Flux-Ratio Theorems for Nonlinear Equations of Generalized Diffusion

Flux-ratio theorems compare the flux of matter through partof the boundary of a medium with spatially inhomogeneous transportproperties incorporating diffusion, migration, and temporarytrapping of the transported substance, with the flux measuredin a complementary experiment and usually through a differentpart of the boundary. Any nonlinearity in the transport equationsleads to a breakdown of the Ussing flux-ratio theorem pertainingto all times. A fluxintegral theorem is proved for the casewhen the nonlinearity is in the kinetics of trapping. New resultson the time-lag constants for the asymptotics of the two complementaryfluxes associated with nonlinear trapping show that one canexpect deviations from the results that would hold if Ussing'sgeneral theorem were true for this case. When the nonlinearityis due both to nonlinear trapping kinetics and to concentrationdependence of the diffusion coefficient, a nonlinear flux-integral-ratiotheorem is shown to hold when the substance diffuses (but doesnot migrate) in a spatially homogeneous medium.     

Heat Conduction and Finite Measures for Transition Times between Steady States

A finite time, the mean action time associated with the conductivetransition from a constant initial temperature to thermal equilibriumat a constant ambient temperature, is related to time lag constants,mean energy residence times, mean first passage times, and Green'sfunction properties for linear equations, and to freezing timesand finite transition times for problems with phase transitions.When the boundary conditions are linear of mixed type and theconductivity is constant, or when they are of Dirichlet typeand the conductivity perhaps temperature dependent, the meanaction time is given by the solution of a linear Poisson problem.It is then easily found and is a useful finite comparative timefor the thermal transition process, giving a measure of itsdependence on size and other geometric factors.     

Flux Theorems for Linear Multicomponent Diffusion

Sten-Knudsen & Ussing (1981) and Bass et al. (1986) haveshown that the flux of matter through membranes is proportionalat all corresponding times to the flux in the opposite directionwhen appropriate boundary conditions are reversed, even if thediffusion-migration parameters and the trapping characteristicsof the material vary arbitrarily with distance normal to themembrane surface. In this paper, we extend this flux-ratio theoremto some cases where the diffusing component is interacting withother diffusing species in the membrane. When the equationsare assumed to be linear, as for tracer experiments, the amountof any component transmitted though any part of the boundaryover the duration of an experiment is determined by the timeintegral of the given boundary conditions and is independentof pulse shape. If the interaction matrix A coupling the transportequations of the components is quasi-symmetric in the sensethat it is the product of a symmetric and a diagonal matrix,the flux-ratio theorem remains valid. Interactions dominatedby a single chemical reaction give rise to quasi-symmetric matricesof this form. An example involving three components illustratesthe structure. It is shown that interaction parameters suchas absolute reaction rates can be deduced for experiments basedon these results.     

Block-size determination in fractured porous media

A flow-reversal method involving the injection and retrievalof fluid from a fractured, porous medium at a different temperatureto that of the system in its undisturbed state is describedand assessed as a potential technique for measuring the thermaltransients of blocks of fractured porous media. Formulae areobtained for the zeroth- and first-order time moments of thethermal history of the fluid when it is well mixed in the injection-retrievalhole, injected at a temperature T₁ (different to the uniformtemperature T₀ of the undisturbed system) at a fixed rate Q,for a time t₁ and then withdrawn through the same drill holeat the same fixed rate Q. These formulae have a potential fordetermining some geometric parameters of the block structure,including the block surface-to-volume ratio and the mean actiontime for conductive heating.     

Some unsteady diffusion and absorption problems

Specific problems of the diffusion type, focusing on the absorptionof heat (or matter) at one or more boundary surfaces after itsrelease within the adjacent medium at a localized source, areconsidered, and the measures of absorption or outflow duringarbitrary intervals of time are directly expressed in termsof auxiliary temperature/matter distributions with given sourceson the pertinent surfaces. Detailed applications of these reciprocalrelations are presented, as is a brief analysis that includessome nonlinear or convective aspects in the basic diffusionequations.