499 research outputs found
An Unusual Moving Boundary Condition Arising in Anomalous Diffusion Problems
In the context of analyzing a new model for nonlinear diffusion in polymers, an
unusual condition appears at the moving interface between the glassy and rubbery phases of the
polymer. This condition, which arises from the inclusion of a viscoelastic memory term in our
equations, has received very little attention in the mathematical literature. Due to the unusual form
of the moving-boundary condition, further study is needed as to the existence and uniqueness of
solutions satisfying such a condition. The moving boundary condition which results is not solvable
by similarity solutions, but can be solved by integral equation techniques. A solution process is
outlined to illustrate the unusual nature of the condition; the profiles which result are characteristic
of a dissolving polymer
Spreading of a density front in the K\"untz-Lavall\'ee model of porous media
We analyze spreading of a density front in the K\"untz-Lavall\'ee model of
porous media. In contrast to previous studies, where unusual properties of the
front were attributed to anomalous diffusion, we find that the front evolution
is controlled by normal diffusion and hydrodynamic flow, the latter being
responsible for apparent enhancement of the front propagation speed. Our
finding suggests that results of several recent experiments on porous media,
where anomalous diffusion was reported based on the density front propagation
analysis, should be reconsidered to verify the role of a fluid flow
Nonlinear diffusion & thermo-electric coupling in a two-variable model of cardiac action potential
This work reports the results of the theoretical investigation of nonlinear
dynamics and spiral wave breakup in a generalized two-variable model of cardiac
action potential accounting for thermo-electric coupling and diffusion
nonlinearities. As customary in excitable media, the common Q10 and Moore
factors are used to describe thermo-electric feedback in a 10-degrees range.
Motivated by the porous nature of the cardiac tissue, in this study we also
propose a nonlinear Fickian flux formulated by Taylor expanding the voltage
dependent diffusion coefficient up to quadratic terms. A fine tuning of the
diffusive parameters is performed a priori to match the conduction velocity of
the equivalent cable model. The resulting combined effects are then studied by
numerically simulating different stimulation protocols on a one-dimensional
cable. Model features are compared in terms of action potential morphology,
restitution curves, frequency spectra and spatio-temporal phase differences.
Two-dimensional long-run simulations are finally performed to characterize
spiral breakup during sustained fibrillation at different thermal states.
Temperature and nonlinear diffusion effects are found to impact the
repolarization phase of the action potential wave with non-monotone patterns
and to increase the propensity of arrhythmogenesis
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Finite element approximation of non-Fickian polymer diffusion
The problem of nonlinear non-Fickian polymer diffusion as modelled by a diffusion
equation with an adjoined spatially local evolution equation for a viscoelastic
stress is considered (see, for example, Cohen, White & Witelski, SIAM J. Appl. Math.
55, pp. 348β368, 1995). We present numerical schemes based, spatially, on the
Galerkin finite element method and, temporally, on the Crank-Nicolson method. Special
attention is paid to linearising the discrete equations by extrapolating the value
of the nonlinear term from previous time steps. Optimal a priori error estimates are
given, based on the assumption that the exact solution possesses certain regularity
properties, and numerical experiments are given to support these error estimates
Coexistence of competitors mediated by nonlinear noise
