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International Journal of Thermal Sciences 78 (2014) 101e110

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International Journal of Thermal Sciences

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Convective heat transfer due to thermal Marangoni flow about twobubbles on a heated wall

Séamus Michael O’Shaughnessy*, Anthony James Robinson**

Department of Mechanical & Manufacturing Engineering, Parsons Building, University of Dublin, Trinity College, Dublin 2, Ireland

a r t i c l e i n f o

Article history:Received 13 September 2012Received in revised form1 October 2013Accepted 7 December 2013Available online 18 January 2014

Keywords:MarangoniThermocapillaryBubblesNumericalHeat transferVortexEnhancement

* Corresponding author. Tel.: þ353 1 896 1061; fax** Corresponding author. Tel.: þ353 1 896 3919; fax

E-mail addresses: oshaughs@tcd.ie (S.M. O’Sh(A.J. Robinson).

1290-0729/$ e see front matter � 2014 Elsevier Mashttp://dx.doi.org/10.1016/j.ijthermalsci.2013.12.007

a b s t r a c t

Three dimensional simulations of thermal Marangoni convection about two bubbles situated on a heatedwall immersed in a liquid silicone oil layer have been performed to gain some insight into the thermaland flow interactions between them. The distance between the two bubbles’ centres was varied betweenthree and twenty five bubble radii to analyse the influence of the inter-bubble spacing on the flow andtemperature fields and the impact upon local wall heat transfer. For zero gravity conditions, it wasdetermined that the local wall heat flux was greatest for the smallest separation of three bubble radii, butthat the increase in heat transfer over the whole domain was greatest for a separation of ten bubble radii.When the effects of gravity were included in the model, the behaviour was observed to change betweenthe cases. At large separations between the bubbles, increasing the gravity level was found to decreasethe local wall heat flux, which was consistent with two-dimensional work. At small separations however,the increase in gravity led to an increase in the local wall heat flux, which was caused by a buoyancy-driven flow formed by the interaction of secondary vortices.

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1. Introduction

Thermal Marangoni convection, also known as thermocapillaryconvection, is caused by surface tension gradients along a gas/liquidor vapour/liquid interface, brought about by temperature gradientsbetween the bubble base and tip. Inmany situations where surfacesare immersed in a liquid layer and a heat flux is applied to thesurface, gas and/or vapour bubbles will form on the surface. As wasevident in the experiments performed by Petrovic et al. [1], it islikely that two or more bubbles are situated near one another onthe heater surface. In this instance the bubbles may exert someinfluence on each other, thereby affecting local heat transfer rates.Some investigations have been conducted on the convection abouttwo bubbles in close proximity on or near a heated or cooled sur-face, such as studies by Kasumi et al. [2,3] Nas [4], Sides [5] andWozniak and Wozniak [6]. However, little to no information isavailable concerning the influence of the thermocapillary convec-tion upon local heat transfer.

For the case of a bubble affixed to a heated surface, Larkin [7] waslikely the first to investigate the contribution of Marangoni

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convection to local heat transfer, obtaining time-dependent dimen-sionless numerical solutions of flow and temperature fields forPrandtl numbers of 1 and 5, and Marangoni numbers in the range0 � Ma � 105. The liquid was seen to move towards the wall beforebeing dragged along the bubble,finally leaving the bubble as a jet, thestrength of which increased with increasing Marangoni number anddecreased with increasing Prandtl number. Larkin investigated theinfluence of the surface-tension driven flow on the Nusselt number,concluding that thermocapillary flow increased heat transfer whencompared to heat transfer by conduction only, but this increase wasmodest. Increasing the Marangoni number increased heat transfer,but theNusselt numberwas insensitive to the Prandtl number. Larkinfound that above a Marangoni number of 105, an increase in the rateof heat transfer of 30% was achievable. It was assumed however thatuntil this critical Marangoni number was reached, thermocapillaryconvection was not an important heat transfer mechanism. Unfor-tunately, due to computational limitations of the time, Larkin wasunable to continue the solution to steady-state.

Straub [8] and Straub et al. [9] also performed numerical sim-ulations on a two-dimensional domain to numerically investigatethe effect of thermocapillary convection and gravity on fluid flowand heat transfer. Two different domains were studied;

� Two-dimensional gas bubble centred in a rectangular cross-section (due to symmetry only half of the problemwasmodelled)

S.M. O’Shaughnessy, A.J. Robinson / International Journal of Thermal Sciences 78 (2014) 101e110102

� Two-dimensional hemispherical bubble sitting on the lowerwall.

