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https://ir.stthomas.edu/cas_mathpub
Recent documents in Mathematics Faculty Publications & Dataen-usSat, 26 Feb 2022 01:54:13 PST3600On the axisymmetric steady incompressible Beltrami flows
https://ir.stthomas.edu/cas_mathpub/201
https://ir.stthomas.edu/cas_mathpub/201Thu, 24 Feb 2022 08:11:52 PST
In this paper, Beltrami vector fields in several orthogonal coordinate systems are obtained analytically and numerically. Specifically, axisymmetric incompressible inviscid steady state Beltrami (Trkalian) fluid flows are obtained with the motivation to model flows that have been hypothesized to occur in tornadic flows. The studied coordinate systems include those that appear amenable to modeling such flows: the cylindrical, spherical, paraboloidal, and prolate and oblate spheroidal systems. The usual Euler equations are reformulated using the Bragg-Hawthorne equation for the stream function of the flow, which is solved analytically or numerically in each coordinate system under the assumption of separability of variables. Many of the obtained flows are visualized via contour plots of their stream functions in the rz-plane. Finally, the results are combined to provide a qualitative quasi-static model for a progression of tornado-like flows that develop as swirl increases. The results in this paper are equally applicable in electromagnetics, where the equivalent concept is that of a force-free magnetic field.
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Pavel Bělík et al.Possible implications of self-similarity for tornadogenesis and maintenance
https://ir.stthomas.edu/cas_mathpub/200
https://ir.stthomas.edu/cas_mathpub/200Thu, 24 Feb 2022 08:11:52 PST
Self-similarity in tornadic and some non-tornadic supercell flows is studied and power laws relating various quantities in such flows are demonstrated. Magnitudes of the exponents in these power laws are related to the intensity of the corresponding flow and thus the severity of the supercell storm. The features studied in this paper include the vertical vorticity and pseudovorticity, both obtained from radar observations and from numerical simulations, the tangential velocity, and the energy spectrum as a function of the wave number. Power laws for the vertical vorticity, pseudovorticity, and tangential velocity obtained from radar observations studied in the literature are summarized. Further support is given to the existence of a power law for vorticity by the analysis of data obtained from a numerical simulation of a tornadic supercell. A possible explanation for an increase in vorticity in a developing tornado is provided, as well as an argument that tornadoes have approximate fractal cross sections and negative temperatures. A power law that relates the increase of the approximate fractal dimension of the cross section of a negative temperature vortex to its energy content is derived, and the possible applicability of the box-counting method to determine this quantity from suitable radar images is demonstrated.
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Pavel Bělík et al.Applications of a vortex gas models to tornadogenesis and maintenance
https://ir.stthomas.edu/cas_mathpub/199
https://ir.stthomas.edu/cas_mathpub/199Thu, 24 Feb 2022 08:11:51 PST
Processes related to the production of vorticity in the forward and rear flank downdrafts and their interaction with the boundary layer are thought to play a role in tornadogenesis. We argue that an inverse energy cascade is a plausible mechanism for tornadogenesis and tornado maintenance and provides supporting evidence which is both numerical and observational. We apply a three-dimensional vortex gas model to supercritical vortices produced at the surface boundary layer possibly due to interactions of vortices brought to the surface by the rear flank downdraft and also to those related to the forward flank downdraft. Two-dimensional and three-dimensional vortex gas models are discussed, and the three-dimensional vortex gas model of Chorin, developed further by Flandoli and Gubinelli, is proposed as a model for intense small-scale subvortices found in tornadoes and in recent numerical studies by Orf et al. In this paper, the smaller scales are represented by intense, supercritical vortices, which transfer energy to the larger-scale tornadic flows (inverse energy cascade). We address the formation of these vortices as a result of the interaction of the flow with the surface and a boundary layer.
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Pavel Bělík et al.Fractal powers in Serrin’s vortex solutions
https://ir.stthomas.edu/cas_mathpub/198
https://ir.stthomas.edu/cas_mathpub/198Thu, 24 Feb 2022 07:22:15 PST
We consider a modification of the fluid flow model for a tornado-like swirling vortex developed by Serrin [Phil. Trans. Roy. Soc. London, Series A, Math & Phys. Sci. 271(1214) (1972), 325–360], where velocity decreases as the reciprocal of the distance from the vortex axis. Recent studies, based on radar data of selected severe weather events [Mon. Wea. Rev. 133(9) (2005), 2535–2551; Mon. Wea. Rev. 128(7) (2000), 2135–2164; Mon. Wea. Rev. 133(1) (2005), 97–119], indicate that the angular momentum in a tornado may not be constant with the radius, and thus suggest a different scaling of the velocity/radial distance dependence.
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Pavel Bělík et al.Modeling the behavior of heat-shrinkable thin films
https://ir.stthomas.edu/cas_mathpub/197
https://ir.stthomas.edu/cas_mathpub/197Thu, 24 Feb 2022 07:22:14 PST
We describe an asymptotic model for the behavior of PET-like heat-shrinkable thin films that includes both membrane and bending energies when the thickness of the film is positive. We compare the model to Koiter’s shell model and to models in which a membrane energy or a bending energy are obtained by Γ-convergence techniques. We also provide computational results for various temperature distributions applied to the films.
