Bob Svendsen

On interfacial transition conditions in two phase gravity flow

A layer of ice and sediment is modelled as a mixture of two nonlinear, very viscous, constant density fluids interacting mechanically via Darcy- and Pick-type forces. An inclined layer of this mixture overlain by a layer of ice modelled as a viscous fluid is considered with boundary conditions of no-slip or viscous sliding at the base and no stress at the free surface. The interface is treated as a singular surface across which the jump conditions of mass and momentum for the constituents are assumed to hold. Furthermore, because the components are viscous fluids, a kinematic condition for the continuity of the tangential velocity is formulated. The momentum jump conditions involve surface production terms requiring additional surfacial constitutive relations.We show that the posed physical problem admits a mathematical solution only in the case that the interface momentum production is non-zero.

On the representation of constitutive relations using structure tensors

Abstract The purpose of this paper is to present a general formulation of the representation of constitutive relations for a material with a given (material) symmetry group in terms of functionals that depend on one or more so-called structure tensors. The principal result in this work takes the form of a relation between the symmetry groups of (1), the material constitutive relation, (2), the corresponding structure tensor(s), and (3), the functional depending on these tensors. In particular, this result contains that of the Rychlewski-Zhang representation theorem (e.g. Zhang and Rychlewski [1]) for anisotropic solids as a special case, and as such clarifies, simplifies and generalizes their formulation.

On the Role of Mechanical Interactions in the Steady-State Gravity Flow of a Two-Constituent Mixture down an Inclined Plane

In this work, we investigate the isothermal gravity-driven Stokes flow of a mixture of two constant true density viscous fluids which are overlain by a (single-constituent) constant density viscous fluid down an inclined plane. The continuum thermodynamical theory for such a system implies that, in the simplest case, the constituents of such a mixture interact mechanically with each other because of (1) friction or drag between the constituents, and (2) the non-uniform (volume) distribution of constituents, in the mixture. The former interaction is proportional to the relative velocity of the two constituents, and the latter to the gradient of the volume fraction. The coefficient of the volume fraction gradient in this latter interaction has the dimensions of pressure, and is usually interpreted as the fluid pressure p in the case of a fluid-solid mixture. More generally, however, this pressure represents that maintaining saturation in the mixture. In this work, we formulate a model for a saturated mixture in which this coefficient takes a slightly more general form, i.e. δp, where δ is a dimensionless constant varying between 0 and 1. In particular, in the context of the thin-layer approximation, analytical solutions of the lowest-order non-dimensionalized constituent momentum balances, under the usual assumption δ=1, yield only pure constituent-1 or pure constituent-2 ‘mixtures’. On the other hand, numerical solution of these momentum balances for δ = 1 yield non-trivial volume fraction variations with depth in the layer, and hence represent true mixture solutions. Applying this model to the case of a sediment-ice mixture, such as that found in a glacier or ice sheet, one obtains good qualitative agreement with observations on the variation of sediment in these bodies with depth for δ > 0.95, i.e. in this case the sediment remains concentrated at the bottom of the layer.

Interaction models for mixtures with application to phase transitions

Continuum mixture theory formulates balances of mass, momentum and energy for a mixture body and non-material singular surfaces. An important component of these balances are the transfers that take place between constituents. These are described by volume interactions within a mixture body and surface interactions at a singular surface. The interactions sum to zero over all the constituents in order that there is no net production of mass, momentum or energy. The principle of Euclidean frame indifference is used to formulate the functional forms of these interactions. A simple set of volume and surface interaction functions are then postulated which satisfy the summation and frame indifference requirements. These partition the mass, momentum and energy transfers into a sum of the interactions between pairs of constituents. Illustrations are presented for a classical phase change front, a phase change front in a binary mixture and two examples of phase change in a tertiary mixture which demonstrate complex reabsorption processes.

On the Constituent Structure of a Classical Mixture

Thiw work is concerned with the formulation of constituent interactions and corresponding balance relations in classical mixture theory as based on a model for the (classical) constituent structure of such a mixture.

