Analysis of the dynamic behavior of suspended mi-crostructures in fluid flows is a classical subject offluid mechanics.1–9 In particular, laser-inducedthermoconvective flow in emulsions was recentlystudied.10–12 Oil-in-water �O�W� emulsions �whoseviscosity is much lower than that of the dispersedphase alone� were developed to allow transportationand commercialization of heavy hydrocarbons.13–17
Besides, O�W emulsions can be burned directly byheating without further treatment. This gave riseto an alternative source of energy, which in Venezu-ela became known as Orimulsion.17 The stability ofO�W emulsions �which in static conditions is cur-rently assured by the addition of surfactants� is de-stroyed in extreme dynamic conditions because of thestrong liquid temperature and velocity gradients. Itis therefore important to avoid emulsion coalescenceand to study physical phenomena giving rise to ag-glomeration and gluing of petroleum droplets sus-pended in the water matrix. Also the inverseproblem �consisting in gathering together petroleumdroplets accidentally dispersed in, for example, oce-anic water� has a definite ecological and environmen-
G. Da Costa �[email protected]�, J. E. Parra, and F. Mosqueda arewith the Departamento de Fısica, Universidad Simon Bolivar,Apartado Postale 89000, Caracas 1080-A, Venezuela.
Received 5 November 2001; revised manuscript received 19 July2002.
tal interest. The aim of the experiments discussedin this paper is twofold.
�a� To record simultaneously the temperature dis-tribution and the liquid flow in laser-heated samplesof crude and refined oils as a function of heating time�Section 2�.
�b� To analyze the laser-induced thermoconvectiveflow in O�W emulsions �Section 3�. In spite of thesteady nature of the heat source, a quasi-periodictime variation of the reflected light beam is observed.A theoretical model presented in Section 4 proposesan explanation for this phenomenon. Additional re-marks and conclusions are presented in Section 5.
2. Laser Heating of Oil Samples
Thermoconvective flow induced in liquid surfaces bylaser heating was studied by several authors.18–29
In particular, experiments with samples of crudeoils heated by He–Ne laser beams have beenpresented.18–22 The main physical phenomenon in-volved in these experiments is the Marangoni ef-fect.1,2 Because the liquid surface tension is adecreasing function of the liquid temperature, a sur-face flow that leads the liquid from warmer to coolerregions increases whenever a temperature gradientis established. As a result the surface profile is de-formed and develops an upward concave shape in theheated region.1,2,18–27 The curved liquid surface actsas a smooth concave mirror for the incoming lightbeam, thus focusing the reflected light beam in itsneighborhood.18,19 This is a typical self-focusing ef-fect of a low-power laser beam that is due to interac-tion with a light-sensitive material medium. Crude
oil samples are particularly sensitive to the Ma-rangoni effect. When the samples are irradiatedwith He–Ne lasers, a reflected beam divergence of theorder of 1 rad is observed.18–22 This is due to thestrong dependence of the surface tension of crude oilson liquid temperature. The temperature distribu-tion in the laser-heated liquid surface was never mea-sured in preceding experiments.18–22 The curvedsurface profile could be measured only by interfero-metric methods21 because the diameter of the heatedregion is smaller than 2 mm. The aim of experi-ments discussed in this section is to extend the pre-ceding results to include other oils less sensitive tothe Marangoni effect, which requires utilization ofmore intense laser beams. In addition, the temper-ature distribution in the liquid surface is measuredfor the first time to our knowledge.
The experiment sketched in Fig. 1 consists of irra-diation of sample S of multigrade motor oil �SUPRASJ SAE 15W-40� with a CO2, cw, 40-W laser �L in Fig.1� working at 10.6 �m. The temperature distribu-tion in the irradiated surface is recorded as a functionof heating time by means of an infrared camera sys-tem �FLIR Systems ThermaCAM SC500, labeled Thin Fig. 1� and processed by ThermaCAM researchsoftware. The large diameter of the deformed liquidregion �approximately 8 cm� allows the surface profileto be coded by classic fringe-projection techniques.A parallel fringe system issued from a slide projectoris projected onto screen F. Fringes are normal to theplane of the drawing. The fringe system is imagedand recorded with a digital video camera Ph thatoperates at 30 frames�s after reflection in the sample-free surface. Therefore, the deformed fringe systemrecorded by the camera follows the correspondingcurvature of the liquid surface. The laser-heated re-gion is circular. The elliptical shape observed in Fig.2 is due to inclination of the photographic and ther-mographic cameras. Horizontal triads of pictures inFig. 2 correspond to different instants �t � �t, �t �
0.5 s� of heating time. Rows �a�–�j� correspond, re-spectively, to t � 0, 0.36, 5.17, 14.04, 39.63, 43.77,44.07, 47.71, 69.40, 112.47 s. The three frames ineach row present, respectively, the fringe-coded sur-face profile, the color-coded temperature distribution,and a plot of the temperature distribution along astraight line that passes through the center of theheated region. In the latter graphs the minimumand maximum attained temperature values are 20 °Cand 160 °C, respectively. The vertical bar on theright-hand side shows the color-coded temperature.The width of each frame corresponds in reality to 12cm on a liquid surface. The laser beam is inter-rupted at time t � 43.70 s, immediately precedingrow �f �. Figures 2�f �–2�j� show the subsequent tem-perature distribution and surface profile relaxation.