In both cases the gaseous phase was not modelled. The fluidproperties were taken to be the same as water at 365 K, with aPrandtl number of Pr ¼ 1.93. With the exception of density,whereby the Boussinesq approximation was implemented, anyvariance in water properties with temperature was not taken intoaccount because the maximum temperature difference wasDTmax¼ 3.4 K. The upper and lowerwalls had constant but differenttemperatures. Simulations were performed for pure thermocapil-lary convection and with buoyancy both assisting and opposingthermocapillary flow.

Straub concluded that at low Marangoni numbers, thermoca-pillary flow could be seen, but had negligible impact on heattransfer, probably due to the moderate fluid velocities. At someMarangoni number, the effect on heat transfer would becomenoticeable, and would increase significantly after this point.Furthermore, for pure thermocapillary convection at larger Mar-angoni numbers (Ma> 105), it was found that the strong increase inheat transfer would reduce due to the very high interface velocities.This would coincide with an oscillatory mode, the onset of whichoccurred at Ma > 2.5 � 105. Straub et al. noted that the maximumReynolds number in the simulations did not exceed Re ¼ 103,meaning the flow should remain laminar. Since numerical errorsfrom round-off, discretization and improper time-steps were alsoruled out, it was concluded that no direct relationship betweenReynolds number or the maximum velocity and the oscillationscould be derived.

Reynard et al. [10] experimentally studied the influence ofgravity level on the periodic thermocapillary convection around abubble. Using shadowgraphy to visual the flow field, experimentswere conducted in reduced and increased gravity conditions. An airbubble was injected onto a downward facing heater surfaceimmersed in a thermally stratified low Prandtl number silicone oil(Pr ¼ 16.7). It was determined that the temperature gradientrequired to drive the flow into the oscillatory mode increased whenthe bubble aspect ratio decreased, and for the same bubble aspectratio the critical temperature gradient (corresponding to a criticalMarangoni number) depended on the Prandtl number. Reynardet al. reasoned that lower Pr fluids, due to their lower viscosity, willinduce greater thermocapillary velocities, thus the threshold ismore rapidly reached.

It was also determined that the onset of the oscillatory modewas not influenced by gravity level; rather that the gravity levelaffected the bubble shape, main vortex size and oscillation period.Under terrestrial conditions, the oscillation was symmetrical, butunder reduced and increased gravity, the oscillation was found tobe asymmetrical. The frequency of oscillationwas also found to be adecreasing function of the gravity level. Increasing gravity levelscaused a flattening of the bubble and a squeezing in size of themainvortex, and the pushing of associated temperature fields towardsthe bubble.

Reynard et al. [11] subsequently continued their research todetermine the critical test conditions for the onset of oscillatorythermocapillary convection. Using a rig similar to that seen inRef. [12] the temperature gradient was fixed at a constant value.The influence of the Prandtl number on the threshold conditionswas investigated by using three different test fluids (silicone oilsPr ¼ 16.7 and Pr ¼ 228 and FC-72, Pr ¼ 12.3). It was discovered thatthe oscillatory thermocapillary convection did not occur for allthree liquids. In the case of the high Pr silicone oil, only the steadystate was observed. In the case of the low Pr silicone oil and theFluorinert FC-72, both steady and oscillatory states were observed.The dimensions of the test cell were alsomodified to investigate the

influence of confinement. Reynard et al. concluded that theoccurrence of the oscillatory mode was independent of theconfinement, and that different thresholds are induced only bythe operating conditions. Furthermore, the occurrence of theoscillation did notmodify the heat transfer. Heat transferwas foundto be modified by changing bubble size, thus inducing a larger/smaller thermocapillary vortex.

More recently, O’Shaughnessy and Robinson [13] investigatedthe influence of Marangoni number on the local wall heat fluxdistributions arising from thermocapillary convection about asingle bubble in zero gravity. Simulations were performed using atwo-dimensional axisymmetric grid for a 1 mm radius air bubblesituated on a heated wall of constant temperature immersed in asilicone oil layer (Pr ¼ 83) of constant depth 5 mm in the range0 � Ma � 915. The velocity and temperature fields were used toexplain the enhancement of local heat transfer in the vicinity ofthe bubble. The increase in the local and surface average heat fluxon the wall to which the bubble was attached was computed andit was determined that, compared with pure conduction, ther-mocapillary convection enhanced the local heat flux on the hotwall to over 65%. Furthermore, the enhanced heat transfer pene-trated a distance of approximately seven bubble radii along thehot wall, and four bubble radii along the cold wall. Cold wall localheat flux values were calculated to be up to 180% greater thanconductive heat flux values. The concept of an area of enhancedheat transfer around the bubble was introduced by the authors.The numerical results indicated that the ratio of Marangoni heattransfer to conduction over a constant area of enhancement dis-played a slightly exponential relationship under zero-gravityconditions. For the range of Marangoni numbers tested, animprovement in the average heat transfer of approximately 20%over an area extending approximately 6 bubble radii surroundingthe bubble along the hot wall was obtained. In subsequent in-vestigations a cold wall heat transfer enhancement of up to 90%was determined.

