thermo capillary

Acta Mechanica 127, 209-224 (1998) ACTA MECHANICA �9 Springer-Verlag 1998

Thermocapillary flow in a liquid layer at minimum in surface tension

S. G. Slavtehev and S. P. Miladinova, Sofia, Bulgaria

(Received June 1, 1996; revised September 3, 1996)

Summary. In the present paper a class of similarity solutions for the two-dimensional Navier-Stokes and energy equations describing thermocapillary flows in a liquid layer of constant width and infinite extent is presented. The layer is bounded by a horizontal rigid plate from one side and opened to the ambient gas from the other one. The physical properties of the liquid are assumed to be constant except the surface tension which varies as a quadratic function with temperature. It is supposed that a constant temperature gradient exists along either the liquid free surface (case 1) or the rigid boundary (case II).

In both cases, by means of a similarity transformation, the equations of motion and energy are reduced to a system of three ordinary differential equations, one for the velocity and two for the temperature. The equation for the velocity can be solved separately from the other equations and its solution, found numerically, exists only for the Marangoni number less than a certain finite value. The solution of the whole system depends also on the Prandtl number. The solution of one of the temperature equations is presented in an analytical form and the other equation is solved numerically. Asymptotic formulas of the functions are also obtained for small and large Marangoni numbers. Flow pattern and temperature fields are presented. One convective roll exists in every semi-infinite layer. Fluid velocities at different points of the free surface are evaluated for an aqueous solution of n-heptanol and compared with those measured in the experiments.

1 Introduction

Most of the liquids have a surface tension that is a monotonically decreasing function of temperature presented usually as a linear one. When a temperature gradient is imposed along a liquid-

gas surface a corresponding surface tension gradient is induced and it causes motion not only at the surface, but, due to viscosity, in the fluid also. That motion is referred to thermocapillary

(Marangoni) convection. The interest of the scientists in studying the Marangoni convection arises from the possiblility of material processing in space crafts, where the gravity force is very small in comparison with the thermocapillary one.

Some dilute aqueous solutions of fatty alcohols, like n-heptanol, n-hexagonal, etc. [1], [2] have a surface tension which is not only decreasing but also increasing function with a minimum at

some temperature. The dependence on the temperature is approximated fairly well by a quadratic function.

The influence of the surface tension minimum on thermocapillary flows in such solutions has been investigated recently both experimentally and theoretically. Experiments with solution of long chain alcohols were conducted in microgravity conditions [3], [4] and on earth [3], [5], [6]. Thermocapillary flows of such solutions in cavities [7], [8], [9], [10] and thin layers [5], [11], [12], [13], [14], [15] were studied by means of numerical and analytical methods.

210 S.G. Slavtchev and S. R Miladinova

Some self-similar solutions of the governing equations for flows in a thin layer and a half space were reported [12], [14], [16]. They exist only if the free surface of the moving liquid is assumed to be flat. It is well known [18], that the effect of the surface deformation is important for Marangoni convection in very thin layers and small cavities, especially near their lateral rigid boundaries. Thus, the obtained solutions for layers of constant depth describe thermocapillary flows in domains situated far from the boundaries.

In [14] a thermocapillary flow in a layer of lateral infinite extent is studied in the case of constant temperature gradient along the rigid bottom of the layer, while the free boundary is assumed to be thermally insulated. The solution of the corresponding system of two ordinary differential equations for the velocity and the temperature is found in series in a small Marangoni number when the Prandtl number is of order of unity. Solutions for large Marangoni and Prandtl numbers are not presented.

In the present paper the problem of existence of similarity solutions of the two- dimensional equations of motion and energy for thermocapillary flows in thin layers of liquids presenting a minimum in the surface tension is considered. Solutions are obtained in the cases of imposed constant temperature gradient along either the free (zero-stress) boundary of the layer or the rigid one, while at the other boundary all possible thermal conditions are assumed. By applying a similarity transformation, the physical problem is reduced to a system of three ordinary differential equations, one for the velocity and two for the temperature. The velocity equation as well as one of the temperature equations are solved numerically, while the solution of the third one is found in an analytical form. It is proved numerically that a unique solution exists for the Marangoni number M within the interval [0,6349] and there is no any other solution for larger M (see definition of M below). The large M structure of the solutions consists of thin dynamic and thermal boundary layers adjacent to the free surface. The streamwise velocity and the temperature functions for various values of the Marangoni and Prandtl numbers are calculated. Flow pattern and temperature fields are presented. In every semi-infinite part of the layer one convective roll exists. The maximum values of the surface velocity evaluated for an aqueous solution of n-heptanol are smaller than those measured in the experiments.

2 Formulation of the problem

Consider a viscous liquid layer of infinite extent and thickness d lying on a horizontal plate and opened to the ambient motionless gas (see Fig. 1). The upper free surface is supposed to be uniform and non-deformable. The physical properties of the fluid are assumed constant, except the surface tension a depending on temperature Tby the expression:

a = a o + ~ ( T - T o ) 2, (1)

where ~, ao and To are positive constants. Function (1) has a minimum value ao at T= To and the influence of this minimum on flows induced in the layer at certain thermal conditions will be studied.