Preceding records reflect the different phases ofdevelopment of the thermoconvective flow.1,2 In theinitial phase a centrifugal stress distribution that isdue to the temperature dependence of the surfacetension grows up around the central hot point. Thisgives rise to a centrifugal flow in the liquid surface.In the following phase the liquid circulates back tothe central region, traveling under the liquid surfaceand thus establishing a closed convective motion �Fig.3�. The cool return flow emerges at the central re-gion, thus counteracting the heating action of thelaser beam. A limited temperature distribution andflow pattern are then asymptotically attained. Thiseffect has been labeled as autoblocking28,29 since itdevelops as a result of self-regulation of the system.The cooling effect of the return flow is clearly notedwhen the laser beam is interrupted �Figs. 2�f � and2�g��. After this instant of time the upwelling coolliquid still flows in the central region for a certainamount of time. Therefore the liquid temperatureat the central region suddenly decreases �Fig. 2�g��and afterward it returns to the equilibrium state�Figs. 2�h�–2�j��.
The second experiment �Fig. 4� consists of irradia-tion of a sample of crude oil with a 30-mW, He–Nelaser beam focused at its open free surface. Notethat crude oil is the dispersed phase of O�W emul-sions. Photographic and thermographic records arepresented in Fig. 5. Horizontal triads of pictures inFigs. 5�a�–5�f � correspond to different instants ofheating time �t � �t, �t � 0.5 s�. Rows �a�–�f � cor-respond, respectively, to t � 0, 0.26, 2.13, 4.83, 18.85,46.24 s. The three frames in each row represent,respectively, the light intensity distribution in theobservation plane �O�, the color-coded temperaturedistribution, and a plot of the temperature distribu-tion along a straight line that passes through thecenter of the heated region. In the latter graphs�third column� the minimum and maximum attainedtemperature values are 24 °C and 44 °C, respectively.The width of each frame in the second and thirdcolumns corresponds to 5 mm on the liquid surface.The diameter of the reflected beam that appears inthe first column of frame �f � corresponds to 5 cm inthe observation plane �O�. The divergence of thereflected light beam is approximately 1�8 rad. The
Fig. 1. Sketch of the experimental setup: S, a motor oil sample;L, a cw CO2 40-W laser beam incident upon the sample surface; Ph,a video camera that works at 30 frames�s to record the image of thefringe system by previous reflection on the liquid surface; Th, athermographic camera that works at 4 frames�s to record thetemperature distribution on the sample surface; F, a screen wherea rectilinear, parallel fringe system is projected by a slide projector.Fringes are normal to the plane of the drawing.
vertical bar at the lower right-hand side representsthe color-coded temperature. Laser heating is inter-rupted immediately after frame �f � �t � 46.31 s�.The three pairs of pictures in rows �g�–�i� follow the
interruption of the laser beam and correspond to t �47.04, 50.85, 57.66 s. Interruption of the laser beamimpedes observation of the subsequent contraction ofthe liquid surface. Bear in mind that the surface
Fig. 2. Horizontal triads of pictures that correspond to different instants of heating time in the setup of Fig. 1. Details on scales andinstants of time that correspond to different frames are given in the text. The frames in each row present, respectively, the fringe-codedsurface profile, the color-coded temperature distribution, and a plot of the temperature distribution along a straight line that passesthrough the center of the heated region. The vertical bar at the right-hand side shows the color-coded temperature. Laser heating isinterrupted at t � 43.70 s immediately before frame �f �.
contraction is observable when an independent ex-ploratory light beam is used �Figs. 2�f �–2�i��. Notethat the sudden temperature decrease following in-terruption of the laser beam in Figs. 2�f � and 2�g� isnot observed in the corresponding frames in Figs. 5�f �and 5�g� of the current experiment. This could bedue to the fact that the viscosity of crude oils is muchhigher than the viscosity of refined oils, which con-siderably slows down the velocity of liquid upwellingat the central hot region.
Thermographic and topographic records in Figs. 2and 5 present for the first time, to our knowledge,simultaneous data sets that relate the temperaturerise and the corresponding surface deformation indifferent kinds of oil heated at different laser power.Similar experiments conducted with O�W emulsionsare presented in Section 3.
3. Laser-Heating of Oil-in-Water Emulsions
In forthcoming experiments samples of Orimulsion17
are irradiated at their open free surface by a 30-mWHe–Ne laser beam. In the first experiment the re-flected light intensity distribution is recorded in asetup similar to the one sketched in Fig. 4, with theonly difference being that the incident laser beam isnow unfocused. The concentration of the emulsionis diminished from the usual commercial value of70% to 20% by the addition of water and a surfactant.The resulting thermoconvective flow then proceeds at
low speed. The resulting photographic sequence inFig. 6 extends along 1 min and then repeats itselfcyclically. Circular light intensity maxima are im-ages of oil droplets driven by the thermoconvectiveflow and slowly upwelling along the laser beam axisin the emulsion bulk. Note the regular geometricpatterns adopted by the droplets. When the drop-lets arrive at the water-free surface �last frame� theycoalesce because of the direct heat from the incominglaser beam, which drastically reduces their viscosityand surface tension, and extend laterally. Theythen contribute to the progressive formation of afloating layer of crude oil.