For a Marangoni number of Ma ¼ 915 using the same fluidproperties, O’Shaughnessy and Robinson [14] analysed the influ-ence of the magnitude of gravitational acceleration on the velocityprofile along the bubble interface and on the location of maximumvelocity. It was found that the gravity level only minimally affectedthe velocity profile on a small portion of the interface approachingthe bubble tip, and that the location of maximum velocity wasalmost independent of gravity level - behaviour also observed byArlabosse et al. [15]. Increasing the gravity level above a certainthreshold caused the formation of secondary vortices which pre-vented the primary vortices growing to full size, and limited theextent of the jet-like flow from the tip of the bubble. Therefore,increasing gravity levels caused a reduction in the effective radiusand area of enhancement. For Ma ¼ 915, maximum enhancementoccurred under zero-gravity conditions. Under terrestrial condi-tions, the improvement in the average heat transfer along the hotwall in the vicinity of the bubble was determined to be 13% lessthan that at zero-gravity. No significant enhancement was observedfor the cold wall above g0 ¼ 0.1 (for this study g0 represents thedimensionless gravity level, g/gearth).

Radulescu and Robinson [16] investigated the effect ofconfinement on thermocapillary convection under both earthgravity and zero gravity conditions. In their simulations, the heattransfer enhancement about a hemispherical bubble on a heatedwall was determined for Marangoni numbers in the range of100 � Ma � 3000 and for channel heights of 1.5 � H/Rb � 7.5. Forthemost confined cases the flow and heat transfer were found to bevery similar for both the g0 ¼ 0 and g0 ¼ 1 cases. The zero gravityresults were found to be highly sensitive to the domain height,which contrasted with the earth gravity results.

S.M. O’Shaughnessy, A.J. Robinson / International Journal of Thermal Sciences 78 (2014) 101e110 103

Raj and Kim [17] numerically investigated thermocapillaryconvection in reduced gravity environments, presenting a qualita-tive study on the effects of dissolved gas content, bubble shape andsize, and heat transfer coefficient on the strength of thermocapil-lary convection. Steady-state simulations were carried out tocalculate the flow around bubbles of various sizes with differentamounts of dissolved gas for subcooled boiling in reduced gravity.For a given heat transfer coefficient with increasing bubble size, itwas found that the strength of the thermocapillary convectionincreased before peaking and then decreasing. Furthermore, thepeakwas observed to occur at lower diameters when increasing theheat transfer coefficient. It was determined that an increase indissolved gas content promoted the interfacial temperaturegradient required for thermocapillary convection, but the simul-taneous increase in bubble size reduced the gradient along theinterface. Raj and Kim concluded that the strength of thermoca-pillary convection is an indirect function of dissolved gas content,which in turn influences variation in heat transfer coefficient, andconsequently the bubble shape and size.

Recently, O’Shaughnessy and Robinson [18] performed a com-bined numerical-experimental study of thermal Marangoni con-vection about a single hemispherical bubble. Tests were performedfor a range of Marangoni numbers (145 � Ma � 915) with varyinglevels of gravitational acceleration between zero gravity and earthgravity in order to quantify the rates of heat transfer. Experimentalvalidation of selected terrestrial gravity numerical results was ob-tained using particle image velocimetry (PIV) for low to moderateMarangoni numbers. For all experiments, steady state Marangoniconvection was observed. The experimental flow patterns showedgood agreement with the numerical solutions.

Most of the above numerical studies considered thermal Mar-angoni convection about a single bubble. This study details some ofthe results obtained from three-dimensional simulations of Mar-angoni convection about two bubbles on a heatedwall immersed insilicone oil. Particular focus is on the flow and temperature fieldsand how these are linked to the wall heat flux distributions.

Table 1Reference parameters.

Length Lref RbTemperature Tref (Th � Tc)(Rb/H)Velocity vref

ðds=dTÞðTh�TcÞRbmH

Pressure pref rv2refTime tref Lref/vref

2. Numerical formulation

The commercially available software package Fluent� version6.3.26 was utilized to solve the system of governing equations.Simulations were performed to investigate the influence of boththe thermocapillary driving potential (Ma) and the buoyancydriving potential (Ra) on the flow and temperature fields as well aslocal heat transfer profiles. For thermocapillary flow around abubble of radius Rb within a channel of height H, the mass and heattransport mechanisms are characterised by the Prandtl, Rayleigh,Marangoni and Bond numbers, defined respectively as [11,12,15,19]:

Pr ¼ y

a(1)

Ra ¼ grbðTh � TcÞR4bmaH

(2)

