Constraining the thermal-rheological influence of open porosity 1

OH-H2O diffusion on magma mobilisation 2

David Heptinstall 3

Abstract 4

Heat flow models can bring new insights into the thermal and rheological 5

evolution of volcanic systems. This study shall investigate the thermal and 6

mechanical processes in a crystallizing, permeable magma column, with a 7

COMSOL heat flow model of Soufriere Hills Volcano (SHV), Montserrat. This 8

study’s objectives are to constrain the partitioning of molecular water (H2O) 9

and hydroxyl molecules (-OH) Water diffusivity in the form of OH- diffusion 10

reactions control the melt viscosity during degassing, this is because 11

molecular water will not react with the silicate structure, restricting the melt 12

from polymerising. Bubble viscous regimes can have a rheological influence 13

of water species (H2O and OH) diffusion in silicate polymerization, where 14

shear viscosity ηs is the most influential rheological property in bubble 15

suspension. The two volatile cooling processes, the work done by bubble 16

expansion (P dV) and the heat of exsolution (dH) are independent of each 17

other, the latter is always in favour of equilibrium conditions. Previous 18

studies have noted that the heat of exsolution (dH) is small above pressures 19

of 100 MPa, whilst at low pressures of 1-2MPa, an albite melt could cool by 20

~8 K/wt% of exsolved water. Furthermore, a saturated albite melt at 100 21

MPa with 3 wt% water can potentially cool a magma by a minimum of 35 K, 22

prior to melt disruption into a spray as gas volume fraction exceeds 0.8. 23

Synchrotron x-ray tomography (microCT) can provide a three-dimensional 24

(3D) visual perspective of the connected porosity, bubble aperture and 25

vesicle volume). We shall couple the diffusivity of molecular and hydroxyl 26

water to the analysis of lava dome and pumice sample permeability. The last 27

objective of this study will be to develop a more advanced model to simulate 28

the multi-component thermal-rheological processes in a magma column, to 29

establish the trends of molecular water degassing, bulk magma viscosity, 30

magma column ascent rate and magma temperatures, over timescales 31

ranging from 4 months to 10 years. 32

33

Keywords 34

X-ray tomography, open porosity, OH-H2O diffusion, bulk viscosity, 2-D 35

COMSOL thermal-rheological simulation model 36

Research justification, aim and objectives 37

Previous studies on degassing have focussed on permeability, volatile 38

diffusivity and external measurements of plumes, however little study has 39

focussed on the thermal influence of degassing compared to the influence of 40

water degassing on magma viscosity. Yet many studies have assumed water 41

degassing is in the form of molecular (H2O) water and not hydroxyl 42

molecules that is responsible for the change in geochemical conditions. Only 43

by constraining the molecule and hydroxyl water degassing on magma 44

thermal and rheological properties, can we model such conditions which 45

may prove invaluable in the field. 46

This study’s aim is to model the thermal and rheological processes of a 47

magma column and lava dome under permeable conditions. To constrain the 48

thermal and rheological conditions, we shall determine the OH-H2O 49

diffusivity and permeability, since the former drives the non-linear rise in 50

viscosity and the latter is responsible for efficient degassing of molecular 51

water. 52

The first objective of this project is to constrain the partitioning of molecular 53

water (H2O) and hydroxyl molecules (-OH), which control the 54

polymerization of silicate molecules as water nucleates from the melt and 55

diffuses through the magmatic system. The second objective of this study is 56

to establish the permeability of pumice and lava deposits, which are 57

representative the fragmentation zone within a magma column and the lava 58

dome respectively, through synchrotron x-ray tomography (microCT). The 59

third objective is to model the bulk viscosity as a function of the crystallinity, 60