Stochastic reaction-diffusion equations are a popular modelling approach for studying interacting populations in a heterogeneous environment under the influence of environmental fluctuations. Although the theoretical basis of alternative models such as Fokker- Planck diffusion is not less convincing, movement of populations is most commonly modelled using the diffusion law due to Fick. An interesting feature of Fokker-Planck diffusion is the fact that for spatially varying diffusion coefficients the stationary solution is not a homogeneous distribution β in contrast to Fickβs law of diffusion. Instead, concentration accumulates in regions of low diffusivity and tends to lower levels for areas of high diffusivity. Thus, we may interpret the stationary distribution of the Fokker-Planck diffusion as a reflection of different levels of habitat quality. Moreover, the most common model for environmental fluctuations, linear multiplicative noise, is based on the assumption that individuals respond independently to stochastic environmental fluctuations. For large population densities the assumption of independence is debatable and the model further implies that noise intensities can increase to arbitrarily high levels. Therefore, instead of the commonly used linear multiplicative noise model, we implement environmental variability by an alternative nonlinear noise term which never exceeds a certain maximum noise intensity. With Fokker-Planck diffusion and the nonlinear noise model replacing the classical approaches we investigate a simple invasive system based on the Lotka-Volterra competition model. We observe that the heterogeneous stationary distribution generated by Fokker-Planck diffusion generally facilitates the formation of segregated habitats of resident and invader. However, this segregation can be broken by nonlinear noise leading to coexistence of resident and invader across the whole spatial domain, an effect that would not be possible in the non-spatial version of the competition model for the parameters considered here
Turbulent mixing
The ability of turbulent flows to effectively mix entrained fluids to a molecular scale is a vital part of the dynamics of such flows, with wide-ranging consequences in nature and engineering. It is a considerable experimental, theoretical, modeling, and computational challenge to capture and represent turbulent mixing which, for high Reynolds number (Re) flows, occurs across a spectrum of scales of considerable span. This consideration alone places high-Re mixing phenomena beyond the reach of direct simulation, especially in high Schmidt number fluids, such as water, in which species diffusion scales are one and a half orders of magnitude smaller than the smallest flow scales. The discussion below attempts to provide an overview of turbulent mixing; the attendant experimental, theoretical, and computational challenges; and suggests possible future directions for progress in this important field
Mathematical models of combustion at high pressure
In this dissertation, we develop new mathematical theories of flame propagation that are valid at elevated, or extreme, pressures. Of particular interest is the regime of burning in which the pressure exceeds the critical pressure of the species undergoing chemical reaction. Fluids and flames are known to behave differently under these extreme conditions as opposed to atmospheric pressure. The focus of this dissertation is to investigate these differences by deriving reduced models that contain the unique features.
In the first part of this dissertation, we analyze the structure of laminar diffusion flames at high pressure in the limit of large activation energy for the particular configuration of a steady flame in counterflow. We consider a dense fluid in which normal Fickian diffusion of the fuel is limited, and thermal diffusion, i.e., the Soret effect, is the dominant mechanism for fuel mass transport. Temperature and species profiles, as well as flame temperature and location, are determined as a function of DamkΓΆhler number and Soret diffusion coefficient. In particular, we find that oxidant is entirely consumed by the flame, while some fuel leaks through. For light fuels, the fuel profile is found to have a local peak on the oxidant side as a result of thermal diffusion. Our analysis includes a description of extinction phenomenon, including explicit criteria in terms of the Soret diffusion coefficient, ratio of temperature of the two streams, and the DamkΓΆhler number at extinction.
