
tangential stress on the interface
characteristic tangential stress
dimensionless surface tension
surface tension
surface tension specified at temperature T0
surface tension temperature dependence coeffi-
cient
(σ ) :
(σ ) :
l
ik
liquid viscous stress tensor
g
ik
gas viscous stress tensor
2
Problem statement and basic equations.
We consider a channel with rectangular cross section,
where the height H is much less than its width B
(H<<B). A layer of viscous incompressible liquid is
moving in the channel under the influence of tangential
stress τ , caused by the gas flow as well as under gravity force for a channel inclined by an angle ϕ with respect to horizontal. A local heat source is situated on the
bottom wall (Fig. 1). The locality of heating means that
the heat flux density is a finite function of variable x . A
two-dimensional flow is considered. Let us choose the
system of Cartesian coordinates (x, y ) so that Oy axis
is orthogonal to the bottom wall of the channel and Ox
axis is directed along the gas flow. The coordinate origin
is situated at the beginning of the heater. It is supposed
that the surface tension depends on temperature (σ0, σT
=const>0)
σ = σ 0 − σ T (T − T0 )
Sub/Superscripts
a
: ambient gas
g
: gas
l
: liquid
Gas
1
n
Introduction
Liquid
The surface tension plays a crucial role in two-phase
flow in mini- and microchannels [1], [2]. The annular
flow regime is one of the basic regimes of flow in miniand microchannels also under microgravity. The laboratory investigation and theoretical modeling of such flow
is possible using liquid film flow caused by a co-current
gas flow [3], [4]. Temperature gradient on gas-liquid
interface and concentration-capillary forces caused by
concentration gradient of multicomponent liquids produce forces (the so-called Marangoni effect), which induce convection and intensive heat and mass transfer.
The Marangoni effect may have an essential influence
on heat transfer intensity and lead to film breakdown,
dry spots formation, burn-out of heater. The aim of the
work is to investigate of the influence of thermocapillary
forces on hydrodynamics and heat transfer of liquid film
flow in a plane channel 2-4 mm high.
Applications of the investigations could be the cooling systems of microelectronic equipment, where heat is
transferred from the integrated circuits to thin liquid film
moving under friction of gas flow in a narrow gap [5].
The heat transfer and hydrodynamics investigations in
such system were performed by Bar-Cohen et al. [6, 7].
The experiments were made under the flowing of Nitrogen and liquid FC-72 in a symmetrically heated 0.508
mm wide channel. For a superficial liquid velocity less
than 1 cm/s the annular flow regime occurs at superficial
gas velocity larger than 0.1 cm/s.
t
g
y
H
ϕ
Fig. 1
nel
x
Sketch of the liquid and gas flow in the chan-
The steady two-dimensional motion of liquid film and
gas in the channel is described by equations:
ρl ( u l u xl + vl u ly ) = µl ( u xxl + u lyy ) − px + ρl g sin ϕ
(1)
ρl ( u l vxl + vl vly ) = µl ( vxxl + vlyy ) − p y − ρl g cos ϕ
(2)
u xl + v ly = 0
(3)
u l Txl + v l Tyl = a (Txxl + Tyly )
(4)
ρ g ( u g u xg + v g u yg ) = µ g ( u xxg + u yyg ) − px + ρ g g sin ϕ
(5)
ρ g ( u g vxg + v g v yg ) = µ g ( vxxg + v yyg ) − p y − ρ g g cos ϕ
(6)
u xg + v yg = 0
(7)
u g Txg + v g Tyg = ag (Txxg + Tyyg )
(8)
The boundary conditions are written in the following way:
u l ( x, 0) = v l ( x, 0) = 0
(9)
u g ( x, H ) = v g ( x , H ) = 0
(10)
are a non-slip conditions
u l ( x, h( x)) = u g ( x, h( x))
(11)
vl ( x, h( x)) = v g ( x, h( x))
(12)
velocity of the liquid is equal to the gas velocity on the
gas-liquid interface
−λTyl ( x, 0) = q0 ( χ ( x) − χ ( x − L) )
(13)
a local heat source is specified
u l ( x, h( x))hx ( x) = v l ( x, h( x))
(14)
is a kinematic condition. The expression
σ ⎤ G ∂σ G
⎡ l
g
σ
σ
n=
t
−
−
ik
ik
⎢
R ⎥⎦
∂s
⎣
(15)
describes the balance of forces acting on the surface of
the liquid.
