501 research outputs found
Infinitely divisible nonnegative matrices, -matrices, and the embedding problem for finite state stationary Markov Chains
This paper explicitly details the relation between -matrices, nonnegative
roots of nonnegative matrices, and the embedding problem for finite-state
stationary Markov chains. The set of nonsingular nonnegative matrices with
arbitrary nonnegative roots is shown to be the closure of the set of matrices
with matrix roots in . The methods presented here employ nothing
beyond basic matrix analysis, however it answers a question regarding
-matrices posed over 30 years ago and as an application, a new
characterization of the set of all embeddable stochastic matrices is obtained
as a corollary
Augmented saddle point formulation of the steady-state Stefan--Maxwell diffusion problem
We investigate structure-preserving finite element discretizations of the
steady-state Stefan--Maxwell diffusion problem which governs diffusion within a
phase consisting of multiple species. An approach inspired by augmented
Lagrangian methods allows us to construct a symmetric positive definite
augmented Onsager transport matrix, which in turn leads to an effective
numerical algorithm. We prove inf-sup conditions for the continuous and
discrete linearized systems and obtain error estimates for a phase consisting
of an arbitrary number of species. The discretization preserves the
thermodynamically fundamental Gibbs--Duhem equation to machine precision
independent of mesh size. The results are illustrated with numerical examples,
including an application to modelling the diffusion of oxygen, carbon dioxide,
water vapour and nitrogen in the lungs.Comment: 27 pages, 5 figure
Structural electroneutrality in Onsager-Stefan-Maxwell transport with charged species
We present a method to embed local electroneutrality within
Onsager-Stefan-Maxwell electrolytic-transport models, circumventing their
formulation as differential systems with an algebraic constraint. Flux-explicit
transport laws are formulated for general multicomponent electrolytes, in which
the conductivity, component diffusivities, and transference numbers relate to
Stefan-Maxwell coefficients through invertible matrix calculations. A
construction we call a `salt-charge basis' implements Guggenheim's
transformation of species electrochemical potentials into combinations
describing a minimal set of neutral components, leaving a unique combination
associated with electricity. Defining conjugate component concentrations and
fluxes that preserve the structures of the Gibbs function and energy
dissipation retains symmetric Onsager reciprocal relations. The framework
reproduces Newman's constitutive laws for binary electrolytes and the
Pollard-Newman laws for molten salts; we also propose laws for salt solutions
in two-solvent blends, such as lithium-ion-battery electrolytes. Finally, we
simulate a potentiostatic Hull cell containing a non-ideal binary electrolyte
with concentration-dependent properties
Finite element methods for multicomponent convection-diffusion
We develop finite element methods for coupling the steady-state Onsager–Stefan–Maxwell (OSM) equations to compressible Stokes flow. These equations describe multicomponent flow at low Reynolds number, where a mixture of different chemical species within a common thermodynamic phase is transported by convection and molecular diffusion. Developing a variational formulation for discretizing these equations is challenging: the formulation must balance physical relevance of the variables and boundary data, regularity assumptions, tractability of the analysis, enforcement of thermodynamic constraints, ease of discretization and extensibility to the transient, anisothermal and nonideal settings. To resolve these competing goals, we employ two augmentations: the first enforces the definition of mass-average velocity in the OSM equations, while its dual modifies the Stokes momentum equation to enforce symmetry. Remarkably, with these augmentations we achieve a Picard linearization of symmetric saddle point type, despite the equations not possessing a Lagrangian structure. Exploiting structure mandated by linear irreversible thermodynamics, we prove the inf-sup condition for this linearization, and identify finite element function spaces that automatically inherit well-posedness. We verify our error estimates with a numerical example, and illustrate the application of the method to nonideal fluids with a simulation of the microfluidic mixing of hydrocarbons
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