VTechWorks Repository :: Browsing by Author "Renardy, Michael J."

Browsing by Author "Renardy, Michael J."

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Asymptotic properties of solutions of a KdV-Burgers equation with localized dissipation
Huang, Guowei (Virginia Tech, 1994-12-05)
We study the Korteweg-de Vries-Burgers equation. With a deep investigation into the spectral and smoothing properties of the linearized system, it is shown by applying Banach Contraction Principle and Gronwall's Inequality to the integral equation based on the variation of parameters formula and explicit representation of the operator semigroup associated with the linearized equation that, under appropriate assumption appropriate assumption on initial states w(x, 0), the nonlinear system is well-posed and its solutions decay exponentially to the mean value of the initial state in H1(O, 1) as t -> +".
Beurling-Lax Representations of Shift-Invariant Spaces, Zero-Pole Data Interpolation, and Dichotomous Transfer Function Realizations: Half-Plane/Continuous-Time Versions
Amaya, Austin J. (Virginia Tech, 2012-04-26)
Given a full-range simply-invariant shift-invariant subspace M of the vector-valued L² space on the unit circle, the classical Beurling-Lax-Halmos (BLH) theorem obtains a unitary operator-valued function W so that M may be represented as the image of of the Hardy space H² on the disc under multiplication by W. The work of Ball-Helton later extended this result to find a single function representing a so-called dual shift-invariant pair of subspaces (M,M^Ã) which together form a direct-sum decomposition of L². In the case where the pair (M,M^Ã) are finite-dimensional perturbations of the Hardy space H² and its orthogonal complement, Ball-Gohberg-Rodman obtained a transfer function realization for the representing function W; this realization was parameterized in terms of zero-pole data computed from the pair (M,M^Ã). Later work by Ball-Raney extended this analysis to the case of nonrational functions W where the zero-pole data is taken in an infinite-dimensional operator theoretic sense. The current work obtains analogues of these various results for arbitrary dual shift-invariant pairs (M,M^Ã) of the L² spaces on the real line; here, shift-invariance refers to invariance under the translation group. These new results rely on recent advances in the understanding of continuous-time infinite-dimensional input-state-output linear systems which have been codified in the book by Staffans.
Bifurcation Analysis and Qualitative Optimization of Models in Molecular Cell Biology with Applications to the Circadian Clock
Conrad, Emery David (Virginia Tech, 2006-04-14)
Circadian rhythms are the endogenous, roughly 24-hour rhythms that coordinate an organism's interaction with its cycling environment. The molecular mechanism underlying this physiological process is a cell-autonomous oscillator comprised of a complex regulatory network of interacting DNA, RNA and proteins that is surprisingly conserved across many different species. It is not a trivial task to understand how the positive and negative feedback loops interact to generate an oscillator capable of a) maintaining a 24-hour rhythm in constant conditions; b) entraining to external light and temperature signals; c) responding to pulses of light in a rather particular, predictable manner; and d) compensating itself so that the period is relatively constant over a large range of temperatures, even for mutations that affect the basal period of oscillation. Mathematical modeling is a useful tool for dealing with such complexity, because it gives us an object that can be quickly probed and tested in lieu of the experiment or actual biological system. If we do a good job designing the model, it will help us to understand the biology better by predicting the outcome of future experiments. The difficulty lies in properly designing a model, a task that is made even more difficult by an acute lack of quantitative data. Thankfully, our qualitative understanding of a particular phenomenon, i.e. the observed physiology of the cell, can often be directly related to certain mathematical structures. Bifurcation analysis gives us a glimpse of these structures, and we can use these glimpses to build our models with greater confidence. In this dissertation, I will discuss the particular problem of the circadian clock and describe a number of new methods and tools related to bifurcation analysis. These tools can effectively be applied during the modeling process to build detailed models of biological regulatory with greater ease.
