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Marangoni instability, effects of tangential surface viscosity on a deformable interface
M.A. Vila, V.A. Kuz, A.N. Garazo, A.E. Rodriguez
To cite this version:
M.A. Vila, V.A. Kuz, A.N. Garazo, A.E. Rodriguez. Marangoni instability, effects of tangential surface viscosity on a deformable interface. Journal de Physique, 1987, 48 (11), pp.1895-1900.
�10.1051/jphys:0198700480110189500�. �jpa-00210632�
Marangoni instability, effects of tangential surface viscosity on a
deformable interface
M. A. Vila (#), V. A. Kuz (*), A. N. Garazo (**) and A. E. Rodriguez (*)
Instituto de Fisica de Liquidos y Sistemas Biológicos, Universidad Nacional de la Plata, C.C.565, La Plata (1900), Argentina
(Requ le 3 f6vrier 1987, révisé le 11 juin 1987, accept6 le 6 juillet 1987)
Résumé. - L’effet de la viscosité de surface sur l’amorce de l’instabilité de tension de surface est analyse à
l’aide de la théorie de la stabilite linéaire, dans le cas d’une couche liquide mince horizontale placée dans un gradient de temperature orthogonal. L’équilibre des forces de contrainte tangentielles est decrit à l’aide des
principes microscopiques de l’hydrodynamique à une interface. L’état neutre est étudié lorsque la surface libre déformable est le bord supérieur ou inférieur de la couche liquide. Dans le premier cas, la viscosité de surface
~s inhibe I’instabilité de Marangoni, tandis qu’elle l’augmente dans le second. Egalement dans le cas du bord inférieur, pour ns~ 0, il n’y a aucune largeur de couche telle que l’amorce de la convection fasse passer d’une structure non périodique à une structure périodique, à la différence du cas ~s = 0.
Abstract. - The effect of surface viscosity on the onset of surface tension driven instability in a horizontal thin
liquid layer under a normal temperature gradient is analysed, using the linear stability theory. The free surface
tangential stress balance is written following the microscopic theory of interfacial hydrodynamics. Then the
neutral state when the upside or the underside of the liquid layer is the deformable free surface has been studied. In the first case the surface viscosity ~s inhibits Marangoni instability, while in the second it induces the contrary effect. Also in the underside case, for ~s ~ 0 there is not any layer width d where the onset of convection changes from non-periodic to periodic structure, as it happens for ~s = 0.
Classification
Physics Abstracts 47 - 25Q
1. Introduction.
The onset of convective instability in a horizontal
fluid layer in the presence of a normal temperature
gradient has been studied by several authors [1-6].
The mathematical treatment of this problem is done
via the resolution of Navier-Stokes and Convection- Conduction equations simultaneously, plus the ade- quate boundary conditions. The method of resol- ution of these partial differential equations, as well
as the boundary conditions used, are of fundamental
importance in the global description of the problem.
(*) Member of the Consejo Nacional de Investigaciones
Cientificas y Tdcnicas (CONICET).
(**) Fellowship of the Consejo Nacional de Inves-
tigaciones Cientificas y Tdcnicas (CONICET).
(*) Fellowship of the Comisi6n de Investigaciones Cien-
tificas Pcia. Bs. As. (CIC).
We will follow the perturbative method proposed originally by Chandrasekar [3].
The boundary conditions play an important role,
as has been shown by Scriven [6]. He found for an upside flat liquid layer that surface viscosity effects,
introduced through a bidimensional model of inter-
face, inhibit Marangoni instability.
Using the microscopic theory of interfacial hyd- rodynamics, Baus et al. [7] obtained a set of bound-
ary conditions. Following this approach we focus on
the effect of surface viscosity in the tangential stress
balance at the free surface and disregard the other dissipative effects in normal stress and thermal balance equations.
Attention will be only centred, in the surface tension driving force instability, often called
« Marangoni instability », and where buoyancy ef- fect, for the thickness layer considered here, is unimportant as has been shown by Pearson [4] and
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:0198700480110189500
1896
Nield [5]. Surface viscosity effect on the onset of convection will be taken into account in the cases
where the deformable liquid-gas surface of a liquid layer (free surface) is upside or underside. A similar treatment was followed by Takashima [8] to account
for the role of surface deformation in the onset of convection.