In the next experiment the emulsion concentrationis 70% and the He–Ne laser beam is focused at theemulsion-free surface. This provokes an intensethermoconvective flow. A large number of oil drop-lets now move at a much higher speed. A photo-graphic sequence of the light intensity distribution inthe reflected light beam is presented in Fig. 7. Thefirst frame in Fig. 7 corresponds to the initial instantof heating time �t � 0�. The central light intensitymaximum corresponds to the laser beam reflected atthe water-free surface. The elapsed time interval inFigs. 7�a�–7�h� is approximately 5 min. Figure 7�c�–7�h� show the sudden appearance of a dynamicspeckle pattern30–36 that is due to coherent superpo-sition of wavelets backscattered from rapidly movingoil droplets. Light speckles follow radial trajectoriesin the observation plane. Each individual lightspeckle has a limited lifetime. Figures 7�i�–7�l� cor-respond to the final heating stage �t 15 min� andare taken at time intervals of approximately 5 min.The peculiar concentric system of interference fringesthat appear in Figs. 7�i�–7�l� remains at rest in thefinal heating stage. Comparison of this interferencepattern with the similar light intensity distributionin the light beam reflected from a sample of crude oil�Fig. 5� proves that a floating oil layer with increasingthickness increases on the emulsion surface. Pro-gressive lateral expansion of the oil layer is also ob-served with the naked eye. Also a periodic timevariation of the oil layer thickness is revealed by theintermittent appearance of circular interferencefringes in the central region of the observation plane�Fig. 8� that are due to coherent superposition of laserbeams reflected at the upper and lower surfaces of thefloating oil layer. Those interference patterns ap-pear only in the initial heating stage, when the thick-ness of the oil layer is still of the order of magnitudeof some micrometers. In the final heating stagelight absorption by the progressively thicker oil layerimpedes efficient reflection of light from the lowersurface. Only the light pattern shown in the lastframes of Fig. 7, which is due to light reflection in theupper deformed surface of the oil layer, remains.
An outstanding feature of the preceding experi-ment is that both the contrast and the brightness ofthe dynamic speckle pattern present an obvious puls-ing behavior when observed with the naked eye. Itseems reasonable to suspect that this could be due toa periodic time variation of both the speed of the
Fig. 3. Schematic representation of the thermoconvective flowinduced in liquid sample S by irradiation with laser beam L in theexperiment of Fig. 1. The liquid flows radially outward from thecentral hot region in the liquid-free surface and returns to thecentral point to close the loop.
Fig. 4. Sketch of the experimental setup: S, a sample of crudeoil; ib, a He–Ne 30-mW laser beam focused on the sample surface;rb, the light beam reflected at the sample surface; M, a mirror; O,an observation plane; Ph, a video camera that works at 30frames�s; Th, a thermographic camera that works at 4 frames�s.
upwelling oil particles and the oil layer thickness.Two sets of measurements were performed to providequantitative support to this statement.
It was shown by several authors that the contrastof a photograph of a dynamic speckle pattern is di-rectly related to the mean speed of the scatteringparticles.30–36 Indeed when scattering particles re-main at rest, so do the reflected light speckles. Inthat case a photograph of the reflected light fieldtaken with a given exposure time is formed by highlylocalized light intensity maxima surrounded by darkregions. In contrast, when scattering particles and
light speckles undergo a given displacement duringthe exposure time, the time-integrated light intensitydistribution is blurred. Consequently the contrast�C� of the speckle pattern �conventionally defined asthe ratio between the standard deviation and themean value of the light intensity distribution, bothmeasured in each photographic frame� is a decreas-ing function of the mean speed of the scatteringparticles. Correspondingly the time variation fre-quency of the speckle pattern contrast is the same asthe time variation frequency of the mean speed of theoil particles. Contrast �C� is numerically calculated
Fig. 5. Horizontal triads of pictures correspond to different instants of heating time in the setup of Fig. 4. The three frames in each rowrepresent, respectively, the light intensity distribution recorded in observation plane O, the color-coded temperature distribution on thesample surface, and a plot of the temperature distribution along a straight line that passes through the center of the heated region on thesample surface. The vertical bar at the lower right-hand side shows the color-coded temperature. Details about the scales and theinstants of time that correspond to different frames are given in the text. Laser heating is interrupted immediately after frame �f � at t �46.31 s. The pairs of pictures in frames �g�–�i� record only temperature data.
by computer software in each film frame and is plot-ted in Figs. 9�a�–9�c� as a function of time.