Ma ¼ ��dsdT

�� ðTh � TcÞ

ma� R2b

H(3)

Bo ¼ � rgbR2bðds=dTÞ (4)

In the above equations the terms r, m, y, a, and b represent thefluid density, dynamic viscosity, kinematic viscosity, thermaldiffusivity and volumetric expansion coefficient respectively. The

term g represents acceleration due to gravity and (ds/dT) thetemperature derivation of surface tension. The numerical modelassumes steady state for an incompressible fluidwith constant fluidproperties (except when using the Boussinesq approximation) andan adiabatic, non-deformable hemispherical bubble interface. Thegoverning equations of continuity, momentum and energy werenon-dimensionalized in accordance with the method used byArlabosse et al. [12] and more recently by O’Shaughnessy andRobinson [18]. The reference parameters are provided in Table 1.

Using these parameters, the steady form of the governingequations of continuity (Eq. (5)), momentum (Eq. (6)) and energy(Eq. (7)) can be expressed in dimensionless form, in which theprime symbol denotes a dimensionless quantity.

V0$v0 ¼ 0 (5)

ðv0$V0Þv0 ¼ �V0p0 þ PrMa

V02v0 þX

F 0 (6)

V0$�v0q0

� ¼ 1Ma

V02q0 (7)

In the dimensionless momentum equation, SF0 represents thesum of all other dimensionless forces acting on the fluid. This termincludes the buoyancy forces due to the presence of a gravitationalfield, which is modelled using the Boussinesq approximation.Consistent with the continuum surface force (CSF) model appliedby Fluent, the effects of surface tension are modelled by includingan extra source term in the momentum equation. Details of the CSFmodel can be found in Brackbill et al. [20]. For this study,

XF 0 ¼ �BoPr

Maq0by þ Pr

Ma

�dsdT

�0dq0

ds0k0bs (8)

In the above equation, by is a unit vector in the y-direction and bsrepresents a unit vector along the interface. The term k0 representsthe dimensionless interface curvature and is equal to kRb. To vali-date the three-dimensional numerical approach, preliminary sim-ulations were performed with a single bubble attached to a heatedwall. These results were compared to the solutions from a 2Daxisymmetric model that had been validated against some externalexperimental results in an earlier study [13,18]. The numericalmodel consisted of a hemispherical bubble of 1 mm radius centredin a cylindrical domain of 20 mm radius and 5 mm height. ForMarangoni numbers Ma < 12,000 it is known that the resultingflow field is symmetric about the vertical bubble axis above thebubble centreline [21,22]. Results from 3D simulations for ther-mocapillary convection about a single bubble reinforced thisapproximation. Fig. 1 plots some liquid pathlines (coloured by ve-locity magnitude) from one of the zero gravity 3D simulations.Some paths have been omitted for display purposes. A singletoroidal vortex develops around the bubble in this scenario.

Of particular interest is the effect of thermocapillary convectionon the local wall heat flux profiles. Fig. 2 compares the 2D and 3Dsolutions for the hot wall heat flux about a single bubble. Resultsare displayed for Ma ¼ 915 and g0 ¼ 1 which correspond respec-tively to the largest thermal gradient and gravitational force

Fig. 1. Typical flow structure for the single bubble zero gravity 3D simulations.

Fig. 2. 3D vs. 2D hot wall heat flux for Pr ¼ 83, Ma ¼ 915 at earth gravity.

S.M. O’Shaughnessy, A.J. Robinson / International Journal of Thermal Sciences 78 (2014) 101e110104

imposed on the system. The 2D axisymmetric model has previouslybeen validated against data from the literature in Ref. [13] andagainst some in-house experimental data in Ref. [18]. From Fig. 2 isit evident that the three-dimensional numerical model is capable ofpredicting the heat transfer characteristics.

Upon successful validation of the three-dimensional approach,the numerical domain was modified to include a second bubble, asshown in Fig. 3. The numerical domain places two bubbles of unitdimensionless radius at the centre of a three-dimensional domainof five bubble radii height and radius of twenty bubble radii.

The upper wall, to which the bubbles are attached has the no-slip velocity condition, and is maintained at the constant dimen-sionless temperature Th ¼ H/Rb for all simulations. This wall istermed the ‘hot’wall hereafter. The lower horizontal wall is also no-slip and the temperature of this wall is maintained at the dimen-sionless temperature of Tc ¼ 0 for all simulations, and is termed the‘cold’ wall henceforward. The bounding side walls of the domainhave no-slip, adiabatic boundary conditions. The bubble interface isadiabatic and non-deformable and the velocity boundary conditionalong the bubble interface is derived from the CSF model. The non-deformable assumption is valid for low Bond numbers and theadiabatic assumption is reasonable considering that the thermalresistance on the air side of the bubble is high since there is nophase change occurring.

