EFFECT OF NONLINEAR TEMPERATURE PROFILES ON T H E ONSET OF CONVECTION DRIVEN BY SURFACE TENSION GRADIENTS A N T O N I O VIDAL’ AND ANDREAS ACRIVOS Department of Chemical Engineering, Stanford University, Stanford, Calif. 94305
The effect of nonlinear preconvective temperature profiles on the magnitude of the critical Marangoni number is considered using linear stability theory for the case of surface tension-driven flow in evaporating liquid layers. The analysis predicts that in the presence of strong nonlinearities in the basic profile the critical Mararigoni number can be much larger than that corresponding to a linear one. Values of the Marangoni number measured at the onset of convection in shallow evaporating layers of propyl alcohol are in good agreement with the analytical predictions.
ECENT
theoretical (Pearson, 1958; Sternling and Scriven,
R 1959) and experimental work (Berg et al., 1966b; Kosh-
mieder, 1966; Orell and Westwater, 1962; Scriven and Sternling, 1960) has firmly established that surface tension gradients along a liquid-liquid or a liquid-vapor interface can often generate pronounced convective motions within such systems. I n fact, such surface tension-driven flows play an important role in various extraction processes and, when present, can lead to much higher rates of interfacial mass transfer than would occur otherwise (Olander and Reddy, 1964). Similarly, the striking convective patterns that have been observed in shallow pools of volatile liquids (Berg et al., 1966b), and whose presence has an important bearing on the rate of evaporation itself, have also been recognized as due to this same surface tension effect brought about by temperature variations along the evaporating surface. I n short, then, not only is it generally accepted that such surface tension-driven motions are of practical significance, but, in addition, the conditions under which they arise are by now well understood. I n view of the importance of this subject it seems surprising, therefore, that the criteria for the onset of this type of convection, derived theoretically by Pearson (1958) and by Sternling and Scriven (1959) on the basis of linearized stability analyses, have not been verified experimentally in a quantitative way. To be sure, there exists sufficient qualitative evidence in support of this theoretical work, but exploratory experiments carried out in evaporating layers of pure liquids have led to Marangoni numbers at the onset of convection that are consistently much larger (Elerg et a/., 1966a) than those calculated from Pearson’s solution, the discrepancy seeming most pronounced for the most volatile liquids. Undoubtedly, the main reason for this quantitative disagreement is that Pearson’s analysis, which set out to predict only which systems would ultimately be stable or unstable, was developed in detail solely for the case of a linear preconvective temperature profile, whereas, in a typical experimental setup involving a free interface, such profiles are seldom linear; in fact, the more volatile the liquid the more nonlinear is 1
Present address, Chicago Bridge and Iron Co., Plainfield, Ill.
the temperature profile at the onset of convection, and the less likely it is that the stability characteristics of such a system would be given quantitatively by the theoretical model considered so far, which, clearly, would no longer be applicable. Hence, it appeared desirable to extend the theory by investigating the effect of nonlinear conductive temperature profiles on the stability of liquid layers subject to surface tension gradients, and to compare these results with those of an experimental study involving some typical shallow evaporating liquid layers, in which the surface tension mechanism provides the principal driving force for any observed convective motion. Stability Analysis
Consider here a liquid layer of infinite horizontal extent, bounded above by an evaporating free surface and below by a surface which is solid and insulating. Initially, it is presumed that the fluid is at rest and a t a constant temperature, To, and that at time zero evaporation begins to take place. T h e system will then be governed by the usual time-dependent equations of change, subject to the following boundary conditions:
T
=
T O .,ii = O a t t 5 0, 0
>z > -d
bT/Bz
=
0, 12 = 0 at z = - d
bT/dz
=
-Q, w
=
0 , -pd2w/bz2 = v12y a t z = 0
where p , p , and K are, respectively, the viscosity, density, and thermal diffusivity of the liquid, and Q is proportional to the heat flux at the evaporating surface assumed nondeformable. T h e last equation with y being the surface tension, w the zcomponent of the velocity, and Vl2 = b2/bx2 b2/by2,represents the balance of shear stress at the free surface. Although this is not essential to the development, Q is further assumed to remain constant, a condition which, as will be seen later, is satisfied approximately in a typical experimental setup. I n the absence of convection, the system admits the simple solution
+
.ii z 0,
T
VOL. 7
NO. 1
= T(z,t) FEBRUARY 1968
(1) 53
where T is the appropriate conductive temperature profile. To test for stability, use is made of the well-known small disturbance analysis which, after linearization, reduces the equations in terms of the perturbationvariables to
with
and, a! modes 1
with F
with
~
where a L I I c . w n v r .~IYIIUI.L U I LLLC U ~ U LV~IILC. marginal state, assumed stationary (p = 0),
(Dz- or9))"fi
= 0,
(D*- or3gl
(D*- orz)2f2
= 0,
(0'- a3gz = 0 for
=
-fi
ncnce, IOT me
for 0 < z c
I-. .. expression tor ^ me. cnaracteristlc . . . equavanisn. ( i n e exp IIC11 tion is given in the A4ppendix.) A numerical solution of Equation 6 yields pairs of values .L- 1 *... .< -,~A-:~~I (m,M) for a given L,. _.
