the role of bubble coalescence in decompression sicknessucbpran/cp3.pdf · bends" is a form of...

Case Presentation 3 • May 2013 • UCL CoMPLEX

The role of bubble coalescence inDecompression Sickness

Mark Ransley

Supervisors: Umber Cheema & Nick OvendenUniversity College London

[email protected]

Abstract

Coalescence - the formation of a larger bubble from the collision of two or more smaller ones - has beensomewhat neglected in the study of Decompression Sickness (DCS), a condition where mathematicalmodels are enabling breakthroughs in both its understanding and prevention. In this report we review thecurrent literature on modelling DCS and bubble coalescence, and address the need and possible approachesfor reconciling the two fields. We also experimentally investigate the effects of sodium chloride on bubblecoalescence during the decompression of 3D collagen hydrogels.

Special thanks to Claire Walsh for providing critical shortcuts through the labyrinth ofdecompression literature, sharing many pages of valuable code and patiently guiding me through

my first real lab session in many years.

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mailto:[email protected]


I. Introduction

DECOMPRESSION SICKNESS or "thebends" is a form of dysbarism sufferedby organisms undergoing changes in

pressure, as the result of bubbles forming frommetabolically inert gasses inside the body. Re-search into its prevention has been of muchinterest to the caisson engineering and aero-nautical industries, though it is DCS in deepwater SCUBA assisted diving that provides themotivation for this report.

As a diver descends, the ambient pressureincreases due to the weight of the water above.This can be simply modelled as

p(z) = ρgz + patm (1)

where water has density ρ kg/m3, z is thedepth in metres, patm is the atmospheric pres-sure at the surface in pascals, and g is the gravi-tational strength, usually taken to be 9.81m/s2.

This acts on various exposed air spacesaround the body such as the chest, and thusbreathing air must be supplied at the depthdependent pressure to counteract its effect onbreathing. The high pressure gasses, mostlyNitrogen or, when Trimix1 is breathed, Helium,become dissolved in the diver’s blood, wherethey are carried around the body via perfusion(blood flow). Microbubbles then form, mostlikely within minuscule hydrophobic crevices[1], which subsequently migrate, accumulateand expand on ascent causing a plethora ofunpleasant and threatening symptoms [2] [3].Once contained in bubbles, the inert gassesbecome harder to eliminate [4].

However, a carefully planned ascent fea-turing "decompression stops" along the waycan allow much of the dissolved gas to leavethe body as partial pressures equilibrate -somewhat like the controlled opening of afizzy drink - thus avoiding DCS. In responsethere has been a rich history of mathemati-cal modelling associated with decompression,pioneered by the work of JS Haldane [5]. Divid-ing the body up into compartments based ontheir rates of gas uptake and release, Haldane

presented the following equation for the timecourse of gas x’s partial pressure in tissue T

dpxT

dt=

pxa − px

TτT

(2)

where pa is the atmospheric partial pressureand

τT =Lx

TQ̇Lx

b(3)

is the gas exchange time constant, a functionof blood flow rate Q̇ and solubility coefficientsin tissue LT and blood Lb.

Representing the body as a set of series andparallel compartments with associated partialpressure limits, algorithms by Haldane andlater Workman and Bühlmann have been usedto produce dive computers and tables withsome success. However, such simple gas ex-change models suffer a number of shortcom-ings including failure to account for repeateddives and other important factors such as exer-cise [5]. They also shed little light onto the stillrather mysterious mechanisms behind DCS.

II. Bubble expansion in tissue

In static bubble modelling, the bubble may ex-pand due to gas diffusing in from the environ-ment as a result of differing partial pressures,and through gas ’evaporating in’ from the sur-rounding liquid, termed the vapour pressure.It may collapse through the inverse of theseprocesses, from the ambient pressure of its sur-rounding medium, and from surface tensionforces acting on the interface. Thus for a bubblein equilibrium we have the Laplace equation[1]

pi + pv = pamb +2σ

R(4)

where R is the bubble’s radius of curvature,and σ is the surface tension coefficient.

With multiple gasses, the partial pressureof each one is summed to give pi, and it is com-monplace to separate diffusible gasses, wherepartial pressures can be expected to evolve overtime, from infinitely diffusible ones such as the

1For deep dives, Nitrogen gas can have intoxicating effects, so a Helium rich mixture is used.