In the second part of this dissertation, we derive an asymptotic theory of laminar premixed flames in high density fluids in the limit of large activation energy. The model is intended to provide insights into the structure and dynamics of deflagration waves in high pressure, dense fluids where normal Fickian diffusion is limited. In such cases, particularly under conditions exceeding the thermodynamic critical point of the fluid, the primary mode of species transport is through thermal diffusion, i.e., the Soret effect. Such a model for diffusive transport is considered, and we derive a model with an explicit dependence on the Soret effect for a one-step overall reaction. The density is assumed sufficiently high to adopt a constant density formulation. The local reaction-diffusion structure is found to be fundamentally different from that of an ideal gas with Fickian diffusion, which results in new conditions relating the equations for thermal and mass transport in the bulk flow. The model is used to investigate the basic structure of planar flames, as well as their stability. Stability boundaries are identified that mark the transition from planar to either steady, spatially periodic structures, or time-dependent modes of propagation. The combined effects of the Soret diffusion coefficient and Lewis number are discussed. Furthermore, a weakly nonlinear analysis of the derived model is carried out, resulting in a modified Kuramoto-Sivashinsky (K-S) equation, accounting for effects of Soret Diffusion. Linear stability analysis shows that the flame front is unstable with respect to long-waves in a range of Soret diffusion coefficient that corresponds to no and weak Soret effect. However, there exists a range of Soret diffusion coefficient for which a flame front is unconditionally stable
Diffusive processes in polyacrylic acid hydrogels
The aim of this work was to characterise the diffusive properties of superabsorbent polymer systems prepared by free radical crosslinking polymerisation of acrylic acid. The polyelectrolyte nature of these hydrogels gives rise to high swelling capacities, and their ability to absorb and retain water is highly dependent on the precise network structure. Modifying the synthesis formulation results in considerable changes to the dynamic and structural properties of these gels, providing the motive for the work presented here. The influence of two factors, namely the extent of monomer neutralisation and the level of crosslinker in the pre-gel solution, were investigated. The dynamic properties of gels were examined using Quasi-Elastic Light Scattering (QELS), from which the cooperative diffusion coefficient and degree of heterogeneity could be determined. The former was found to increase linearly with neutralisation, due to the introduction of electrostatic interactions. The diffusion coefficient initially remained constant with the addition of crosslinker, due to the dominating influence of physical entanglements, but increased above a threshold crosslinking degree, corresponding to a reduction of the network mesh size. The extent of large-scale inhomogeneity increased for higher ionisations, as both the crosslinker solubility and the efficiency of monomer-crosslinker reaction decreased. However, there was a tendency for concentration fluctuations to be minimised for higher neutralisations, making the gel more microscopically homogeneous. Kinetics of swelling experiments were used to investigate gels of varying composition. The macroscopic diffusion coefficient was found to increase rapidly with increasing neutralisation until the monomer was approximately 35% neutralised, beyond which point counterion condensation caused insignificant variation. This trend was also reflected in the equilibrium swelling ratios, and mode of diffusion. For the majority of gels, the diffusion process was characterised as case II transport. Variation of crosslinking degree caused an overall increase in the diffusion coefficient, reflecting the trend observed in the QELS studies. Nuclear Reaction Analysis (NRA) was used to probe the penetration of heavy water into dry network slabs. The concentration-depth profiles revealed a discontinuity in the diffusion coefficient, corresponding to the transition between glassy and rubbery states, for which the diffusivities differed by several orders of magnitude. The kinetics of plasticisation was assumed to be the rate determining factor in the swelling process, on the timescale of the NRA experiments. The diffusion coefficient for the swollen rubbery region, representative of the macroscopic diffusion process, was found to increase linearly with neutralisation, and decrease with crosslinking degree. The latter observation was explained as due to a reduction in the free volume available for solvent diffusion with higher levels of crosslinker
Effects of Soret diffusion on premixed counterflow flames
Abstract Soret diffusion is a secondary mass diffusion and it affects laminar flames with very light or heavy species and large temperature gradient. To get a general understanding of Soret effects on stretched flame, we conduct theoretical analysis on premixed counterflow flame with Soret diffusion. A deliberately idealized premixed counterflow flame model is analyzed asymptotically within the framework of large activation energy, potential flow, and thermaldiffusive model. A correlation among flame stretch rate, flame position, and flame temperature is derived and is used to assess the effects of Soret diffusion on counterflow flame structure, Markstein length, and extinction stretch rate. Results show that Soret diffusion quantitatively affects the premixed counterflow flame and that the influence of Soret diffusion strongly depends on Lewis number and stretch rate. For light fuel, the premixed counterflow flame becomes stronger after including Soret diffusion. The opposite trend occurs for heavy fuel. The influence of Soret diffusion is found to increase with the stretch rate. A linear change between normalized Markstein length and Soret diffusion coefficient is observed, indicating that flame becomes more sensitive to stretch rate after including Soret diffusion. Furthermore, Soret diffusion is shown to greatly increase the extinction stretch rate of light fuels. These results indicate that for highly-stretched premixed flames containing very light species, the impact of Soret diffusion cannot be neglected
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