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Thermocapillary convection in a liquid layer resting on a nonisothermal plate
Adv. Space Res. Vol. 11, No. 7, pp. (7)125—(7)128 1991 Printed inGreat Britain. All rights reserved.
0273—1177/91 $0.00 + .50 Copyright © 1991 COSPAR
THERMOCAPILLARY CONVECTION IN A LIQUID LAYER RESTING ON A NONISOTHERMAL PLATE J. M. Floryan and S. Krol Department of Mechanical Engineering, University of Western Ontario, London, Ontario, Canada, N6A 5B9
Abstract Thermocapillary convection in a thin liquid layer resting on a nonisothermal plate and in the absence of gravity has been analyzed. It has been demonstrated that steady convection and interface deformation occur only if the temperature field satisfies restrictive constraint conditions. When these conditions are satisfied, the topology of the flow can be represented in terms of a superposition of toroidal rolls wrapped around the local interface temperature minima. The layer acts as a filter that eliminates high and small wave number components from the convection field. There exist a finite wave number that leads to the most intensive convection. 1. Introduction We consider dynamics of a thin liquid layer resting on a nonisothermal plate in the absence of gravity. The flow is driven by thermocapillary effect induced by temperature gradient vector parallel to the interface. This problem can be viewed as an extension of the classical Benard /1 / and Marangoni /2/ problems, where the temperature gradient vector acts in the direction normal to the interface. 2. Problem Formulation Consider a liquid layer of thickness infinite in the x and y directions, bounded by a solid plate of known temperature T~(x,y)from below and a gas of known temperature T9 = const from above, as shown in Fig. 1. The liquid is incompressible, Newtonian, has density p, thermal conductivity k, specific heat per unit mass c, thermal diffusivity ,~= k/pc, kinematic viscosity v and dynamic viscosity ji. The free surface, described by z h(x,y), is bounded by a passive gas of negligible density and viscosity. This free surface is associated with surface tension a, which depends on the local temperature. ~,
The steady motion of the liquid is governed by the Navier-Stokes, continuity and energy equations subject to the following boundary conditions: z=O:u=v=w=o,T=t~(x,y) z
=
h(x,y):
w
S~n1 t~ a~ (id),
=
(la)
uh~+ vh~ (ib), S1, n1 t~1= ~
s
~n1 n1
(le)
,
=
2 oA, kT0
(ic) +
H (T T9) -
=
0. (if)
In the above, u,v,w are components of the velocity vector in the x, y, z directions, respectively, p is the pressure of the liquid, I is the temperature of the liquid, the subscripts x, y, z denote partial derivatives 8/dx, d/ay, 8/3z, respectively, and the subscripts n, sx, sy denote normal, tangential in the (x,z) -plane and tangential in the (y,z) -plane derivatives at the interface, respectively. Equation (1 b) is the kinematic condition at the liquid-gas interface. The stress balances at the interface in the normal direction and tangential in the x-direction and tangential in the y-direction are given by Eqs. (ic), (id) and (ie), respectively. In these equations, the S11 are the components of the stress tensor of the liquid. The unit outward normal vector n, the unit tangent vector in the (x,z) -plane t)t and the unit tangent vector in the (y,z) -plane t~are defined as follows: n=(-h5i-h~j+~)/N, tx=(i+hx~)INX, tY=(j+h~l~)/N~ (2) 2 2~)”2, Nx = (1 + h2 1~, N1’ = (1 + h2~)’~, (3) with N = (1 ÷h 5 + h 5) and i, j, k being the unit vectors in the x, y, z directions, respectively. The mean curvature A of the interface in Eq. (1 c) has the definition (7)125
(7)126
J. M. Flotyan and S. Krol
(4) h~2) 2h5h1’h,~,÷h~(1+ h~2)]N4. The thermal boundary condition at the interface is given by Eq. (if) in which k is the thermal conductivity of liquid and H is the heat transfer coefficient in the gas. The thermal boundary condition at the wall is given by (1 a), where we represent I,,~(x,y) as A
=
-1/2 nil
=
t~(x,y)
1/2 [h,~(1
+
T~(x,y)
+
=
-
B
(5)
where B is a constant and T~(x,y)is either periodic or decays to zero as x,y ± Apart from boundary conditions (1), the liquid must also satisfy the mass conservation constraint, ie. ~.
V
const.