To formulate the corresponding two-dimensional problem Cartesian coordinates x, y are chosen with x as a streamwise coordinate and y measured from the plate. The layer is supposed to be under various thermal conditions. In one case (numbered as I) a constant temperature gradient A is imposed along the liquid free surface and the rigid boundary is considered as

Thermocapillary flow in a liquid layer

C&se I

Y T~-To+ Ax

il,,-IJl////lllllll/llllllllllllllltlllliil' X aT - - = 0 (la) or T=T~ (Ib) &

Fig. 1. Schematic diagram for the liquid layer

1 d

211

Case II

_k~ ay Y

H/J/1////// 1/1 / / / / ) / / / /1 / /Hl / / / I / / / / / / x

T = Two+ Ax

thermally insulated (case I a) or isothermic (case I b) (see Fig. 1). In other case (numbered as II) the temperature gradient is applied along the plate and the heat balance at the free boundary is assumed to obey Newton's law.

The equations of continuity, momentum and energy for an incompressible viscous fluid are respectively

(V.v) = 0, (2)

Q(v.V) u = - - [7p -~-/2[72 v q- 0 g , (3)

(v.VT) = KV2T. (4)

Here v(u, v) is the fluid velocity, p the pressure, # the dynamic viscosity, Q the density, K the thermal diffusivity and g(0, - g ) the gravitational vector.

The boundary conditions for the velocity are: (i) no-slip condition at the rigid plate

v = 0 , at y = 0 , (5)

(ii) kinematic condition for a non-deformable surface

v = 0 , at y = d (6)

(iii) stress balance at the free boundary

~u ~a ~T # ~y 3x 6(T-- To) -~x' at y = d. (7)

The boundary conditions for the temperature are as follows:

a) Case I

ST - - = 0 (Ia) or T = Tw (Ib), at y = 0 ; T= To +Ax , at y = d, (8) Oy

where A > 0 and Tw is the wall temperature assumed to be different from To.


b) Case II

0T T = T w o + A x , at y = 0 ; -k~yy=h(T- -Tg) , at y = d , (9)

where Two is the temperature of the plate at the origin of the coordinate system, T o the gas temperature, k the thermal conductivity and h the surface thermal conductance of the liquid.

The flow is to be symmetric about the axis y, but this property could be violated for the temperature field. Hence, it is necessary to consider both positive and negative x. We seek similarity solutions for the velocity and the temperature in the following form:

X y g)A 2 d 2 (~A 2 d 2 d ' )7 = ~, u ~f'(3~), v f(35), # #

P - - P g = b A 2 d [ ~22 + P ( ; ) l ' (10)

T= rlOl(y) + A d f~ 0(•),

where 2 is unknown constant, Po is the gas pressure, f, P, 01 and 0 are functions of/f only and the prime denotes differentiation with respect to the independent variable. The constant T1 is equal

to To in case I a, Tw in case I b and Two in case II. Substituting (10) in x-momentum equation of (3) leads to the equation

f ' " + M( f f " _f ,2 ) = )~, (11)

with boundary conditions

f(0) =f ' (0) = 0, f(1) = 0, f"(1) = 02(1), (12)

where M = o~A 2 d3/# 2 is the Marangoni number based on the second derivative of the surface tension with respect to the temperature. Equation (11) is third order with four boundary conditions. The problem is however not over-specified, because the fourth boundary condition enables the pressure coefficient 2 to be determined.

From y-momentum equation of (3) after integration one obtains the expression

P(3~) = - M f 2 ( f ) - i f ( y ) + i f ( l ) + Bd(1 - y), (13)

where Bd---0g/6A 2 is the modified dynamic Bond number standing for the ratio of the hydrostatic pressure to the thermocapillary force. It is negligibly small for liquid layers under

microgravity conditions. Substitution of expression (10) into energy equation (4) results in two equations

01" + M Pr f01' = 0, (14)

0" + M Pr(fO'- frO) = 0, (15)

where Pr = #/oK is the Prandtl number. The boundary conditions for the temperature become

as follows (i) Case I a

0,'(0) = O, if(O) = O, 01(1) = 1, 0(1) = 1; (16)

Thermocapillary flow in a liquid layer 213

(ii) Case Ib

01(0) = 1,

(iii) Case II

TO 8(0) = O, 81(1) = ~ , 8(1) = 1; (17)

81(0)-- 8(0)= 1, 81'(1) + Bi [ 8 a ( 1 ) - T~ool = 0, 8'(1) + Bi 8(1)= 0, (18)

where Bi = hd/k is the Biot number. In the last case the surface temperature Tat x = 0 is to be equal to the temperature To from (1). Hence, Two is assumed to be defined by the expression T~,o = To/01(1), which is necessary to keep the flow symmetric about the axis y.