Light intensity is spatially integrated in a rectan-gular region that covers the central light intensitymaximum and is represented as a function of time inFigs. 10�a�–10�c�. Note the spiking, quasi-periodicstructure of the plots in Figs. 9 and 10. The number�N� of bursts that appear per unit of time is repre-sented in Fig. 11 as a function of previous heatingtime. Finally, time-resolved records of the temper-ature distribution in the emulsion surface are shownin Fig. 12. The horizontal pairs of pictures in Figs.12�a�–12�f � correspond to different instants of heat-ing time �t � �t, �t � 0.5 s�. Figures 12�a�–12�f �
correspond, respectively, to t � 0, 5, 15.01, 44.04,47.64, 50.45 s. They were recorded in the setup ofFig. 4, where �S� is an Orimulsion sample with a 70%concentration and the laser beam is focused on theliquid surface. The frames in each row of Fig. 12represent the color-coded temperature distributionand a plot of the temperature distribution along astraight line that passes through the center of theheated region. In the latter graphs, the minimumand the maximum temperature values are 22 °C and35 °C, respectively. The width of each frame corre-
Fig. 6. Negative photographic sequence �1 frame�5 s� recorded ina setup similar to the one in Fig. 4, except that the 30-mW He–Nelaser beam is unfocused and S is a sample of O�W emulsion with20% concentration. The round intensity maxima with regulargeometric patterns are images of oil droplets that emerge from thelaser-heated region. Coalescence and lateral expansion of oildroplets occur when the droplets arrive on the water-free surface�frame L� because of the direct heating action of the laser beam.The sequence then restarts as in frame A.
Fig. 7. Positive photographs recorded in the setup of Fig. 4, wheresample S is an O�W emulsion with 70% concentration. Frames�a�–�h� cover the initial heating stage �heat time t � 5 min� atapproximately equally spaced time intervals. Frames �i�–�l� coverthe final heating stage �t 15 min� at 5-min intervals betweenconsecutive frames.
Fig. 8. Photographic sequence �taken at 30 frames�s in the setupof Fig. 4� of the interference fringe system periodically reflectedback from the O�W emulsion with 70% concentration.
Fig. 9. Contrast of the speckle pattern backscattered from theO�W emulsion with 70% concentration in the experiment of Fig. 4.The sequences in �a�–�c� start at heating times of t � 0, 15, and 30min, respectively. The horizontal axis in each figure representsthe number of video frames �at 30 frames�s� that have elapsedafter the initial heating time.
sponds to 5 mm of the liquid surface. Laser heatingis interrupted immediately before frame �e� at t �47.04 s.
Preceding experiments proved that O�W emul-sions as well as crude oil samples are extremely sen-sitive to laser heating, even at the low-powerconcentration attained with He–Ne lasers. A com-parison of Figs. 5 and 12 shows that the time requiredfor a rise in temperature to its final value on the onehand and its later decay to its initial value on theother hand are of the same order of magnitude �ap-proximately 40 and 10 s, respectively� in crude oilsand O�W emulsions, which is due to formation of alayer of crude oil on the emulsion surface. The tem-perature distribution in Fig. 12 really corresponds toan upper oil layer, not to the underlying water sur-face. This statement reinforces the idea that the oillayer plays a major role in observed phenomena,which are theoretically discussed in Section 4.
4. Theoretical Interpretation of Observed Phenomena
Certain mechanically or thermally forced hydrody-namic systems exhibit periodic behavior when thereis no obvious related periodicity in the forcing pro-cess. Correspondingly, some systems of differentialequations designed to represent hydrodynamic sys-tems possess periodic solutions when the forcing is
strictly constant.37,38 A typical phenomenon of thiskind was discussed in Section 3. The intensity andcontrast of the speckle pattern backscattered from anO�W emulsion show a spiking, quasi-periodic behav-ior in spite of the steady nature of the external heat-ing process. It is thus worth it to study theunderlying physical processes that give rise to spon-taneous oscillation. In addition, we bear in mindthat surface light scattering �the basic experimentaltechnique used in Sections 2 and 3� is a powerful toolrecently used by other authors to study the dynamicbehavior of liquid interfaces.39–42 It allows nonin-vasive measurement of interfacial parameters andpropagation of capillary waves,39 even in crude oils42
of the kind studied in this paper.In our actual case and based on the experiments
presented in Section 3, it seems reasonable to inferthat intermittent spikes in the speckle pattern inten-sity and contrast are closely related to two phenom-ena: �a� the time variation of the floating oil layerthickness and �b� the discrete nature of the arrivalof oil droplets on the water-free surface. Analysis ofthe latter aspect requires a study of the motion ofindividual oil particles �of different diameters andplaced at different initial depths� driven by the con-vective flow, and a statistical analysis of their arrivaltimes at the water surface, which is beyond the scopeof this paper. Only the role played by time variationof the oil layer thickness is considered. The up-welling oil flow is thus considered as a continuousvariable. The following recurrent mechanism is pro-posed within this restricted frame. In the earlystage heat absorption from the laser beam produces athermoconvective flow that drives oil particles up-ward. At the water-free surface their coalescencegives rise to a floating oil layer with time-increasedthickness. In the following stage partial absorptionof the laser beam energy by the oil layer diminishesthe intensity of light that passes into the underlyingemulsion and consequently reduces the intensity ofthe upwelling flow. The process thus slows down.But lateral spreading of the oil layer �because of grav-ity capillary waves and the Marangoni effect, both
Fig. 10. Light intensity �arbitrary units� in the neighborhood of the central point of the backscattered light beam as a function of time.The sample is the O�W emulsion with 70% concentration in the setup of Fig. 4. Light intensity is spatially integrated in a rectangularregion centered in the central point of the light pattern. The sequences in �a�–�c� start at heating times of t � 0, 15, and 30 min,respectively. The horizontal axis in each figure represents the number of frames �at 30 frames�s� that have elapsed after the initialheating time.
