June  2013, 6(3): 677-685. doi: 10.3934/dcdss.2013.6.677

Simulation of lava flows with power-law rheology

1. 

Istituto Nazionale di Geofisica e Vulcanologia, Sez. di Catania, Piazza Roma 2, I-95152 Catania, Italy

2. 

Dipartimento di Geologia e Geofisica, Università di Bari, Via Edoardo Orabona 4, I-70125 Bari, Italy

3. 

Dipartimento di Fisica, Università di Bologna, Viale Carlo Berti Pichat 8, I-40127 Bologna, Italy

Received  March 2010 Revised  November 2010 Published  December 2012

In this work we studied the effect of a power-law rheology on a gravity driven lava flow. Assuming a viscous fluid with constant temperature and constant density and assuming a steady flow in an inclined rectangular channel, the equation of the motion is solved by the finite volume method and a classical iterative solutor. Comparisons with observed channeled lava flows indicate that the assumption of the power-law rheology causes relevant differences in average velocity and volume flow rate with respect to the Newtonian rheology.
Citation: Marilena Filippucci, Andrea Tallarico, Michele Dragoni. Simulation of lava flows with power-law rheology. Discrete & Continuous Dynamical Systems - S, 2013, 6 (3) : 677-685. doi: 10.3934/dcdss.2013.6.677
References:
[1]

N. Bagdassarov and H. Pinkerton, Transient phenomena in vesicular lava flows based on laboratory experiments with analogue materials J. Volcanol. Geotherm. Res. 132 (2004), 115-136. Google Scholar

[2]

R. Champallier, M. Bystricky and L. Arbaret, Experimental investigation of magma rheology at 300 MPa: From pure hydrous melt to 75 vol. of crystals Earth Planet. Sc. Lett. 267 (2008), 571-583. Google Scholar

[3]

M. Capobianchi, Pressure drop predictions for laminar flows of extended modified power law fluids in rectangular ducts Int. J. Heat Mass Transfer, 51 (2008), 1393-1401. Google Scholar

[4]

M. Dragoni, M. Bonafede and E. Boschi, Downslope flow model of a Bingham liquid: Implications for lava flows J. Volcanol. Geotherm. Res. 30 (1986), 305-325. Google Scholar

[5]

M. Dragoni and A. Tallarico, The effect of crystallization on the rheology and dynamics of lava flows J. Volcanol. Geotherm. Res. 59 (1994), 241-252. Google Scholar

[6]

M. Dragoni, A. Piombo and A. Tallarico, A model for the formation of lava tubes by roofing over a channel J. Geophys. Res. 100 (1995), 8435-8447. Google Scholar

[7]

M. Dragoni, I. Borsari and A. Tallarico, A model for the shape of lava flow fronts J. Geophys. Res. 110 (2005), B09203. Google Scholar

[8]

C. Ferlito and J. Siewert, Lava Channel Formation during the 2001 Eruption on Mount Etna: Evidence for Mechanical Erosion Phys. Rev. Lett. 96 (2006), 028501. Google Scholar

[9]

J. H. Ferziger and M. Peric, "Computational Methods for Fluid Dynamics," Springer-Verlag, Berlin,, 2002., ().  doi: 10.1007/978-3-642-97651-3.  Google Scholar

[10]

M. Filippucci, A. Tallarico and M. Dragoni, A three dimensional dynamical model for channeled lava flow with non-linear rheology, J. Geophys. Res. 115, 115 ().   Google Scholar

[11]

R. C. Gupta, On developing laminar non-Newtonian flow in pipes and channels Nonlin. Anal.: Real World Appl. 2 (2001), 171-193. doi: 10.1016/S0362-546X(00)00109-7.  Google Scholar

[12]

A. J. L. Harris and S. K. Rowland, FLOWGO: A kinematic thermo-rheological model for lava flowing in a channel Bull. Volcanol. 63 (2001), 20-44. Google Scholar

[13]

A. J. L. Harris, J. Bailey, S. Calvari and J. Dehn, Heat loss measured at a lava channel and its implications for down-channel cooling and rheology in "Kinematics and Dynamics of Lava Flows" (eds. M. Manga and G. Ventura) Geol. Soc. of Am. Special Paper, 2005, 396, 125-146. Google Scholar

[14]

K. Hon, J. Kauahikaua, R. Denlinger and K. McKay, Emplacement and inflation of pahoehoe sheet flows: Observations and measurements of active flows an Kilauea Volcano, Hawaii Geol. Soc. Am. Bull. 106 (1994), 351-370. Google Scholar

[15]

Y. Lavallée, K.U. Hess, B. Cordonnier and D. B. Dingwell, Non-Newtonian rheological law for highly crystalline dome lavas Geology 9 (2007), 843-846. Google Scholar

[16]

