Our experiments in thin liquid layers (approximately 0.1 mm thick) heated from below reveal a well-defined long-wavelength instability: at a critical temperature difference across the layer, the depth of the layer in the center of the cell spontaneously decreases until the liquid-air interface ruptures and a dry spot forms. The onset of this critical instability occurs at a temperature difference across the liquid layer that is 35% smaller than that predicted in earlier theoretical studies of a single layer model. Our analysis of a two-layer model yields predictions in accord with the observations for liquid layer depths greater than or equal to 0.15 mm, but for smaller depths there is an increasing difference between our predictions and observations (the difference is 25% for a layer 0.06 mm thick). In microgravity environments the long-wavelength instability observed in our terrestrial experiments is expected to replace cellular convection as the primary instability in thick as well as thin liquid layers heated quasistatically from below

McCormick, W. D.

Schatz, Michael F.

Swift, Jack B.

Swinney, Harry L.

VanHook, Stephen J.

English

NASA Technical Reports Server

ILONG-WAVELENGTH INSTABILITY IN MARANGONI CONVECTION
Stephen J. Van Hook, Michael F. Schatz, Jack B. Swift,
W. D. McCormick and Harry L. Swinney
Center for Nonlinear Dynamics and Department of Physics
The University of Texas at Austin
Austin, TX 78712 ...,: ,
ABSTRACT
Our experiments in thin liquid layers (,._0.1 mm thick) heated from below reveal a well-defined
long-wavelength instability: at a critical temperature difference across the layer, the depth of the
layer in the center of the cell spontaneously decreases until the liquid-air interface ruptures and a
dry spot forms. The onset of this critical instability occurs at a temperature difference across the
liquid layer that is 35% smaller than that predicted in earlier theoretical studies of a single layer
model. Our analysis of a two-layer model yields predictions in accord with the observations for
liquid layer depths > 0.15 mm, but for smaller depths there is an increasing difference between
our predictions and observations (the difference is 25% for a layer 0.06 mm thick). In microgravity
environments the long-wavelength instability observed in our terrestrial experiments is expected to
replace cellular convection as the primary instability in thick as well as thin liquid layers heated
quasistatically from below.
INTRODUCTION
An understanding of thermocapillary flows and other surface-tension-driven phenomena will
be critical for successful containerless processing in microgravity. Surface-tension-driven B6nard
(Marangoni) convection (see Fig. 1), in which a liquid with a free upper surface is heated from
below, is a classic example of such thermocapiltary flows. The temperature gradient on the free
surface that initially drives the fluid flow can originate from both temperature fluctuations and
deformational perturbations. The former are stabilized by diffusion, the latter by gravity. When
gravity is sufficiently strong to stabilize deformational perturbations, the primary instability leads
to the short-wavelength hexagonal convection cells first observed by B6nard [1]. This instability
forms when the Marangoni number (M -- a w /k Td/pv,_), the non-dimensionalized AT across
the liquid, exceeds a critical value Me. Recent experiments have yielded Mc = 83 [2], in accord
with linear stability theory [3]. In microgravity, however, another instability can arise; the liquid
is particularly vulnerable to deformational instabilities since gravity (characterized by the Galileo
number, G - gda/vg, where g is the gravitational acceleration) is not strong enough to prevent
thermocapillarity (characterized by M) from pulling liquid from warm, depressed regions of the
interface to cool, elevated regions. The suppression of curvature by surface tension causes this
deformational instability to appear with a long wavelength.
This long-wavelength instability was predicted to exist in the 1960's [4, 5], but the present
experiments are the first to demonstrate the existence of this instability [6]. Both linear [7-9] and
nonlinear [10-13] theories have used a one-layer model, with the effect of the air layer modeled by use
of a heat transfer coefficient, the Biot number, which in the long-wavelength limit is H = k_d/kd_,,