Fig. 11. Number N of bursts counted in each plot of the kindshown in Figs. 9 and 10 �thus corresponding to 1 min of recordingtime� as a function of an initial heating time �t� represented inminutes on the abscissa axis.
illustrated in Section 2� now reduces the oil layerthickness, thus again increasing the intensity of lightthat enters the emulsion bulk, and the whole processrestarts from similar initial conditions. The math-ematical expression of this self-sustained process is
�h� x, t���t � � x, t� � tA� x�, (1a)
� � x, t���t � � exp��h� x, t����, (1b)
where
A� x� � � exp��� x�a�2�. (1c)
The structure of Eqs. �1a�–�1c� is discussed below.
�a� In Fig. 13 h�x, t� is the oil layer surface profilemeasured from the underlying water surface �Ox axisin Fig. 13� and ��, �� are positive constants, freeparameters of the model.
�b� Function �x, t� represents the upwelling flowof oil droplets, which are represented as filled circlesin Fig. 13.
�c� Equations �1a� and �1b� constitute a system ofcoupled, ordinary differential equations with un-known quantities h, and an independent variable t.Abscissa x in Fig. 13 plays the role of a parameter.
�d� Quantity � characterizes light absorption by theoil sample. In the current experiment we have � �1 �m.
�e� The time variation of h�x, t� in Eq. �1a� is due tocompetition between �x, t� �representing the up-welling flow� and the depleting term ��tA�x��. Thelatter represents lateral spreading of the oil layerthat is due to gravity capillary surface waves and tocentrifugal thermocapillary flow. All these phenom-ena �illustrated in Section 2� evolve at a higher speed,the lower the oil viscosity, which is in turn a steeplydecreasing function of the oil temperature. Theterm t exp���x�a�2� is a first-order approximation tothe temperature distribution that exists in a laser-heated slab of a solid material �see Appendix A�.
�f � The upwelling flow �x, t� in Eq. �1b� is assumedto grow in time proportional to the intensity of lightthat enters the emulsion, which is in turn propor-tional to exp��h�x, t����.
Replacing �x, t� from Eq. �1a� into Eq. �1b� thefollowing ordinary, second-order differential equationfor h�x, t� results:
h � � exp��h��� � A� x�. (2)
In Eq. �2� the dots over h represent time derivatives,and h stands for h�x, t�. At each surface point, thatis, for each value of abscissa x, Eq. �2� coincides with
Fig. 12. Left column: color-coded thermographic record of thetemperature distribution on the surface of the O�W emulsion with70% concentration heated by a He–Ne 30-mW laser beam accord-ing to the setup of Fig. 4. Right column: plot of the temperaturedistribution along a straight line that passes through the center ofthe heated region. The minimum and maximum temperaturevalues are 22 °C and 35 °C, respectively. The width of each framecorresponds to 5 mm of the liquid surface. The vertical bar rep-resents the color-coded temperature. Heating is interrupted im-mediately before frame �e� at t � 47.04 s.
Fig. 13. Black horizontal strip represents the oil layer that floatsupon the O�W emulsion. Dispersed oil droplets are representedby filled circles in the underlying region. Point O is the centralhot point, Oxh is a rectangular coordinate system, the abscissa axisOx lies upon the emulsion surface, and h�x, t� is the oil-layer-freesurface profile as a function of time.
the Newton equation that rules the one-dimensionalmotion of a point particle with unit mass and abscissah submitted to the action of a conservative force F�h�� � exp��h��� � A�x�. The corresponding potentialenergy is
U�h� � Ah � �� exp��h��� � A��1 � log���A��, (3)
where we set A � A�x�. Function U�h� is repre-sented in Fig. 14. The fictitious particle oscillates ina potential well between the extreme values of h � 0and h � hmax, which are the roots of the transcen-dental equation
Ah � ���1 � exp��h����. (4)
The oscillation period is
T � �2 �0
h max du����1 � exp��u���� � Au�1�2 . (5)
As A � A�x� the oil layer thickness oscillates at adifferent frequency �1�T in Eq. �5�� and with a differ-ent amplitude �hmax�2 from Eq. �4�� at different sur-face points. Equation �2� is solved with initialconditions h�x, 0� � h�x, 0� � 0 �equivalent to h�x, 0�� �x, 0� � 0 in Eqs. �1a� and �1b�� for differentvalues of abscissa x, thus obtaining h�x, t�. Figure15 shows the oil-layer-free surface profile at differentinstants of time. Surface waves are seen to propa-gate radially outward from the central hot point. Alight interference effect between light beams re-flected in both faces of a thin film now takes place ateach surface point. The field distribution E�x� at theoil-layer-free surface, resulting from interference be-tween �a� the light beam directly reflected at the oil–air interface and �b� the corresponding light beamreflected at the oil–water interface �thus passingtwice through the oil layer�, is numerically calcu-lated.