Fig. 3. Slice plane showing the three-dimensional domain and boundary conditionsfor the two bubble simulations.

For the two bubble model, the spacing between the bubbles’centres, Sb, was varied between simulations to study the influenceof Marangoni convection and of the spacing between the bubbleson the local flow and temperature fields, as well as analysing anyenhancement of heat transfer. Although entrainment of bubbles viathermocapillary convection has been observed in some experi-ments such as those in Ref. [6], the bubbles in this study were fixedat specific locations for each simulation. This is consistent with theobservations of Petrovic et al. [1] where many naturally occurringbubbles, some of which were quite close to one another, remainedanchored in place in the presence of intense Marangoni flow.

To ensure accurate convergence of the solution, several ‘monitors’were performed. Strict convergence criteria for the continuity, mo-mentum and energy residuals were set, and the total rate of massflow and heat transfer were scrutinised closely. From the softwaredocumentation, it was noted that the calculation of surface tensioneffects on triangular and tetrahedral meshes is not as accurate as onquadrilateral and hexahedral meshes. Therefore, the mesh shown inFig. 4 consisted of hexahedral and tetrahedral mesh volumes whereapplicable, and was refined in the immediate vicinity of the bubble.Approximately 7 � 105 cells were required for a grid independentsolution. The test fluid was selected to have the same properties assilicone oil of kinematic viscosity 7.5 cSt, as provided in Table 2.

The inclusion of gravity caused some instability in the solution.These instabilities were addressed by under-relaxing the bodyforces term in the momentum equation, and slightly under-relaxing the energy equation. This increased the time required fora solution, but provided improved convergence of the residuals,and accurate tracking of the mass flow and heat transfer rates.

3. Results and discussion

Simulations were performed initially under zero gravity condi-tions for Marangoni numbers of Ma ¼ 183, 366, 550, 732 and 915respectively. The inter-bubble spacing, Sb, was measured as thedistance between the bubbles’ centres, and was varied betweensimulations. Solutions were computed for separations of 3Rb, 4Rb,5Rb, 7Rb, 10Rb, 15Rb, 20Rb and 25Rb.

3.1. Influence of Ma under zero gravity

Expectedly, thermocapillary convection increased in strengthwith increasingMa at each inter-bubble spacing under zero gravityconditions. The heat transfer characteristics were consistent withthe single bubble results published in Ref. [13]. For each bubbleseparation, the Ma ¼ 915 case resulted in the greatest heat transferenhancement. A sample plot for Sb ¼ 5Rb is provided in Fig. 5. Moreinteresting behaviour was observed when the bubble separationwas varied at a particular value of Ma, and with the inclusion ofgravity. These results will be discussed in turn.

Fig. 4. Plan view of the 3D grid & mesh refinement in the region of the bubbles.

Table 2Properties of silicone oil.

Dimensional Dimensionless

r 930 (kg/m3) r0 1y 7.5e-6 (m2/s) y0 (1/r0)*(Pr/Ma)k 0.125 (W/mK) k0 1Cp 1480 (J/kg K) Cp0 Mab 1.08e-3 (1/K) b0 (1/r0g0)*(BoPr/Ma)(ds/dT) �5.8e-5 (N/mK) (ds/dT)0 Pr/Mag 0 / 9.81 (m/s2) g0 0 / 1T 0 / 50 (�C) q0 0 (T � Tc)/(Th � Tc)*(H/R)

Fig. 6. Contours of hot wall heat flux enhancement at different bubble separations forPr ¼ 83, Ma ¼ 915, Bo ¼ Ra ¼ 0.

S.M. O’Shaughnessy, A.J. Robinson / International Journal of Thermal Sciences 78 (2014) 101e110 105

3.2. Influence of separation distance under zero gravity

Since the behaviour in response to increasing Marangoninumber (effectively increasing the temperature gradient in adimensional scenario) was determined to be similar for each bub-ble separation, the Ma ¼ 915 case was chosen to compare the localwall heat flux for each spacing arrangement. Fig. 6 shows the hotwall heat flux enhancement contours. The same scale is applied toeach plot in the figure. Enhancement of local heat flux is calculatedas q00total/q00cond, where q00cond represents the conductive heat fluxthat would exist in the absence of the bubbles. As a means ofinvestigating the range of influence of the bubbles, line plots of heatflux in the x direction are provided in Fig. 7. Fig. 8 shows the coldwall heat flux enhancement for each separation. From these figures,it can be seen that at small spacings such as Sb ¼ 3Rb, the closeproximity of both bubbles causes a region of high enhancementbetween them, yet the area over which the wall heat flux is affectedis not very large. As the distance between the bubbles increases, theregion corresponding to the greatest heat flux becomes confined toa region in the immediate vicinity of the gas/liquid interface.Significantly, the bubbles seem to affect heat transfer in the regionbetween them above a spacing of Sb ¼ 10Rb and up to a spacing ofSb ¼ 15Rb (see Fig. 7). From the single bubble axisymmetric simu-lations [13,14], a zero gravity hot wall effective radius asymptoti-cally approaching 7Rb was predicted, which is in agreement withresults from this study.