=
(6)
where V denotes total volume of the liquid. We shall use a linear equation of state for surface tension, ie. a(T) = ~i’(T T~) (7) -
-
where 00 is the surface tension of the liquid at the reference temperature T~and the constant ‘y is the negative of the derivative of the surface tension with respect to temperature. We scale the problem using thickness of the layer E as a length scale, u~= ~ (Tm~ Tmin) / ~i as a velocity scale, ~u.I ~ as a pressure scale and T T~= (Tm,~, Tmin) T’ as the temperature definition. In the above, T’ denotes dimensionless temperature and Tm~and Tmjn stand for a measure of a maximum and a minimum of the temperature of the wall, respectively. We assume for simplicity that B= T 9 = I~.Presentation of all results will be done in terms of dimensionless quantities. -
-
3. Results We consider small deformation of the interface which corresponds to small amplitude of surface tension variation. Because the liquid layer is assumed to be thin, convective effects in momentum and energy transport equations become negligible. The resulting problem is completely linearized and can be solved using Fourier transforms. The results are +~ +10
I(x,y,z)
=~—
J J J ~
[cos~z~- ~
+Lcosh(k~sinh(kz)] T~(k1,k2) e -i(k1 x + k2y) dk1dk2
J J
+10+10
u(x,y,z)
=
~-
+10+10
k~f(k;z) r(k1,k2) dk1dk2
(8b),
v(x,y,z)
=
~-
+10+10
w(x,y,z)
=
_i_
J J
(8a),
k~f(k;z)
r(k1,k2)dk1dk2
(Sc),
+10+10
g(k;z) r(k1,k2) dk1dk2
J J
(Sd)
p(x,y,z)
=
- ~-
J J
s(k;z) r(k1,k2)dk1dk2 (Se),
+10 +10
h(x,y)
=
-
2~r
sinh(2k) 2k r(k,,k2) dk1dk2
where 1{( 1/2 f(k;z) = k
+
g(k;z)
1/2 k(z-1)] sinh[k(z-1)]
=
(e2(k)
-
-
k ~,(k) (2-1)] sinh(k(z-1)]
-
(Sf)
[e
2(k) ~,(k) 1/2 k(z-1)] cosh[k(z-1)]}, -
-
-
k e,(k) (z-i) cosh(k(z-1)],
k sinh[k(z-1)] + 2 k ç,(k) cosh[k(z-1)], r(k1,k2) = e ~Qc, X + kEY) I (k,,k21), 2(k) [sinh(2k) 2ky’, e 2 [sinh(2k) 2k]-’. ~ (k) = sinh 2(k) = k In the above T(k~,k 2 + k~)/2, 2Lz) =andHf/k T~(k,,k2) denotes denote Biot Fourier numbertransforms and the location of T (x,y,z)of and the T~(x,y), interfacerespectively, has been k = (k, represented as h(x,y) = 1 + F~(x,y).Application of the volumetric constraint condition (6) leads to the following constraint condition on the acceptable wall temperature distribution s(k;z)
=
-
-
Therinocapillary Convection in a Liquid Plate
(7)127
+10+10
JJ
sinh(2k) 2k T(k,,k21) 6(k,) 8(k~Jdk1dk2=0 -
(9) where 8 denotes Dirac del~afunction. The above satisfied only if T~ (x,y) is either periodic 2) when k 0. condition The latterisimplies that in at least one direction or T~= o(k -,
+10
JJ
+10
G(x,y) T~(x,y)dxdy
=
0 (10)
“10-10
= 1, x, y, x2, xy and y2. This demonstrates that the steady interface deformation, and therefore steady convection, cannot exist for arbitrary temperature distributions.
for G(x,y)
To illustrate the character of the convection when the existence condition (10)is satisfied, we consider temperature of the wall to be in the form T~(x,y)= T~sin(ax) sin(fty)
(11)
This particular temperature distribution can be viewed as representing a typical Fourier mode. The temperature, velocity and deformation fields are expressed in the following form: T (x,y,z) =T 0 [cosh(Az)
-
u(x,y,z)
=
-
A
T0 a f(A;z) cos(ax) sin(fiy),
w(x,y,z) =T0 g(A;z) sin(ax) sin($y), l~(x,y,z)
=
()sinh(Az)] v(x,y,z)
=
p(x,y,z)
=
-
-
sin(ax) sin(py),
T.~$ f(A;z) sin(ax) cos(fiy), T0 s(A;z) sin(ax) sin(.$y),
(12)
-2AT0 [sinh(2A)-2A]’sin(ax) sin(fty)
2 + $2) where T0 = T0 A [Acosh(A) + L sinh (A)]”, A = (a The qualitative character of the temperature, flow and deformation fields Is illustrated in Figs. 2,3 and 4 for a = ~r,fi = 2~r,L = 1 and T 0 selected in such a way that It makes = 1. It can be seen that the liquid is drawn towards surface cold spots and away from surface hot spots. The flow field consists of toroidal rolls wrapped around axis passing through surface cold spots. The dynamic pressure field associated with the convection bulges Interface out around the cold spots ~.
and in around the hot spots.