melt viscosity and bubble deformation. The fourth objective will be the 61

development of a thermal-rheological model in COMSOL, using the 62

diffusivity, viscosity and permeability data to simulate the magma column 63

rheological and thermal processes over timescales of 4 months to 10 years. 64

65

Research Context 66

67

Mechanisms of water diffusivity 68

Water diffusivity in the form of OH- reactions control the melt viscosity 69

during degassing, this is because molecular water will not react with the 70

silicate structure, restricting the melt from polymerising. Once the OH- free 71

radical reacts with a silicate-oxygen bond, the -OH will replace the O-in the 72

silicate molecule, with the oxygen free radical reacting with molecular water 73

to form two more -OH free radicals. As more H2O molecules nucleate from 74

the melt and react with -O free radicals to form -OH free radicals, a greater 75

concentration of SiO3-OH molecules can polymerize, increasing the bulk 76

viscosity. 77

There are many different diffusion mechanisms that contribute to the 78

transport of water in silicate melts, of these we will look more closely at 79

three; diffusion of OH groups bound by tetrahedral cations; direct jumps of 80

water molecules from one site into a neighbouring site without interaction 81

with the silicate framework; and the reaction of water molecules with 82

bridging oxygens by formation OH group pairs (Behrens et al 1997). 83

84

1) Diffusion of OH groups bound by tetrahedral cations 85

86

Behrens et al (1997) noted at the time that experimental data for the 87

diffusivity of OH- groups (DOH) was not available, so as a raw 88

approximation the Eyring equation was used to estimate DOH from viscous 89

measurements; 90

91

D=kT/η λ 92

93

where k is the Boltzmann constant, T the absolute temperature, η the 94

viscosity and λ the diffusive jump distance between tetrahedral cations. 95

Watson and Baker (1991) noted that the Eyring equation can predict the 96

diffusivities of non-alkalies during interdffusion within the range of 10-11 to 97

10-7 cm2/s with a factor of 3 (Behrens et al 1997). The application of this 98

mechanisms has been implied by Shimizu & Kushiro (1984) to be limited to 99

polymerized melts, given the breaking of Si-O and Al-O bonds is fundamental 100

of viscous flow. The diffusion of isolated OH- groups within polymerized 101

melts was assumed by Behrens et al (1997) to be similar to bridging oxygens. 102

This exchange of bonds between silicic tetrahedral groups would relate both 103

viscous flow and OH diffusion. The contribution of isolated OH groups to the 104

transport of chemical component water was estimated by Behrens et al 105

(1997) to be for water contents <10 ppm. 106

2) Diffusion-reaction mechanism and 3) inter-conversion-diffusion 107

mechanism 108

109

The diffusion of H2O molecules has been inferred to be the fundamental 110

process for the transport of water in silicate glasses and melts by numerous 111

studies (Doremus., 1995; Zhang et al., 1991). The diffusion-reaction 112

mechanism denoted by Doremus (1969, 1995), assumes that H2O molecules 113

move through the silicate network by direct jumps from one cavity to 114

another without reaction with oxygen in the silicate network, occasionally 115

being trapped and immobile by hydroxyl groups. Comparatively, the inter-116

conversion-diffusion mechanism assumes that H2O molecules react with 117

bridging oxygens during the movement of water (Tomozawa., 1985). The 118

H2O molecule will jump from one site to another through reaction with the 119

oxygen bridge, followed by bridge re-formation (Roberts & Roberts., 1969). 120

121

At temperatures above the glass transition for the diffusion-related 122

mechanism, the structural relaxation of the melt is assumed to be fast 123

enough to establish local equilibrium between water species during 124

diffusion. Thus if H2O molecule reaction with bridging oxygens results in an 125

immobilization, then for low water contents, the effective diffusion 126

coefficient is proportional to the concentration of water (Behrens et al., 127

1997; Zhang et al., 1991b). This was found by Doremus (1995) to be 128

dependent on trace water diffusion in silica glass and for diffusion of water 129

up to 3 wt% in haplogranitic and quartz-feldspathic melts. At melt 130

temperatures far below the glass transition, structural relaxation is too slow 131

to achieve equilibrium of hydrous species (Behrens et al., 1997). 132

133

In order to gain a more detailed perspective of water diffusion by the inter-134

conversion-diffusion mechanism, between water molecules and bridging 135

oxygens, a study must distinguish between OH singletons and OH pairs 136

(Behrens et al 2007). 137

138

H2O + O = [OH—HO] 139

140

An OH pair would indicate interaction and thermodynamic between the 141

two OH groups and associated Al or Si groups. 142

143

[OH—HO] = 2OH 144

145

Zhang et al (1995) found that the dissociation of an OH pair is very slow, as 146

it is controlled by diffusion of OH singletons, with the timescale of this 147

reaction being several minutes at 400-600oC in rhyolitic glasses with 0.5–2.3 148

wt% water. 149

Effective diffusion of water in the melt can be described as the sum of 150

contributions from, ‘n’ reaction between molecular H2O, OH pairs and OH 151

singletons (Zhang et al 1991). Decreasing water content in the melt will also 152

result in the average distance between OH singletons becoming larger, 153

resulting in longer timescales for bridging OH singletons to form a mobile 154

OH pair. Thus a change in the dependence of water diffusivity (Dwater) on 155

water content (Cwater), is related to a critical distance of OH groups for which 156

the dissociation reaction does not significantly affect the transport of water 157