Figures 16�a�–16�f � represent the square module ofthe Fourier transform of E�x�, that is, the intensitydistribution of diffracted light in the far field at dif-ferent instants of time. The plots in Fig. 16 could becompared with the photographic sequence in Fig. 8.Figure 17�a� represents h�0, t�, the oil layer thicknessat the central hot point as a function of time. Figure17�b� represents the time variation of light intensityat the central point of the diffracted light beam.This plot could be compared with the experimentalplots in Figs. 10�a�–10�c�.
We conclude that the above model correctly pre-views spontaneous development of oscillations of thesystem in spite of the steady nature of the heatingprocess. On the other hand, although the model pre-views the development of strictly periodic oscillations�Fig. 17�b��, experimental data �Figs. 10�a�–10�c��show a quasi-periodic variation of reflected light in-tensity. Indeed, light intensity spikes in Figs.10�a�–10�c� are not shape invariant in time. How-ever, the Fourier transform of signals represented in
Fig. 14. Potential energy U�h� corresponds to fictitious force F�h��right-hand side of Eq. �2�� for the values of � � 100, � � 50, and� � 1. A fictitious particle �whose abscissa coincides with the oillayer thickness h oscillates within the potential well between theextreme values of h � 0 and h � hmax. At these points the kineticenergy is null and U�h� attains its maximum value. The abscissaof the equilibrium point, where F�h� � 0, is h � � log���A�, whichcorresponds to the ordinate of the inflection points of h�t� in Fig.17�a�.
Fig. 15. Oil-layer-free surface profile h�x� calculated from Eq. �2�at different instants of time �t� for the values of � � 100, � � 50,and � � 1. Only surface points with x 0 are represented. Thesurface profile is symmetric with respect to point x � 0. Note thatsurface waves propagate outward from the central point.
Fig. 16. Numerically calculated light intensity distribution �ar-bitrary vertical scale� that appears in the observation plane duringthe growth and early propagation of a wave packet in the liquidsurface �Fig. 15�. The sequence extends to 0.25 s at regular timeintervals, which is the numerical simulation of the transient fringesystem that appears in Fig. 8.
Figs. 9 and 10 show a pronounced maximum at acertain basic frequency, superposed with randomlyvarying, smaller amplitude harmonics. The dis-crepancy between exact periodicity predicted by themodel and quasi-periodicity shown by the experimen-tal data could be due to the fact that the model doesnot take into account the discrete nature of the up-welling oil flow. Bear in mind that �x, t� in Eq. �1�is assumed to be a continuous variable in the preced-ing analysis. In reality, oil arrives at the free sur-face in discrete packets, formed by droplets oraggregates. Model improvement taking this aspectinto account is the subject of current research.
4. Conclusions
The results of experiments performed with laser-heated samples of crude oil and O�W emulsions werepreviously published.12,18–22 We have reported thefollowing new results:
�a� Time-resolved thermographic analysis of sam-ples of crude oil and O�W emulsions heated by a cw30-mW He–Ne laser beam.
�b� Time-resolved thermographic and topographicanalysis of samples of motor oil heated by a cw 40-WCO2 laser beam.
�c� Time-resolved analysis of intensity and contrastof the speckle pattern backscattered from an O�Wemulsion.
�d� Photographic records of interference patternsreflected from the emulsion surface proved the for-mation of a floating oil layer with increased thicknessand lateral extension.
�e� Formulation of a theoretical model that showsthat coupling between the intensity of an upwellingoil flow and the oil layer thickness gives rise to prop-agation of surface waves in the oil-layer-free surfaceand could thus be responsible for observed oscillation
of the reflected light beam intensity. In close rela-tion with this result, it can be noted that active re-search on capillary wave propagation in liquidinterfaces is frequently reported in the currentliterature.43–46
The experiments we have discussed are a part ofthe research on physical phenomena that gives rise tophase separation in O�W emulsions. Our resultsclearly show that a homogeneous layer of crude oil isaccumulated on the emulsion surface because of thelaser-induced thermoconvective flow. Although thisconclusion cannot be extrapolated directly to morecomplex industrial setups, it strongly suggests thatspecial precautions should be taken to avoid coales-cence of the emulsion that is due to temperature andvelocity gradients that currently exist in petroleumpipelines and containers.
Conversely, the experiment also proved that laser-induced thermoconvective flow produces migration ofsuspended oil particles to the water-free surface andcould therefore be a useful technique to separate oilparticles from a water matrix.
Appendix A
The temperature distribution in a slab of materialheated by a cw Gaussian laser beam is given by19
T� x, t� � T0 �� x�a�2
1�t�t0
� x�a�2 duu exp�u�
� Ei���xa�
2� � Ei��� x�a�2
1 � t�t0� , (A1)
where Ei is the exponential integral function, a is thelaser beam radius, and T0, t0 are, respectively, thecharacteristic temperature and time constants of theheating process. The Taylor–McLaurin expansionof T�x, t� in the neighborhood of �t � 0� is
T� x, t� � T0
tt0
exp��� x�a�2� � O�t2�. (A2)
This research was conducted within the frameworkof Agenda Petroleo project 97003569 funded by theConsejo Nacional de Investigaciones Cientıficas yTecnologicas �CONICIT�, Universidad Simon Bolivar�USB�, and Petroleos de Venezuela �PDVSA�.