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metabolic gasses, where pressures may be as-sumed to equilibrate immediately. Solving (4)for R we obtain the critical radius Rc, a thresh-old below which the bubble will collapse.

For a diffusive gas x, the concentration canbe described by

∂cx

∂t= Dx∇2cx (5)

where D is the diffusion constant. Henry’s lawdescribes the relationship between concentra-tion and partial pressure, and thus is essentialfor combining these equations. It is given as

cx =Lx

RTpx (6)

with temperature T and gas constant R.Gaseous pressure changes within the bub-

ble come about through diffusion across thesurface membrane, as described by the Fickequation

1RT

ddt(px

i V) = ADx ∂cx

∂r

∣∣∣∣r=R

(7)

where R is the radius of the bubble (as opposedto the distance r from its centre), and V and Aare the volume and surface area, given by

V =4π

3R3 A = 4πR2 (8)

Given boundary conditions including ini-tial bubbles with R ≥ Rc this system can besolved numerically together with the Haldaneequation (2) using finite difference methods toyield expressions for growth and shrinkage ofbubble radii over time at a given pressure andgas mixture. Numerical schemes and parame-ter values are given by O’Brien in [5].

There are a few differences in O’Brien’smodel; due to the non-aqueous medium,vapour pressure was not considered in (4), butinstead there was a term for tissue elasticity,describing the tissue medium opposing thebubble volume. Also since O’Brien’s researchfocused on a given volume of tissue rather thanthe body as a whole, (2) was not required.

A fairly recent set of NASA sponsored pa-pers by Srinivasan et al. have considered ways

of making the solutions more computationallytractable. The term ’well stirred’ is used todescribe tissue, or a region thereof, where dif-fusion does not occur due to the gases alreadybeing in equilibrium, and hence (5) can be setto zero leaving all environmental changes de-pendent on the perfusion term (2). Conversely,’unstirred’ regions must be equilibrated by dif-fusion. In [6] the group asked whether, fora single bubble, the model should consist ofonly two regions, bubble and unstirred butuniformly perfused tissue, or whether the un-stirred region should be confined to a thirdboundary layer around the bubble, such thatthe remainder of tissue is well stirred (Fig. 1).The diffusion equation (5) was rewritten interms of distance r from the bubble centre.

�

Figure 1: Comparing the two region model (left) wherediffusion affects the entire tissue beyond the bubble, andthe three region (right), where diffusive effects are con-fined to a boundary layer of fixed width h.

For both models, Srinivasan and colleaguesused the quasi-static approximation - that is,assuming bubble dynamics occur on a far morerapid timescale than environmental pressurechanges, so that ∂p/∂t may be set to zero. Sub-sequently they were able to obtain the follow-ing solutions for a single gas to the two andthree region models respectively.

dRdt

=LtDt

(λ + 1

R

)(pt − pi)− R

3dpamb

dt

pamb +4σ3R

dRdt

=LbDt

(1h + 1

R

)(pt − pi)− R

3dpamb

dt

pamb +4σ3R

where subscripts t and b represent the differ-ent solubilities of the gas in tissue and blood.Analogous solutions for multiple gases were

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also provided. Evidently, the models are struc-turally very similar, however the two regionmodel involves an infinite volume and thus isonly applicable when the bubble to tissue sizeratio is very large.

In [7] the group revised the model, notingthat the three region model with fixed h vio-lates Henry’s law since, as the bubble expands,the volume of the boundary layer (and hencegaseous partial pressure) increases whilst thegas concentration does not. Thus, an expres-sion for h was derived in terms of bubble vol-ume, gas contents of both the bubble and tis-sue, and partial pressures such that Henry’slaw was obeyed.

In [8] the two region model was adapted topermit finite tissue volumes through variableperfusion, allowing for the tissue to again beeventually well stirred and hence finite in vol-ume. This approach was generalised in [9] toallow for multiple bubbles. Interestingly, an ex-pression was derived for the maximum numberof bubbles a given volume could take, howeverunfortunately for this report coalescence effectswere neglected.