Contact of orthotropic laminates with a rigid spherical indentor
Chen, Chun-Fu (Virginia Tech, 1991-04-29)
Three dimensional contact problems of square orthotropic laminates indented by a rigid spherical indenter are solved. Simplified problems of indentations of beam and isotropic square plate are studied first to develop an efficient numerical technique and to gather the knowledge of the shape of the contact area in order to solve for the three dimensional orthotropic cases. The approach combines an exact solution method in conjunction with a simple discretization numerical scheme. Numerical sensitivity due to the ill-posed nature of the problem was experienced but was cured by enhancing the numerical approach with a least square spirit. Well agreement is obtained by comparing the results of these simplified studies with available published solutions. For isotropic plate, contact area is found to be either a circle or a hypotrochoid of four lobes featured with a shorter length of contact along the through-the- corner directions of the plate. Hertz's theory fails earlier than assuming the contact area to be a circle. In-plane dependence of the contact stress is presented to illustrate the difference of contact behavior between a square plate and a circular plate. Load-indentation relation reveals indenting a square plate is harder than indenting a circular plate of a diameter equal to the side length of the square plate. Solutions of multi-layered orthotropic cases are achieved by employing a modified analytical approach with the same numerical method. Three different configurations of plate are implemented for the orthotropic case, namely, a single layered magnesium (Mg) plate, which is slightly orthotropic, and a single and double layered plates of graphite-epoxy (G-E), which are highly orthotropic. Results for the (Mg) plate agrees with the previous isotropic case. Concept of modifying the previous hypotrochoids is introduced to seek for the contact stresses for comparatively large indentation conditions. Single-layered (G-E) plate was implemented for small indentations. The result supports the validity of Hertz's theory for small indentation and shows a relatively longer contact length in the direction of less stiffness. Two layered (G-E) plate illustrates similar distributions for the contact stresses along both of the in-plane directions with a smaller range of validity of Hertzian type behavior than the previous cases. The boundary effect prevails at the initial stage of indentation but is overcome by the effect of material orthotropy as the indentation proceeds. Thus, the contact area for small indentation appears to be the same kind of hypotrochoids as located in the isotropic case but changes to be the other type of hypotrochoids as the indentation advances.
Controllability of the Stresses in Multimode Viscoelastic Fluid of Upper Convected Maxwell Type
Savel'ev, Evgeny (Virginia Tech, 2009-06-19)
Viscoelastic fluids, or Non-Newtonian fluids, are those that do not have a linear algebraic relation between the velocity field and the stresses arising in the media. Such fluids exhibit properties of both solids and liquids, and therefore cannot be modeled with methods of elasticity or Newtonian fluid mechanics. The popular models of viscoelasticity differ from each other only by the differential equation that describes the constitutive law for the fluid. Also, the media can have several relaxation modes, such as fluid mixes. This means that the stresses are determined as the sum of the stresses for each individual relaxation mode, which are described by corresponding differential equations evolving independently. The question of controllability of the equations that describe the evolution of viscoelastic fluids is largely open. The presence of the non-algebraic constitutive relation makes the analysis unfeasible in general setup. The presence of several relaxation modes makes the problem even more complicated. Another issue is the necessity of controlling the stresses, since they are not determined by the momentary velocity field, thus they need to be included as the controlled states. In this work we are concentrating on the controllability of the stresses arising in the viscoelastic fluid that has its motion constrained to be of the shearing type. This restriction allows us to concentrate on the stresses only and assign the shearing rate to be the control. We consider only the Upper Convected Maxwell fluid which has several relaxation modes present. The results demonstrate that contrary to the one relaxation mode case the normal stresses cannot be driven arbitrary close to the exponentially decaying regime, unless the shearing stresses satisfy certain requirements, while the shear stresses remain exactly controllable.
