Showing papers on "Mixture theory published in 2001"

Flow of variably fluidized granular masses across three‐dimensional terrain: 1. Coulomb mixture theory

Abstract: Rock avalanches, debris flows, and related phenomena consist of grain-fluid mixtures that move across three-dimensional terrain. In all these phenomena the same basic forces govern motion, but differing mixture compositions, initial conditions, and boundary conditions yield varied dynamics and deposits. To predict motion of diverse grain-fluid masses from initiation to deposition, we develop a depth-averaged, three-dimensional mathematical model that accounts explicitly for solid- and fluid-phase forces and interactions. Model input consists of initial conditions, path topography, basal and internal friction angles of solid grains, viscosity of pore fluid, mixture density, and a mixture diffusivity that controls pore pressure dissipation. Because these properties are constrained by independent measurements, the model requires little or no calibration and yields readily testable predictions. In the limit of vanishing Coulomb friction due to persistent high fluid pressure the model equations describe motion of viscous floods, and in the limit of vanishing fluid stress they describe one-phase granular avalanches. Analysis of intermediate phenomena such as debris flows and pyroclastic flows requires use of the full mixture equations, which can simulate interaction of high-friction surge fronts with more-fluid debris that follows. Special numerical methods (described in the companion paper) are necessary to solve the full equations, but exact analytical solutions of simplified equations provide critical insight. An analytical solution for translational motion of a Coulomb mixture accelerating from rest and descending a uniform slope demonstrates that steady flow can occur only asymptotically. A solution for the asymptotic limit of steady flow in a rectangular channel explains why shear may be concentrated in narrow marginal bands that border a plug of translating debris. Solutions for static equilibrium of source areas describe conditions of incipient slope instability, and other static solutions show that nonuniform distributions of pore fluid pressure produce bluntly tapered vertical profiles at the margins of deposits. Simplified equations and solutions may apply in additional situations identified by a scaling analysis. Assessment of dimensionless scaling parameters also reveals that miniature laboratory experiments poorly simulate the dynamics of full-scale flows in which fluid effects are significant. Therefore large geophysical flows can exhibit dynamics not evident at laboratory scales.

A solid-fluid mixture model allowing for solid dilatation under external pressure

Abstract: A sponge subjected to an increase of the outside fluid pressure expands its volume but nearly mantains its true density and thus gives way to an increase of the interstitial volume This behaviour, not yet properly described by solid-fluid mixture theories, is studied here by using the Principle of Virtual Power with the most simple dependence of the free energy as a function of the partial apparent densities of the solid and the fluid The model is capable of accounting for the above mentioned dilatational behaviour, but in order to isolate its essential features more clearly we compromise on the other aspects of deformation Specifically, the following questions are addressed: (i) The boundary pressure is divided between the solid and fluid pressures with a dividing coefficient which depends on the constituent apparent densities regarded as state parameters The work performed by these tractions should vanish in any cyclic process over this parameter space This condition severely restricts the permissible constitutive relations for the dividing coefficient, which results to be characterized by a single material parameter (ii) A stability analysis is performed for homogeneous, pressurized reference states of the mixture by postulating a quadratic form for the free energy and using the afore mentioned permissible constitutive relations It is shown that such reference states become always unstable if only the external pressure is sufficiently large, but the exact value depends on the interaction terms in the free energy The larger this interaction is, the smaller will be the critical (smallest unstable) external pressure (iii) It will be shown that within the stable regime of behaviour an increase of the external pressure will lead to a decrease of the solid density and correspondingly an increase of the specific volume, thus proving the wanted dilatation effects (iv) We close by presenting a formulation of mixture theory involving second gradients of the displacement as a further deformation measure (Germain 1973); this allows for the regularization of the otherwise singular boundary effects (dell'Isola and Hutter 1998, dell'Isola, Hutter and Guarascio 1999)

Separation of Sources Based on the Partitioning of the Space of Observations

Abstract: The techniques of Blind Separation of Sources (BSS) are used in many Signal Processing applications in which the data sampled by sensors are a mixture of signals from different sources, and the goal is to obtain an estimation of the sources from the mixtures This work shows a new method for blind separation of sources, based on geometrical considerations concerning the observation space This new method is applied to a mixture of two sources and it obtains the coefficients of the unknown mixture matrix A and separates the unknown sources, So Following an introduction, we present a brief abstract of previous work by other authors, the principles of the method and a description of the algorithm, together with some simulations

Vertical flow of a multiphase mixture in a channel

Abstract: The flow of a multiphase mixture consisting of a viscous fluid and solid particles between two vertical plates is studied. The theory of interacting continua or mixture theory is used. Constitutive relations for the stress tensor of the granular materials and the interaction force are presented and discussed. The flow of interest is an ideal one where we assume the flow to be steady and fully developed; the mixture is flowing between two long vertical plates. The non-linear boundary value problem is solved numerically, and the results are presented for the dimensionless velocity profiles and the volume fraction as functions of various dimensionless numbers.

A new method of solving the basic plane boundary value problems of statics of the elastic mixture theory

Abstract: The basic plane boundary value problems of statics of the elastic mixture theory are considered when on the boundary are given: a displace- ment vector (the first problem), a stress vector (the second problem); dier- ences of partial displacements and the sum of stress vector components (the third problem). A simple method of deriving Fredholm type integral equa- tions of second order for these problems is given. The properties of the new operators are established. Using these operators and generalized Green for- mulas we investigate the above-mentioned integral equations and prove the existence and uniqueness of a solution of all the boundary value problems in a finite and an infinite domain.

Mixed finite element modelling of cartilaginous tissues

Abstract: Swelling and shrinking of cartilaginous tissues is modelled by a four-component mixture theory. This theory results in a set of coupled non-linear partial differential equations for the electrochemical potentials and the displacement. For the sake of local mass conservation these equations are discretised in space by a mixed finite element method. Integration in time by backward Euler leads to a non-linear system of algebraic equations. A subtle solution strategy for this system is proposed and tested for one-dimensional situations.