(Behrens et al., 2007). 158

Oxygen is present within magma in three dominant species, molecular water 159

(H2O), OH and oxygen atoms connected to polymers (eg. SiO2) in the form of 160

bridging or non-bridging oxygen, referred also as dry oxygen. Dry oxygen 161

species can contribute to the oxygen 18O flux, along with other oxygen 162

species H3O+, CO2 and O2. This will lead to spurious 18O diffusion coefficients 163

not associated with hydrogen, hence unconnected to hydrothermal water 164

diffusion (Zhang et al., 1991). Zhang et al (1991) further noted that in 165

hydrous systems, it is important to determine the dominant diffusion 166

species, and to relate the apparent diffusion coefficient to the concentrations 167

and diffusion coefficients of these species, such as 18O and molecular water, 168

potentially enhancing oxygen diffusivity. This study concluded that oxygen 169

transport depends on the presence of water and generally depends of water 170

fugacity. There is a relationship between the effective oxygen diffusion 171

coefficient and total water diffusion coefficient, which is a function of the 172

water concentration of silicates at low water content. 173

174

175

Bubble suspension rheology 176

Volatile nucleation and bubble growth will have a profound impact on 177

magma rheology, namely the bulk viscosity. We shall view bubble viscous 178

regimes independently of the rheological influence of water species (H2O 179

and OH) diffusion on silicate polymerization. The rheological property of 180

bubble suspensions that is most influential on bulk viscosity is the shear 181

viscosity ηs (Llewellin & Manga., 2005). The shear viscosity is then 182

normalised to the melt viscosity µ0 and relative viscosity ηr. 183

µ0= ηs* ηr 184

We can then establish the conditions of the bubble relaxation time λ, which 185

is the timescale a bubble requires to respond to changes in shear viscosity. 186

λ = µ0a / Γ, 187

where ‘a’ is the undeformed bubble radius and Γ is the bubble-liquid 188

interfacial tension. 189

Llewellin & Manga (2005) noted that the viscous regime is controlled by the 190

capillary number Ca, which indicates if a flow is steady or unsteady. 191

Ca = λγ*, 192

where γ* is the shear strain rate. 193

Thus we can determine if shear viscosity increases with increasing gas 194

volume fraction (Ca<<1), or if shear viscosity decreases with increasing gas 195

volume fraction (Ca>>1). If the Ca<<1, interfacial tension forces will 196

dominate and bubbles will be spherical, this will have the effect of decreasing 197

the bubble free-slip surface area within the bubble suspension and 198

increasing the flow-line distortion and relative viscosity. Alternatively, if 199