T l ( x, h( x)) = T g ( x, h( x))
(16)
and
λTnl ( x, h( x)) = λ g Tng ( x, h( x))
(17)
are the temperature and heat flux continuity conditions.
T l (−∞, y ) = T0
(18)
is the given initial temperature.
T g ( x, H ) = Tw = const
(19)
this condition takes place because of heat-transfer coefficient to the top wall is small. Here
⎛ 2u l
1 0
(σ ) = − p ⎛⎜ 0 1 ⎞⎟ + µ ⎜⎜ u l +xvl
⎝
⎠
⎝ y x
l
ik
⎛ 2u g
1 0
(σ ) = − pa ⎜⎛ 0 1 ⎟⎞ + µ ⎜⎜ u g +xv g
⎝
⎠
x
⎝ y
g
ik
u ly + vxl ⎞
⎟
2v ly ⎟⎠
,
u yg + vxg ⎞
⎟
2v yg ⎟⎠
are the viscous stress tensors, R is the curvature radius
of the free surface of the liquid, which is determined as
hxx
1
,
=
R (1 + h 2 )3 2
x
G ⎛ 1
hx
,
t =⎜
⎜ 1 + h2 1 + h2
x
x
⎝
⎞ G ⎛
hx
1
⎟, n = ⎜−
,
⎟
⎜ 1 + h2 1 + h2
x
x
⎠
⎝
⎞
⎟
⎟
⎠
are the unit vectors of the tangent plane and of the outward normal to the surface of the liquid.
For simplicity we divide the problem (1)-(18) into
three subproblems. The first one is finding initial film
thickness and constant tangential stress on liquid-gas
interface for rigid isothermal laminar film flow and
co-current gas flow [8]. It can be done out of problem
(1)-(3), (5)-(12), (15), taking into account that the liquid
density is much higher than the gas density ρ g << ρl .
The second subproblem is solving a heat transfer problem for rigid liquid film flow. And the third one is finding film deformations using results of first and second
subproblems. Moreover, heat transfer in gas is simplified
and changed by Newton-Richman law. It is considered
that film deformations do not change the value of the
heat transfer coefficient. Condition of heat exchange on
the film surface can be written in the following form:
−λTnl ( x, h( x)) = α (T l ( x, h( x)) − Ta )
(20)
Let the tangential stress τ and initial film thickness
h0 be done [8]. Taking into account that the initial film
thickness h0 is much less than the height of channel H
(h0<<H) and velocity of gas flow is much higher than
velocity of the film surface we change the expression
(15) into
G σ G
G G
⎡⎣ pa + σ ikl ⎤⎦ n = n − σ T Tt l t + τ t
R
(21)
where τ is a constant tangential stress caused by gas
flow.
Making use of the scalar products of (21) by unit
vectors of tangent plane and of outward normal to the
surface, we obtain boundary conditions for the normal
and tangential stresses on the free surface of the liquid:
p − pa = −
σ hxx
(1 + h )
2 32
x
+
(
2 µ vly − hx ( u ly + vxl ) + hx2 u xl
)
1+ h
2
x
(
µ 2hx ( vly − u xl ) + (1 − hx2 )( vxl + u ly )
) +σ T
T
1+ h
2
x
t
l
−τ = 0
(22)
(23)
G Txl ( x, h( x)) + Tyl ( x, h( x))hx
T ( x)
Tt l ( x, h( x)) = ∇T l , t =
= x
2
1 + hx
1 + hx2
(
)
where T ( x) = T l ( x, h( x)) .