In case I 0(1) = 1, the problem (11)-(12) can be solved separately from the equations (14)-(17). For case II the following transformation

f(Y) = a2f(Y), 81(y) = O l ( y ) ,

8(.9) = if(y), M = - - 2 = ,[a 2 , (19) a 2 '

where a = 0(1) is a non-zero constant, keeps the form of all equations and boundary conditions, except for the fourth condition (12) which becomes

f"(1) = 1, (20)

as in case I. Hence, in both cases it is possible first to solve Eq. (11) and then, to find the temperature field in the layer based on the solution of (14) and (15).

3 Similarity solutions of the problem

Provided that the function f(37) is known, it is easy to obtain the general solution of (14), namely

z 81(37) = C1 + C2go(37), go(Y) = S exp [ - M Pr q(z)] dz, q(z) = ~ f (y ) dy, (21)

0 0

where constants C1 and C 2 are to be determined from the appropriate boundary condition (16), (17) or (18). The function 81(37) is expressed by

1, (case Ia)

01(Y) = 1 + [ ~ - 1 1 go(Y)go(1)

1 - B i l l - ~o~ go(Y)

[ - M Pr q(1)] + B/go(l)'

(case Ib) . (22)

(case II)

It is constant in cases I a and II for Bi = 0 or T o = T,,o, but, as we shall see below, it varies in a different way in case I b and II when Bi is not zero.


The other Eqs. (11) and (15) are to be solved numerically. We apply first the t ransformation used in [17] for solving an equation identical to (11), namely

t / = M'/f , f(37) = M ~- t~b(t/), 0(.f) -= ~(t/), (23)

where y is yet unknown constant. This t ransformation allows to eliminate the Marangoni number in the equations and to arrive at the following problem

~b'" + ~bq~" - ~,2 = 2", (24)

q)(O) = r = O, ~(M') = O, 4~"(M') = M 1 -3 , , (25)

~ " + Pr (q~7 t' - ~b'~) = 0, (26)

tP(O) = 1, t/"(1) + Bi M-r iP( l ) = O, (27)

The Eq. (24) is derived for various physical problems concerning viscous flows. It is very similar to the Falkner-Skan equation describing a flow in the boundary layer on a rigid wedge and coincides with that studied first by Hiemenz (see, for example, [19]). The same equation is

solved for a flow in a channel with injection or suction through the channel porous walls [20], [21], [22] and in a channel with free accelerating boundaries [17]. The existence of multiple solutions for laminar flows is numerically proved.

Here, a solution of Eq. (24) satisfying the boundary conditions (25) which differ from those in the corresponding channel problems is obtained. It is solved as an initial-value problem for a given value/3 = ~"(0), with .~* as a parameter, until ~b becomes zero at some point t/o. That quanti ty and the value ~ = ~b"(t/o) serve to determine the unknown parameters

M = ~rlo 3 , ). = 2* r/~. (28)

As positive values of the Marangoni number are of interest, solutions of (24) passing the abscissa > 0 with positive second derivative at the crossing point are considered.

It is easy to show that any solution ~(y; 2*,/3) of the initial-value problem is invariant to the

t ransformation of affinity [23]

c I ) (~ ;b42* ,b3 /3 ) = bq) ( Y ; 2* , / 3 ) , (29)

with arbitrary non-zero constant b. Using this relation, one can easy prove that the parameters M and 2 calculated from (28) on the base of the solution for given 2* and/~, do not differ from the corresponding values determined from the solution for b42 * and b3/3, i.e. they do not depend on b. Hence, in case of non-zero values of/~ it is sufficient to choose/3 equal to 1 and - 1 (corresponding to the values ofb in (29) equal to/~- 1/3, if/3 > 0, and ( - / 3 ) - 1/3 for/3 < 0) and to change A* over all real numbers. For/~ = 0 and any non-zero 2" the constant b in (29) could be determined through 2*, thus reducing the problem to seeking solutions for 2* = 1 and 2* = - 1 only.

Setting ~"(0) equal to + 1, 0, - 1, one considers three different slopes of the streamwise velocity profile at the plate surface. If/3 = 0, the stress is zero at y = 0, which is not true for the rigid wall, but this value is included for completeness.

After determination of M and 2 from (28), the velocity and the pressure are calculated from (23), (13) and (10). The fourth-order Runge-Kutta method was used for solving the initial-value

problem.


With known function ~, Eq. (26) is solved for fixed Bi and various values of the Prandtl number. A finite-difference method is used and the solution of the corresponding algebraic systems is found by the Gauss elimination method with a choice of the main element of the matrices [24]. The function 01(~) is calculated from (22).

4 Discussion of the results

Following the procedure outlined above, the value of q~"(0) has been fixed and Eq. (24) has been solved for many values of the parameter 2* within the interval ( - 0% + oo). We have found that the solutions of the first group at ~"(0)= 1 do not correspond to the problem under consideration because the function �9 does not vanish for positive )~ and tends to infinity with increasing the independent variable.

The solutions of the second group at ~"(0) = 0 present only one pair of values M = 6 349 and 2 = -0.03561 calculated for any 2* > 0. The solutions for 2" < 0 give negative values of the Marangoni number and they were disregarded.