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Pavel Bělík et al.A stochastic model for PSA levels: behavior of solutions and population statistics
https://ir.stthomas.edu/cas_mathpub/196
https://ir.stthomas.edu/cas_mathpub/196Thu, 24 Feb 2022 07:22:13 PST
This paper investigates the partial differential equation for the evolving distribution of prostate-specific antigen (PSA) levels following radiotherapy. We also present results on the behavior of moments for the evolving distribution of PSA levels and estimate the probability of long-term treatment success and failure related to values of treatment and disease parameters. Results apply to a much wider range of parameter values than was considered in earlier studies, including parameter combinations that are patient specific.
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Pavel Bělík et al.Time averaging, hierarchy of the governing equations, and the balance of turbulent kinetic energy
https://ir.stthomas.edu/cas_mathpub/195
https://ir.stthomas.edu/cas_mathpub/195Thu, 24 Feb 2022 07:22:12 PST
We discuss a connection between governing equations, constitutive theory, and closure problem for atmospheric boundary layer. Such a connection is of prime importance in building algorithms for numerical simulations. We consider averaging in time and its relation to the Boussinesq approximation of the governing equations. We introduce a notion of instantaneous turbulent kinetic energy and derive a new balance equation for its material derivative.
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Douglas P. Dokken et al.A stochastic model for prostate-specific antigen levels
https://ir.stthomas.edu/cas_mathpub/194
https://ir.stthomas.edu/cas_mathpub/194Thu, 24 Feb 2022 07:22:11 PST
We introduce a continuous stochastic model for the prostate-specific antigen (PSA) levels following radiotherapy and derive solutions for the associated partial differential (Kolmogorov–Chapman) equation. The solutions describe the evolution of the time-dependent density for PSA levels which take into account an absorbing condition along the boundary and various initial conditions. We include implications for single-dose and multi-dose radiation treatment regimens and discuss parameter estimation and sensitivity issues.
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PWA Dayananda et al.Characterizing slop in mechanical assemblies via differential geometry
https://ir.stthomas.edu/cas_mathpub/193
https://ir.stthomas.edu/cas_mathpub/193Thu, 24 Feb 2022 07:22:10 PST
Slop (or backlash) in mechanical assemblies is often present and is usually undesirable from both craftsmanship and performance points of view. It is our belief that this phenomenon is not that well understood and that current methods of assessment are based largely on only qualitative, common-sense approaches. The focus of this paper is on developing an analytical theory for accurately characterizing slop, and on presenting an illustrative example. As one might expect, in principle, with a better understanding of slop, CAD (computer-aided-design) software package designers can create more refined software tools, mechanical engineers can design better products, and manufacturing engineers can be prepared to measure and improve craftsmanship levels. The underlying theory is based on combining concepts from differential geometry, including envelopes, constrained piecewise-smooth sweeps, and sweep vector fields (SVFs), along with basic configuration space (C-space) methods. In essence, the volumetric (or areal) error, which is generated as the movable part in an assembly is swept throughout its complete constrained volume (or area), may be viewed as a quantitative manifestation of craftsmanship errors. A 2-dimensional (2D) idealization of a common assembly that often suffers from poor craftsmanship due to slop, i.e., a doorknob assembly with exaggerated slop, is analyzed. The swept area is calculated using both traditional and SVF methods with the aid of Mathematica™. High quality Mathematica™ visualization of interesting sweeps along the bounding edges of the nonlinear slop constraint region, including generation of all of the envelope curves, is done. Finally, this work attempts to serve as a paradigm for characterizing slop based on engineering criteria.
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Michael P. Hennessey et al.Creation and propagation of oscillations in one-dimensional elasticity with surface energy
https://ir.stthomas.edu/cas_mathpub/192
https://ir.stthomas.edu/cas_mathpub/192Thu, 24 Feb 2022 07:22:09 PST
Creation and propagation of oscillations are studied on an evolutionary model for phase transtitions involving surface energy contributions and memory effects.
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Irene Fonseca et al.Oscillations in one-dimensional elasticity with surface energy
https://ir.stthomas.edu/cas_mathpub/191
https://ir.stthomas.edu/cas_mathpub/191Thu, 24 Feb 2022 07:22:08 PST
The characterization of the oscillatory behavior of solutions of a semilinear equation in one space dimension is obtained. In this work the model equation for a material undergoing a phase transition encompasses a surface energy term and first-order memory effects.
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Irene Fonseca et al.A note on the thermomechanics of curvature flow in IR<sup>3</sup> and on surfaces in IR<sup>3</sup>
https://ir.stthomas.edu/cas_mathpub/190
https://ir.stthomas.edu/cas_mathpub/190Thu, 24 Feb 2022 07:22:07 PST
Equations for the evolution of curves in IR^{3} and on surfaces in IR^{3} are derived from a configurational force balance, a mechanical version of the second law, and suitable constitutive assumptions. Both the isotropic and anisotropic cases are considered.