Similar behaviour is observed for the cold wall, shown in Fig. 8.The bubbles appear to act as isolated bubbles from a separationdistance of 10Rb. This is also consistent with the earlier axisym-metric work since a cold wall effective radius of 4Rb was predictedfrom the single bubble simulations.

Upon close inspection of the hot wall heat flux plots provided inFig. 6, it is noticed that between the bubbles there exists a smallregionwhere the heat flux is reduced. For the smaller separations ofSb¼ 3Rb and Sb¼ 4Rb, the heat flux in this area between the bubblesis less than other regions which are similar distances from the gas/liquid interface. To explain this phenomenon, the flow structures of

Fig. 5. Hot wall heat flux enhancement in response to increasing Marangoni numberfor the Sb ¼ 5Rb separation.

the liquid are analysed. By taking a cross-section of the domain inthe x-plane aligned with the central axis of both bubbles, it ispossible to analyse the flow profiles in the region.

Fig. 9 plots the flow field for the simulation inwhich the bubbleswere separated by a distance of 3Rb. In this figure the vectors arecoloured by fluid temperature. Interestingly, a pair of small vorticeshas developed between the bubbles. These vortices are essentiallythe primary vortex for each bubble, and rotate in the opposite di-rection to one another. The proximity of the bubbles chokes theflow of liquid between them, preventing the warm fluid beingexpelled from this region and also the entrainment of cooler fluidfrom above. The formation of this small vortex pair is likelyresponsible for the local drop in heat flux seen in Fig. 6. Thebehaviour at small spacings contrasts that at larger spacings, suchas the Sb ¼ 10Rb case shown in Fig. 10. At this separation, the two

Fig. 7. X-axis line plots of hot wall heat flux enhancement at different bubble separations for Pr ¼ 83, Ma ¼ 915, Bo ¼ Ra ¼ 0.

S.M. O’Shaughnessy, A.J. Robinson / International Journal of Thermal Sciences 78 (2014) 101e110106

bubbles and hence the major vortices are sufficiently far apart thatthey can develop to almost full size, and impact positively on wallheat transfer.

Fig. 11 plots the maximum dimensionless heat flux obtained forthe hot wall for each separation. From the graph it is evident thatwith increased distance between the bubbles, the peak heat fluxmagnitude diminishes until it reaches a similar value to that ob-tained in the single bubble simulations at approximately Sb ¼ 15Rb.This is further evidence to suggest that the bubbles’ convective flowpatterns act independently from this separation outwards.

From the heat flux contour plots provided previously it is difficultto ascertain whether there is a particular spacing arrangement thatprovides maximum heat transfer from the hot wall. Since the two-bubble solutions are not axisymmetric, the enhancement radiusconcept as discussed in Refs. [13,14] is not employed in this scenario.Instead, the enhancement of heat transfer over the entire domain iscomputed. Since there was no mass flow in or out of the computa-tional domain, the heat transferred from the hot wall is equal to theheat transferred to the cold wall in all cases, and this value wasmonitored closely during the solution process. Fig. 12 plots the

increase in heat transfer caused byMarangoni convection about bothbubbles compared to pure molecular diffusion without any bubble.This is expressed as a Nusselt number, which is computed as the totalrate of heat transfer in the presence of thermocapillary convectiondivided by the total rate of heat transfer in the absence of the bub-bles. The graph shows that the presence of two bubbles increasesheat transfer compared to a single bubble scenario. This behaviour isobviously expected, yet the plot still shows some interesting trends.At the small separations of Sb¼ 3Rb and Sb¼ 4Rb the highest heat fluxvalues were computed. However, overall heat transfer is confined toa smaller area surrounding both bubbles. As the bubbles are movedfurther apart, the vortices are able to develop to greater size betweenthe bubbles, thus increasing the area over which they can influenceheat transfer. From the figure it would appear that for the separa-tions simulated, a spacing of 10Rb corresponds to maximum heattransfer for this configuration. Although the difference between thecases is not profound, one must consider that the bubbles occupy asurface area approximately 1/200th of the hot wall surface area, soany increase in overall heat transfer across the entire area issignificant.

Fig. 8. Contours of cold wall heat flux enhancement at different bubble separations forPr ¼ 83, Ma ¼ 915, Bo ¼ Ra ¼ 0.