Under these assumptions the influence of surface
viscosity is studied on the mapping of the neutral stability states. Both cases (upside and underside)
will be treated separately.
2. Surface properties and tangential stress balance at
the liquid-gas interface.
A fluid-fluid interface of contaminated [9] or pure
liquids [10a] has intrinsic surface viscous properties,
as been shown by microscopic and macroscopic experiments [10b, c].
Baus and Tejero [7] from a microscopic theory of
interfacial hydrodynamics, found within Goodrich’s
approximation [11-13], the following boundary con-
dition for the tangential stress component :
where v = (vx, vy, vz ) is the fluid velocity ; bT is the
surface excess of bT, bT = - ( a cr ae being the
variation of surface tension with temperature 0. q is the bulk shear viscosity, fiT and IT are the
tangential shear and dilational surface viscosities
respectively. [ ]XI indicates the difference between values assumed by a property in each phase, i.e.
liquid and gas respectively. V2 = V2 + V2 indicates
the horizontal Laplacian. The upper dash is a
notation for the excess of a property A at the interfacial region defined as :
with AG(z) = All H(z) + AI(1- H(z)),-H(z) being
the Heaviside step function.
Equation (2.1), which includes the effect of surface
viscosities, will be used as a boundary condition at
the liquid-gas interface.
3. Marangoni instability : formulation of the prob-
lem.
Let us consider an infinite horizontal liquid layer of depth d, where z indicates the direction normal to the liquid layer. The temperature profile is con-
tinuous ; at z = 0 the liquid layer is in contact with a
solid conducting plate, at a temperature Ow, while at
z = d the temperature at the free surface is ° A .
As we said previously, buoyancy effect will be
disregarded since layer depth is less than 0.1 cm.
The basis equations are :
where a is the thermal diffusivity, p the pressure and p the density of the fluid. Care must be taken with the sign of g, when the underside of the liquid layer
is the free surface.
As we mentioned in the introduction, the differen-
tial equations will be solved with a perturbative
method [3, 8] proceeding as follows. A set of steady
solutions of the above equations are :
where pA is the atmospheric pressure and 8 =
(8w-8A)/d.
If this steady state is slightly perturbed, equations (3.2) and (3.3) become :
Primes indicate the perturbed quantities ; z, the
normal coordinate to the interface, is represented by
where is small compare with d. The perturbed velocity v’ at z = d +, is v’ = a, .
By perturbing equation (2.1), for the case of an incompressible fluid (V . v = 0), the tangential boundary condition at z = d becomes :
where the vapor shear viscosity is assumed to be
negligible, and ’ij s is a linear combination of
fiT and §T.
The other boundary conditions at z = d (8) are the
normal stress balance :
and the thermal balance equation :
where k is the thermal conductivity. At z = 0 (solid wall) the boundary conditions are :
In order to solve this system of perturbed equations, we introduce dimensionless units by choosing d, d2la, a /d and P d as unit of length, time, velocity and temperature respectively. The
solution in terms of normal modes is :
where w is the time constant (generally a complex number). Its imaginary part corresponds to the
oscillation frequency of the system, while the real part accounts for the amplitude variation in time. So if Re (w > 0 the amplitude increases ; inversely if
Re (w ) 0, ax, ay are dimensionless wave numbers.
In this paper we are concerned with the neutral
stability states, so Re (w ) = 0, and we shall restrict ourselves to the stationary case, i.e. 1m (Cù) = o.
The occurrence of overstability (Im (w ):A 0 will
not be considered here.
Equations (3.7) and (3.8), for w = 0, become :
where D = d/dz is the differential operator in z and
a2 = a; + a;.
The tangential boundary condition at z = 1 in
dimensionless units is :
The other boundary conditions at z = 1 are [8] :
and at z = 0 are :
where :
Bo > 0 and Bo 0 indicate whether the upside or
the underside of the liquid layer is the free surface respectively. When 8w::> 8a M::> 0 and when 8w - 9a, M0.
Equations (3.15-3.21) constitute an eigenvalue problem.
