Continuum theories of mixtures: basic theory and historical development
01 May 1976-Quarterly Journal of Mechanics and Applied Mathematics (Oxford University Press)-Vol. 29, Iss: 2, pp 209-244
About: This article is published in Quarterly Journal of Mechanics and Applied Mathematics.The article was published on 1976-05-01. It has received 666 citations till now.
Citations
More filters
TL;DR: In this paper, a simple model that satisfies most of these criteria uses depth-averaged equations of motion patterned after those of the Savage-Hutter theory for gravity-driven flow of dry granular masses but generalized to include the effects of viscous pore fluid with varying pressure.
Abstract: Recent advances in theory and experimen- tation motivate a thorough reassessment of the physics of debris flows. Analyses of flows of dry, granular solids and solid-fluid mixtures provide a foundation for a com- prehensive debris flow theory, and experiments provide data that reveal the strengths and limitations of theoret- ical models. Both debris flow materials and dry granular materials can sustain shear stresses while remaining stat- ic; both can deform in a slow, tranquil mode character- ized by enduring, frictional grain contacts; and both can flow in a more rapid, agitated mode characterized by brief, inelastic grain collisions. In debris flows, however, pore fluid that is highly viscous and nearly incompress- ible, composed of water with suspended silt and clay, can strongly mediate intergranular friction and collisions. Grain friction, grain collisions, and viscous fluid flow may transfer significant momentum simultaneously. Both the vibrational kinetic energy of solid grains (mea- sured by a quantity termed the granular temperature) and the pressure of the intervening pore fluid facilitate motion of grains past one another, thereby enhancing debris flow mobility. Granular temperature arises from conversion of flow translational energy to grain vibra- tional energy, a process that depends on shear rates, grain properties, boundary conditions, and the ambient fluid viscosity and pressure. Pore fluid pressures that exceed static equilibrium pressures result from local or global debris contraction. Like larger, natural debris flows, experimental debris flows of ;10 m 3 of poorly sorted, water-saturated sediment invariably move as an unsteady surge or series of surges. Measurements at the base of experimental flows show that coarse-grained surge fronts have little or no pore fluid pressure. In contrast, finer-grained, thoroughly saturated debris be- hind surge fronts is nearly liquefied by high pore pres- sure, which persists owing to the great compressibility and moderate permeability of the debris. Realistic mod- els of debris flows therefore require equations that sim- ulate inertial motion of surges in which high-resistance fronts dominated by solid forces impede the motion of low-resistance tails more strongly influenced by fluid forces. Furthermore, because debris flows characteristi- cally originate as nearly rigid sediment masses, trans- form at least partly to liquefied flows, and then trans- form again to nearly rigid deposits, acceptable models must simulate an evolution of material behavior without invoking preternatural changes in material properties. A simple model that satisfies most of these criteria uses depth-averaged equations of motion patterned after those of the Savage-Hutter theory for gravity-driven flow of dry granular masses but generalized to include the effects of viscous pore fluid with varying pressure. These equations can describe a spectrum of debris flow behav- iors intermediate between those of wet rock avalanches and sediment-laden water floods. With appropriate pore pressure distributions the equations yield numerical so- lutions that successfully predict unsteady, nonuniform motion of experimental debris flows.
2,426 citations
Cites background from "Continuum theories of mixtures: bas..."
...CONSERVATION IN DEBRIS FLOW MIXTURES Mass and linear momentum balances for debris flows can be borrowed with only minor modification from the relatively mature field of continuum mixture theory [Atkin and Craine, 1976]....
[...]
...The granular temperature may be interpreted as twice the fluctuation kinetic energy per unit mass of granular solids and defined as T 5 ^v9s2& 5 ^vs 2 v̄s!...
[...]
...…be substituted into (61), yielding = ? k~1 2 yf! mC =yf < Dyf Dt (65) This may be interpreted as an advection-diffusion equation for yf, in which the advection velocity is the velocity vs of the reference frame for the material time derivative Dyf/Dt [Iverson, 1993] [cf. Atkin and Craine, 1976]....
[...]
...If the solid and fluid densities are assumed constant, the mass-conservation equation for solids (22a) can be manipulated to yield = z vs 5 (21/ys)(Dys/Dt), in which D/Dt designates the material time derivative following the motion of the solids [Bird et al., 1960; Atkin and Craine, 1976]....
[...]
...However, an absolute distinction between solid and fluid constituents is necessary for application of formal mixture theories [Atkin and Craine, 1976] and can be deduced if time as well as length scales are considered....
[...]
01 Jan 1993
TL;DR: In this paper, the authors focus on the fundamentals of poroelasticity, and discuss the formulation and analysis of coupled deformation-diffusion processes, within the framework of the Biot theory of pore elasticity.