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Paolo Cermelli et al.Configurational forces and the dynamics of planar cracks in three-dimensional bodies
https://ir.stthomas.edu/cas_mathpub/189
https://ir.stthomas.edu/cas_mathpub/189Thu, 24 Feb 2022 07:22:06 PST
This paper develops a three-dimensional framework for the evolution of planar cracks, concentrating on the derivation of balances and constitutive equations that describe the motion of the crack tip. The theory is based on the notion of configurational forces in conjunction with a mechanical version of the second law.
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Morton E. Gurtin et al.Coexistent phases in nonlinear thermoelasticity: radially symmetric equilibrium states of aeolotropic bodies
https://ir.stthomas.edu/cas_mathpub/188
https://ir.stthomas.edu/cas_mathpub/188Thu, 24 Feb 2022 07:22:05 PST
We determine the detailed qualitative behavior of radially symmetric equilibrium states having coexistent phases for general classes of aeotropic nonlinearly thermoelastic materials. We treat both structured and non-structured interfaces. The aeolotropy is responsible for many novel effects. We show that linearly elastic materials cannot sustain coexistent radially symmetric phases unless the interfaces are structured. Our analysis is largely elementary, being based on a combination of geometric constructions with phase-plane methods. A few results, however, depend on our development of appropriate versions of the theory of asymptotically autonomous ordinary differential equations.
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Mikhail M. Shvartsman et al.The shrink-fit problem for aeolotropic nonlinearly elastic bodies
https://ir.stthomas.edu/cas_mathpub/187
https://ir.stthomas.edu/cas_mathpub/187Thu, 24 Feb 2022 07:17:01 PST
In the classical shrink-fit problem of linear elasticity (cf. Timoshenko [4, Sec. 31]) an annulus of natural inner radius a and natural outer radius (scaled to be) 1 is expanded (e.g., by heating) and then allowed to shrink down upon a disk of natural radius b, which is greater than a. See Fig. 1.1. Here we solve generalizations of this axisymmetric problem for aeolotropic, nonlinearly elastic bodies of different constitution subject to arbitrary axisymmetric boundary conditions on the outer edge of the annulus. We show that these problems exhibit a remarkable richness of physical phenomena and we show how easy it is to determine the detailed qualitative properties, the existence or nonexistence, and the uniqueness or multiplicity of all equilibrium states. We also show how to construct solutions for problems in which there are several annular layers.
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Stuart S. Antman et al.Phase boundaries in anisotropic elastic materials
https://ir.stthomas.edu/cas_mathpub/186
https://ir.stthomas.edu/cas_mathpub/186Thu, 24 Feb 2022 07:17:00 PST
This dissertation treats exact axisymmetric steady-state solutions of the governing equations of a two-phase anisotropic nonlinearly elastic disk and gives a full qualitative description of such solutions for very general classes of materials. The special feature of these solutions is that they admit non-planar interfaces due to non-constant deformation gradients. In particular, this work treats the classical mechanical shrink-fit problem (including linear stability) and the isothermal steady-state phase-change free-boundary problem for both coherent and noncoherent interfaces. It also presents some aspects of the steady-state phase-change free-boundary problem for the coherent interface in thermoelastic case. As an auxiliary result it provides the analog of the Maxwell condition for noncoherent interfaces.
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Mikhail M. ShvartsmanMechanism of zinc transfer in molten slag
https://ir.stthomas.edu/cas_mathpub/185
https://ir.stthomas.edu/cas_mathpub/185Thu, 24 Feb 2022 07:16:59 PST
The mechanism and dynamics of zinc transfer in molten slag are discussed. Studies were made of the kinetics of zinc voltailization from slags made by smelting copper-zinc ore to a white matte, and the kinetics of copper, nickel and cobalt extraction from slags formed during the conversion of copper-nickel ore to rich mattes. The important points in the present report concern the following experimentally established kinetic laws of slag stripping: a) the concentrations of dissolved metals and sulfur in the reducing and melting zones of the furnace coincide throughout the entire process (established by chemical and X-ray microprobe analyses); b) in both reduction methods of slag reduction, conditions are found in which the rate of zinc volatilization is independent of the amount present in the slag, over the range 10-12 down to 0. 5-1%; c) at zinc concentrations below 0. 5-1%, the volatilization rate under these conditions becomes possible to the residual zinc content of the slags.
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Mikhail M. Shvartsman et al.Stress-strain state in an elastic medium containing an inclusion of a new phase
https://ir.stthomas.edu/cas_mathpub/184
https://ir.stthomas.edu/cas_mathpub/184Thu, 24 Feb 2022 07:16:58 PST
The problem of elastic equilibrium of an isotropic solid phase with a melt is examined in connection with a study of problems of magma generation and volcanic earthquakes. For the case of a spherical inclusion and zero pressure at infinity the stress-strain state is calculated to terms of the second order of smallness (with respect to the relative difference of densities of the phases).
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Michael A. Grinfeld et al.