Fig. 9. Pathlines in the x-plane for the Sb ¼ 3Rb separation, coloured by fluid tem-perature, with g0 ¼ 0 and Ma ¼ 915.

Fig. 10. Pathlines in the x-plane for the Sb ¼ 10Rb separation, coloured by fluid tem-perature, with g0 ¼ 0 and Ma ¼ 915.

Fig. 11. Influence of bubble separation on the peak heat flux enhancement, with g0 ¼ 0and Ma ¼ 915.

S.M. O’Shaughnessy, A.J. Robinson / International Journal of Thermal Sciences 78 (2014) 101e110 107

3.3. Influence of gravity magnitude

The influence of the magnitude of gravitational accelerationwasalso investigated. Since the zero gravity results for the two bubblesimulations highlighted that the bubbles behaved as isolated bub-bles above an inter-bubble spacing of Sb¼ 15Rb, andmaximum heattransfer was achieved at a spacing of Sb ¼ 10Rb, gravity-includedsimulations were carried out for bubble separations of 3Rb, 4Rb,5Rb, 7Rb, and 10Rb.

Fig.13 shows the contours of hot wall heat flux enhancement forthe Sb ¼ 3Rb case at Ma ¼ 915. Noticeably, the area over which heattransfer is affected decreases with increasing gravity level. Thisbehaviour is expected and consistent with the single bubble nu-merical results in Ref. [14]. For all separations, the inclusion ofgravity above a certain magnitude instigates the formation of sec-ondary, counter-rotating vortices beneath the primary vortices.These secondary vortices can appear from gravity levels less thang0 ¼ 0.1 and act to push the thermocapillary rolls closer to thebubbles. Further increases in gravity magnitude strengthen thebuoyancy-driven vortices and prevent the warm fluid fromescaping to the cold wall by attempting to restore the thermalstratification that would exist without Marangoni convection. From

Fig. 12. Heat transfer computed over the entire hot wall for various bubble spacings,with g0 ¼ 0 and Ma ¼ 915.

S.M. O’Shaughnessy, A.J. Robinson / International Journal of Thermal Sciences 78 (2014) 101e110108

Fig. 13, the area of enhancement is dramatically different for theg0 ¼ 0 and g0 ¼ 1 cases. The secondary vortices are also responsiblefor the lack of heat flux enhancement upon the cold wall - for allseparations simulated, there is no significant cold wall local heatflux enhancement above a gravity level of g0 ¼ 0.01.

The reduction in wall heat transfer in the small region betweenthe bubbles for the g0 ¼ 0 case is clear and the cause of this localdrop in heat transfer was described previously. As the gravity levelis increased, this small region of lower local heat flux remains untilapproximately g0 ¼ 0.25. Above this gravity level, the heat flux inthe region between the bubbles escalates. Indeed, at terrestrialgravity (g0 ¼ 1), this region becomes the area of greatest heat flux asshown in Fig. 13h.

To investigate whether or not the hot wall heat flux profileswere similar for each spacing, the data for Sb ¼ 5Rb is provided inFig. 14. Once again, the area over which Marangoni flow influences

Fig. 13. Hot wall heat flux enhancement at different gravity levels for 3Rb spacing withMa ¼ 915.

Fig. 14. Hot wall heat flux enhancement at different gravity levels for 5Rb spacing withMa ¼ 915.

heat transfer decreases with increasing gravity level. However, instark contrast to the Sb ¼ 3Rb case, the maximum heat fluxes forSb ¼ 5Rb were found for the g0 ¼ 0 simulation. As described byFig. 14h, with increasing gravity level the maximum heat flux wasobserved to decrease rather than increase. It was expected that thesmaller spacing would produce an area of greater heat flux as thisbehaviour was evident from the zero gravity plots in Fig. 6, but thediffering responses to increasing gravity level are peculiar.

To explain the changing heat transfer behaviour on the hot wallit is necessary to analyse the flow and temperature fields in theregion. It is known from the g0 ¼ 0 flow structure provided in Fig. 9that the proximity of the bubbles caused the vortices between themto be smaller in size. From the two-dimensional axisymmetricsingle bubble simulations performed in Ref. [14], it was shownusing cross-sectional plots that secondary vortices appeared on allsides of the bubble centreline. In three-dimensional space thismeans that the flow structure consisted of two toroidal vortices:

Fig. 15. Vertical cross-section showing the flow structure in the region of the bubblesfor the Sb ¼ 3Rb case at Ma ¼ 915 and g0 ¼ 1.

Fig. 16. Vertical cross-section showing the flow structure in the region of the bubblesfor the Sb ¼ 5Rb case at Ma ¼ 915 and g0 ¼ 1.

Fig. 17. Total rate of heat transfer enhancement vs. gravity level at Ma ¼ 915 forSb ¼ 3Rb and Sb ¼ 5Rb.

