Preface The basics of asymptotics Perturbation theory Model examples Models of asymptotic approximation of the Navier-Stokes model Asymptotic approximation of the Boltzmann model for small and large mean free path Other models of asymptotic approximation References Index

We discuss the asymptotic behavior of a solute plume undergoing reversible sorption governed by a Freundlich isotherm after single-pulse injection. Our analysis predicts that the concentration at a fixed position decays asymptotically like a power law with an exponent α = 1/(1 − n) where n is the Freundlich exponent. Correspondingly, the shape of tail is time invariant. The results are checked by comparison with numerical solutions for one-dimensional transport in a homogeneous medium. Some further asymptotic results for this case are derived. The power law behavior provides an alternative way to derive the Freundlich n parameter from breakthrough curves in comparison to the use of inverse estimation methods. This is especially the case when evaluating breakthrough curves obtained in two- or three-dimensional flow domains, for which indirect estimation of parameters becomes very difficult.

Asymptotic Treatment of Differential Equations

Asymptotic analysis of nonlinear equilibrium solute transport in porous media

[1] We introduce a trajectory-based method for the inversion of flow and transport observations. The approach operates on the output of a standard numerical simulator and is applicable under very general conditions. Using only head and concentration histories, we efficiently construct the model parameter sensitivities required for solving the inverse problem. A single simulation or forward calculation is required in order to take a step in the inversion. Our formulation, based on asymptotic solutions for flow and transport, partitions the inverse problem into an arrival time and an amplitude-matching problem. Because the model parameter sensitivities are defined along trajectories, both inverse problems scale very well with respect to model size. At the migration experimental site of the Grimsel Rock Laboratory in Switzerland we utilized transient head and tracer arrival times to constrain the hydraulic conductivity between a suite of boreholes. The squared data misfit is reduced by over an order of magnitude in 30 iterations of our inversion algorithm. Each iteration requires two runs of our numerical simulator TOUGH2, one for each experiment, to construct the trajectories and associated model parameter sensitivities. We find a high-conductivity channel curving in a roughly east-west direction, in agreement with two previous studies. Estimates of model parameter resolution indicate that we can image the large-scale conductivity variations within the array of boreholes.

Numerical trajectory calculations for the efficient inversion of transient flow and tracer observations

The main steps used in deriving the mathematical models governing the fluid flows are shown. The fluids and their motions presented are more complicate than those studied by fluid dynamics. However, the relative position of fluid dynamics vs. thermodynamics of fluids is analyzed. The mathematical technicalities are avoided but the rigor was kept by the attentive explanation of the mathematical meaning involved in the concepts and results. The mathematical tools used are described separately from the physical hypotheses. Nevertheless, the mathematical content of basic physical notions is carefully revealed.

[ DOI : 10.1685/SELN09002] About DOI

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https://cab.unime.it/journals/index.php/lecture/article/download/351/255

Thermodynamics of fluids

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Perturbed Differential Equations with Singular Points

Geometric quantum hydrodynamics merges geometric hydrodynamics with quantum hydrodynamics to study the geometric properties of vortex structures in superfluid states of matter. Here the vortex line acts as the fundamental building block and is a topological defect of the fluid medium about which the otherwise irrotational fluid circulates. In this thesis, we show that except for the simplest fluid flows, a vortex line seeks to decompose localized regions of curvature into helical configurations. The simplest flow, known as the local induction approximation, is also an integrable one. Integrability makes the transfer of energy into helical modes impossible. In the following, we demonstrate that any arclength conserving correction to this approximation defines a non-Hamiltonian evolution of the vortex geometry, which is capable of supporting dissipative solitons and helical wavefronts. Quantum turbulence in ultracold vortex tangles relies on energy transfer between helical or Kelvin modes to decay. Thus, models of vortex lines beyond the lowest order integrable cases are vitally important to our mathematical description of free decay of turbulent tangles. To motivate the results of this thesis we connect our theory of vortex line dynamics to continuum fluid mechanics. The Navier-Stokes equations are a statement of momentum balance for a fluid whose response to shear stress is proportional to the fluid's velocity gradients. Building off of Onsager's rather obscure work in fluid turbulence, others have shown that solutions to Navier-Stokes limit to Euler evolutions of distributional velocity profiles for large Reynolds number. Using this as our context, we show that the Euler equation can be transformed, in an inverse Madelung sense, to the Gross-Pitaevskii equation associated with the mean-field quantum dynamics of a dilute Bose gas. This theory predicts that an irrotational fluid is capable of circulating around regions of density depletion known as vortex lines. Furthermore, if a vortex line is used as the ansatz for the Gross-Pitaevskii equation then it is possible to show that the Biot-Savart integral over the vortex source results. This connection between the vortex line geometry and the induced velocity field provides the basis for the application of geometric quantum hydrodynamics to Bose-Einstein condensates. If the Biot-Savart integral is the basis of geometric quantum hydrodynamics, then Hasimoto's transformation is the structure built on top. The fundamental theorem of space curves states that up to rotations and translations, a curve is defined by its curvature and torsion. Through the Hasimoto transformation, it is possible to map flows defined by the Biot-Savart integral to scalar partial differential equations evolving the curvature and torsion of a vortex line. In this thesis, we conduct an asymptotic expansion of the Biot-Savart integral and apply the Hasimoto transformation to show that vortex lines prefer to relax curved abnormalities through the excitation of helical waves along the vortex. Thus, our…

Asymptotic Treatment of Differential Equations

Citations

Asymptotic analysis of nonlinear equilibrium solute transport in porous media

Numerical trajectory calculations for the efficient inversion of transient flow and tracer observations

Thermodynamics of fluids

Perturbed Differential Equations with Singular Points

Geometric quantum hydrodynamics and Bose-Einstein condensates: non-Hamiltonian evolution of vortex lines

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