S. V. Patankar, "Numerical Heat Transfer and Fluid Flow," Series in Computational Methods in Mechanics and Thermal Sciences, McGraw-Hill, 1980 Google Scholar

[17]

H. Pinkerton and R. S. J. Sparks, Field measurements of the rheology of lava Nature 276 (1978), 383-385. Google Scholar

[18]

H. Pinkerton and G. Norton, Rheological properties of basaltic lavas at sub-liquidus temperatures: Laboratory and field measurements on lavas from Mount Etna J. Volcanol. Geotherm. Res. 68 (1995), 307-323. Google Scholar

[19]

H. Pinkerton and R. Stevenson, Methods of determining the rheological properties of magmas at sub-solidus temperatures J. Volcanol. Geotherm. Res. 53 (1992), 47-66. Google Scholar

[20]

H. R. Shaw, T. L. Wright, D. L. Peck and R. Okamura, The viscosity of basaltic magma: An analysis of field measurements in Makaopuhi lava lake Hawaii. Am. J. Sci. 266 (1968), 255-264. Google Scholar

[21]

J. V. Smith, Textural evidence for dilatant (shear thickening) rheology of magma at high crystal concentrations J. Volcanol. Geotherm. Res. 99 (2000), 1-7. Google Scholar

[22]

I. Sonder, B. Zimanowski and R. Büttner, Non-Newtonian viscosity of basaltic magma Geoph. Res. Lett. 33 (2006), L02303. Google Scholar

[23]

J. Spera, A. Borgia, J. Strimple and M. Feigenson, Rheology of melts and magmatic suspensions 1. design and calibration of concentric cylinder viscosimeter with application to rhyolitic magma J. Geophys. Res. 93 (1988), 273-10. Google Scholar

[24]

D. J. Stein and F. J. Spera, Rheology and microstructure of magmatic emulsions: Theory and experiments J. Volcanol. Geotherm. Res. 49 (1992), 157-1742. Google Scholar

[25]

S. Syrjälä, Finite-element analysis of fully developed laminar flow of power-law non-Newtonian fluid in a rectangular duct Int. Commun. Heat Mass Transfer 22 (1995), 549-557. Google Scholar

[26]

A. Tallarico and M. Dragoni, Viscous Newtonian laminar flow in a rectangular channel: Application to Etna lava flows Bull. Volcanol. 61 (1999), 40-47. Google Scholar

[27]

A. Tallarico and M. Dragoni, A three-dimensional Bingham model for channeled lava flows J. Geophys. Res. 105 (2000), 969-980. Google Scholar

[28]

A. Tallarico, M. Dragoni and G. Zito, Evaluation of lava effusion rate and viscosity from other flow parameters J. Geophys. Res. 111 (2006). Google Scholar

[29]

H. C. Weed, F. J. Ryerson and A. J. Piwinskii, Rheological properties of Molten Kilauea Iki Basalt containing suspended crystals in "Mineral Matter and Ash in Coal" (eds. E.H. Zarantonello and Author 2) ACS Symposium Series, 301, 1986, 223-233. Google Scholar

show all references

References:
[1]

N. Bagdassarov and H. Pinkerton, Transient phenomena in vesicular lava flows based on laboratory experiments with analogue materials J. Volcanol. Geotherm. Res. 132 (2004), 115-136. Google Scholar

[2]

R. Champallier, M. Bystricky and L. Arbaret, Experimental investigation of magma rheology at 300 MPa: From pure hydrous melt to 75 vol. of crystals Earth Planet. Sc. Lett. 267 (2008), 571-583. Google Scholar

[3]

M. Capobianchi, Pressure drop predictions for laminar flows of extended modified power law fluids in rectangular ducts Int. J. Heat Mass Transfer, 51 (2008), 1393-1401. Google Scholar

[4]

M. Dragoni, M. Bonafede and E. Boschi, Downslope flow model of a Bingham liquid: Implications for lava flows J. Volcanol. Geotherm. Res. 30 (1986), 305-325. Google Scholar

[5]

M. Dragoni and A. Tallarico, The effect of crystallization on the rheology and dynamics of lava flows J. Volcanol. Geotherm. Res. 59 (1994), 241-252. Google Scholar

[6]

M. Dragoni, A. Piombo and A. Tallarico, A model for the formation of lava tubes by roofing over a channel J. Geophys. Res. 100 (1995), 8435-8447. Google Scholar

[7]

M. Dragoni, I. Borsari and A. Tallarico, A model for the shape of lava flow fronts J. Geophys. Res. 110 (2005), B09203. Google Scholar

[8]

C. Ferlito and J. Siewert, Lava Channel Formation during the 2001 Eruption on Mount Etna: Evidence for Mechanical Erosion Phys. Rev. Lett. 96 (2006), 028501. Google Scholar

[9]

