15,871 research outputs found
Eigenvalue Comparisons for Second-Order Linear Equations with Boundary Value Conditions on Time Scales
This paper studies the eigenvalue comparisons for second-order linear equations with boundary conditions on time scales. Using results from matrix algebras, the existence and comparison results concerning eigenvalues are obtained
Unifying discrete and continuous Weyl-Titchmarsh theory via a class of linear Hamiltonian systems on Sturmian time scales
In this study, we are concerned with introducing Weyl-Titchmarsh theory for a
class of dynamic linear Hamiltonian nabla systems over a half-line on Sturmian
time scales. After developing fundamental properties of solutions and regular
spectral problems, we introduce the corresponding maximal and minimal operators
for the system. Matrix disks are constructed and proved to be nested and
converge to a limiting set. Some precise relationships among the rank of the
matrix radius of the limiting set, the number of linearly independent square
summable solutions, and the defect indices of the minimal operator are
established. Using the above results, a classification of singular dynamic
linear Hamiltonian nabla systems is given in terms of the defect indices of the
minimal operator, and several equivalent conditions on the cases of limit point
and limit circle are obtained, respectively. These results unify and extend
certain classic and recent results on the subject in the continuous and
discrete cases, respectively, to Sturmian time scales.Comment: 34 page
A multiple scales approach to evaporation induced Marangoni convection
This paper considers the stability of thin liquid layers of binary mixtures of a volatile (solvent) species and a non-volatile (polymer) species. Evaporation leads to a depletion of the solvent near the liquid surface. If surface tension increases for lower solvent concentrations, sufficiently strong compositional gradients can lead to Bénard-Marangoni-type convection that is similar to the kind which is observed in films that are heated from below. The onset of the instability is investigated by a linear stability analysis. Due to evaporation, the base state is time dependent, thus leading to a non-autonomous linearised system, which impedes the use of normal modes. However, the time scale for the solvent loss due to evaporation is typically long compared to the diffusive time scale, so a systematic multiple scales expansion can be sought for a finite dimensional approximation of the linearised problem. This is determined to leading and to next order. The corrections indicate that sufficient separation of the top eigenvalue from the remaining spectrum is required for the validity of the expansions, but not the magnitude of the eigenvalues themselves. The approximations are applied to analyse experiments by Bassou and Rharbi with polystyrene/toluene mixtures [Langmuir 2009 (25) 624–632]
Numerical comparison between a Gyrofluid and Gyrokinetic model investigating collisionless magnetic reconnection
The first detailed comparison between gyrokinetic and gyrofluid simulations
of collisionless magnetic reconnection has been carried out. Both the linear
and nonlinear evolution of the collisionless tearing mode have been analyzed.
In the linear regime, we have found a good agreement between the two approaches
over the whole spectrum of linearly unstable wave numbers, both in the drift
kinetic limit and for finite ion temperature. Nonlinearly, focusing on the
small- regime, with indicating the standard tearing
stability parameter, we have compared relevant observables such as the
evolution and saturation of the island width, as well as the island oscillation
frequency in the saturated phase.The results are basically the same, with small
discrepancies only in the value of the saturated island width for moderately
high values of . Therefore, in the regimes investigated here, the
gyrofluid approach can describe the collisionless reconnection process as well
as the more complete gyrokinetic model.Comment: Accepted for publication on Physics of Plasma
Elastic Wave Eigenmode Solver for Acoustic Waveguides
A numerical solver for the elastic wave eigenmodes in acoustic waveguides of
inhomogeneous cross-section is presented. Operating under the assumptions of
linear, isotropic materials, it utilizes a finite-difference method on a
staggered grid to solve for the acoustic eigenmodes of the vector-field elastic
wave equation. Free, fixed, symmetry, and anti-symmetry boundary conditions are
implemented, enabling efficient simulation of acoustic structures with
geometrical symmetries and terminations. Perfectly matched layers are also
implemented, allowing for the simulation of radiative (leaky) modes. The method
is analogous to eigenmode solvers ubiquitously employed in electromagnetics to
find waveguide modes, and enables design of acoustic waveguides as well as
seamless integration with electromagnetic solvers for optomechanical device
design. The accuracy of the solver is demonstrated by calculating
eigenfrequencies and mode shapes for common acoustic modes in several simple
geometries and comparing the results to analytical solutions where available or
to numerical solvers based on more computationally expensive methods
Theory of weakly nonlinear self sustained detonations
We propose a theory of weakly nonlinear multi-dimensional self sustained
detonations based on asymptotic analysis of the reactive compressible
Navier-Stokes equations. We show that these equations can be reduced to a model
consisting of a forced, unsteady, small disturbance, transonic equation and a
rate equation for the heat release. In one spatial dimension, the model
simplifies to a forced Burgers equation. Through analysis, numerical
calculations and comparison with the reactive Euler equations, the model is
demonstrated to capture such essential dynamical characteristics of detonations
as the steady-state structure, the linear stability spectrum, the
period-doubling sequence of bifurcations and chaos in one-dimensional
detonations and cellular structures in multi- dimensional detonations
Surface Impedance Determination via Numerical Resolution of the Inverse Helmholtz Problem
Assigning boundary conditions, such as acoustic impedance, to the frequency
domain thermoviscous wave equations (TWE), derived from the linearized
Navier-Stokes equations (LNSE) poses a Helmholtz problem, solution to which
yields a discrete set of complex eigenfunctions and eigenvalue pairs. The
proposed method -- the inverse Helmholtz solver (iHS) -- reverses such
procedure by returning the value of acoustic impedance at one or more unknown
impedance boundaries (IBs) of a given domain, via spatial integration of the
TWE for a given real-valued frequency with assigned conditions on other
boundaries. The iHS procedure is applied to a second-order spatial
discretization of the TWEs on an unstructured staggered grid arrangement. Only
the momentum equation is extended to the center of each IB face where pressure
and velocity components are co-located and treated as unknowns. The iHS is
finally closed via assignment of the surface gradient of pressure phase over
the IBs, corresponding to assigning the shape of the acoustic waveform at the
IB. The iHS procedure can be carried out independently for different
frequencies, making it embarrassingly parallel, and able to return the complete
broadband complex impedance distribution at the IBs in any desired frequency
range to arbitrary numerical precision. The iHS approach is first validated
against Rott's theory for viscous rectangular and circular ducts. The impedance
of a toy porous cavity with a complex geometry is then reconstructed and
validated with companion fully compressible unstructured Navier-Stokes
simulations resolving the cavity geometry. Verification against one-dimensional
impedance test tube calculations based on time-domain impedance boundary
conditions (TDIBC) is also carried out. Finally, results from a preliminary
analysis of a thermoacoustically unstable cavity are presented.Comment: As submitted to AIAA Aviation 201
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