These papers are by no means the extent ofthe tissue modelling literature, and indeed themodels contained therein were based in partupon earlier ones discussed in more detail inthe thesis of Michael Chappell [1], which fo-cused largely on bubble nucleation during de-compression. Chappell investigated the theorythat bubbles are formed within hydrophobiccrevices, most likely located in points of cellu-lar contact and clefts between the endotheliaof blood vessels. A range of crevice geometrieswere modelled, and simulations conducted todetermine under what conditions bubbles mayform and detach. It was suggested that the ob-served delay between a diver’s ascent and theonset of DCS symptoms could be due to gasstored in bubbles being slowly released andthen available for further nucleation, althoughthe parameter values required for this are atthe limits of the plausible ranges. However,importantly Chappell stated

"It is likely that the interaction between bubblesand the vasculature and other bubbles may intro-

duce further time delays for the arrival of bubblesat observation sites."

Potential DCS related problems such as theocclusion of blood vessels are likely to requirelarger bubbles than those possible from decom-pression alone. Indeed, in vivo observation ofdecompression bubbles has in the past onlybeen able to detect those > 10µm in diameter[10], a figure only achieved in very aggressivesimulated ascent profiles in O’Brien’s model[5], where coalescence is assumed to occur in-stantaneously upon two bubbles making con-tact. However, the coalescence literature paintsa more complicated picture.

III. Coalescence

Figure 2: Bubble coalescence observed in the veins of amouse following decompression. Taken from Lever et al.

In the simplest terms, the description of co-alescence between two bubbles is stated in [12]among other works as:

1. Bubbles collide.

2. A small amount of the liquid between thebubbles is trapped and drains gradually.

3. The liquid film between the bubblesreaches a critical thickness and a film rup-ture occurs due to van der Waals forces,resulting in coalescence.

The value of critical thickness seems fairlystable from author to author; in [13] they claim

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it to be 100Å, whilst in [14] it is claimed to be5-10nm. Coalescence has been observed in vivoas a direct result of decompression (Fig. 2),though the modelling literature on this topictends to be either purely academic in nature,or related to industrial processes, with littleattention given to biological cases. In the statictissue models covered in Section II coalescencewould occur due to the expansion of bubbles,yet in most scenarios collisions are the result offluid motion as described by the Navier-Stokesequations for incompressible fluids:

∇ ·U = 0 (9)

ρDUDt

= −∇p +∇ · T + f (10)

where U is the fluid velocity, f represents bodyforces, usually gravity (in which case f = ρg),and T represents stress factors. Many vari-ants exist depending on the scenario, for ex-ample problems involving viscosity may have∇ · T = µ∇2U, and problems involving multi-phase fluids (in our case bubbles) may haveρ = ρ f − ρg to represent the phase depen-dent densities whilst surface tension effectsare incorporated into T. Hence after non-dimensionalisation the systems in bubble prob-lems will often feature the following importantdimensionless quantities:

The Reynolds number [15],

Re =ρ f g1/2R3/2

0µre f

(11)

where ρ f is the fluid density (gas density as-sumed to be zero), and R0 and µ are charac-teristic scales for distance and viscosity usedin the non-dimensionalisation. Re gives a mea-sure of the importance of inertial versus viscos-ity forces, and as such Re ≈ 0 corresponds to arelatively viscous fluid.

The Bond number [16] occurs in rising bub-ble problems

Bo =ρ f gR2

0

σ(12)

since it compares buoyancy with surface ten-sion.

For the collision of two bubbles with rela-tive velocity V, the Weber number [17]

We =ρV2Req

σ(13)

is often used to measure inertia against surfacetension. Here we define

Req =1

2(R−11 + R−1

2 )(14)

as the equivalent radius in the two-bubble sys-tem.

Another governing principle in fluid dy-namics is the continuity equation

∂ρ

∂t= −∇ · (ρU) (15)

which is used for conserved quantities, in thiscase density.

Figure 3: Chesters and Hoffman’s formulation of thebinary collision system. Note how the bubbles’ sphericalsymmetry has been exploited to reduce the dimensionalityof the problem. Taken from [13].

In [13] a binary bubble collision was mod-elled in coordinates centred at the point ofcontact (Fig. 3) in a gravity free environment.Writing (10) with pressure in terms of gas pres-sure and surface tensions across the boundariesof both bubbles and using the continuity equa-tion to constrain the conservation of bubblevolume, the equations were solved numericallyfor film thickness h to show that the minimumthickness occurs at some r > 0, indicating theformation of a dimple. This corroborates phase(2) of the coalescence description given at thestart of this section, whereupon a small quan-tity of liquid becomes trapped at the interface.