Analysis of solution (12) reveals that the strength of convection decreases when A 0 and A -i.. In the former case a decrease of A leads to a reduced temperature gradient along the Interface and thus a less intensive convection. While an increase of A increases the temperature gradient in the latter case, there is an associated drop in the temperature amplitude due to a very rapid drop of the temperature level across the layer. Both processes are illustrated in Fig. 5 which displays variations of the amplitude of the interface velocity vector V~as a function of A for a = $, L = 1, T0 = 1. This amplitude is defined as 2(A) A2] [sinh(2A)-2A]~’ [Acosh(A) + L sinh(A)]~’ (13) V, = 21~A[sinh It can be shown analytically that V, = 0(A) when A 0, and V~= 0 (eL) when A -‘., ie. the layer acts -~
-
as a filter that eliminates both high and low wavenumber components from the convection field. Acknowledaements This work has been supported by the Natural Sciences and Engineering Research Council of Canada. References 1. Chandrasekhar, S., Hvdrodvnamic end Hvdromaanetic Instability. Clarendon Press, Oxford, 1961. 2. Pearson, J.R.A., “On Convection Cells Induced by Surface Tension”, J. Fluid Mach., #4, 489, 1958.
J. M. Floryanand S. Krol
(7)128
Gas
T
=const
K
M
_
Figure 1. Sketch of a liquid layer with interface deformed by thermocapillary effects.
Figure 2. Temperature distribution corresponding to a typical Fourier mode (Eq.11) In the wall temperature distribution I (x,y). Here, ~x-O.5, ~=y-O.25,a=f ,p=2Y a)~d plot extends to ~2.O and y=1.O.
~
Figure 3. Figure 4. Topology of the flow field induced by the tern- Surface deformation induced by flow and tempe— perature field displayed in Fig.2. Dash lines rature fields displayed in Figs. 2 and 3. identify locations where the x—component of the surface stress is zero, while the dash—dot lines identify locations where the y-component of the surface stress is zero. Intersections of the dash and dash-dot lines identify locations of the nodal and saddle (stagnation) points.
Figure 5. Distribution of the amplitude V of the surface velocity vector as a function of A
0273—1177/91 $0.00 + .50 Copyright © 1991 COSPAR
THERMOCAPILLARY CONVECTION IN A LIQUID LAYER RESTING ON A NONISOTHERMAL PLATE J. M. Floryan and S. Krol Department of Mechanical Engineering, University of Western Ontario, London, Ontario, Canada, N6A 5B9
Abstract Thermocapillary convection in a thin liquid layer resting on a nonisothermal plate and in the absence of gravity has been analyzed. It has been demonstrated that steady convection and interface deformation occur only if the temperature field satisfies restrictive constraint conditions. When these conditions are satisfied, the topology of the flow can be represented in terms of a superposition of toroidal rolls wrapped around the local interface temperature minima. The layer acts as a filter that eliminates high and small wave number components from the convection field. There exist a finite wave number that leads to the most intensive convection. 1. Introduction We consider dynamics of a thin liquid layer resting on a nonisothermal plate in the absence of gravity. The flow is driven by thermocapillary effect induced by temperature gradient vector parallel to the interface. This problem can be viewed as an extension of the classical Benard /1 / and Marangoni /2/ problems, where the temperature gradient vector acts in the direction normal to the interface. 2. Problem Formulation Consider a liquid layer of thickness infinite in the x and y directions, bounded by a solid plate of known temperature T~(x,y)from below and a gas of known temperature T9 = const from above, as shown in Fig. 1. The liquid is incompressible, Newtonian, has density p, thermal conductivity k, specific heat per unit mass c, thermal diffusivity ,~= k/pc, kinematic viscosity v and dynamic viscosity ji. The free surface, described by z h(x,y), is bounded by a passive gas of negligible density and viscosity. This free surface is associated with surface tension a, which depends on the local temperature. ~,
The steady motion of the liquid is governed by the Navier-Stokes, continuity and energy equations subject to the following boundary conditions: z=O:u=v=w=o,T=t~(x,y) z
=
h(x,y):
w
S~n1 t~ a~ (id),
=
(la)
uh~+ vh~ (ib), S1, n1 t~1= ~
s
~n1 n1
(le)
,
=
2 oA, kT0
(ic) +
H (T T9) -
=
0. (if)