In the problem (1)-(4), (9), (13), (14), (20), (18),
(22), (23) we change over to dimensionless variables
X , Y , U , V , h , Τ , θ , θ , θ a , P with the help
of the formulas:
X = x L , Y = y h0 , U = u l u , V = vl v , h = h h0 ,
Τ = τ τ 0 , θ = (T l − T0 ) ∆T , θ = (T l ( x, h( x)) − T0 ) ∆T ,
θ a = (Ta − T0 ) ∆T , P = ( p − pa ) p0 , τ 0 = µ u h0 ,
∆T = q0 L c p Γ
,
p0 = µ u ε h0
,
ε = h0 L
,
Thermal Science & Engineering Vol. 12, No.1 (2004)
h
εσ ∆T
Ma = T
µu
v = εu ,
1 0
Marangoni number, u = ∫ u l ( y )dy ,
h0 0
C=
uµ
ε −3
σ0
C = O(1) when ε → 0 .
analog of capillar number,
∞
When the velocity profile in the film is linear the average velocity in film is expressed with the help of the
tangential stress u = τ h0 2 µ , and dimensionless tangential stress is Τ = τ / τ 0 = 2 . Physically, such a situation
can occur in one of two ways. First, if the channel is
oriented normal to any acting body force; for example, if
channel is placed horizontally in a gravitational field.
The second situation involves the complete absence of a
body force as would occur in a nonrotating space vehicle
(microgravity condition).
Ignoring the terms of the order ε 2 and higher the
system of equations (1)-(4), (9), (13), (14), (20), (18),
(22), (23) in the terms of dimensionless variables becomes:
0 = εU YY − ε PX +
0 = − PY −
−
λ
cpΓ
ρ gh0
ρ gh0
p0
p0
sin ϕ
(24)
cos ϕ
(25)
U X + VY = 0
(26)
ε Pe (Uθ X + V θ Y ) = θ YY
(27)
U ( X , 0) = V ( X , 0) = 0
(28)
θ Y ( X , 0) = ε ( χ ( X ) − χ ( X − 1) )
P = −C −1 h XX
(
θY ( X , h( X )) = − Bi θ ( X , h( X )) − θ a
θ (−∞, Y ) = 0
V ( X , h( X ))
U ( X , h( X ))
(29)
θ ( X , Y ) = θ 0 ( X , Y ) + ∑ Giψ i (Y ) exp(−ξ i2 X )
i =1
(32)
(33)
(34)
where
⎛
⎝
θ 0 ( X , Y ) = ⎜1 +
Temperature distribution
We consider a heat transfer problem for rigid liquid
film, i.e., h = h0 = const , when the velocity profile in
the film is linear and v = 0 . The existence of a local
heat source with the constant heat flux density on the
bottom wall of the channel is supposed by condition
(13).
1
⎞ cpΓ
− Y ⎟ε
η( X ) + θ a
Bi
⎠ λ
is a solution of the following problem:
θ YY0 ( X , Y ) = 0
−
λ
cpΓ
(36)
θ Y0 ( X , 0) = εη ( X )
(37)
θY0 ( X , 0) = − Bi (θ 0 ( X ,1) − θ a )
(38)
3
2
2ε Peξi Y 2 ) , ξi , i = 1,..., ∞
−
3 3
are eigenfunctions and eigenvalues of Sturm-Liouville’s
problem:
and ψ i (Y ) = Y J 1 (
d 2ψ i (Y )
+ ξ i2ε Pe 2Yψ i (Y ) = 0
dY 2
(39)
dψ i
(0) = 0
dY
(40)
dψ i
(1) + Biψ i (1) = 0
dY
(41)
Jν (ξ ) is the Bessel function of the first kind, coefficients Gi are defined by formula:
Gi =
2ξi
⎛ 2 ⎞
⎛
⎞
∂ψi
⎜⎜ ∂ ψi ⎟⎟
⎜⎜ ∂ψi (1)⎟⎟
(1)
(1)
ψ
−
i
⎜⎝⎜ ∂Y ∂ξ ⎠⎟⎟
⎜⎝ ∂ξ
⎠⎟ξ =ξ ∂Y
ξ =ξ
i
3
ξi2
(35)
(31)
)
1
X
⎡
cpΓ
⎛
⎞⎤
∂η ( X )
θ
ψ
ε
ψ
η
−
+
Bi
dX ⎟ ⎥
(1)
(0)
(0)
exp(ξ i2 X )
⎢ a i
⎜
i
∫
λ
∂X
0
⎝
⎠ ⎦⎥
⎣⎢
(30)
U Y + Maθ X − Τ = 0
hX =
Let us denote η ( X ) = χ ( X ) − χ ( X − 1) .