J. H. Ferziger and M. Peric, "Computational Methods for Fluid Dynamics," Springer-Verlag, Berlin,, 2002., ().  doi: 10.1007/978-3-642-97651-3.  Google Scholar

[10]

M. Filippucci, A. Tallarico and M. Dragoni, A three dimensional dynamical model for channeled lava flow with non-linear rheology, J. Geophys. Res. 115, 115 ().   Google Scholar

[11]

R. C. Gupta, On developing laminar non-Newtonian flow in pipes and channels Nonlin. Anal.: Real World Appl. 2 (2001), 171-193. doi: 10.1016/S0362-546X(00)00109-7.  Google Scholar

[12]

A. J. L. Harris and S. K. Rowland, FLOWGO: A kinematic thermo-rheological model for lava flowing in a channel Bull. Volcanol. 63 (2001), 20-44. Google Scholar

[13]

A. J. L. Harris, J. Bailey, S. Calvari and J. Dehn, Heat loss measured at a lava channel and its implications for down-channel cooling and rheology in "Kinematics and Dynamics of Lava Flows" (eds. M. Manga and G. Ventura) Geol. Soc. of Am. Special Paper, 2005, 396, 125-146. Google Scholar

[14]

K. Hon, J. Kauahikaua, R. Denlinger and K. McKay, Emplacement and inflation of pahoehoe sheet flows: Observations and measurements of active flows an Kilauea Volcano, Hawaii Geol. Soc. Am. Bull. 106 (1994), 351-370. Google Scholar

[15]

Y. Lavallée, K.U. Hess, B. Cordonnier and D. B. Dingwell, Non-Newtonian rheological law for highly crystalline dome lavas Geology 9 (2007), 843-846. Google Scholar

[16]

S. V. Patankar, "Numerical Heat Transfer and Fluid Flow," Series in Computational Methods in Mechanics and Thermal Sciences, McGraw-Hill, 1980 Google Scholar

[17]

H. Pinkerton and R. S. J. Sparks, Field measurements of the rheology of lava Nature 276 (1978), 383-385. Google Scholar

[18]

H. Pinkerton and G. Norton, Rheological properties of basaltic lavas at sub-liquidus temperatures: Laboratory and field measurements on lavas from Mount Etna J. Volcanol. Geotherm. Res. 68 (1995), 307-323. Google Scholar

[19]

H. Pinkerton and R. Stevenson, Methods of determining the rheological properties of magmas at sub-solidus temperatures J. Volcanol. Geotherm. Res. 53 (1992), 47-66. Google Scholar

[20]

H. R. Shaw, T. L. Wright, D. L. Peck and R. Okamura, The viscosity of basaltic magma: An analysis of field measurements in Makaopuhi lava lake Hawaii. Am. J. Sci. 266 (1968), 255-264. Google Scholar

[21]

J. V. Smith, Textural evidence for dilatant (shear thickening) rheology of magma at high crystal concentrations J. Volcanol. Geotherm. Res. 99 (2000), 1-7. Google Scholar

[22]

I. Sonder, B. Zimanowski and R. Büttner, Non-Newtonian viscosity of basaltic magma Geoph. Res. Lett. 33 (2006), L02303. Google Scholar

[23]

J. Spera, A. Borgia, J. Strimple and M. Feigenson, Rheology of melts and magmatic suspensions 1. design and calibration of concentric cylinder viscosimeter with application to rhyolitic magma J. Geophys. Res. 93 (1988), 273-10. Google Scholar

[24]

D. J. Stein and F. J. Spera, Rheology and microstructure of magmatic emulsions: Theory and experiments J. Volcanol. Geotherm. Res. 49 (1992), 157-1742. Google Scholar

[25]

S. Syrjälä, Finite-element analysis of fully developed laminar flow of power-law non-Newtonian fluid in a rectangular duct Int. Commun. Heat Mass Transfer 22 (1995), 549-557. Google Scholar

[26]

A. Tallarico and M. Dragoni, Viscous Newtonian laminar flow in a rectangular channel: Application to Etna lava flows Bull. Volcanol. 61 (1999), 40-47. Google Scholar

[27]

A. Tallarico and M. Dragoni, A three-dimensional Bingham model for channeled lava flows J. Geophys. Res. 105 (2000), 969-980. Google Scholar

[28]

A. Tallarico, M. Dragoni and G. Zito, Evaluation of lava effusion rate and viscosity from other flow parameters J. Geophys. Res. 111 (2006). Google Scholar

[29]

H. C. Weed, F. J. Ryerson and A. J. Piwinskii, Rheological properties of Molten Kilauea Iki Basalt containing suspended crystals in "Mineral Matter and Ash in Coal" (eds. E.H. Zarantonello and Author 2) ACS Symposium Series, 301, 1986, 223-233. Google Scholar

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