### Abstract

The purpose of this paper is to derive modeling equations for debris flows on real terrain. Thus, we use curvilinear coordinates adapted to the topography as introduced, e.g., by Bouchut and Westdickenberg [F. Bouchut and M. Westdickenberg, Commun. Math. Sci., 2(3), 359-389, 2004], and develop depth-averaged models of gravity-driven saturated mixtures of solid grains and pore fluid on an arbitrary rigid basal surface. First, by only specifying the interaction force and ordering approximations in terms of an aspect ratio between a typical length perpendicular to the topography, and a typical length parallel to the topography, we derive the governing equations for the shallow flow of a binary mixture, driven by gravitational force. In doing so, the non-uniformity through the avalanche depth of the constituent velocities and of the solid volume fraction is accounted for by coefficients of Boussinesq type. Then, the material behaviour peculiarities of both constituents properly enter the theory. One constituent is a granular solid. For its stresses we propose three models, one of them of Mohr-Coulomb type. The other constituent is a Newtonian/non-Newtonian fluid with small viscosity, obeying a viscous bottom friction condition. The final governing equations for the shallow flow of the mixture, incorporating the constitutive assumptions, are deduced, and the limiting equilibrium is then investigated.

Original language | English |
---|---|

Pages (from-to) | 1-36 |

Number of pages | 36 |

Journal | Communications in Mathematical Sciences |

Volume | 7 |

Issue number | 1 |

Publication status | Published - 2009 Jan 1 |

### Fingerprint

### All Science Journal Classification (ASJC) codes

- Mathematics(all)
- Applied Mathematics

### Cite this

*Communications in Mathematical Sciences*,

*7*(1), 1-36.

}

*Communications in Mathematical Sciences*, vol. 7, no. 1, pp. 1-36.

**Modeling shallow gravity-driven solid-fluid mixtures over arbitrary topography.** / Luca, Ioana; Tai, Yih-Chin; Kuo, Chih Yu.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Modeling shallow gravity-driven solid-fluid mixtures over arbitrary topography

AU - Luca, Ioana

AU - Tai, Yih-Chin

AU - Kuo, Chih Yu

PY - 2009/1/1

Y1 - 2009/1/1

N2 - The purpose of this paper is to derive modeling equations for debris flows on real terrain. Thus, we use curvilinear coordinates adapted to the topography as introduced, e.g., by Bouchut and Westdickenberg [F. Bouchut and M. Westdickenberg, Commun. Math. Sci., 2(3), 359-389, 2004], and develop depth-averaged models of gravity-driven saturated mixtures of solid grains and pore fluid on an arbitrary rigid basal surface. First, by only specifying the interaction force and ordering approximations in terms of an aspect ratio between a typical length perpendicular to the topography, and a typical length parallel to the topography, we derive the governing equations for the shallow flow of a binary mixture, driven by gravitational force. In doing so, the non-uniformity through the avalanche depth of the constituent velocities and of the solid volume fraction is accounted for by coefficients of Boussinesq type. Then, the material behaviour peculiarities of both constituents properly enter the theory. One constituent is a granular solid. For its stresses we propose three models, one of them of Mohr-Coulomb type. The other constituent is a Newtonian/non-Newtonian fluid with small viscosity, obeying a viscous bottom friction condition. The final governing equations for the shallow flow of the mixture, incorporating the constitutive assumptions, are deduced, and the limiting equilibrium is then investigated.

AB - The purpose of this paper is to derive modeling equations for debris flows on real terrain. Thus, we use curvilinear coordinates adapted to the topography as introduced, e.g., by Bouchut and Westdickenberg [F. Bouchut and M. Westdickenberg, Commun. Math. Sci., 2(3), 359-389, 2004], and develop depth-averaged models of gravity-driven saturated mixtures of solid grains and pore fluid on an arbitrary rigid basal surface. First, by only specifying the interaction force and ordering approximations in terms of an aspect ratio between a typical length perpendicular to the topography, and a typical length parallel to the topography, we derive the governing equations for the shallow flow of a binary mixture, driven by gravitational force. In doing so, the non-uniformity through the avalanche depth of the constituent velocities and of the solid volume fraction is accounted for by coefficients of Boussinesq type. Then, the material behaviour peculiarities of both constituents properly enter the theory. One constituent is a granular solid. For its stresses we propose three models, one of them of Mohr-Coulomb type. The other constituent is a Newtonian/non-Newtonian fluid with small viscosity, obeying a viscous bottom friction condition. The final governing equations for the shallow flow of the mixture, incorporating the constitutive assumptions, are deduced, and the limiting equilibrium is then investigated.

UR - http://www.scopus.com/inward/record.url?scp=64549115456&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=64549115456&partnerID=8YFLogxK

M3 - Article

VL - 7

SP - 1

EP - 36

JO - Communications in Mathematical Sciences

JF - Communications in Mathematical Sciences

SN - 1539-6746

IS - 1

ER -