In the above, u,v,w are components of the velocity vector in the x, y, z directions, respectively, p is the pressure of the liquid, I is the temperature of the liquid, the subscripts x, y, z denote partial derivatives 8/dx, d/ay, 8/3z, respectively, and the subscripts n, sx, sy denote normal, tangential in the (x,z) -plane and tangential in the (y,z) -plane derivatives at the interface, respectively. Equation (1 b) is the kinematic condition at the liquid-gas interface. The stress balances at the interface in the normal direction and tangential in the x-direction and tangential in the y-direction are given by Eqs. (ic), (id) and (ie), respectively. In these equations, the S11 are the components of the stress tensor of the liquid. The unit outward normal vector n, the unit tangent vector in the (x,z) -plane t)t and the unit tangent vector in the (y,z) -plane t~are defined as follows: n=(-h5i-h~j+~)/N, tx=(i+hx~)INX, tY=(j+h~l~)/N~ (2) 2 2~)”2, Nx = (1 + h2 1~, N1’ = (1 + h2~)’~, (3) with N = (1 ÷h 5 + h 5) and i, j, k being the unit vectors in the x, y, z directions, respectively. The mean curvature A of the interface in Eq. (1 c) has the definition (7)125
(7)126
J. M. Flotyan and S. Krol
(4) h~2) 2h5h1’h,~,÷h~(1+ h~2)]N4. The thermal boundary condition at the interface is given by Eq. (if) in which k is the thermal conductivity of liquid and H is the heat transfer coefficient in the gas. The thermal boundary condition at the wall is given by (1 a), where we represent I,,~(x,y) as A
=
-1/2 nil
=
t~(x,y)
1/2 [h,~(1
+
T~(x,y)
+
=
-
B
(5)
where B is a constant and T~(x,y)is either periodic or decays to zero as x,y ± Apart from boundary conditions (1), the liquid must also satisfy the mass conservation constraint, ie. ~.
V
const.
=
(6)
where V denotes total volume of the liquid. We shall use a linear equation of state for surface tension, ie. a(T) = ~i’(T T~) (7) -
-
where 00 is the surface tension of the liquid at the reference temperature T~and the constant ‘y is the negative of the derivative of the surface tension with respect to temperature. We scale the problem using thickness of the layer E as a length scale, u~= ~ (Tm~ Tmin) / ~i as a velocity scale, ~u.I ~ as a pressure scale and T T~= (Tm,~, Tmin) T’ as the temperature definition. In the above, T’ denotes dimensionless temperature and Tm~and Tmjn stand for a measure of a maximum and a minimum of the temperature of the wall, respectively. We assume for simplicity that B= T 9 = I~.Presentation of all results will be done in terms of dimensionless quantities. -
-
3. Results We consider small deformation of the interface which corresponds to small amplitude of surface tension variation. Because the liquid layer is assumed to be thin, convective effects in momentum and energy transport equations become negligible. The resulting problem is completely linearized and can be solved using Fourier transforms. The results are +~ +10
I(x,y,z)
=~—
J J J ~
[cos~z~- ~
+Lcosh(k~sinh(kz)] T~(k1,k2) e -i(k1 x + k2y) dk1dk2
J J
+10+10
u(x,y,z)
=
~-
+10+10
k~f(k;z) r(k1,k2) dk1dk2
(8b),
v(x,y,z)
=
~-
+10+10
w(x,y,z)
=
_i_
J J
(8a),
k~f(k;z)
r(k1,k2)dk1dk2
(Sc),
+10+10
g(k;z) r(k1,k2) dk1dk2
J J
(Sd)
p(x,y,z)
=
- ~-
J J
s(k;z) r(k1,k2)dk1dk2 (Se),
+10 +10
h(x,y)
=
-
2~r
sinh(2k) 2k r(k,,k2) dk1dk2
where 1{( 1/2 f(k;z) = k
+
g(k;z)
1/2 k(z-1)] sinh[k(z-1)]
=
(e2(k)
-
-
k ~,(k) (2-1)] sinh(k(z-1)]
-
(Sf)
[e
2(k) ~,(k) 1/2 k(z-1)] cosh[k(z-1)]}, -
-
-
k e,(k) (z-i) cosh(k(z-1)],
k sinh[k(z-1)] + 2 k ç,(k) cosh[k(z-1)], r(k1,k2) = e ~Qc, X + kEY) I (k,,k21), 2(k) [sinh(2k) 2ky’, e 2 [sinh(2k) 2k]-’. ~ (k) = sinh 2(k) = k In the above T(k~,k 2 + k~)/2, 2Lz) =andHf/k T~(k,,k2) denotes denote Biot Fourier numbertransforms and the location of T (x,y,z)of and the T~(x,y), interfacerespectively, has been k = (k, represented as h(x,y) = 1 + F~(x,y).Application of the volumetric constraint condition (6) leads to the following constraint condition on the acceptable wall temperature distribution s(k;z)
=
-
-
Therinocapillary Convection in a Liquid Plate
(7)127
+10+10
JJ
sinh(2k) 2k T(k,,k21) 6(k,) 8(k~Jdk1dk2=0 -
(9) where 8 denotes Dirac del~afunction. The above satisfied only if T~ (x,y) is either periodic 2) when k 0. condition The latterisimplies that in at least one direction or T~= o(k -,
+10
JJ
+10
G(x,y) T~(x,y)dxdy
=
0 (10)
“10-10
= 1, x, y, x2, xy and y2. This demonstrates that the steady interface deformation, and therefore steady convection, cannot exist for arbitrary temperature distributions.