We assume that the thin layer approximation, i.e.,
ε = h0 L 1 , is valid. Using the method described in
[9] we find the solution of the dimensionless problem
(27), (29), (32), (33) for V = 0 :
.
i
Let us notice that the solution (35) takes place for
any functions η , having an integrable distributional
derivative.
cient on the film surface from 1.02 W/m2K to 203.1
W/m2K. The picture shows qualitatively a possible effect
of evaporation of the liquid.
1.2
Re=1,
h=0.17 mm
Re=5,
h=0.35 mm
Re=10,
h=0.47 mm
(T-T0)/∆T
1
0.8
0.6
4 The solution of linearized equation for film
thickness
0.4
0.2
0
-1
Fig. 2
0
1
x/L
2
3
The temperature distribution on the film surface,
FC-72, Ta=T0=17o C, L=6.7 mm, ϕ=0o, α=20
W/Km2, Reg=242,8 the heater is situated from 0
to 1 along Ox-axis.
1.2
(T-T0 )/∆T
0.8
0.6
h( X )
∫
U ( X , Y )dY = 0
(
−1
εU YY = −ε C h XXX + ε
0
0
)
Using last expression the equation (24) can be written in the following form
0.2
-1
(42)
0
P = −C −1 h XX + ρ gh0 h − Y cos ϕ p0
0.4
Fig. 3
∂
∂X
The pressure out of (25) and boundary condition
(30) expressed as
Bi=0.005
Bi=0.05
Bi=0.1
Bi=1
1
The non-zero temperature gradient on the film surface
leads to appearance of the thermocapillary effect. A tangential stress on the film surface deforms the film by
reason of Marangoni effect. If the temperature distribution on the liquid surface is known, for instance out of
analytical solution (35) or experiment, one can find a
value of the deformations.
The equation of continuity (26) and kinematic condition (34) give
1
x/L
2
3
Biot numbers effect on temperature distribution
on the film surface, FC-72, Ta=T0=30o C, L=10
mm, ϕ=0o, Re=4.9, τ=0.025 kg/s2m.
p0
cos ϕ h X −
ρ gh0
p0
sin ϕ
(43)
The integration of equation (43) gives the equation
for film thickness
3
⎞
ρ gh0
ρ gh0
h ⎛ −1
cos ϕ h X +
sin ϕ ⎟ +
⎜ ε C h XXX − ε
3⎝
p0
p0
⎠
2
+
The calculated temperature distribution on the film
surface of FC-72 for different liquid flowrates is presented in Fig. 2. The liquid and size of the heater have
been chosen analogously to the works [4], [10], where
the deformation and the interface temperature of locally
heated liquid film falling down under the action of gravity have been studied theoretically and experimentally.
There is a practically linear increase of temperature in
the heater area. The influence of convective heat transfer
mechanism is becoming stronger with increasing Re
number. The length of the thermal boundary layer also
increases with increasing Re number. The temperature
reduction beyond the heater is explained by convective
heat exchange between surface of the film and gas flow.
The reduction of temperature is more noticeable at
smaller Re numbers.
The influence of heat exchange between surface of
the film and gas flow on temperature of the liquid is
shown in Fig. 3. A variation of Biot number in the range
0.005-1 corresponds to variation of heat-transfer coeffi-
ρ gh0
(
(44)
)
h
ε Τ − Maθ X = εγ
2
where γ is dimensionless flow rate.
Let h( X ) = 1 + h1 ( X ) , where h1 << 1 . Linearizing
equation (44) we obtain
⎛
⎞
ρ gh0
cos ϕ h1 X + 3C ⎜ Τ − Maθ X +
sin ϕ ⎟ h1 =
p0
p0ε
⎝
⎠
ρ gh0
3
C sin ϕ − C Τ − Maθ X
= 3Cγ −
p0ε
2
h1 XXX − C
ρ gh0
(
)
(45)
Neglecting the component εh1 Maθ X =O(ε3) in
left-hand member, i. e. assuming the thermocapillary
tangential stress is much less than tangential stress
caused by gas flow (Ma<<1), we come to the following
linearized equation
Thermal Science & Engineering Vol. 12, No.1 (2004)
h1 XXX − C
ρ gh0
⎛
⎞
ρ gh0
cos ϕ h1 X + 3C ⎜ Τ +
sin ϕ ⎟ h1 = f ( X )
ε p0
⎝
⎠
(46)
p0
(
when D > 0
F(X ) =
c
X
⎛
− cX
⎜ χ ( X )e + χ ( − X )e 2
⎝
)
Here f ( X ) = 3C γ − Τ 2 + Maθ X 2 − ρ gh0 C sin ϕ ε p0 .