The solutions of the third group at ~"(0) -- - 1 determine the Marangoni number within the interval 0 < M < 6 349. When 2* varies from - o o to zero, the Marangoni number increases from zero to 1150. The other values of M correspond to positive values of this parameter and at 2* ~ + oo the Marangoni number tends to the value coinciding with that obtained from the second group solutions. It is proved numerically that the problem (11), (12) has only one solution for the values of the Marangoni number from the interval [0, 6 349] and a solution for other positive values of M does not exist.

The non-existence of solution for sufficiently large Marangoni numbers seems to come from the fact that it is not possible to impose quite large temperature gradients at a liquid free surface without disturbing its flat shape. Hence, the constrain of the formulated problem relating to the non-deformability of the upper boundary is probably the reason for this feature of the solution.

The non-existence of solution of the present problem for the Marangoni number larger than some finite value is in contrast to the very similar problem for two-dimensional flows in a channel with accelerating free boundaries [17] studied on the base of the same dynamic equation, with the Reynolds number Re instead of M. The solution of that problem exists for any value of the Reynolds number and it is even multiple for large Re, due to the possibility to have different directions of the fluid velocity on the zero-stress center line of the channel. In the case of thermocapiUary flow the velocity is zero at the rigid boundary of the layer and the slope of the streamwise velocity (i.e. the second derivative of the similarity function) has only one (negative) sign. Thus, the solutions of the initial-value problem for positive slopes are excluded.

Some results of the numerical integration of the Eq. (11) are exposed in Figs. 2, 3 and 4 a, b. The decrease of 2 from 1.5 to - 0.035 61 with increasing M is shown on Fig. 2. The quantityf'(1) presenting the fluid surface velocity is plotted against the Marangoni number on Fig. 3. It also decreases monotonically from the maximum value 0.25 to 0.053 5.

Typical profiles of the streamwise velocity are presented on Fig. 4 a and Fig. 4 b for various values of the Marangoni number corresponding to positive and negative values of 2 respectively. Flow pattern for two values of M are presented on Fig. 5. As expected, the liquid on the free surface and in a sublayer below it moves in both directions of increasing surface tension, starting from the point x = 0 (where the surface tension has a minimum). Due to continuity, reverse flows occur in the lower part of the layer and in every semi-infinitive part of the layer only one, very long, roll exists. The intensity of the flow decreases with increasing the Marangoni number.

216

1.5

X 1.0

0.5

0,0

-0.5 0 2000 40'00 60'00 M

Fig. 2. The coefficient 2 as a function of M

S. G. Slavtchev and S. R Miladinova

0.25 f '(1)

0.20

0.15

0.10

0,05

0.00 o, 2000 4000 60'00 M

Fig. 3. The surface veloci tyf ' (1) as a function of M

0.2 ~ . / i !

0._0~. 1 ' ~ 0 . . . . 0.~1 . . . . 0.r2 ' f'(y/d)

(a)

1 . 0 - - -

y/d ~ ~ 0.8

0.6 -

o 4

0.2 -

='o,o~ 0.00 o.o2 o.o3 0.05 0.06 i f (y /0)

(b) Fig. 4. Longitudinal velocity profiles at M equal to: a - 0.000 432 (1), 138 (2), 1032 (3); b - 3105 (1), 6 345 (2)

y / d 1.0

0.8

0.6

0.4

0.2

0"02.0 - 1 . 0 O, 0.0 1.0 x / d 2.0

(a)

Y / d . 8

0.6

0.4

0.2

0.0 - 2 , 1

i i i - i i r

- 1 . 0 0.0 1.0 2.0 x / d

(b) Fig. 5. F low pattern at M equal to: a - 11, b - 1032

1.0 y / d

0.8

2 I 0.6

0.4 / I

0.2 , , I I

o.o . . . . . . . . . . . . . . . . . i J - o , ~ o.o 0.5 ~.o

0

Fig. 6. Distribution of 0 for case I a at Pr = 1, and M equal to 0.00043 (1), 11 (2), 138 (3). 1032 (4), 3105 (5)

y /a 1.0

0.8

0.6

0.4

0.2

0.0 , ,

-0 .5 0.0 0.5 1.0 0

Fig. 7. Distribution of 0 for case Ia at M = 138, and Pr equal to 1 (1), 4 (2), 7 (3)


The thickness of the sublayer in which the liquid goes from the axis y has the largest value; 0.33 d for the Marangoni number being almost zero and it decreases monotonically with increasing M. At large values of M the solution involves into flow with a boundary-layer type structure, e.g. a thin boundary layer with a thickness of order M - 1/3 exists next to the free surface

and hence, we have

f(37) = M - 2 / 3 F ( y ) , 37 = 1 - M - 1 / 3 Y , (30)

where F ( Y ) satisfies the equation

2 F " ' - F F " + F '2 - - - - - - (31)

M 1 / 3

and the boundary conditions

F(0) = 0, F"(0) = 1. (32)

The third boundary condition must match the function with the solution outside of the boundary layer as Y ~ + oo. As the right-hand side of(31) is small for large M, the leading-order solution is the function

F o ( Y ) = - C[1 - exp ( - C Y ) ] , (33)

satisfying the reduced equation. This solution is first found in [25]. In the case under consideration the constant C = 1.