A local frame formulation of dual stress-strain pairs and time derivatives

SummaryThis work is concerned with a formulation of the “dual” stress-strain pair concept of Haupt and Tsakmakis in terms of the (spatially) local form of the differential geometric concept of a frame field. Among other things, such a formulation yields additional insight into the mathematical structure and physical interpretation underlying the concept of “dual”, which turns out to be based intimately on that of a time-dependent frame.To facilitate the approach taken in this work, the (usual) referential kinematic setting (i.e., relative to a reference configuration of the material body) used by Haupt and Tsakmakis is first generalized to a material setting (i.e., relative to the material body itself) with the help of Nolls concept of a body element. From this point of view, the formulation of dual stress-strain pairs relies essentially upon an equivalence class of body element placements, or equivalently, material frames. In particular, the notion of a material strain tensor is dependent on the metric tensor induced by such an equivalence class. Euclideanrepresentations (e.g., referential or spatial) of the basic material deformation, stress and strain tensors are induced by any motion of the body element in Euclidean (vector) space with respect to the corresponding induced time-dependent Euclidean frame, yielding in particular the two families of strain and stress tensors discussed by Haupt and Tsakmakis.The fast part of this work investigates the consequences of extending the notion of a Euclidean representation of the basic material deformation, strain and stress tensors (induced by any motion of the body element) to certain physical quantities depending on these tensors and their time derivatives, i.e., the material stress power density and its incremental forms. In the work of Haupt and Tsakmakis, such a representation is achieved by requiring each stress-strain pair in each family to be conjugate with respect to the stress power density, as well as its appropriate incremental form. This is possible in general iff the tensortime derivatives involved transform in the same fashion as the tensors themselves, yielding the notion ofdual time derivatives of these tensors. As shown in the last part of this work, such derivatives represent a type of covariant derivative based on the connection of a time-dependent frame. From this point of view, one can show, among other things, that the well-known problem with “oscillating stresses” arising for certain hypoelastic constitutive relations (e.g., for a Maxwell fluid) can most likely be ascribed to an extra time-dependent frame-dependence in the Jaumann form of the constitutive relation, something that is avoided in the Oldroyd form.

A thermodynamic formulation of the equations of motion and buoyancy frequency for Earth's fluid outer core

This work presents a precise formulation of the balance and constitutive relations appropriate to the modeling of the motion of the fluid outer core, including a full thermodynamic analysis and derivation of the buoyancy frequency, its role in the equations of motion for the fluid outer core, and the dependence of this equation on the thermodynamic state of the outer core. It is shown that the equation of motion is controlled by a single parameter, the buoyancy frequency. Its definition is general, so as to include also thermodynamic effects in various forms. The dependence on concentration or entropy-gradients is of importance for motions on longer time scales but also influences core undertones, the oscillations of the Earths outer core, in regions of strong concentration gradients. By appropriate scaling a variety of different approximations arise from this formulation using the thermodynamic definiton of the buoyancy frequency. Special interest is put on the scaling for the outer core eigenmodes yielding consistent formulations for the Boussinesq- and the subseismic approximation.

A local superposed-constituent volume-fraction mixture theory based on relative motion

Abstract This work presents a local formulation of kinematics and balance relations for a mixture composed of superposed-constituents as based on their motion relative to a model of the mixture as a moving region, with the motion of this region being determined by that of all the constituents. The relative motion of each constituent with respect to this moving mixture region can be interpreted as its diffusive motion in the mixture. A number of constituent kinematic properties in the mixture , in particular the constituent volume density (i.e. the infinitesimal volume-fraction) and constituent diffusion velocity, are determined by the constituent diffusive motion, and arise naturally in a formulation based upon it. In addition, this formulation yields in a natural fashion an evolution relation for the constituent volume density relative to the moving mixture region. The consequences of this kinematic model for constituent and mixture local balance relations, as well as the sum relations, are investigated in the last part of the work.