Deformation of a hydrophobic ferrofluid droplet suspended in a viscous medium under uniform magnetic fields
Afkhami, Shahriar; Tyler, A. J.; Renardy, Yuriko Y.; Renardy, Michael J.; St Pierre, T. G.; Woodward, R. C.; Riffle, Judy S. (Cambridge University Press, 2010-11)
The effect of applied magnetic fields on the deformation of a biocompatible hydrophobic ferrofluid drop suspended in a viscous medium is investigated numerically and compared with experimental data. A numerical formulation for the time-dependent simulation of magnetohydrodynamics of two immiscible non-conducting fluids is used with a volume-of-fluid scheme for fully deformable interfaces. Analytical formulae for ellipsoidal drops and near-spheroidal drops are reviewed and developed for code validation. At low magnetic fields, both the experimental and numerical results follow the asymptotic small deformation theory. The value of interfacial tension is deduced from an optimal fit of a numerically simulated shape with the experimentally obtained drop shape, and appears to be a constant for low applied magnetic fields. At high magnetic fields, on the other hand, experimental measurements deviate from numerical results if a constant interfacial tension is implemented. The difference can be represented as a dependence of apparent interfacial tension on the magnetic field. This idea is investigated computationally by varying the interfacial tension as a function of the applied magnetic field and by comparing the drop shapes with experimental data until a perfect match is found. This estimation method provides a consistent correlation for the variation in interfacial tension at high magnetic fields. A conclusion section provides a discussion of physical effects which may influence the microstructure and contribute to the reported observations.
Draw resonance revisited
Renardy, Michael J. (Siam Publications, 2006)
We consider the problem of isothermal fiber spinning in a Newtonian fluid with no inertia. In particular, we focus on the effect of the downstream boundary condition. For prescribed velocity, it is well known that an instability known as draw resonance occurs at draw ratios in excess of about 20.2. We shall revisit this problem. Using the closed form solution of the differential equation, we shall show that an infinite family of eigenvalues exists and discuss its asymptotics. We also discuss other boundary conditions. If the force in the. lament is prescribed, no eigenvalues exist, and the problem is stable at all draw ratios. If the area of the cross section is prescribed downstream, on the other hand, the problem is unstable at any draw ratio. Finally, we discuss the stability when the drawing speed is controlled in response to changes in cross section or force.
Effect of upstream boundary conditions on stability of fiber spinning in the highly elastic limit
Renardy, Michael J. (AIP Publishing, 2002-07)
We consider fiber spinning for the upper-convected Maxwell fluid in the limit of a high Deborah number. We compare several choices of boundary conditions that may be imposed. In addition to the takeup speed and the upstream flow rate, we consider four different boundary conditions: the upstream velocity, upstream elastic stress, the force in the fiber, and the ratio of stress to the square of the velocity (the latter can be motivated by a limit of vanishing retardation time). We find that the effect of the boundary condition on stability is crucial; in one case we even find an instability even though the draw ratio is 1.
Elongational Flows in Polymer Processing
Hagen, Thomas Ch. (Virginia Tech, 1998-12-01)
The production of long, thin polymeric fibers is a main objective of the textile industry. Melt-spinning is a particularly simple and effective technique. In this work, we shall discuss the equations of melt-spinning in viscous and viscoelastic flow. These quasilinear hyperbolic equations model the uniaxial extension of a fluid thread before its solidification. We will address the following topics: first we shall prove existence, uniqueness, and regularity of solutions. Our solution strategy will be developed in detail for the viscous case. For non-Newtonian and isothermal flows, we shall outline the general ideas. Our solution technique consists of energy estimates and fixed-point arguments in appropriate Banach spaces. The existence result for a simple transport equation is the key to understanding the quasilinear case. The second issue of this exposition will be the stability of the unforced frost line formation. We will give a rigorous justification that, in the viscous regime, the linearized equations obey the ``Principle of Linear Stability''. As a consequence, we are allowed to relate the stability of the associated strongly continuous semigroup to the numerical resolution of the spectrum of its generator. By using a spectral collocation method, we shall derive numerical results on the eigenvalue distribution, thereby confirming prior results on the stability of the steady-state solution.
