Rigorous justification of Taylor dispersion via center manifolds and hypocoercivity

Taylor diffusion (or dispersion) refers to a phenomenon discovered experimentally by Taylor in the 1950s where a solute dropped into a pipe with a background shear flow experiences diffusion at a rate proportional to $1/\nu$, which is much faster than what would be produced by the static fluid if its viscosity is $0 < \nu \ll 1$. This phenomenon is analyzed rigorously using the linear PDE governing the evolution of the solute. It is shown that the solution can be split into two pieces, an approximate solution and a remainder term. The approximate solution is governed by an infinite-dimensional system of ODEs that possesses a finite-dimensional center manifold, on which the dynamics correspond to diffusion at a rate proportional to $1/\nu$. The remainder term is shown to decay at a rate that is much faster than the leading order behavior of the approximate solution. This is proven using a spectral decomposition in Fourier space and a hypocoercive estimate to control the intermediate Fourier modes.Comment: 37 pages, 0 figure

Investigation of interface properties of sputter deposited TiN/CrN superlattices by low-angle X-ray reflectivity

Approximately 1.8 m thick nanolayered multilayer coatings of TiN/CrN (also known as superlattices) were deposited on silicon (100) substrates at different modulation wavelengths (4.6–12.8 nm), substrate temperatures (50-400 °C) and substrate bias voltages (-50 to -200 V) using a reactive direct current magnetron sputtering system. X-ray reflectivity (XRR) technique was employed to determine various properties of the multilayers such as interface roughness, surface roughness, electron density, critical angle and individual layer thicknesses. The modulation wavelengths of the TiN/CrN superlattice coatings were calculated using modified Bragg’s law. Furthermore, the experimental X-ray reflectivity patterns were simulated using theoretically generated patterns and a good fit was obtained for a three layer model, i.e., (1) top surface roughness layer, (2) TiN/CrN multilayer coating (approximately 1.8 m) and (3) Ti interlayer (~ 0.5 m) at the film-substrate interface. For the superlattice coatings prepared at a modulation wavelength of 9.7 nm, a substrate bias of -200 V and a substrate temperature of 400 C the XRR patterns showed Bragg reflections up to 5th order, indicating well-defined periodicity of the constituent layers and relatively sharp interfaces. The simulation showed that the superlattice coatings prepared under the above conditions exhibited low surface and interface roughnesses. We also present the effect of substrate temperature and substrate bias, which are critical parameters for controlling the superlattice properties, onto the various interface properties of TiN/CrN superlattices

Multiphoton Bloch-Siegert shifts and level-splittings in spin-one systems

We consider a spin-boson model in which a spin 1 system is coupled to an oscillator. A unitary transformation is applied which allows a separation of terms responsible for the Bloch-Siegert shift, and terms responsible for the level splittings at anticrossings associated with Bloch-Siegert resonances. When the oscillator is highly excited, the system can maintain resonance for sequential multiphoton transitions. At lower levels of excitation, resonance cannot be maintained because energy exchange with the oscillator changes the level shift. An estimate for the critical excitation level of the oscillator is developed.Comment: 14 pages, 3 figure

Level Splitting in Association with the Multiphoton Bloch-Siegert Shift

We present a unitary equivalent spin-boson Hamiltonian in which terms can be identified which contribute to the Bloch-Siegert shift, and to the level splittings at the anticrossings associated with the Bloch-Siegert resonances. First-order degenerate perturbation theory is used to develop approximate results in the case of moderate coupling for the level splitting.Comment: 8 pages, 2 figure

Sexting and Mental Health: A School-based Longitudinal Study Among Youth in Texas

Background: Sexting has emerged as a common socio-cultural problem in our society today. Few studies have estimated the prevalence of sexting among younger middle school youth and even fewer have assessed the relationship between sexting and mental health outcomes like anxiety and depression symptoms among middle school youth. Objectives: To estimate the prevalence of sexting among sixth and seventh-grade middle school students in a large urban school district in Southeast Texas and to assess its relationship with mental health outcomes (both anxiety and depression) among these youth. Methods: A retrospective analysis of an existing three-year randomized, two-arm, nested longitudinal study was conducted. Associations between sexting and depression symptoms; and sexting and anxiety symptoms were assessed via univariate and multivariate logistic analysis. Results: The prevalence of sexting among sixth graders was found to be 12%. Compared to youth who were not engaged in sexting, engagement in sexting was associated with significantly increased odds of depression and anxiety symptoms. Conclusion: Sexting is common among youth and is associated with poorer mental health outcomes such as anxiety and depression among these youth, but further validation of these findings is needed

Optimal Control Promotional Policy for a New Product Incorporating Repeat Purchase in Segmented Market: A Control Theoretic Approach

This chapter considers an optimal control model to obtain dynamic promotional policies for a product considering a segmented market where first-time and additional repeat purchase sales are assumed to be generated through mass and differentiated promotions. Mass promotion is carried out in the whole market which reaches each segment with a fixed spectrum, and differentiated promotion is catered to each segment individually. The firm’s finite promotional resources are to be allocated for promoting a product at mass and segment levels of the market in a finite time period. The formulated control problem obtains optimal promotional effort policy for each segment using the maximum principle. The applicability of the proposed control model is illustrated through a numerical example by discretizing the model

Heat Transfer for Laminar Flow Through Parallel Porous Disks

The problem of temperature distribution and heat transfer for laminar flow through two parallel porous disks, has been investigated when the flow is entirely due t injection or suction at the two disks. Viscous dissipation terms have been included in the energy equation and the uniform injection/suction velocities at the two disks, are assumed to be small. The boundaries are maintained at constant temperature. The variation of temperature and Nusselt numbers at the two disks, is shown graphically, for various of the injection/suction velocities

On the Flow of a Visco-Elastic Fluid between Two Rotating Non-Concentric Cylinders for Small Eccentricity

The motion of a visco elastic fluid contained between two rotating circular cylinders whose axes are set slightly apart is considered. An approximate solution of the Navier Stokes equations is obtained by a perturbation method for the case of small eccentricity. The expansions contain a perturbation parameter Epsional related to the eccentricity e which is assumed to be small. The method entails the employment of asymptotic expansions of Bessel functions and the results thus obtained are valid for finite gap between the two cylinders. The leading terms of the expansions are the exact solution to the Couette flow of Newtonian Fluid between concentric rotating cylinders. The Transverse velocity profiles are presented for small eccentricity and gap, when both the cylinders are rotating with the same angular velocities in the same direction and compared with the corresponding transverse velocity profile in Newtonian flow