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one thermocapillary-driven torus above a buoyancy-driven torus.However, when a second bubble is included in the model, theaxisymmetric flow structure no longer remains.

Taking a vertical cross-section of the 3D domain in the x-plane,Fig. 15 plots the flow structure (coloured by temperature) for theSb ¼ 3Rb case at Ma ¼ 915 and g0 ¼ 1. The inclusion of gravitationalforces results in the secondary vortices squeezing the primaryvortices closer to the bubbles. These upward-pushing buoyancydriven vortices also reduce the size of the primary vortices on theleft and right of the image. The cause of the increase in local heatflux between the bubbles at higher gravity levels is obviouslyinfluenced by the flow of liquid in this region. For this two bubblesimulation, the distance between the bubbles is too small to allowthe formation of secondary vortices between the bubbles. Instead,another jet-like flow is formed, onewhich appears to entrain colderfluid from the lower regions of the cell and transports it toward theheated part of the wall between the bubbles. This ‘pump’ thereforeacts in the opposite direction to the Marangoni ‘jet’. This constantflow of colder liquid toward the hot wall increases the local heatflux to values greater than any other location on the hot wall for aspacing of Sb ¼ 3Rb. The liquid flowing toward the hot wall leaves ina direction perpendicular to the plane shown in Fig. 15 (i.e., in andout of the plane), and is responsible for the hourglass-shapedcontours of enhanced heat transfer between the bubbles in Fig. 13.

For the Sb ¼ 5Rb spacing atMa ¼ 915 and g0 ¼ 1 the two bubblesare sufficiently far apart to allow the development of secondaryvortical structures in between them, as evidenced by Fig. 16.Indeed, at this separation and gravity level a new stagnation pointforms in the central region at the intersection of the four vortices.This may prevent much of the hot fluid escaping from the regionbetween the bubbles and is most likely responsible for the rela-tively poor heat transfer observed at the centre of the image pro-vided in Fig. 14g.

To quantify the contribution to heat transfer, the total rate ofheat transfer over the entire hot wall was computed for each of thesimulations. Fig.17 plots this enhancement for selected separations.Clearly, the trend changes between the cases. At the small spacingsof Sb ¼ 3Rb, the g0 ¼ 0 area of enhancement is relatively smallcompared with larger spacings. The total rate of heat transfer fallsbriefly with the instigation of gravity before rising again. Eventhough the area of enhancement is reduced further with increasinggravity level, the corresponding increase in heat flux around thebubbles compensates for this. It is expected that the gravitationalpump effect described above is responsible for the inflection in the

curve. For the spacing Sb ¼ 5Rb, a different profile is observed. Thebubbles are sufficiently far apart so that the buoyancy pump effectdoes not occur, and heat transfer decreases with increasing gravitylevel because the area affected also decreases. This behaviour issimilar to that described in the single bubbleMa ¼ 915 simulationsin Ref. [14]. As the spacing is increased above Sb ¼ 5Rb the total rateof heat transfer improvement is found to be very similar.

4. Conclusions

Three-dimensional simulations of thermal Marangoni convec-tion about two bubbles situated on a heated wall immersed in aliquid silicone oil layer (Pr ¼ 83) of depth 5 mm have been per-formed to gain some insight into the thermal and flow interactionsbetween them. The distance between the two bubbles’ centres wasvaried between 3 and 25 bubble radii to analyse the influence of theinter-bubble spacing on the flow and temperature fields and theimpact upon local and global wall heat transfer. For zero gravityconditions, it was determined that the local wall heat flux reached amaximum value for the smallest separation of three bubble radii,but that the increase in heat transfer over the entire hot wall wasgreatest for a separation of ten bubble radii. This was the result of asmall pair of vortices forming between the bubbles at small sepa-rations. These vortices were unable to recirculate cold fluid towardsthe heated wall. As the bubbles were moved further apart, thevortices could develop to greater size, thus increasing the area overwhich they can positively influence heat transfer.

S.M. O’Shaughnessy, A.J. Robinson / International Journal of Thermal Sciences 78 (2014) 101e110110

When the effects of gravity were included in the model, thebehaviour was observed to change between the cases. At largeseparations between the bubbles, increasing the gravity level wasfound to decrease the maximum value of the local hot wall heatflux. At small separations however, the increase in gravity level ledto an increase in the local hot wall heat flux, which was caused by abuoyancy-driven flow of cooler liquid toward the heated wall. Thiscool flow was formed by the amalgamation of secondary vortices.When the heat transfer was computed over the entire heated wall,increasing the gravity level was observed to augment overall wallheat transfer for the smallest bubble separation. This behaviourcontradicts that observed at the larger bubble spacings and that ofsingle bubble systems.

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