By considering the energy required to de-form the bubble surfaces in terms of We, andevaluating the time at which coalescence oc-curs, the pair showed that there exists a critical

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Weber number, Wec, above which the bubbleswill convert all their kinetic energy to free en-ergy at the surface, arresting motion and caus-ing the bubbles to bounce rather than coalesce.It was also shown that in the viscous case lowReynolds numbers correspond to a lower ∂h/∂tand thus a slower coalescence.

Figure 4: The conical (left) and exponential (right) trail-ing wakes contemplated by de Nevers. Taken from [18].

A different class of coalescence problem,that due to buoyancy, was first modelled byde Nevers and Wu [18]. It was observed thatwhen two bubbles ascend in series the secondone gains additional velocity from the upward-moving wake left by the first, eventually result-ing in collision. Using hemispherical bubbles,the only two forces considered were buoyancyand drag, equilibrated for each bubble as(

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πR3)

ρg = CDU2

2Ap (16)

where CD is the drag coefficient and Ap is theprojected area of the leading edge. Two wakegeometries were formulated; conical and expo-nential (Fig. 4). For the second bubble, dragwas reduced by subtracting the area inside thefirst’s wake from Ap, from which expressionsfor velocity and collision time were solved. Theconical model has a finite wake, shown on thediagram to be of length Lc, so if the separationdistance between bubbles is greater than thisthey will not coalesce. However the exponen-tial model trails an infinite wake, and also hap-

pens to provide a (slightly) better fit to experi-mental data in glycerine. Interestingly, whentesting with distilled water the authors choseto add sodium ethyl xanthate to reduce surfacetension and promote coalescence. These sim-ple models were unable to explain an observedfurther increase in velocity of both bubbles justprior to collision.

Considerably more complex was the buoy-ancy actuated model by Chen et al. [19], devel-oped to provide 3D simulations over a range ofReynolds and Bond numbers to observe quali-tative differences in the two-bubble dynamics.

The model was shown to match experimen-tal footage with impressive accuracy (Fig. 5),though in this study the liquid jet formed be-hind the leading bubble at high Reynolds num-bers was found to hinder coalescence, seem-ingly in contrast to the wake-accelerated modeldiscussed previously. However, as in de Nev-ers’ experiment, Chen’s simulations perhapsunsurprisingly showed a lower σ value aidedcoalescence.

The system used 3D Navier-Stokes equa-tions with viscosity and surface tension terms,and employed volume fraction function F, gov-erned by the continuity equation, to representthe liquid and gas phase’s separate density andviscosity properties, as well as the discontinu-ities at the fluid interface.

ρ(x, t) = F(x, t)ρ f + [1− F(x, t)ρg] (17)

µ(x, t) = F(x, t)µ f + [1− F(x, t)µg] (18)

where F = 0 within the bubble, F = 1 withinthe liquid and F ∈ (0, 1) on the interface. Fwas also used to localise the surface tension asa volume force.

The system was solved over a cylindricalvolume of stationary liquid, illustrated in Fig-ure 5, for an initially spherical bubble restingalong the central axis. The highly advanced nu-merical scheme was based on one developedin [20] and consisted of many sub-schemes.The Semi-Implicit Method for Pressure LinkedEquations (SIMPLE) was used to link pres-sure and velocity, the Rhie-Chow interpolationmethod permitted edge dependent terms to be

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used whilst allowing the mesh to remain col-located, a third-order upwind biased schemewas used for convection and second-order cen-tral differencing for diffusion, whilst a 27-pointstencil was employed for surface-tension forcelinearisation. The transport functions (17) &(18) were solved with modified line-constantVolume of Fluid methods.

The author of this report made an attemptto understand this set of schemes and adaptthem for modelling coalescence in decom-pressed tissue, but found the task hopelesslydifficult, inspiring an examination of the nextclass of coalescence models.

a)

b)

c)

Figure 6: Illustration of the three collision scenarios out-lined in [21]. a) turbulence, b) buoyancy, c) laminarshear.

Prince and Blanch [21] developed a proba-

bilistic model for coalescence in large groupsof bubbles, building on the work of Friedlan-der (1977). Observing that the coalescence rateswere linked intrinsically with the collision rates,they categorised collision by type2 into:

• Turbulence (Fig. 6a) - bubbles are pro-pelled into one another by liquid eddycurrents.