where k and k_, respectively, are the liquid and air thermal conductivities_ and d and d_ are the
layer thicknesses. The one-layer model predicts onset of the instability at M -- (2/3)G(1 + H). To
study this long-wavelength instability in terrestrial experiments, we achieve small (7 by using very
thin depths, d _ 0.1 mm; a 0.I-ram-thick layer of liquid in terrestrial gravity has the same G as a
1.0-mm-thick layer for g = 10-3gE .
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https://ntrs.nasa.gov/search.jsp?R=19970000401 2020-06-18T00:08:29+00:00Z
EXPERIMENTAL METHODS
We study a thin layer of silicone oil that lies on a heated, gold-plated aluminum mirror and is
bounded above by an air layer (see Fig. 1). A single-crystal sapphire window (0.3-cm-thick) above
the air is cooled by a temperature-controlled chloroform bath. The temperature drop across the
liquid layer is calculated assuming conductive heat transport and is typically 0.5-5 °C. We use a
distilled polydimethylsiloxane silicone oil with a viscosity of 10.2 cS at 50 °C [14]. The circular
cell (3.81 cm inner diameter) has aluminum sidewalls whose upper surface is made non-wetting
with a coating of Scotchgard. The liquid layers are sufficiently thin that buoyancy is negligible;
the experiments are performed with 0.005 cm < d < 0.025 cm (with aspect ratios ranging from
150 to 750) and 0.023 < dg < 0.080 cm (typically dg -- 0.035 cm); the corresponding fundamental
wavevectors are in the range 0.008 < q = 21rd/L < 0.040 (<< 1). We visualize the layer using
interferometry, shadowgraph and infrared imaging (256 × 256 pixel InSb staring array, sensitive
in the range 3-5 #m). The gap between the sapphire window and the mirror is uniform to 1%, as
verified interferometrically. The liquid surface is initially fiat and parallel to the mirror to 1% in
the central 90% of the cell, with a boundary region near the sidewalls due to contact line pinning.
This initial depth variation is accentuated by thermocapillarity as AT is increased; measurements
of depth variation show a 10% surface deformation in the central 90% of the cell at 3% below onset.
EXPERIMENTAL RESULTS
Instability Onset Above a critical AT, the liquid layer becomes unstable to a long-wavelength
draining mode that eventually forms a dry spot [see Figs. 2(left) and 5(left)]. The drained region
takes several hours (of order a horizontal diffusion time, L2/_) to form. Although we refer to the
drained region as a "dry spot", it is not completely dry since an adsorbed layer ,._ 1 #m thick
remains. The size of the drained region is typically one-quarter to one-third the diameter of the
entire cell. As Fig. 3(left) shows, our measurements of onset are consistently 35% below in AT (or
M) the prediction of linear stability for the one-layer model that appears in the literature [7-13]. We
do not believe this discrepancy is due to systematic errors in the characterization of our experiment
(e.g., geometry, fluid properties) or nonuniform heating of the liquid because experiments in the
same convection cell using thicker liquid layers find onset of hexagons uniform across the cell at a
AT that agrees with another experiment [2] and linear stability theory [3].
Mode Competition The long-wavelength and the hexagonal modes become simultaneously
unstable at a critical liquid depth de. Near this critical depth, both modes of instability compete
and influence the formation of the pattern. The two modes are not mutually exclusive, but there
is a fundamental imbalance in their relationship: the presence of the hexagons suppresses the long-
wavelength mode, while the presence of long-wavelength deformation may induce the formation of
hexagons. One-layer linear stability theory predicts dc = (120vu/g) 1/3 = 0.023 :t: 0.001 cm; we
observe the exchange of primary instabilities at dc = 0.025 ± 0.001 cm. For d > de, hexagons
are the primary instability [see Fig. 2(right)l; the hexagons smooth the large-scale temperature
variations that would allow formation of the long-wavelength mode as a secondary instability when
M exceeds Me.
For d < de, the long-wavelength mode is the primary instability. The liquid expelled from
the forming dry spot increases the local height h(x,y,t) and thus the local Marangoni number
[(x h(x, y,t)] in the newly formed elevated region; for 0.017 cm < d < de, hexagons form in the
elevated region since the local M in this region exceeds Mc for the onset of hexagons [see Fig.