It is worth noting that in the boundary layerf'(37) is of order of M - 1/3 and this explains the decrease of the free surface velocity as well as the flow intensity with increasing the Marangoni number.

For small values of the Marangoni number the solution of (11) can be found in a regular expansion in M and for example, in Case I, when f"(1) = 1, the zero-order term is (see [14])

j72 3 (34) y(37) = ~ - ( ; - 1), ;T= ~-.

As f '(1) = 0.25, this function presents the flow with the largest surface velocity. Let consider now the solutions of the temperature equations (14) and (15) in each case.

0.6 i l 0.4

0"220 ~ o o

t

-1.0

y / d O.B

0.6

0,4

0,2

0.0_2.0 ' o.o .o 2.0 o.0 x/d

(~) (b)

- I

- 1 . 0 1.0 2.0 x/d

Fig. 8. Isotherms for case Ia at P r = 1, A d / T o = 0.1 and M equal to a -- 11, b -- 1032


a) Case I a

Here 01 is equal to 1. The function 0 is plotted on Fig. 6 for various values of the Marangoni

number and P r = 1. The same function is also presented on Fig. 7 for different Prandtl numbers and M = 138. The both numbers appear as a product in (15) and the behavior of 0(37) for

moderate and large M depends mostly on the values of that product, if the Prandtl number is of order of 1. For large values of the product the function varies slightly in the lower part of the

layer, due to the weak flow there, being negative near the wall.

For relatively small M and the Prandtl number of order of unity the solution of (15) can be found in a series in M. By use of (34), one obtains the first two terms

Pr 0(37) = 1 + Mff(37), ff = ~ (3)7 ~ - - 437 3 + 1). (35)

The function ff(37) satisfies the equation and the boundary conditions

if" = P r f ' , ~(0) = 0, 0(1) = 0. (36)

When the Marangoni number is sufficiently large, it is reasonable to expect the existence of a thin

thermal boundary layer adjacent to the free surface. At Pr = 1 the thickness of the layer should be of order of that of the dynamic boundary layer, i.e. of O ( M - 1/3). Presenting 0(37) in a series

0(37) = ~po(Y) + M - 1 / 3 t p l ( Y ) + . . . . (37)

and substituting (37), (30) and (33) into (15), one obtains, to leading order, the equation

Wo" + C(1 - e - c r ) tpo' - C2e-Cr~po = 0. (38)

Two linearly independent solutions of (38) are

fe - Z

Z = e - c r and e - z + Z dz (39) Z

from which the general solution can be constructed. In the case under consideration (0(1) = i and

C = 1) the function ~po(Y) = ~o(Z) has the form

e - Z ~ 2 o ( Z ) = Z - C 1 --Z _ e - Z + Z - - d z . (40)

e z z

with a constant C1, which is to be determined from the matching of ~o(Y) with 0(37) outside of the layer as Y ~ + oe. The very quick variation of ~o(Y) in the boundary layer layer next to the liquid

surface causes some difficulty in the numerical integration of (15).

The temperature fields, i.e. isotherms T/To = const, for two values of M and Pr = 1 are presented on Fig. 8. The deviation of the isotherms from the vertical lines in the upper part of the

layer is due to the flow going out of the axis y. The temperature fields are symmetric about this axis in a sense that at any level the temperatures at symmetric points deviate from that at x = 0 with the same quantity, but with different signs. For small and moderate M the dimensionless

temperature is larger (smaller) than 1 for positive (negative) x. For large M the thermal field is divided into four domains by two (vertical and horizontal) isotherms at T/To = 1. For x > 0 the dimensionless temperature in the lower part of the layer becomes smaller than 1, because the reverse flow is intensive mostly in its upper part and the warmer fluid going back to the axis y remains at middle levels. For negative x the colder fluid also returns at the middle levels of the layer.


1.0 y/d

0.8

0.6

0.4

0.2

0.0 0.5 0.'6 0.'7 O J8 0.'9 1.0 1.1

01

1.0 y/d

0.8

0.6

0.4

0.2

0.0 - 0 . 3 0.0 0.2 0.5 0.8 0 1.0

(b) Fig. 9. Distribution of 01 (a) and 0 (b) for case Ib at Pr = 1, To/T ~ = 0.5 and M equal to 0.00043 (1), 11 (2), 138 (3), 1032 (4), 3105 (5)

1.0 y / d

0.8

0.6

0.4

0.2

0.0 - 0 . 2 0.0 0.'2 . . . . . 0.4 0.6 0.'8 1.0

0

Fig. 10. Distribution of 0 for case I b at M = 138, and Pr equal to 1 (1), 4 (2), 7 (3)

, / d , / d

0.4 0.4

0.2 0.2 T

0"o-2.0 -1 .0 0.0 1.0 x / d 2.0 0"O-2.0 ' - i ' .0 ' 0.0 i.'0 x / d 2.0

(b) Fig. 11. Isotherms for case Ib at Pr = 1, To/T ~ = 0.8, A d / T = 0.1 and M equal to a - 11, b - 1032

b) Case I b

In this case the funct ion 01 depends on y which means that the temperature varies from Tw to

T~ a long the axis y. This is well seen on Fig. 9 a presenting the funct ion for different Marangon i

numbers and To/Tw = 0.5.