for G(x,y)
To illustrate the character of the convection when the existence condition (10)is satisfied, we consider temperature of the wall to be in the form T~(x,y)= T~sin(ax) sin(fty)
(11)
This particular temperature distribution can be viewed as representing a typical Fourier mode. The temperature, velocity and deformation fields are expressed in the following form: T (x,y,z) =T 0 [cosh(Az)
-
u(x,y,z)
=
-
A
T0 a f(A;z) cos(ax) sin(fiy),
w(x,y,z) =T0 g(A;z) sin(ax) sin($y), l~(x,y,z)
=
()sinh(Az)] v(x,y,z)
=
p(x,y,z)
=
-
-
sin(ax) sin(py),
T.~$ f(A;z) sin(ax) cos(fiy), T0 s(A;z) sin(ax) sin(.$y),
(12)
-2AT0 [sinh(2A)-2A]’sin(ax) sin(fty)
2 + $2) where T0 = T0 A [Acosh(A) + L sinh (A)]”, A = (a The qualitative character of the temperature, flow and deformation fields Is illustrated in Figs. 2,3 and 4 for a = ~r,fi = 2~r,L = 1 and T 0 selected in such a way that It makes = 1. It can be seen that the liquid is drawn towards surface cold spots and away from surface hot spots. The flow field consists of toroidal rolls wrapped around axis passing through surface cold spots. The dynamic pressure field associated with the convection bulges Interface out around the cold spots ~.
and in around the hot spots.
Analysis of solution (12) reveals that the strength of convection decreases when A 0 and A -i.. In the former case a decrease of A leads to a reduced temperature gradient along the Interface and thus a less intensive convection. While an increase of A increases the temperature gradient in the latter case, there is an associated drop in the temperature amplitude due to a very rapid drop of the temperature level across the layer. Both processes are illustrated in Fig. 5 which displays variations of the amplitude of the interface velocity vector V~as a function of A for a = $, L = 1, T0 = 1. This amplitude is defined as 2(A) A2] [sinh(2A)-2A]~’ [Acosh(A) + L sinh(A)]~’ (13) V, = 21~A[sinh It can be shown analytically that V, = 0(A) when A 0, and V~= 0 (eL) when A -‘., ie. the layer acts -~
-
as a filter that eliminates both high and low wavenumber components from the convection field. Acknowledaements This work has been supported by the Natural Sciences and Engineering Research Council of Canada. References 1. Chandrasekhar, S., Hvdrodvnamic end Hvdromaanetic Instability. Clarendon Press, Oxford, 1961. 2. Pearson, J.R.A., “On Convection Cells Induced by Surface Tension”, J. Fluid Mach., #4, 489, 1958.
J. M. Floryanand S. Krol
(7)128
Gas
T
=const
K
M
_
Figure 1. Sketch of a liquid layer with interface deformed by thermocapillary effects.
Figure 2. Temperature distribution corresponding to a typical Fourier mode (Eq.11) In the wall temperature distribution I (x,y). Here, ~x-O.5, ~=y-O.25,a=f ,p=2Y a)~d plot extends to ~2.O and y=1.O.
~
Figure 3. Figure 4. Topology of the flow field induced by the tern- Surface deformation induced by flow and tempe— perature field displayed in Fig.2. Dash lines rature fields displayed in Figs. 2 and 3. identify locations where the x—component of the surface stress is zero, while the dash—dot lines identify locations where the y-component of the surface stress is zero. Intersections of the dash and dash-dot lines identify locations of the nodal and saddle (stagnation) points.
Figure 5. Distribution of the amplitude V of the surface velocity vector as a function of A