4. Numerical results and conclusions.
The numerical solution for the onset of stationary Marangoni convection (Eqs. (3.15 and 3.16) and boundary conditions, Eqs. (3.17-3.21)) was obtained
for the following values of dimensionless numbers : Cr = 1.7 x 10- 6/d, Bo = 31.0 x d2. They corre-
spond approximately to a mercury-air interface.
Since layer depth d ranges from 0.01 cm to 0.1 cm, Cr and Bo are bounded by: 1.7 x 10- 4 Cr ,-
1.7 x 10-4 and 31 x 10-4 Bo 31 x 10-2 , re- spectively. We will examine the stability domains for different values of the surface viscosity coefficient
fj s’ or its associated dimensionless coefficient Ns.
First we study those cases where the free surface is the upper side of the liquid layer (Bo ::. 0 ).
The neutral stability curves M = M (a ), for diffe-
rent values of the interfacial width d and Ns =
0, are shown in figure 1. Here, surface viscosity is
considered negligible, although recent experiments
Fig. 1. - Neutral stability curves for different values of the interfacial width d (cm) indicated on each curve. The
case when the upside of the layer is a free surface
(Bo :> 0) with null surface viscosity (Ns = 0 ) is represen- ted.
1898
of light scattering showed that a clean mercury-air
interface could have a surface viscosity different
from zero [10b, 10c]. The curves divide the plane
M - a into two parts, the upper and the lower
regions, associated with unstable and stable solutions
respectively. The coordinates of the lowest point of
each curve M(a) give the critical Marangoni and
wave number values, Me and ac respectively, at
which the onset of convection occurs.
In figure 2 we see the neutral stability curves for a
fixed interfacial width d for different values of surface viscosity. The critical Marangoni number
increases with increasing values of q, ; it implies that
the onset of convection requires a larger temperature
gradient, i.e. surface viscosity inhibits the instability.
Fig. 2. - Neutral stability curves in the case Bo > 0 with a
fixed interfacial width d = 0.07 cm for different values of surface viscosity ff s (sP) indicated in the figure.
The procedure by which the critical Marangoni
number is obtained resembles that of finding the equilibrium points of a mechanical system for a given potential ; it also resembles the study of phase
transitions of a simple fluid in thermal equilibrium.
So we could think that similarly to a mechanical or a
thermodynamical system, there exist, in convective
instability, « metastable » and « unstable » states associated with the extremum points of the neutral
stability curve M (a ). A « metastable » state could be attained in a system having at least two minima : the onset of convection will occur at the second minimum of M (a ) by the application of a thermal gradient higher than critical. We do not know
whether any experiment of this kind has been
performed.
For the sake of simplicity we indicate with
Mo, M1 and M2 the ordinates of the origin, the
maximum and the local minimum, of M (a ), respect- ively. In figure 3 values of Mo, M1 and M2 are
Fig. 3. - Values of Mo, M1 and M2 represented as
functions of d for Bo > 0. The critical Marangoni number Me lies on the branches DA and AE. For d = dA the onset
of convection changes from non-periodic to periodic
structure. On AB and AC lie « metastable » points Mo and M2, respectively, while on BC « unstable » points Ml. Curves I and II correspond to null viscosity and ff, = 10 sP, respectively.
represented as functions of d. The points Mc, where
convection occurs, lie on the branches DA and AE. Points on DA correspond to Mo = Mc while
those on AE to M2 = Mc. The loop ABC is as-
sociated with « unstable » and « metastable » states.
The branch AB represents « metastable » states
M0(d) for M0 > Mc = M2. The other branch AC
corresponds to « metastable » states M2(d) for M2 > Mc = Mo. Finally the top of the loop BC is
associated with the unstable points M1 (d). Curve II
shows the effect of surface viscosity. The stability
domain increases with increasing values of surface
viscosity. Figure 3 resembles to the Gibbs free
energy of a simple fluid as a function of pressure.
In figure 4 we represent the dimensionless waive
numbers a corresponding to the abscissae of points Mo, M1 and M2 of M (a ), as functions of d, for the
same values of surface viscosity shown in figure 3.