Erratum: "derivation of amplitude equations and analysis of side-band instabilities in two-layer flows" (vol 5, pg 2738, 1993)
Renardy, Michael J.; Renardy, Yuriko Y. (AIP Publishing, 1994-10)
In the amplitude equations, one further term should have been taken into account. This term is formally of higher order when the original scaling of variables is used. However, the analysis of sideband instabilities involves various rescalings, and for one of the cases the term becomes of the same order as others. It therefore affects the criteria for sideband instability.
Exponential Stability for a Diffusion Equation in Polymer Kinetic Theory
Mulzet, Alfred Kenric (Virginia Tech, 1997-04-22)
In this paper we present an exponential stability result for a diffusion equation arising from dumbbell models for polymer flow. Using the methods of semigroup theory, we show that the semigroup U(t) associated with the diffusion equation is well defined and that all solutions converge exponentially to an equilibrium solution. Both finitely and infinitely extensible dumbbell models are considered. The main tool in establishing stability is the proof of compactness of the semigroup.
Field-induced motion of ferrofluid droplets through immiscible viscous media
Afkhami, Shahriar; Renardy, Yuriko Y.; Renardy, Michael J.; Riffle, Judy S.; St Pierre, T. (Cambridge University Press, 2008-09)
The motion of a hydrophobic ferrofluid droplet placed in a viscous medium and driven by an externally applied magnetic field is investigated numerically in an axisymmetric geometry. Initially, the drop is spherical and placed at a distance away from the magnet. The governing equations are the Maxwell equations for a non-conducting flow, momentum equation and incompressibility. A numerical algorithm is derived to model the interface between a magnetized fluid and a non-magnetic fluid via a volume-of-fluid framework. A continuum-surface-force formulation is used to model the interfacial tension force as a body force, and the placement of the liquids is tracked by a volume fraction function. Three cases are Studied. First, where inertia is dominant, the magnetic Laplace number Is varied while the Laplace number is fixed. Secondly, where inertial effects are negligible, the Laplace number is varied while the magnetic Laplace number is fixed. In the third case, the magnetic Bond number and inertial effects are both small, and the magnetic force is of the order of the viscous drag force. The time taken by the droplet to travel through the medium and the deformations in the drop are investigated and compared with a previous experimental study and accompanying simpler model. The transit times are found to compare more favourably than with the simpler model.
Forced Capillary-Gravity Waves in a 2D Rectangular Basin
Brunnhofer, Harald Michael (Virginia Tech, 2005-04-20)
This dissertation concerns capillary-gravity surface waves in a two-dimensional rectangular basin that is partially filled with water. To generate the surface waves, a harmonic forcing is applied to the vertical side walls of the basin. The dissertation consists of four parts which work with different assumptions on the frequencies of the forcing. The first part discusses the linearized model with Hocking's edge condition and gives an eigenvalue equation and an asymptotic expansion for the eigenvalues. Then, for the nonlinear problem, it is assumed that the frequency of the forcing is close to an eigenfrequency and the solution has an asymptotic expansion using a two time-scales approach. Under an edge condition, the first- and second-order approximations of the solution and a solvability condition from the third-order equations yield an ordinary differential equation for the amplitude of the solution. In part two, it is assumed that the frequency of the forcing applied to the boundary is close to the sum of two eigenfrequencies. In this case, the solvability conditions give a system of two differential equations for the complex valued amplitudes of the two eigenmodes. The system can be reduced to one real-valued differential equation. Its solutions yield the solutions of the original system and their properties. A condition for the existence of homoclinic orbits connecting the trivial equilibrium is obtained. These results are confirmed by numerical experiments. The third part is based on the results in the second part. Here, one of the eigenfrequencies is chosen to be much larger than the other one, and different orders of the amplitudes of the eigenmodes are assumed. The orders of the coefficients of the system found in the second part are obtained, and the resulting special case is discussed in detail. In particular, numerical examples of orbits that can be associated with homoclinic orbits connecting nontrivial equilibria are given. The behavior of solutions close to those orbits is demonstrated. In the fourth part, an additional frequency for the forcing terms given in parts two and three is introduced. In each situation, the modified systems are presented and discussed.