Ca>>1, then viscous forces will dominate and bubbles will be elongate, which 200

will increase the bubble free-slip surface area and decreasing the flow-line 201

distortion and relative viscosity (Llewellin & Manga., 2005) 202

203

Volatile thermodynamics 204

Studies on the thermal effect of degassing has received less attention than 205

the thermal effect of latent heat of crystallization, which may offset any 206

degassing induced cooling, as noted by Sahagian and Proussevitch (1996). 207

Sahagian and Proussevitch (1996) stated that the major thermal effects of 208

degassing are the heat of vaporization and the work of bubble expansion, 209

where magma system cooling can occur through water exsolution from the 210

melt or gas expansion and work done against external forces, such as magma 211

viscosity. The study by Sahagian and Proussevitch (1996) used a numerical 212

assessment of these effects on cooling of a bubbly magma of albite 213

composition, that the magma exsolves volatiles at equilibrium or with a 214

degree of oversaturation. The equilibrium or oversaturation style of 215

degassing depends on decompression history and degassing kinetics. 216

Typically basaltic and low decompression silicic eruptions will have 217

reversible equilibrium degassing, whilst irreversible oversaturation 218

degassing is common in explosive silicic eruptions. This is because the 219

diffusive magma properties cannot offset the extreme expansion of the 220

erupting magma column. 221

The work of bubble expansion refers to the relationship of the change in 222

internal energy (dU) with magma pressure and the change in magma 223

volume, for both reversible and irreversible processes. Adiabatic reversible 224

processes through equilibrium degassing will result in small pressure 225

changes to maintain water saturation in the melt. Whilst for both reversible 226

and irreversible processes, the change in internal energy will approximately 227

equal the sum of magma property (melt, gas and water) temperatures, 228

pressures and the sum of chemical potential and the number of moles of 229

water. The adiabatic irreversible process of oversaturation degassing is 230

characterized by increasing entropy dS>0, these processes differ from 231

reversible processes through a disequilibrium in water vapour in the magma 232

with dissolved water in the melt. 233

The study by Sahagian and Proussevitch (1996) indicated that cooling 234

through an irreversible process results in less cooling than a reversible 235

process, since the two cooling processes (PdV and dH) are independent and 236

the heat of exsolution (dH) is always in favour of equilibrium conditions. 237

Sahagian and Proussevitch (1996) noted that the heat of exsolution (dH) is 238

small at pressure above 100 MPa, whilst at low pressures of 1-2MPa the 239

cooling of an albite melt could be by ~8 K/wt% of exsolved water. 240

Furthermore, a saturated albite melt at 100 MPa with 3 wt% water can 241

potentially cool a magma by a minimum of 35 K, prior to melt disruption into 242

a spray as gas volume fraction exceeds 0.8. 243

The results of the study by Sahagian and Proussevitch (1996) concluded that 244

the bubble cooling rate may be significant during rapid magma ascent at the 245

volcanic vent, leading to extreme cooling at the bubble-melt interface if not 246

equilibrated by magma diffusivity. As the bubble-melt interface enters a 247

glass transition, volatile diffusion is limited to thermal diffusivity rather than 248

chemical diffusion, as oversaturation degassing can lead to fragmentation of 249

the magmatic foam into fine ash through brittle failure. Alternatively, bubble 250

wall cooling prior to glass formation can reduce the diffusive volatile flux 251

into bubbles, decreasing the system cooling and lead to solidification of 252

oversaturate magmas. 253

254

Magma permeability 255

The magma permeability can be estimated through various techniques, such 256

as through helium permeameters or through synchrotron x-ray tomography 257

(microCT). In this study, we shall use the latter to gain a three-dimensional 258

(3D) visual perspective of the connected porosity, bubble aperture and 259

vesicle volume (Degruyter et al., 2009). We shall couple the diffusivity of 260

water, which is the most influential volatile to magma rheology with this 261

study on the permeability within lava dome and pumice samples. Whilst 262

other techniques, such as helium pycnometers can establish sample open 263

and closed porosity, the microCT technique will provide more detail as to the 264

internal sample vesicle dimensions. Mueller (2005) suggested that magma 265

permeability is affected by the geometry and distribution of vesicles 266

generating the connected flow pathways, rather than solely the bulk volatile 267

content. 268

As noted by Wright et al (2007), the physical movement of volatiles in 269

volcanic systems will control most aspects of volcanic behaviour, however 270

physical constraints on degassing is elusive. Through this microCT method 271

we can establish the vesicle connectivity and dimensions, such as calculating 272

permeability in perpendicular directions within in tube pumice (Wright et 273

al., 2007). Through synchrotron x-ray micro-tomography, images of lava 274

and pumice samples can reveal pore fine scale topology, including the pore 275

shape, pore size and degree of vesicle anisotropy. (Wright et al., 2007). 276

Once we have imaged the lava and pumice samples to understand the pore 277

size, shape and anisotropy, we will need to understand the mechanics of 278

volatile flow. We will calculate the simple Poiseuille flow conditions using 279

low Reynolds number (laminar) flow in the direction of pore elongation, and 280

empirical Kozeny-Carman approximation for circular cylinder-shaped pore 281

space. Any offset between these two permeability equations may be due to a 282

change in pore diameter, a change in tortuosity or a change in the cross-283

sectional pore geometry away from circular shapes (Wright et al., 2007). 284

Lattice-Boltzmann flow conditions will slightly overestimate the 285

permeability for tube pumices compared to simple Poiseuille flow 286

calculations, because large bubbles increase permeability compared to that 287

of vesicle diameters with small volumes. 288

289

Thermal-Rheological modelling 290

This study is a continuation of a previous COMSOL model that simulated the 291

cooling timescales of a static, impermeable magma column, took into 292

account the latent heat of crystallization which was determined through a 293

MELTS analysis of a Couch et al (2003) Soufriere Hills Volcano groundmass 294

sample. Previous studies on temperature-crystallinity evolution have been 295

on degassing-induced crystallization kinetics of crystal nucleation rates and 296