2(center)]. For d < 0.017 cm, hexagons do not form at the onset of the long-wavelength mode [see
Fig. 2(left)l, but can form in the elevated region for AT sufficiently far above onset. Similar mode
266
competition phenomena have been studied theoretically for solutocapillary convection [15]. Rapid
instead of quasistatic ramping can lead to the formation of the hexagons where the long-wavelength
is predicted to be the primary instability since the timescale for formation of hexagons (the vertical
diffusion time, d2/_) is much shorter than the timescale for formation of the long-wavelength mode
(the horizontal diffusion time, L2/t_). When d << de, increasing AT far above ATe increases the
area of the dry spot until the remaining layer is thick enough to form hexagons, which then halts
any further advance of the dry spot. At fixed AT, the dry spot is stable.
TWO-LAYER MODEL
The evolution of the instability can be studied using a nonlinear evolution equation derived from
the Navier-Stokes equations in the limit of long-wavelengths. Several authors [10-13] have derived
evolution equations using a one-layer model. We have developed a two-layer model which takes into
account the change in the temperature profile in the air due to deformation of the interface. The
appropriate heat transfer coefficient then becomes F = (d/d_ - H)/(1 + H). The two-layer model
reduces to the one-layer model in the limit d/d_ ----+ O. The two-layer evolution equation for the
surface height h(x, y, t) is:
3 {3M (l+F)h2vh _h 3 (2Ld)2h3 V2 }vh+- vh =0
where time is scaled by d2/_ and BL is a modified static Bond number, BL -- pg(L/27r)2/a > O.
Bz _ 30 for our experiments. The first term in curly brackets describes the effect of thermocapil-
larity; the second, gravity; and the third, surface tension.
Analysis A linear stability analysis of the above equation predicts onset of the instability at
2G
M=
3(l+F)'
which is smaller than predicted by the one-layer model. As d/d_ increases, the difference between
the two theories becomes more pronounced. A weakly nonlinear analysis of the evolution equation
reveals that the bifurcation is subcritical for all BL and F, with squares being more unstable than
rolls. We solved for all the steady states in one dimension (rolls) and discovered that no stable,
deformed states exist; that is, the bifurcation curve continues backwards in M, never turning over
in a saddle-node bifurcation. The shape of the (unstable) steady states changes at F = 1/2, and
thus the two-layer model suggests a qualitative change in behavior at F = 1/2. The two-layer model
reduces to the one-layer model when d/d_ = 0 and thus F _< 0, so the one-layer model cannot make
this prediction.
Numerical Simulations We performed numerical simulations of the evolution equation, both
with a one-dimensional and a two-dimensional surface. The simulations employed a pseudospectral
method to handle the extreme nonlinearity of the equations. Because of the fourth-order nonlinearity,
a 2/5 rule was required to prevent aliasing (e.g., for 256 spatial locations in one dimension, only
103 spectral modes from -51 to +51 were used). At each time-step, the power in the remaining
3/5 of modes was exponentially damped. We used periodic boundary conditions (a square box in
two-dimensions) with a Fourier series as a basis set. The simulations find the same bifurcation curve
as the analytical theory found. When the system is unstable, it forms either a dry spot (F < 1/2)
or an elevated region (F > 1/2), as Fig. 4 illustrates. The evolution equation gives good qualitative
agreement to the experiments for F < 1/2 (see Fig. 5).
267
CONCLUDING REMARKS
Terrestrial experiments produce the first observation of a long-wavelength deformational insta-
bility in surface-tension-driven Marangoni convection. The experimental results do not agree with
a one-layer model in the literature, but do agree with a two-layer model for certain liquid depths.
The source of the deviation from the two-layer model is being investigated.
The two-layer model predicts a qualitatively different state, one with an elevated region, for
sufficiently large d/da (F > 1/2); we are currently testing this prediction. Employing gases other
than air (e.g., helium) allows us to vary F without changing the geometry of the experiment or the
liquid layer thickness.
ACKNOWLEDGMENTS