O n Fig. 9 b 0 is plotted for some values of M and Pr = 1. The influence of the Prandtl number

on this funct ion is shown on Fig. 10. It is seen that its behavior is very different for small and large

M ar angon i numbers. For small M the funct ion has the presentat ion

e r 0(37) = 37 + Mff(37), 0"= 2~6 37(6374 + 5373 - 11) (41)

2 2 0 J S. G. Slavtchev and S. R M i l a d i n o v a

1.0 y / d

0.8

0.6

0.4

0.2

0,0 0.95 'o.9~ 'o.97 0 . 9 8 0.99 011'00

t .0 y / d

0.8

0.6

0.4

0.2

0.0 - 5 . 0

J

- 2 . 5 0,0 0

(b)

Fig. 12. D i s t r i b u t i o n of 01 (a) and 0 (b) for case II at Pr = 1, Tg/Two = 0.5, Bi = 0.1 and M equal to 0.00043

(1), 11 (2), 138 (3), 1032 (4), 3105 (5)

1.0 y / d

0.8 1

0.6

0.4

0.2

0.0 - 5 . 5 - 4 . 5 - 3 . 5 - 2 . 5 - 1 . 5 - 0 . 5 0.5 1.5

0

Fig. 13. D i s t r i b u t i o n of 0 for case II at M = 138, Bi = 0.1

and P r equa l to 1 (1), 4 (2), 7 (3)

1.0

y / d 0.8

0.6

0.4

0.2

0.0 - 2 . 1

-4 -4 "

- 1 . 0 0.0 1.0 x/d

1.0

y / d 0.8

0.6

0.4

0.2

_J ~ 1 7 6 -~0 g.0 1.o 20 2.0 x/d

(b)

Fig. 14. Isotherms for case II at Pr = 1, Bi = 0.1, To~Two = 1, Ad/Two = 0.1 a n d M equal to a - 11, b - 1032

and varies almost linearly, specially in the lower part of the layer. For large M the variation of 0 is considerable in the upper part of the layer.

Isotherms T/Tw = const are plotted on Fig. 11 for two Marangoni numbers with Pr = 1, To/Tw -- 0.8 and Ad/T~ = 0.1. The temperature fields are not symmetric about the axis y and the temperature at some point with x > 0 is higher than that in the symmetric point.


c) Case I I

This case is the most relevant to the experiments because in experimental conditions the temperature of the layer bottom is usually well controlled, while the liquid surface temperature may change in a different way due to the fluid motion.

At Bi = 0 the function 01 = 1 and this particular case is similar to case Ia, but the temperature conditions at the layer boundaries are exchanged. Moreover, 0(1) is now not known in advance.

The influence of the heat tansfer through the free surface on the dynamic and thermal fields in the layer is studied for Bi = 0.1. The function is plotted on Fig. 12a for To/T,~ o = 0.5. For small M it decreases almost linearly, while for large M the function 01 is about 1 in the lower part of the layer and decreases very weakly in the upper one reaching a minimum at the free surface. So, the surface temperature To = Tow0(1) is less than the wall temperature at x = 0.

The function 0, presented on Fig. 12 b for various values of the Marangoni number at fixed Pr and on Fig. 13 for some Prandtl numbers and fixed M, changes considerably for different values of the product Pr M. For small and moderate values of this quantity it is positive while for large values the function is negative almost in the whole layer. By use of (34), for small M, with Pr = O(1) and 0(1) = O(1), the function 0 is expressed by

o(;) = 1 - B y + M g ( ; ) , B = 1 -7--Bi '

g = ~ 3 ~ - ~ - + ( B + 3 ) -43~ 2 + 3 B 1 + Bi , (42)

where g satisfies

~' = Pr [f' + B ( f - ff ')], 8(0) = 0, 0~(1) + Big(l) = 0. (43)

The function 0 is positive everywhere and the value 0(1) is less than 1, decreasing very slowly with increasing M (see curves 1 and 2 on Fig. 12b). Thus, the temperature gradient along the free surface has the same direction as that of the gradient at the wall.

For large Marangoni numbers 0(1) becomes negative and the surface temperature gradient changes its direction. It means that the temperatures at the surface for x < 0 are higher than those for x > 0.