Abstract: Publisher Summary
This chapter focuses on fundamentals of poroelasticity. The presence of a freely moving fluid in a porous rock modifies its mechanical response. Two mechanisms play a key role in the interaction between the interstitial fluid and the porous rock: (i) an increase of pore pressure induces a dilation of the rock; and (ii) compression of the rock causes a rise of pore pressure, if the fluid is prevented from escaping the pore network. These coupled mechanisms bestow an apparent time-dependent character to the mechanical properties of the rock. If excess pore pressure, induced by compression of the rock, is allowed to dissipate through diffusive fluid mass transport, further deformation of the rock progressively takes place. The rock is more compliant under drained conditions than undrained ones. The chapter discusses the formulation and analysis of coupled deformation–diffusion processes, within the framework of the Biot theory of poroelasticity. The Biot model of a fluid-filled porous material is constructed on the conceptual model of a coherent solid skeleton and a freely moving pore fluid.
1,056 citations
TL;DR: In this article, a physically motivated regularization of the Euler equations is proposed to allow topological transitions to occur smoothly, where the sharp interface is replaced by a narrow transition layer across which the fluids may mix.
Abstract: One of the fundamental problems in simulating the motion of sharp interfaces between immiscible fluids is a description of the transition that occurs when the interfaces merge and reconnect. It is well known that classical methods involving sharp interfaces fail to describe this type of phenomena. Following some previous work in this area, we suggest a physically motivated regularization of the Euler equations which allows topological transitions to occur smoothly. In this model, the sharp interface is replaced by a narrow transition layer across which the fluids may mix. The model describes a flow of a binary mixture, and the internal structure of the interface is determined by both diffusion and motion. An advantage of our regularization is that it automatically yields a continuous description of surface tension, which can play an important role in topological transitions. An additional scalar field is introduced to describe the concentration of one of the fluid components and the resulting system of equations couples the Euler (or Navier–Stokes) and the Cahn–Hilliard equations. The model takes into account weak non–locality (dispersion) associated with an internal length scale and localized dissipation due to mixing. The non–locality introduces a dimensional surface energy; dissipation is added to handle the loss of regularity of solutions to the sharp interface equations and to provide a mechanism for topological changes. In particular, we study a non–trivial limit when both components are incompressible, the pressure is kinematic but the velocity field is non–solenoidal (quasi–incompressibility). To demonstrate the effects of quasi–incompressibility, we analyse the linear stage of spinodal decomposition in one dimension. We show that when the densities of the fluids are not perfectly matched, the evolution of the concentration field causes fluid motion even if the fluids are inviscid. In the limit of infinitely thin and well–separated interfacial layers, an appropriately scaled quasi–incompressible Euler–Cahn–Hilliard system converges to the classical sharp interface model. In order to investigate the behaviour of the model outside the range of parameters where the sharp interface approximation is sufficient, we consider a simple example of a change of topology and show that the model permits the transition to occur without an associated singularity.
878 citations
Cites background or methods from "Continuum theories of mixtures: bas..."
...We remark that the above model of the mixing layer is different from the more traditional models of homogeneous mixtures (see Atkin & Crane (1976) and Bedford & Drumheller (1983) for comprehensive reviews)....
[...]
...the conserved order parameter (Cahn-Hilliard diffusion) was suggested in Truskinovsky (1993) (see Appendix A)....
[...]
TL;DR: In this article, a depth-averaged, three-dimensional mathematical model that accounts explicitly for solid and fluid-phase forces and interactions was developed to predict motion of diverse grain-fluid masses from initiation to deposition.
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.
810 citations
TL;DR: In this article, a review emphasizes models in which debris behavior evolves in response to changing pore pressures and granular temperatures, and quantifies how pore pressure and temperature can influence the behavior of debris flows.
Abstract: ▪ Abstract Field observations, laboratory experiments, and theoretical analyses indicate that landslides mobilize to form debris flows by three processes: (a) widespread Coulomb failure within a sloping soil, rock, or sediment mass, (b) partial or complete liquefaction of the mass by high pore-fluid pressures, and (c) conversion of landslide translational energy to internal vibrational energy (i.e. granular temperature). These processes can operate independently, but in many circumstances they appear to operate simultaneously and synergistically. Early work on debris-flow mobilization described a similar interplay of processes but relied on mechanical models in which debris behavior was assumed to be fixed and governed by a Bingham or Bagnold rheology. In contrast, this review emphasizes models in which debris behavior evolves in response to changing pore pressures and granular temperatures. One-dimensional infinite-slope models provide insight by quantifying how pore pressures and granular temperatures c...
764 citations
References
More filters