Geometric Properties of Over-Determined Systems of Linear Partial Difference Equations
Boquet, Grant Michael (Virginia Tech, 2010-02-19)
We relate linear constant coefficient systems of partial difference equations (a discretization of a system of linear partial differential equations) satisfying some collection of scalar polynomial equations to systems defined over the coordinate ring of an algebraic variety. This motivates the extension of behavioral systems theory (a generalization of classical systems theory where inputs and outputs are lumped together) to the setting where the ring of operators is an affine domain and the signal space is restricted to signals which satisfy the same scalar polynomial equations. By recognizing the role of the kernel representation's Gröbner basis in the Cauchy problem, we extend notions of controllability from the classical behavioral setting to accommodate this generalization. We then address the question as to when an autonomous behavior admits a Livšic-system state-space representation, where the state update equations are overdetermined leading to the requirement that the input and output signals satisfy their own compatibility difference equations. This leads to a frequency domain setting involving input and output holomorphic vector bundles and a transfer function given by a meromorphic bundle map. An analogue of the Hankel realization theorem developed by J. Ball and V. Vinnikov then leads to a Livšic-system state-space representation for an autonomous behavior satisfying some natural additional conditions.
Glass transition seen through asymptotic expansions
Olivier, J.; Renardy, Michael J. (Siam Publications, 2011)
Soft glassy materials exhibit the so-called glassy transition, which means that the behavior of the model at a low shear rate changes when a certain parameter (which we call the glass parameter) crosses a critical value. This behavior goes from a Newtonian behavior to a Herschel-Bulkley behavior through a power-law-type behavior at the transition point. In a previous paper we rigorously proved that the Hebraud-Lequeux model, a Fokker-Planck-like description of soft glassy material, exhibits such a glass transition. But the method we used was very specific to the one-dimensional setting of the model, and as a preparation for generalizing this model to take into account multidimensional situations, we look for another technique to study the glass transition of this type of model. In this paper we shall use matched asymptotic expansions for such a study. The difficulties encountered when using asymptotic expansions for the Hebraud-Lequeux model are that multiple ansaetze have to be used, even though the initial model is unique, due to the glass transition. We shall delineate the various regimes and give a rigorous justification of the expansion by means of an implicit function argument. The use of a two parameter expansion plays a crucial role in elucidating the reasons for the scalings which occur.
Hardy-space Function Theory on Finitely Connected Planar Domains
Guerra Huaman, Moises Daniel (Virginia Tech, 2008-04-17)
Hardy space scalar theory on the disk is now classical. Some extensions have been done, one of them is the approach done by Donald Sarason using Laurent series. We present the more complicated function theory, without the use of either power series or Laurent series, for finitely-connected planar domains.
Initial Value Problems for Creeping Flow of Maxwell Fluids
Laadj, Toufik (Virginia Tech, 2011-02-24)
We consider the flow of nonlinear Maxwell fluids in the unsteady quasistatic case, where the effect of inertia is neglected. We study the well-posedness of the resulting PDE initial-boundary value problem. This well-posedness depends on the unique solvability of an elliptic boundary value problem. We first present results for the 3D case, locally and globally in time, with sufficiently small initial data, and for a simple shear flow problem, locally in time with arbitrary initial data; after that we extend our results to some 3D flow problems, locally in time, with large initial data. Additionally, we present results for models of White-Metzner type in 3D flow, locally and globally in time, with sufficiently small initial data. We solve our problem using an iteration between elliptic and hyperbolic linear subproblems. The limit of the iteration provides the solution of our original problem.