References1. L. Landau and E. Lifschitz, Mechanics of Fluids �Addison-
Wesley, Reading, Mass., 1959�.2. G. Levich, Physicochemical Hydrodynamics �Prentice-Hall,
Englewood Cliffs, N.J., 1962�.3. A. J. Szeri, S. Wiggins, and L. G. Leal, “On the dynamics of
suspended microstructure in unsteady spatially inhomoge-neous, two dimensional fluid flows,” J. Fluid Mech. 228, 207–241 �1991�.
4. T. Elperin, N. Kleerin, and I. Rogachevskii, “Self-excitation offluctuations of inertial particle concentration in turbulent fluidflow,” Phys. Rev. Lett. 77, 5373–5376 �1996�.
5. T. Elperin, N. Kleerin, and I. Rogachevskii, “Turbulent diffusionof small inertial particles,” Phys. Rev. Lett. 76, 224–227 �1996�.
Fig. 17. �a� Thickness h�0, t� of the oil layer at the central pointx � 0 as a function of time, as calculated from Eq. �2� for the valuesof � � 100, � � 50, and � � 1. �b� Light intensity �in arbitraryunits� that results from coherent superposition of the light beamreflected at the oil-layer-free surface and the corresponding lightbeam reflected at the oil–water interface. The time scale is thesame as in �a�. The calculation was performed at point x � 0.This plot is the numerical simulation of the corresponding exper-imental plots in Figs. 10�a�–10�c�.
6. B. Van Haarlem, B. J. Boersma, and F. T. M. Nieuwstadt,“Direct numerical simulation of particle deposition onto a free-slip and no-slip surface,” Phys. Fluids 10, 2608–2620 �1998�.
7. J. S. Marshall, “A model of heavy particle dispersion by orga-nized vortex structures wrapped around a columnar vortexcore,” Phys. Fluids 10, 3236–3238 �1998�.
8. M. Tirumkudulu, A. Tripathi, and A. Acrivos, “Particle segre-gation in monodisperse sheared suspensions,” Phys. Fluids 11,507–509 �1999�.
9. J. Blawzdziewicz, E. Wajnryb, and M. Loewenberg, “Hydrody-namic interactions and collision efficiencies of spherical dropscovered with an incompressible surfactant film,” J. FluidMech. 395, 29–59 �1999�.
10. N. V. Tabiryan and W. Luo, “Soret feedback in thermal diffu-sion of suspensions,” Phys. Rev. E 57, 4431–4440 �1998�.
11. W. Schaertl and C. Roos, “Convection and thermodiffusion ofcolloidal gold tracers by laser light scattering,” Phys. Rev. E60, 2020–2028 �1999�.
12. G. Da Costa, “Laser-induced coalescence of petroleum-in-water emulsions,” Opt. Commun. 149, 239–244 �1998�.
13. M. Salou, B. Siffert, and A. Jada, “Study of the stability ofbitumen emulsions by application of DLVO theory,” ColloidsSurf. A 142, 9–16 �1998�.
14. M. C. Sanchez, M. Berjano, A. Guerrero, E. Brito, and C.Gallegos, “Evolution of the microstructure and rheology ofO�W emulsions during the emulsification process,” Can.J. Chem. Eng. 76, 479–485 �1998�.
15. M. Van den Temple, “Stability of oil-in-water emulsions: mech-anism of the coagulation of an emulsion,” Recueil 72, 433–441�1953�.
16. G. Urbina-Villalba and M. Garcıa-Sucre, “Brownian dynamicssimulation of emulsion stability,” Langmuir 16, 7975–7985�2000�.
17. G. Urbina-Villalba and M. Garcıa-Sucre, “Effect of non-homogeneous spatial distributions of surfactants on the stabilityof high-content bitumen-in-water emulsions,” Interciencia 25,415–422 �2000�.
18. G. Da Costa and J. Calatroni, “Self-holograms of laser-inducedsurface depressions in heavy hydrocarbons,” Appl. Opt. 17,2381–2385 �1978�.
19. G. Da Costa and J. Calatroni, “Transient deformation of liquidsurfaces by laser-induced thermocapillarity,” Appl. Opt. 18,233–235 �1979�.
20. G. Da Costa, “Competition between capillary and gravitywaves in a viscous liquid film heated by a Gaussian laserbeam,” J. Phys. �Paris� 43, 1503–1508 �1982�.
21. J. Calatroni and G. Da Costa, “Interferometric determinationof the surface profile of a liquid heated by a laser beam,” Opt.Commun. 42, 5–9 �1982�.
22. G. Da Costa, “Optical visualization of the velocity distributionin a laser-induced thermocapillary liquid flow,” Appl. Opt. 32,2143–2151 �1993�.
23. T. R. Anthony and H. E. Cline, “Surface rippling induced bysurface-tension gradients during laser surface melting andalloying,” J. Appl. Phys. 48, 3888–3894 �1977�.
24. G. G. Gladush, L. S. Krasitskaya, E. B. Levchenko, and A. L.Chernyakov, “Thermocapillary convection in a liquid underthe action of high-power laser radiation,” Sov. J. QuantumElectron. 12, 408–412 �1982�.