The thermocapillary force ought not be neglected in
right-hand member since this member has a less order
ε Maθ X /2=O(ε ).
4
9c + 4b 2
2
F(X ) =
4
9c 2
c
X
⎛
− cX
⎜ χ ( X )e + χ ( − X )e 2
⎝
F(X ) =
The solution of equation (46) is presented as convolution of function F ( X ) and f [11]:
∞
∫
⎛ 3c ⎞ ⎞
⎜1 − X ⎟ ⎟
2 ⎠⎠
⎝
F ( X − ξ ) f (ξ )dξ
+
(47)
Here F ( X ) is the fundamental solution of operator
χ (− X sgn s1 )e s X
1
( s1 − s2 )( s1 − s3 )
χ (− X sgn s2 )e s X
2
( s2 − s1 )( s2 − s3 )
+
χ (− X sgn s3 )e s X
( s3 − s1 )( s3 − s2 )
2
⎛ ρg
⎞ ⎛ 3 ⎛τ
⎞⎞
−⎜
cos ϕ ⎟ + ⎜
⎜ + ρ g sin ϕ ⎟ ⎟⎟ > 0 . Let us
⎜
⎝ 3σ 0
⎠ ⎝ 2σ 0 h0 ⎝ h0
⎠⎠
3
⎛
⎞
ρ gh0
ρ gh0
d
d3
d
−C
+ 3C ⎜ Τ +
) :=
cos ϕ
sin ϕ ⎟
3
ε p0
dX
p0
dX
dX
⎝
⎠
denote lσ =
The characteristic polynomial of the operator Α is
condition D > 0 equivalents to the condition
ρ gh0
p0
s1 = −c , s2,3 =
⎛
⎞
ρ gh0
cos ϕ s + 3C ⎜ Τ +
sin ϕ ⎟ = 0
ε p0
⎝
⎠
c
± ib are the roots of the polynomial,
2
where
c=
+3
b=
3
⎞
ρ gh0
3 ⎛
C ⎜Τ +
sin ϕ ⎟ − D +
2 ⎝
ε p0
⎠
⎞
ρ gh0
3 ⎛
C⎜Τ+
sin ϕ ⎟ + D
2 ⎝
ε p0
⎠
⎞
ρ gh0
3⎛
3 ⎛
⎜ 3 − C ⎜Τ +
sin ϕ ⎟ + D −
ε p0
2 ⎜
2 ⎝
⎠
⎝
⎞
⎞
ρ gh0
3 ⎛
−3 − C ⎜Τ +
sin ϕ ⎟ − D ⎟
⎟
ε p0
2 ⎝
⎠
⎠
⎛ ρ gh0 C
⎞ ⎛3 ⎛
⎞⎞
ρ gh0
D = −⎜
cos ϕ ⎟ + ⎜⎜ C ⎜ Τ +
sin ϕ ⎟ ⎟⎟
ε p0
⎝ 3 p0
⎠ ⎝2 ⎝
⎠⎠
3
σ0
capillary length. We obtain, that the
ρg
τ + ρ gh0 sin ϕ > ρ gh0
2 h0
3 lσ
⎛ cos ϕ ⎞
⎜
⎟
⎝ 3 ⎠
32
(51)
In this case the damped perturbations of free surface
before the bump up to flow appear. If the film moving
forces overbalance the hydrostatic forces, the damped
perturbations of free surface exist. The perturbations
always exist when angle of channel inclination is high
(ϕ≈90°). The capillary perturbations are more appreciably under small Reynolds numbers (Re=0.04) and relatively intense influence of gas flow (Fig. 4). The discriminant of characteristic polynomial is positive at the
absence of gravity, and fundamental solution of equation
(46) has the form of (48). In this case the damped perturbations of free surface take place at any τ >0, but they
can be inconspicuous (Fig. 5).