• Buoyancy (Fig. 6b) - already described.

• Laminar shear (Fig. 6c) - parallel streamsof differing flow velocity could cause bub-bles to interact with potential for coales-cence. Currents need not be directly op-posed, for example in pipe flow wherethe velocity is radially dependent.

They defined within a bubble mixture eachtype of collision rate between bubbles of radiusi and j. Using buoyancy as an example:

θBij = ninjSij(uri − urj) (19)

where uri is the rise velocity, a function of vol-ume and by implication R, and S is the collisioncross-sectional area defined as

Sij =π

4(ri + rj)

2 (20)

Similar though more complicated rateswere formulated for turbulence (θT) and lam-inar shear (θLS). Assuming the bubbles col-lide, they must remain in contact long enoughfor the film to rupture according to the filmthinning equations developed by Chesters andHoffman [13]. Defining the collision efficiencyas

λij = exp(tij/τij) (21)

the authors totalled the three different colli-sion types with coalescence efficiencies, thensummed over all the possible discretised radiicombinations to obtain the total coalescencerate:

ΓT =12

ΣiΣj[θTij + θB

ij + θLSij ]× λij (22)

Decoalescence or break-up was examinedalso, and both models were then shown to beable to describe experimental data.

2The author would like to point out that our research concerns a fourth type of collision - that driven by expansion.

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Figure 5: Above: Comparison of model and experiment, adapted from [19]. Below: The cylindrical discretisation usedfor solving, taken from [20].

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The work of Prince and Blanch gave riseto a number of modified models from variousauthors, which are aggregated in a recentlypublished paper by Nguyen et al. (2013) [12].The group adapted Prince and Blanch’s modelto investigate the effect of bubbly flow induc-ing and suppressing turbulence, which as wehave seen itself then influences the bubble dy-namics.

All the coalescence models thus far havebeen Euler-Euler models in that both phaseshave been considered Eulerian fluids. Anotherrecent paper by Mattson and Mahesh (2012)[22] used a probabilistic model but in an Euler-Lagrange system where the bubbles were con-sidered hard spheres obeying Lagrangian par-ticle system rules.

It has been noted in many works that thepresence of electrolytes in solution reduces co-alescence, with the foamy nature of seawaterserving as a textbook example. In a letter toNature [14] Craig et al. described an experi-ment where bubbles were passed through anilluminated water column and the escapinglight recorded by a photodetector. The largerbubbles emerging from coalescence would al-low maximum light through. The group thenadded increasing concentrations of NaCl andobserved a logistic type decrease in transmis-sion, indicating inhibition of coalescence. Sys-tematically trying a number of different salts,the group then noticed not all of them had thiseffect, but fell into categories α and β such thatanion cation pairings from the same categorywould inhibit coalescence, however alternatepairings had no effect whatsoever. The au-thors could provide no explanation for this phe-nomenon asides a possible mechanism throughthe (at the time of publishing) poorly under-stood long-range hydrophobic interaction.

In [23] Weissenborn and colleagues sug-gested the inhibition could also be a resultof electrical repulsive forces or the Gibbs-Marangoni effect, concerning mass transfer be-tween the fluid-bubble interface due to a sur-face tension gradient. They also pointed out alink between Craig’s electrolyte pairings andoxygen solubility.

Interestingly, in another article [24] Craig re-marks that the electrolyte concentration mark-ing maximum coalescence inhibition coincideswith the NaCl level in the human body, 1.1E-1M. He suggests that electrolytes in vivo playan essential role in preventing coalescence ofever-present microbubbles that would lead toDCS symptoms even at atmospheric conditions.The author of this report speculates that areasexhibiting the earliest symptoms of DCS couldwell have lower electrolyte concentrations thanthe rest of the body, though no study data hasbeen yet found to corroborate this.

Salt inhibition was discussed in [21], withthe suggested mechanism being immobilisa-tion of the gas-liquid interface and subsequentincreased required contact time tij. The criticalelectrolyte concentration was formulated as

ct = 1.18(

Bσ

R

)1/2RgT

(∂σ

∂c

)−2(23)

where B is the retarded van der Waals coeffi-cient.

Clearly a coalescence based model woulddo well to take salt concentrations into account,through (23) or otherwise.

