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Modelling shallow gravitydriven solidfluid mixtures over arbitrary Topography
http://sms.cam.ac.uk/media/524465
Ioana Luca (Division of Mechanics, Research Center for Applied Sciences, Academia Sinica, Taipei, Taiwan; on leave from Department of Mathematics, University Politehnica, Bucharest, Romania), YihChin Tai (Department of Civil Engineering, National Chi Nan University, Puli, Nantou, Taiwan) and ChihYu Kuo (Division of Mechanics, Research Center for Applied Sciences, Academia Sinica, Taipei, Taiwan)
Wednesday 7 January 2009, 10.0510.30
40
Modelling shallow gravitydriven solidfluid mixtures over arbitrary Topography
http://sms.cam.ac.uk/media/524465
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Modelling shallow gravitydriven solidfluid mixtures over arbitrary Topography
ucs_sms_521697_524465
http://sms.cam.ac.uk/media/524465
Modelling shallow gravitydriven solidfluid mixtures over arbitrary Topography
Ioana Luca (Division of Mechanics, Research Center for Applied Sciences, Academia Sinica, Taipei, Taiwan; on leave from Department of Mathematics, University Politehnica, Bucharest, Romania), YihChin Tai (Department of Civil Engineering, National Chi Nan University, Puli, Nantou, Taiwan) and ChihYu Kuo (Division of Mechanics, Research Center for Applied Sciences, Academia Sinica, Taipei, Taiwan)
Wednesday 7 January 2009, 10.0510.30
Thu, 29 Jan 2009 16:06:43 +0000
Ioana Luca
YihChin Tai
ChihYu Kuo
Isaac Newton Institute for Mathematical Sciences
Ioana Luca
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Ioana Luca (Division of Mechanics, Research Center for Applied Sciences,...
Ioana Luca (Division of Mechanics, Research Center for Applied Sciences, Academia Sinica, Taipei, Taiwan; on leave from Department of Mathematics, University Politehnica, Bucharest, Romania), YihChin Tai (Department of Civil Engineering, National Chi Nan University, Puli, Nantou, Taiwan) and ChihYu Kuo (Division of Mechanics, Research Center for Applied Sciences, Academia Sinica, Taipei, Taiwan)
Wednesday 7 January 2009, 10.0510.30
Cambridge University
1104
http://sms.cam.ac.uk/media/524465
Modelling shallow gravitydriven solidfluid mixtures over arbitrary Topography
Ioana Luca (Division of Mechanics, Research Center for Applied Sciences, Academia Sinica, Taipei, Taiwan; on leave from Department of Mathematics, University Politehnica, Bucharest, Romania), YihChin Tai (Department of Civil Engineering, National Chi Nan University, Puli, Nantou, Taiwan) and ChihYu Kuo (Division of Mechanics, Research Center for Applied Sciences, Academia Sinica, Taipei, Taiwan)
Wednesday 7 January 2009, 10.0510.30
The description of geophysical flows over arbitrary terrain is a topic comprising many challenging problems. One of them is the incorporation in the modelling equations of the influences due to the geometry of the basal surface. This issue was revigorated by the relatively recent introduction due to Bouchut and Westdickenberg of curvilinear coordinates, able to properly account for the arbitrariness of the bottom topography and for the shallowness of the avalanche mass. These coordinates are: two surface parameters on the bed surface, and the distance between the point in question and its orthogonal projection onto the topography; the depth of the avalanche is measured along the normal direction to the bottom surface.
We use these coordinates and develop depthaveraged models of gravitydriven saturated mixtures of solid grains and pore fluid on an arbitrary rigid basal surface. First, by only specifying the interaction force (as deduced by Schneider and Hutter within a thermodynamic analysis, and which is different from that used by Pitman and Le) and ordering approximations in terms of an aspect ratio between a typical length perpendicular to, and a typical length parallel to the topography, we derive governing equations for the shallow flowing mixture. In doing so, the nonuniformity through the avalanche depth of the constituent velocities and of the solid volume fraction is accounted for by coefficients of Boussinesq type.
Then, the rheology of both constituents is specified. One constituent is a Newtonian/nonNewtonian fluid with a viscosity so small, that in the governing equations only the basal shear stress survives, and this is parameterized by a viscous friction law. For the bulk stresses of the solid constituent we propose three models: one reduces to the inviscid fluid, and the other two are topography adapted versions of IversonDenlinger and SavageHutter models. Common to the proposed models is the assumption that two of the principal directions of the mean stress tensor are aligned with the principal directions of the mean surface stretching. The basal shear stress is parameterized by a Coulomb friction law.
With these mixture models at hand, we derived equations governing the motion of two shallow layers on arbitrary rigid basal surface: the layer near the bottom topography is essentially one of the three models of solidfluid mixture, and the upper layer is the Newtonian/nonNewtonian fluid, present in the mixture layer. The interface between the two layers is a material surface for the solid constituent, and across it the jump conditions of mass and momentum balance equations corresponding to each constituent are envisaged.
Numerical results concerning the derived model equations are planned to be obtained in future.
Acknowledgements The authors are very grateful to Prof. K. Hutter for his neverending support and advice; the topographyadapted version of the SavageHutter model was developed following his idea.
A seminar from the Institute of Mathematics and its Applications (IMA) conference on Dense Granular Flows.
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