0.003
0.002
2
If the discriminant of the characteristic polynomial
D > 0 , then it has one real and two complex conjugated
roots; if D = 0 , then all three roots are real, and two of
its are congruent; if D < 0 , then all three roots are real
and different. The fundamental solution has the following form [12], [3]:
F(x/L)
s3 − C
(50)
3
Let us analyze the condition D > 0 . We have
−∞
Α(
(49)
when D < 0
h1 XXX − C ρ gh0 h1 X p0 + 3C Τh1 = 3CMaθ X 2 .
h1 ( X ) =
(48)
when D = 0
2
For horizontally located channel γ = 1 and Τ = 2 ,
and equation (46) takes form
3c
⎛
⎞⎞
⎜ cos(bX ) − sin(bX ) ⎟ ⎟
2b
⎝
⎠⎠
0.001
0
-0.001
-0.002
-4
-2
0
2
x/L
Fig. 4
Fundamental solution for FC-72, T0=17o C,
Re=0.04, ϕ=5o, τ=0.024 kg/s2m.
1.8
1.6
h/h0
1.4
1.2
1
0.8
0.6
-2
0
2
x/L
4
6
Fig. 5
Film deformation for FC-72, Ta=T0=17o C,
Re=1, L=6.7 mm, α =100 W/m2K , g=0, q=1000 W/m2,
h0=0.00017 m, τ=0.024 kg/s2m .
1.6
α, W/m2K
α=20
α=100
α=345
h/h0
1.4
1.2
4
1
0.8
-2
2
4
6
x/L
Film thermocapillary deformation for different
heat-transfer coefficient: FC-72, Ta=T0=17o C,
L=6.7 mm, Re=1, q=1000 W/m2, ϕ=0o,
h0=0.00017 m, τ=0.024 kg/s2m.
Fig. 6
0
1.6
τ, kg/s2m
τ=0.024
τ=0.012
τ=0.0028
h/h0
1.4
1.2
0.8
Fig. 7
0
2
x/L
4
Conclusions
It is shown that the temperature on the film surface
reaches the maximum beyond the border of the heater.
The position of the maximum of temperature is replacing
downstream with increasing Re number. When Biot
number is growing up the maximum temperature on the
film surface is going down.
The formation of a liquid bump caused by the thermocapillary effect in the region where thermal boundary
layer reaches the film surface is obtained. It may exist
damped oscillations of free surface before the bump up
to flow according to the obtained solution. It depends on
the forces balance in the film.
References
1
-2
channel for different heat-transfer coefficients and for
different tangential stresses are presented in Fig. 6, 7.
The positive temperature gradient in the heater area
causes the thermocapillary tangential stress directed toward to the main flow and therefore the increasing of the
film thickness is observed in the heating area. The free
surface temperature is decreased outside of the heater
(Fig. 2,3) along the flow. The thermocapillary force is
directed streamwise, therefore the decreasing of the film
thickness relatively to the initial film thickness is observed. The minimum of film thickness occurs under
maximum of heat-transfer coefficients. The present region is most risky for a film breakdown. The damped
upstream perturbations of free surface are missing in Fig.
6, 7 since the condition (51) is not satisfied.
Under constant Re number with tangential stress increasing, that correspond to gas flow rate increasing, the
initial film thickness is decreasing that enhance the thermocapillary effect. The calculation predicts the formation of thermocapillary thickening up to 30-50% compare with the initial film thickness at low heat transfer to
the gas phase and relatively high heat flux density on the
heater. A thermocapillary bump of analogous size of the
order of magnitude has been observed in experiments [4],
[10] at local heating of falling liquid film under gravity
along a vertical plate.
6
Film thermocapillary deformation for different
tangential stresses: FC-72, Ta=T0=17o C, L=6.7
mm, Re=1, q=1000 W/m2, ϕ=0o, α =20
W/m2K.
The calculated relative film thicknesses along the
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