25. J. Hartikainen, J. Jaarinen, and M. Luukkala, “Deformation ofa liquid surface by laser heating: laser-beam self-focusingand generation of capillary waves,” Can. J. Phys. 64, 1341–1344 �1985�.
26. S. A. Vizniuk, S. F. Rastopov, and A. T. Sukhodol’skii, “Onthermocapillary aberrational transformation of laser beams,”Opt. Commun. 71, 239–243 �1989�.
27. N. Postacioglu, P. Kapadia, and J. Dowden, “Capillary waves
on the weld pool in penetration welding with a laser,” J. Phys.D 22, 1050–1061 �1989�.
28. Al. A. Kolomenskii and H. A. Schuessler, “Nonlinear excitationof capillary waves by the Marangoni motion induced with amodulated laser beam,” Phys. Rev. B 52, 16–19 �1995�.
29. Al. A. Kolomenskii and H. A. Schuessler, “Excitation of capil-lary waves in strongly absorbing liquids by a modulated laserbeam,” Appl. Opt. 38, 6357–6364 �1999�.
30. J. W. Goodman, “Statistical properties of laser speckle pat-terns,” in Laser Speckle and Related Phenomena, J. C. Dainty,ed., Vol. 9 of Topics in Applied Physics �Springer-Verlag, NewYork, 1984�, pp. 9–74.
31. T. Okamoto and T. Asakura, “The statistics of dynamic speck-les,” in Progress in Optics, E. Wolf, ed. �North-Holland, Am-sterdam, 1995�, Chap. 3, pp. 183–248.
32. B. Ruth, “Superposition of two dynamic speckle patterns,” J.Mod. Opt. 39, 2421–2436 �1992�.
33. G. Da Costa, “Optical remote sensing of heartbeats,” Opt. Com-mun. 117, 395–398 �1995�.
34. J. E. Parra and G. Da Costa, “Optical remote sensing of heart-beats,” in Visualization of Temporal and Spatial Data for Ci-vilian and Defense Applications, G. O. Allgood and N. L. Faust,eds., Proc. SPIE 4368, 113–121 �2001�.
35. J. D. Briers and S. Webster, “Quasi real-time digital version ofsingle-exposure speckle photography for full-field monitoringof velocity of flow fields,” Opt. Commun. 116, 36–42 �1995�.
36. A. F. Fercher and J. D. Briers, “Flow visualization by means ofsingle-exposure speckle photography,” Opt. Commun. 37,326–330 �1981�.
37. E. N. Lorenz, “Deterministic nonperiodic flow,” J. Atmos. Sci.20, 130–141 �1963�.
38. N. Minorsky, Nonlinear Oscillations �Krieger, Huntington,N.Y., 1974�.
39. W. V. Meyer, G. H. Wegdam, D. Fenistein, and J. A. Mann, Jr.,“Advances in surface-light-scattering instrumentation andanalysis: noninvasive measuring of surface tension, viscos-ity, and other interfacial parameters,” Appl. Opt. 40, 4113–4133 �2001�.
40. J. A. Mann, Jr., P. D. Crouser, and W. V. Meyer, “Surfacefluctuation spectroscopy by surface-light-scattering spectros-copy,” Appl. Opt. 40, 4092–4112 �2001�.
41. D. Fenistein, G. H. Wegdam, W. V. Meyer, and J. A. Mann, Jr.,“Capillary waves on an asymmetric liquid film of pentane inwater,” Appl. Opt. 40, 4134–4139 �2001�.
42. R. B. Dorshow and R. L. Swofford, “Application of surface lightscattering spectroscopy to photoabsorbing systems: the mea-surement of interfacial tension and viscosity in crude oil,”J. Appl. Phys. 65, 3756–3759 �1989�.
43. H. Linde, P. D. Weidman, and M. G. Velarde, “Marangoni-driven solitary waves,” in Capillarity Today: Proceedings of anAdvanced Workshop on Capillarity, G. Petre, R. Kippenhahn, H.Araki, W. Beiglbock, D. Ruelle, R. L. Jaffe, J. Ehlers, K. Hepp,and A. Sanfeld, eds., Vol. 386 of Springer Lecture Notes inPhysics �Springer-Verlag, New York, 1991�, pp. 261–267.
44. M. G. Velarde, X. L. Chu, and A. N. Garazo, “Solitons and otherinterfacial waves excited and sustained by capillarity,” in Cap-illarity Today: Proceedings of an Advanced Workshop onCapillarity, G. Petre, R. Kippenhahn, H. Araki, W. Beiglbock,D. Ruelle, R. L. Jaffe, J. Ehlers, K. Hepp, and A. Sanfeld, eds.,Vol. 386 of Springer Lecture Notes in Physics �Springer-Verlag, New York, 1991�, pp. 108–112.
45. X. L. Chu and M. G. Velarde, “Nonlinear transverse oscillatorymotions at the open surface of a liquid layer subjected to theMarangoni effect,” Phys. Lett. A 136, 126–130 �1989�.
46. B. M. Grigorova, S. F. Rastopov, and A. T. Sukhodol’skii, “Co-herent correlation spectroscopy of capillary waves,” Sov. Phys.Tech. Phys. 35, 374–376 �1990�.