We thank S.H. Davis and R.E. Kelly for helpful discussions. This research is supported by
the NASA Microgravity Science and Applications Division (Grant No. NAG3-1382). S.J.V.H. is
supported by the NASA Graduate Student Researchers Program.
REFERENCES
[1] H. B4nard, Rev. G4n. Sci. Pure Appi. 11, 1261 (1900).
[2] M. F. Schatz, S. J. VanHook, W. D. McCormick, J. B. Swift and H. L. Swinney, Phys. Rev.
Lett. 75, 1938 (1995).
[3] J. R. A. Pearson, J. Fluid Mech. 4, 489 (1958).
[4] L. E. Scriven and C. V. Sternling, J. Fluid Mech. 19, 321 (1964).
[5] K. A. Smith, J. Fluid Mech. 24, 401 (1966).
[6] S. J. Van Hook, M. F. Schatz, W. D. McCormick, J. B. Swift, and H. L. Swinney, Phys. Rev.
Lett. 75, 4397 (1995).
[7] M. Takashima, J. Phys. Soc. Japan 50, 2745 (1981).
[8] D. A. Goussis and R. E. Kelly, Int. J. Heat Mass Trans. 33, 2237 (1990).
[9] C. Pfirez-Garc_a and G. Carneiro, Phys. Fluids A 3, 292 (1991).
[10] S. H. Davis, in Waves on Fluid Interfaces, edited by R. E. Meyer (Academic Press, New York,
NY, 1983), p. 291.
[11] S. H. Davis, Annu. Rev. Fluid Mech. 19, 403 (1987).
[12] B. K. Kopbosynov and V. V. Pukhnachev, Fluid Mech. Soy. Res. 15, 95 (1986).
[13] A. Oron and P. Rosenau, J. Phys. II (France) 2, 131 (1992).
[14] M. F. Schatz & K. Howden, Exp. in Fluids 19,359 (1995).
[15] A. A. Golovin, A. A. Nepomnyashchy and L. M. Pismen, Phys. Fluids 6, 34 (1994).
268
Agz
Gas cold _dg
....... .......-----ti
Liquid hot h(x,y,t) |
._ , ................. d
_, L ""
Fig. 1. Sketch of Mara, ngoni convection experiment. Silicone oil (depth d) is healed from below
and is bounded above by an air hayer (depth d,). The air layer is cooled from above and the mean
temperature drop across the liquid layer is AT. In the presence o[ deformation, a temperature-
gradient-induced surface-tension gradient pulls liquid against gravity up the interface. The liquid
has density p, kinematic viscosity u, thermal conductivity k, thermal diffusivity _, and temperature
coefficient of surface tension Cry - Idcr/dT {.
increasing G = gd3/m_
l"ig. 2. Infrared images from experiments with increasing liquid depth (increasing G). Left: For d
= 0.011 cm, the long-wavelength mode is the primary instability. The dark region is the dry spot.
Center: At d = 0.022 cm, the long-wavelength and hexagonal modes coexist. A droplet (white circle)
is trapped within the dry spot (dark oval); the liquid layer is strongly deformed (white annulus)
between the dry spot and the hexagonal pattern. Right: For d > 0.025 cm (in air), hexagons are
the primary instability (here d = 0.033 cm).
269
0.8
M
G( I+H)
0.4
' ' ' ' I ' ' ' ' I ' ' ' ' I ' ' ' '
• ° ........................................... "
0.8
M(I+F)
G
O4
0 .... I .... I .... I .... 0
0.05 0.10 0.15 0.20 0.25 0.05
'''' I ' ' ' ' I ' ' ' ' I ' ' '+
,I , , , , I,,,,I .... 11111
O.lO 0.15 0.20 ).25
d (mm) d (mm)
Fig. 3. Comparison of the observed instability onset to both (left) one- and (right) two-layer linear
stability theory (dashed line). Circles correspond to dry spots, triangles to dry spots with hexagons.
For d > 0.18 mm, tile circles are from experiments in which tie, not air, was used as the gas. The
data agree well with the two-layer theory for thick liquid depths, but not for thin liquid depths.
2
0 0
0 L/2 x L 0 L/2 x L
Fig. 4. The height of the layer as a function of position in the cell as determined by one-dimensional
numerical simulations for two values of 1,' at 0.33% above the onset of linear instability. Loft:
F = 1/3, BL = 30. A dry spot fornls where the <tepth of lhe li<lui<t nearly approaches zero in the
drained region. Right: P' = 2/3, I_L = 30. The system forms a lo<'alize<l elevate<l region instead of
a dry spot. It continues lo evolve until it 'hits' the top plale.
Experiment Simulation
Fig. 5. l)ry SF,Ol in t,oth eXF,<,rintetlt and tlum+,t'i<'al situ,tlati_m. Tile, exl:,+'rimeltlal pi<ttlr+, shrw,:s
the measured brightness temF,eralure as a function of l:,¢_siti¢>tla]oI,_ lit<' itJt_r'rfa¢',_,about 10</, above
onset of instability; d = 0.011 <'m (1",-+ 0.2). Numerical sitll,tlati¢m ofa I<>tig-waw,h,llgth evoluli<>n
e<lttation (1" = O, I_t+ --- 30) shows the depth as a ['lrtwti¢m CJ['p.+iti<m for 1_/+ab<>ve ott,_<,t ¢>1"litwar
instabilily. The depth of tit<' liquid apF,r<:,acheszero ill the _lraiT,e<Jtee)on.
270


Long-Wavelength Instability in Marangoni Convection

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