Mathematical Analysis on the PEC model for Thixotropic Fluids
Wang, Taige (Virginia Tech, 2016-05-03)
A lot of fluids are more complex than water: polymers, paints, gels, ketchup etc., because of big particles and their complicated microstructures, for instance, molecule entanglement. Due to this structure complexity, some material can display that it is still in yielded state when the imposed stress is released. This is referred to as thixotropy. This dissertation establishes mathematical analysis on a thixotropic yield stress fluid using a viscoelastic model under the limit that the ratio of retardation time versus relaxation time approaches zero. The differential equation model (the PEC model) describing the evolution of the conformation tensor is analyzed. We model the flow when simple shearing is imposed by prescribing a total stress. One part of this dissertation focuses on oscillatory shear stresses. In shear flow, different fluid states corresponding to yielded and unyielded phases occur. We use asymptotic analysis to study transition between these phases when slow oscillatory shearing is set up. Simulations will be used to illustrate and supplement the analysis. Another part of the dissertation focuses on planar Poiseuille flow. Since the flow is spatially inhomogeneous in this situation, shear bands are observed. The flow is driven by a homogeneous pressure gradient, leading to a variation of stress in the cross-stream direction. In this setting, the flow would yield in different time scales during the evolution. Formulas linking the yield locations, transition width, and yield time are obtained. When we introduce Korteweg stress in the transition, the yield location is shifted. An equal area rule is identified to fit the shifted locations.
Mathematical Models of the Alpha-Beta Phase Transition of Quartz
Moss, George W. (Virginia Tech, 1999-07-27)
We examine discrete models with hexagonal symmetry to compare the sequence of transitions with the alpha-inc-beta phase transition of quartz. We examine a model by Parlinski which employs interactions of nearest and next-nearest neighbor atoms. We numerically determine the configurations which lead to minimum energy for a range of parameters. We then use Golubitsky's results on systems with hexagonal symmetry to derive the bifurcation diagram for Parlinski's model. Finally, we study a large class of modifications to Parlinski's model and show that all such modifications have the same bifurcation picture as the original model.
Nonhomogeneous Initial Boundary Value Problems for Two-Dimensional Nonlinear Schrodinger Equations
Ran, Yu (Virginia Tech, 2014-05-08)
The dissertation focuses on the initial boundary value problems (IBVPs) of a class of nonlinear Schrodinger equations posed on a half plane R x R+ and on a strip domain R x [0,L] with Dirichlet nonhomogeneous boundary data in a two-dimensional plane. Compared with pure initial value problems (IVPs), IBVPs over part of entire space with boundaries are more applicable to the reality and can provide more accurate data to physical experiments or practical problems. Although there is less research that has been made for IBVPs than that for IVPs, more attention has been paid for IBVPs recently. In particular, this thesis studies the local well-posedness of the equation for the appropriate initial and boundary data in Sobolev spaces H^s with non-negative s and investigates the global well-posedness in the H^1-space. The main strategy, especially for the local well-posedness, is to derive an equivalent integral equation (whose solution is called mild solution) from the original equation by semi-group theory and then perform the Banach fixed-point argument. However, along the process, it is essential to select proper auxiliary function spaces and prepare all the corresponding norm estimates to complete the argument. In fact, the IBVP posed on R x R+ and the one posed on R x [0,L] are two independent problems because the techniques adopted are different. The first problem is more related to the initial value problem (IVP) posed on the whole plane R^2 and the major ingredients are Strichartz's estimate and its generalized theory. On the other hand, the second problem can be studied as an IVP over a half-line and periodic domain, which is established on the analysis for series inspired by Bourgain's work. Moreover, the corresponding smoothing properties and regularity conditions of the solution are also considered.

Browsing by Author "Renardy, Michael J."

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