# American Institute of Mathematical Sciences

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 and 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. [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. [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. [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. [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. [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. [7] M. Dragoni, I. Borsari and A. Tallarico, A model for the shape of lava flow fronts J. Geophys. Res. 110 (2005), B09203. [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. [9] J. H. Ferziger and M. Peric, "Computational Methods for Fluid Dynamics," Springer-Verlag, Berlin,, 2002., ().  doi: 10.1007/978-3-642-97651-3. [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 (). [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. [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. [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. [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. [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. [16] S. V. Patankar, "Numerical Heat Transfer and Fluid Flow," Series in Computational Methods in Mechanics and Thermal Sciences, McGraw-Hill, 1980 [17] H. Pinkerton and R. S. J. Sparks, Field measurements of the rheology of lava Nature 276 (1978), 383-385. [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. [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. [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. [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. [22] I. Sonder, B. Zimanowski and R. Büttner, Non-Newtonian viscosity of basaltic magma Geoph. Res. Lett. 33 (2006), L02303. [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. [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. [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. [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. [27] A. Tallarico and M. Dragoni, A three-dimensional Bingham model for channeled lava flows J. Geophys. Res. 105 (2000), 969-980. [28] A. Tallarico, M. Dragoni and G. Zito, Evaluation of lava effusion rate and viscosity from other flow parameters J. Geophys. Res. 111 (2006). [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.

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##### 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. [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. [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. [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. [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. [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. [7] M. Dragoni, I. Borsari and A. Tallarico, A model for the shape of lava flow fronts J. Geophys. Res. 110 (2005), B09203. [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. [9] J. H. Ferziger and M. Peric, "Computational Methods for Fluid Dynamics," Springer-Verlag, Berlin,, 2002., ().  doi: 10.1007/978-3-642-97651-3. [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 (). [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. [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. [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. [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. [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. [16] S. V. Patankar, "Numerical Heat Transfer and Fluid Flow," Series in Computational Methods in Mechanics and Thermal Sciences, McGraw-Hill, 1980 [17] H. Pinkerton and R. S. J. Sparks, Field measurements of the rheology of lava Nature 276 (1978), 383-385. [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. [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. [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. [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. [22] I. Sonder, B. Zimanowski and R. Büttner, Non-Newtonian viscosity of basaltic magma Geoph. Res. Lett. 33 (2006), L02303. [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. [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. [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. [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. [27] A. Tallarico and M. Dragoni, A three-dimensional Bingham model for channeled lava flows J. Geophys. Res. 105 (2000), 969-980. [28] A. Tallarico, M. Dragoni and G. Zito, Evaluation of lava effusion rate and viscosity from other flow parameters J. Geophys. Res. 111 (2006). [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.
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