growth (Hort & Spohn, 1991, Hort, 1997, Melnik & Sparks., 2002), and the 297

behaviour of conduit geometry and elasticity during an eruption (Costa et 298

al., 2007). COMSOL was previously used by Bea (2010), who modelled 299

convective cooling induced crystallization in magma chambers at different 300

temperatures, between 1000oC and 800oC, over timescales of 1500 years. 301

Through the previous study, we have a MELTS database, which consists of 302

the latent heat of crystallization, specific heat of melts and crystals and the 303

melt and crystal phase viscosities. 304

We shall develop a more advanced model to establish trends in molecular 305

water degassing, bulk viscosity, magma column ascent rate and magma 306

temperature trends over timescales ranging from 4 months to 10 years. This 307

will require numerous interpolations, which are data sets as a function of 308

magma temperature, lithostatic pressure or dissolved volatile content. 309

COMSOL has numerous thermal and mechanical modules, the previous 310

model used a heat flow in closed system module, and this new module will 311

use the permeable heat flow module. The disadvantage of previous versions 312

of COMSOL has been that the model does not work well with more than one 313

module, this may be overcome with the assistance of COMSOL support to 314

allow the model to work effectively with thermal and mechanical modules, 315

however the permeable heat flow module can support 2 dimensional magma 316

velocity and melt viscosity data. A mechanical module would be useful for 317

understanding how the combined role of volatile degassing, crystallization 318

and heat flow influence a non-linear rise in bulk viscosity and brittle failure 319

of the magma column. To effectively develop a mechanical module for the 320

COMSOL model, this study will require interpolations to characterise bulk 321

viscosity from the melt viscosity, bubble deformation induced shear 322

viscosity and crystal phase viscosity through H2O-OH diffusivity and X-ray 323

tomography research methods. 324

325

Research questions 326

327

How does the characteristics of connected porosity vary in lava and pumice 328

samples, as shown by X-ray tomography? 329

How does the partitioning of H2O-OH control bulk-viscosity? 330

How does the open porosity influence the thermal processes within the 331

magmatic system? 332

Through the use of a thermal-rheological 2-dimensional model, what does 333

it tell us about the timescales of magma mobilization through degassing or 334

decompression? 335

336

Research Methods 337

338

X-Ray Tomography 339

Synchrotron X-ray computed micro-tomography (microCT) can provide us 340

with a 2D tomographic image that corresponds to different sample rotation 341

angles, which can be processed to reconstruct a 3D volume (Polacci et al 342

2006). This three-dimensional (3D) volume can show us a visual image of 343

the open and closed porosity, including the vesicularity (%), vesicle number 344

density, volume and connectivity. This technique will allow us to attain 345

micro-scale, high resolution, three dimensional data in a short time period. 346

Sample preparation is short, with the sample only requiring cutting to fit into 347

the microCT sample holder; the microCT is also non-destructive, so does not 348

alter the internal and external sample dimensions for future study (Polacci 349

et al., 2006). 350

Previous studies using the microCT have been to characterize vesicles in 351

basaltic rock (Song et al., 2001) and a imaging the volumetric bubble size 352

distributions of synthetic and natural silicate glass foams (Robert et al., 353

2004). Polacci et al (2006) used this microCT method to investigate the 3D 354

structure of pumice and scoria deposits to visualize the deposit vesicle 355

content, geometry and clast volume. Computated tomography (CT) 356

apparatus that can generate 3D high resolution images is the GE Phoenix 357

v/tome/x s micro-CT scanner, which uses a high-power nanofocus X-ray 358

tube with has pixel/vowel resolution down to 2 µm depending on sample 359

size. 360

361

H2O-OH Diffusivity 362

Fourier-transform infrared spectroscopy (FTIR) was the technique utilised 363

by Zhang (1991) to determine the water concentration profiles through a 364

basaltic melt. This technique is used to study the infrared spectrum of 365

absorption, emission, photoconductivity and Raman scattering of a solid, 366

liquid or gas sample. The advantage of the FTIR spectrometer is that it can 367

simultaneously collect high spectral resolution data over a wide spectral 368

range, it also has a high sensitivity and low noise level, mechanical reliability 369