An attempt has been made to understand the behavior of 0 at large M, when the termal boundary layer is expected to exist next to the free surface. If the function F in (30) is given by (33) and 0 is presented by (37), the function ~Po(Y) = ~o(Z) is found in the form

e-~ ~ o ( Z ) = C2 e - Z - z - - d z . (44)

Z

z

It satisfies, to the first order, the boundary condition (18) for 0 at the free surface, which reduces to 0'(1) = 0. The constant C2 is related to C by the equation C22 = C a obtained from the fourth condition. Matching of ~Po(Y) with 007) outside of the boundary layer could yield the exact value of C2. But, it is easy seen that if this constant approaches zero (from any side), the flow motion at the surface will stop, nevertheless what is the value of M. Then, Eq. (11) has simply the trivial solution. Moreover, for two values of 0(1) which are equal in absolute value and different in sign, the flow pattern is the same (without changing the direction of motion along the free surface).


Isotherms T/Two = const are plotted on Fig. 14 for Pr = 1, To~Two = 1 and Ad/T,,,o = 0.1. The values of M are chosen to present temperature fields for both signs of 0(1). The thermal field at M = 1032 shows the possibility to have a flow with the opposite temperature gradient along the free surface.

5 Conclusions

An exact solution to the Navier-Stokes equations and the temperature equation for the thermocapillary flow in a thin layer of a liquid presenting the surface tension minimum is obtained. The solution is found for the Marangoni number within a finite interval, when a constant temperature gradient is imposed along either the liquid free surface or the wall on which the layer lies. Flow pattern and thermal fields are presented for different Marangoni numbers and Pr = 1. One convective roll exists in every semi-infinite layer. The flow is more intensive near the free surface and its intensity decreases with increasing M. It is shown that in the case of constant temperature gradient along the rigid boundary there is a possibility to have a temperature gradient along the free surface in the opposite direction.

The existence of one-roll flows in cavities is predicted in some previous theoretical studies ignoring the gravity [8], [9]. Such flows are also observed in very thin layers of aqueous n-heptanol solutions [5], [6]. But, in the reported experiments a flow with an "opened" free surface has not occurred and the liquid always moves from one lateral wall of the channel to the other one in direction of increasing surface tension, even when the lateral walls are kept at temperatures, one being larger and the other smaller than the temperature of the surface tension minimum. In microgravity conditions the "opening" of the fluid surface has not been detected either.

For this reason the results obtained here can not be compared directly with the available experimental data. But, the present solution has a property which differs from the previous theoretical results for thin layers of such liquids. It is obtained for the surface tension gradient calculated directly from expression (1), without supposing it to be constant locally at any point, as is assumed, for example, in [5]. This advantage of the solution makes reasonable to evaluate the free surface velocity for a given liquid and to compare it with measured values.

The calculations were made for 6.24 x 10- 3 molal aqueous solution of n-heptanol used in the experiments reported in [6]. The values of the slope ~a/~ T, which is equal to ~ae/O T, where ae is the equilibrium surface tension measured in an isothermal system, are presented in Table 1. They are taken from Table 2 in [6]. To compare with the experimental results reported there, we define the quantity

U s d 2 aft as* . . . . . i f ( l ) , (45)

A # OT

which presents the ration of the surface velocity u, at some point to the temperature gradient at the liquid surface. The values of the surface tension gradient in Table 1 allow us to consider the change of the surface temperature from To = 40~ to the value 8.5~ for negative x and to 51~ for positive ones. As the velocity depends linearly on x, the mean surface velocity for a distance xo is equal to a half of the velocity at that point. So, the mean value of e~*, say c~,,, is equal to ~**/2. In Eq. (45) the viscosity is taken to be equal to that of pure water and the layer thickness d is 1.82 x 10 - 3m [6]. To estimate the maximum surface velocities, we take f'(1) = 0.25.

The predicted values of ~,, for some values of the surface temperature are presented in Table 1, together with the experimental ones from [6] measured at the mean temperature of the liquid


Table 1

0a T3~ ~ [Nm - 1K- 1] ~[rn2sec- 1K - 1] %[m2sec - 1K - 1]

0T

8.5 -2 .64 • 10 -4 0.9 • 10 -7 -2 .19 • 10 -7 20 -1 .68 x 10 -4 0.5 • 10 -6 -1 .39 • 10 -7 31 -0 .69 • 10 - 4 2.4 x 10 -6 -0 .57 • 10 -7 40 N0 4.6 • 10 -6 N0 51 +0.76 • 10 -4 13 • 10 -6 +0.63 • 10 -7

(denoted there by T3). I t is wor th to m e n t i o n that our results are approx imate ly 103 smaller t han

the theoret ical ones presented in [6] on the base of the fo rmula

V~ d 0or ~s = . . . . . (46)

A 4# 0 T

The absolu te values of ~ are closer to the exper imenta l ones, being most ly less t han them. The

rat io of the abso lu te value Of~m to ~ varies from 1.21 for To = 8.5 ~ to 1.1 • 10 -2 at To = 31~

The surface velocity nea r the axis y is a lmost zero because the slopes of the surface tens ion in the

vicini ty of the t empera ture To are very small.

Acknowledgement

This work is partially supported by the Bulgarian National Foundation for Scientific Research under

contract TN-402/94.

References

[1] Vochten, R., Prtrr, G. J.: Study of the heat of reversible adsorption at the air-solution interface. J. Colloid Interface Sci. 42, 320-327 (1973).