To summarise the coalescence modellingsection, it is clear that many different tech-niques have been used throughout the litera-ture. All the models reviewed concern fluids inmotion and some, particularly the model devel-oped by Chen et al., are very computationallyintensive and far too detailed to be necessaryin a diving algorithm. As the next section willshow, bubbles do coalesce in tissue but not allthe time, even when they start off touching,so more modelling is indeed required. Theauthor would suggest using the Chesters andHofman method, where relative velocity u isreplaced by the relative expansion rate dR/dtto obtain pressure, distance and time rangeswhere coalescence is possible. Coalescence hasbeen observed in more than two bubbles simul-taneously [25], though this author has beenunable to find any work on the modelling ofsuch scenarios, so would suggest coalescingbubbles individually in silico.

Following coalescence of identical bubbles,

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Figure 7: Bubble coalescence in collagen hydrogel observed with (top) and without (bottom) success at UCL Institute ofOrthopaedics and Musculoskeletal Science. Decompression cycle lasted 5 seconds.

the resulting one would be centred on the pointof contact (r = 0 in Figure 3) and have radius2R. For non equal bubble radii and approachvelocities, the resulting radius would be

rn = (r31 + r3

2)1/3 (24)

though calculating the resulting centre hasproved more difficult and seems to requiresolving numerically.

IV. Coalescence in collagen

hydrogels

Collagen hydrogels have allowed the engineer-ing of tissues containing cells and other bio-logical structures in 3D configurations. Theyhave proved very useful for studying bubblesunder decompression due to being transpar-ent, firm enough to hold their contents in afixed position and yet still respond well topressure changes in a vacuum chamber, andrelatively simple to make given their biologicalsemblance.

The hydrogel consists of intact collagen fib-rils, essential media containing salt concentra-tions, pH indicator, and NaOH to neutralisethe acetic acid used to extract and store the fib-rils [26]. By shaking the mixture before settingat 37◦C, the hydrogel can be set with microbubbles scattered throughout.

In Figure 7, stills from videos taken at UCLStanmore Campus illustrate the fact bubbles

may coalesce in tissue under depressurisation,but may also just expand even in close con-tact whilst remaining separate. An experimentwas attempted to examine the effect of NaClconcentration on coalescence. The results werenot in line with those reported in the literature,so the experiment almost certainly containedmistakes, however the details shall be includedhere for completeness.

Bubbly hydrogels were produced with es-sential media NaCl concentrations 0, 1.5, 4 & 10Mols/l. 3 gels of each concentration were madeso the experiment could be repeated. Photoswere then taken at 3 random locations on eachof the 12 gels before and after decompressionto 0.1 atm followed by return to ambient pres-sure, the theory being that the average bubblenumber would decrease following coalescence,and a relationship should emerge with NaClcontent. Photos of an Optik Labor 0,0025mm2

cell sizer were taken also to provide scale refer-ence.

Bubble counting was automated using Mat-lab’s image processing functions, most notably’imfindcircles’. Bubble diameters ranged from0.02mm - 0.96mm. Results are shown in Fig-ure 8. Surprisingly, many of the bubble countsseemed to increase following decompression,quite the opposite of what we were expecting.Note that at no point during decompressionwas decoalescence observed. The inconclusiveresults could be down to a number of factors.

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First the method of obtaining bubble count byaveraging over 3 photos could be at fault. Reli-able results would need very large numbers toconstitute a representative average, especiallygiven that reported densities can be as high as104 bubbles/ml [5]. Additionally the mixingprocess for introducing bubbles was carried outby a human and thus has none of the consis-tency of the bubble apparatus described in theliterature review section of this report. Finallyit is quite possible that the low pressure envi-ronment was not maintained for long enough.Much of the bubble modelling literature hasplaced emphasis on the time taken for the liq-uid film to drain during coalescence. Grantedthe times are often of order milliseconds, butthe conditions have been wildly different dueto the viscous nature of the medium we used,and the inflation based collisions.

Figure 8: NaCl concentration versus change in bubblenumbers following decompression

References

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[25] Postema, Michiel, et al. "Ultrasound-induced microbubble coalescence." Ultra-sound in medicine & biology 30.10 (2004):1337-1344.

[26] Cheemer, U. "3D collagen biomatrix devel-opment." UCL lecture notes (2012).

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the role of bubble coalescence in decompression sicknessucbpran/cp3.pdf · bends" is a form of...

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