and internal self-calibration. The FTIR spectrometer is developed by 370

numerous suppliers around the world, including the Thermo-Nicolet 371

Corporation (mmrc.caltect.edu). 372

The FTIR spectrometer passes IR radiation through a sample, some of this 373

radiation is absorbed by the sample and some transmits through the sample, 374

the spectrum received at the detector represents the molecular absorption 375

and transmission of the sample (mmrc.caltec.edu). Zhang (1991) assumed 376

that only molecular H2O was diffusing and that there was a local equilibrium 377

between H2O molecules and OH groups. This was due to an inadequate 378

model of FTIR water profiles on the basis of constant water diffusivity, this 379

study will need to model such profiles to investigate water diffusivity in 380

more detail. 381

382

Thermal analysis of gases 383

Thermal analysis of volcanic gases can be achieved through the use of a 384

Differential Scanning Calorimeter, which measures the thermodynamic 385

properties of an unknown sample compared to a reference material as a 386

function of temperature. Two different calorimeter apparatus include the 387

NETZSCH DSC 404 oC Pegasus, which has a maximum temperature range of 388

1650oC operating at an ambient pressure; and the PSETRAM SENSYS EVO 389

DSC, which has a maximum temperature range of 830oC operating within a 390

pressure of up to 40MPa. The gas flow within both types of ambient and 391

pressure calorimeter operates using Argon at a rate of 25 cm3/min. 392

393

Thermal-rheological COMSOL model 394

The COMSOL Multiphysics finite element software is employed to link a 395

specific geometry, with multiple partial differential equations to construct a 396

two-dimensional simulation of a vertical magma conduit (Bea, 2010). The 397

model components include tracer points which estimate the temperature for 398

a specific geometrical point in the conduit, conduit walls and host rock. The 399

optimum mesh layout will need to be extremely fine within and around the 400

conduit, to allow the model to converge with greater accuracy (Heptinstall 401

et al., 2015). The COMSOL model software allows different thermal and 402

viscosity interpolations to operate within a three dimensional geometry, 403

such as a latent heat of crystallization interpolation, thermal and mechanical 404

modules will be required for such a model which may require assistance 405

from COMSOL support to make the model accept two different modules. 406

407

Significance of Research 408

The vision of this research is to understand the thermal and rheological 409

implications of water degassing within a multi-component andesitic magma, 410

in particular the geochemical interactions of water species on melt 411

polymerization and the heat capacity of such species. Our previous model 412

only estimated heat flow within a closed system, with a simple empirical 413

solution for the specific heat capacity of liquid water at constant pressure 414

(Di Genova et al., 2014). Further work to establish heat flow within a 415

permeable system, will need to estimate the work of bubble expansion and 416

the latent heat of vapourisation, as volatiles, such as water, are removed 417

from the melt. 418

To accomplish this aim, this research will need to appreciate the 419

mechanisms of water diffusion in order to understand the thermal and 420

rheological processes on the melt. Likewise, this research will need to 421

understand the controls on bulk viscosity and the development of open 422

porosity from bubble coalescence. 423

The ambition of this study is to incorporate developments in the thermal and 424

rheological processes on magmatic systems from water degassing, into a 3D 425

simulation model. This may have the benefit of establishing an 426

interpretation of detectable thermal and rheological characteristics during 427

phases of volcanic eruptions. 428

429

Further Expansion of project 430

Constraining the thermal-rheological role of 3-Dimensional stress using 431

uniaxial and tri-axial pressure experimentation on remelted samples during 432

rotary shear conditions. 433

Developing a linking rheological model to determine the role of 434

crystallization on a non-linear rise in viscous, how does the bulk viscosity 435

vary in a magma when modelling crystal clusters or homogenous crystal 436

distribution. 437

Constraining the magma ascent profile through consideration of conduit 438

geometry and different types of magma velocity profile and magma 439

pressure distribution. 440

441

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