The inclusion of « metastable » and « unstable »
states generates a continuous curve. If we would have only considered those points corresponding to
the lowest M values of figure 3, the resultant curve
would be discontinuous at d = dA, analogously to
that obtained by Takashima [8]. The interfacial width dA is found making Mo (d ) = M2(d) = Mc (see Fig. 3). Here the « metastable » and « unstable »
states are indicated by AB, A’C and BC, respect- ively.
In real experiments the dimensionless wave
number ac cannot reach zero, due to the present of lateral walls [4, 5].
Fig. 4. - Dimensionless wave numbers a, corresponding to the abscissae of Mo, M1 and M2 of M (a ), represented as
functions of d in the case Bo > 0. « Metastable » and
« unstable » states are indicated by AB, A’C and BC, respectively. dA and the viscosity values considered coin- cide with figure 3.
It follows from figure 4 that, at the interfacial width dA, the structure of the onset of convection changes from non-periodic (ac - 0) to periodic. The
value dA increases when surface viscosity is con-
sidered (see curve II). It also follows that, within the periodic region, the critical wave number ac de-
creases with increasing viscosity.
Second, we analyse the case Bo 0, that is the
case where the free surface is the underside of the
liquid layer. The neutral stability curves M = M (a ) for fig = 0 (see Fig. 5) divide the M - a plane in a
Fig. 5. - Neutral stability curves for different values of the interfacial width d (cm). The case when the underside of the layer is a free surface (Bo 0) is considered and the surface viscosity is taken null (Ns = 0 ).
stable lower region and an unstable upper region.
Some of these curves have two local minima and one
local maximum. As before, the coordinates of the lowest point define the critical values M, and
ac.
In general, the onset of convection happens at the
lowest minimum, which is always negative (M 0).
The temperature of the solid wall is lower than that
of the free surface. The positive minimum could, in
the sense discussed before, correspond to a « metas-
table » state.
The addition of surface contaminants (ff s :0 0) changes qualitatively the behaviour observed in figure 5. For a given value of d, some extreme points disappear (see Fig. 6). When q, increases the diffe- rent curves show a lowering of the critical Marangoni number ; to prevent the onset of convection, it is
necessary to cool down the solid wall.
Fig. 6. - Neutral stability curves in the case Bo 0 for a
fixed interfacial width d = 0.03 cm. The different values of surface viscosity if s (sP) are indicated on each curve.
In figure 7, we represent those values of M
corresponding to the absolute minimum of M (a ), as
a function of d. The curve lies fully in the negative region of M, hence the stable states correspond to
o w 6a. (Takashima arrives at the same conclusion
in the case ff s = 0.) When ifs = 0, the transition
from non-periodic to periodic structure of the onset
of convection (curve I), is indicated by dA. For
d> dA, (periodic structure), the critical Marangoni
number corresponds to a local minimum of M(a),
while for d dA, (non-periodic), Me coincides with the ordinate to the origin.
Finally in figure 8 we show the dimensionless wave
numbers ac, as a function of d. Surface viscosity
modifies the behaviour of this function when a, is close to zero. When ifs = 0 the curve shows an abrupt slope at d = dA, while for increasing values of
ff,, a, goes asymptotically to zero. In this last case
1900
Fig. 7. - Values of Me for Bo 0, represented as a
function of d. In curve I (’ij = 0 sP ) for d dA the points correspond to the ordinate to the origin, while for d:> dA to the first local minimum of M(a). In curve II ( ’ij s = 10 sP) Ate correspond to the first minimum of
M(a).
the onset of convection will occur with a periodic
structure. For values of the interfacial width d >
0.09 cm, the surface viscosity turns out to be irrelev- ant on the onset of convection.
Fig. 8. - Dimensionless wave numbers ac represented as a
function of d for Bo : 0. In curve I (fi = 0 sP ) is shown
an interfacial width dA (same that of Fig. 7) where the
structure of convection changes from non-periodic to periodic, while in curves II (ff, = 0.5 sP) and III (ff, = 10 sP) this transition does not appear ; the onset of convection is always periodic.
In short, surface viscosity modifies the onset of Marangoni convection whether the free deformable surface is the upside or the underside of the liquid layer.
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