[2] Legros, J. C.: Problems related to non-linear variations of surface tension. Acta Astron. 13, 697-703 (1986).

[3] Limbourg-Fontaine, M. C., Prtr6, G., Legros, J. C.: Effects of a surface tension minimum on thermocapillary convection. PCH Physico Chem. Hydrodyn. 6, 3 0 1 - 310 (1985).

[4] Legros, J. C., Limbourg-Fontaine, M. C., Prtr6, G.: Surface tension minimum and Marangoni convection. In: Fluid dynamics and space, Proceedings of ESA Symp., Rhode-Saint-Genese, 2 5 - 2 6 June (ESA SP-265) pp. 137-143 (1986).

[5] Platten, J. K., Villers, D.: On thermocapillary flows in containers with differentially heated side walls. In: Physicochemical hydrodynamics. Interfacial phenomena (Velarde, M. G., ed.), pp. 3 l l - 336. New York: Plenum Press 1988.

[6] Prtrr, G., Azouni, M. A., Tshinyama, K.: Marangoni convection at alcohol aqueous solution-air interfaces. Appl. Sci. Res. 50, 97--106 (1993).

[7] Napolitano, L. G., Golia, C., Viviani, A.: Effects of non-linear tension on combined free convection in cavities. L'Aerotechnica 63, 2 9 - 3 6 (1984).

[8] Villers, D., Platen, J. K.: Marangoni convection in systems presenting a minimum in surface tension. PCH Physico Chem. Hydrodyn. 6, 435-451 (1985).

[9] Dubovik, K. G., Slavtchev, S. G.: Numerical modeling of thermocapillary convection in a liquid layer at a non-finear dependence of the surface tension on temperature. Mech. Zhidkosti Gaza, 1, 138-143 (1991) (in Russian).

224 S.G. Slavtchev and S. R Miladinova: Thermocapillary flow in a liquid layer

[10] Slavtchev, S. G., Dubovik, K. G.: Thermocapillary convection in a rectangular cavity at minimum of surface tension. Theor. Appl. Mech. (Sofia) 23, 8 5 - 9 0 (1992).

[11] Pukhnatchov, V. V.: Display of anomalous thermocapillary effect in a thin liquid layer. In: Hydrodynamics and heat mass transfer in liquids with free surfaces, (Schreiber, I. R., ed.), pp. 119 - 127. Novosibirsk: Inst. Teplofiziki, 1985 (in Russian).

[12] Pukhnatchov, V. V.: Thermocapillary convection in weak force fields. Preprint No. 178-88, Inst. Teplofiziki, Novosibirsk, 1988 (in Russian).

[13] Cloot, A., Lebon, G.: Marangoni convection induced by a nonlinear temperature-dependent surface tension. J. Phys. 47, 2 3 - 2 9 (1986).

[14] Gupalo, Yu. R, Ryazantsev, Yu. S.: Thermocapillary motion of a liquid with free surface with non-linear dependence of the surlhce tension on the temperature. Fluid Dyn. 23, 752-757 (1989).

[15] Oron, A., Rosenau, Ph.: On a nonlinear thermocapillary effect in thin liquid layers. J. Fluid Mech. 273, 361 - 374 (1994).

[16] Gupalo, Yu. R, Ryazantsev, Yu. S., Skvortsova A. u Effect of thermocapillary forces on free surface fluid motion. Fluid Dyn. 24, 657-661 (1990).

[17] Brady, J. F., Acrivos, A.: Steady flow in a channel or tube with an accelerating surface velocity. An exact solution to the Navier-Stokes equations with reverse flow. J. Fluid Mech. 112, 127-150 (1981).

[18] Davis, S. H.: Thermocapillary instabilities. Annu. Rev. Fluid Mech. 19, 403-435 (1987). [19] Schlichting, H.: Boundary layer theory. New York: McGraw-Hill, 1968. [20] Terril, R. M.: Laminar flow in a uniformly porous channel. Aeron. Q. 15, 299-310 (1964). [21] Terril, R. M., Thomas, R W.: On laminar flow through a uniformly porous pipe. Appl. Sci. Res. 21,

3 7 - 6 7 (1969). [22] Robinson, W. A.: The existence of multiple solutions for laminar flow in a uniformly porous channel

with suction at both walls. J. Eng. Math. 10, 2 3 - 4 0 (1976). [23] Szaniawski A.: Quasi-isobaric solutions of the Hiemenz equation. Arch. Mech. 45, 689-725 (1993). [24] Samarsky, A., Nikolaev, E.: Methods for solving finite-difference equations. Moscow: Nauka 1978 (in

Russian). [25] Riabouchinsky, D.: Quelques considerations sur les mouvements plans rotationnels d'un liquide. C. R.

Acad. Sci. 179, 1133-1136 (1924).

Authors' address: S. G. Slavtchev and S. R Miladinova, Institute of Mechanics, Bulgarian Academy of Sciences, Acad. G. Bontchev Str., B1. 4, 1113 Sofia, Bulgaria

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