Longitudinal Coordinate Research Articles

A unified analytical solution of the steady-state atmospheric diffusion equation

A unified analytical solution of the steady-state atmospheric diffusion equation for a finite and semi-infinite/infinite media was developed using the classic integral transform technique (CITT) which is based on a systematized method of separation of variable.The solution was obtained considering an arbitrary mean wind velocity depending on the vertical coordinate (z) and a generalized separable functional form for the eddy diffusivities in terms of the longitudinal (x) and vertical coordinates (z).The examples described in this article show that the well known closed-form analytical solutions, available in the literature, for both finite and semi-infinite/infinite media are special cases of the present unified analytical solution. As an example of the strength of the developed methodology, the Copenhagen and Prairie Grass experiments were simulated (finite media with the mean wind speed and the turbulent diffusion coefficient described by different functional forms). The results indicate that the present solutions are in good agreement with those obtained using other analytical procedures, previously published in the literature. It is important to note that the eigenvalue problem is associated directly to the atmospheric diffusion equation making possible the development of the unified analytical solution and also resulting in the improvement of the convergence behavior in the series of the eigenfunction-expansion.

A Novel Beam Model for Mode-II Delamination Analysis of Composite Laminates

A two dimensional beam model is proposed for delamination analysis of composite laminates, where Timoshenko's plane cross-section assumption is abandoned, and the longitudinal displacement of the beam is proposed as continuous functions of the transverse and the longitudinal coordinates, to match shear deformations. Moreover, the composite laminate is modeled as upper and lower sub-beam through the delamination interface, and corresponding conditions for interface sliding are satisfied through a generalized variational theory. The accuracy of the results of this analytical model is verified by comparison with those from three dimensional finite element models. Furthermore, the influences of delamination depth, length, central position and modulus mismatch ratio ( E1/ E2) of the upper and lower sublaminate to the energy release rate are discussed.

Noise in Submicron Metal–Oxide–Semiconductor Field Effect Transistors: Lateral Electron Density Distribution and Active Trap Position

Experiments were carried out for the n-channel devices, processed in a 0.3 µm spacer less complementary metal–oxide–semiconductor technology. Random-telegraph-signal measurements were performed for the constant gate voltage. It is supposed that electron concentration in the channel decreases from the source to the drain contact. Lateral component of the electric field is inhomogeneous in the channel and it has a minimum value near the source and reaching the maximum value near the drain electrode. Drain current is given by two components – diffusion and drift ones. Diffusion current component is independent on the x-coordinate and it is equal to the drift current component for the low electric field. The model explaining the experimentally observed capture time constant dependence on the lateral electric field and the trap position is given. From the dependence of the capture time constant τc on the drain current could be calculated longitudinal coordinate of the trap position.

Noise in Submicron Metal–Oxide–Semiconductor Field Effect Transistors: Lateral Electron Density Distribution and Active Trap Position

Experiments were carried out for the n-channel devices, processed in a 0.3 µm spacer less complementary metal–oxide–semiconductor technology. Random-telegraph-signal measurements were performed for the constant gate voltage. It is supposed that electron concentration in the channel decreases from the source to the drain contact. Lateral component of the electric field is inhomogeneous in the channel and it has a minimum value near the source and reaching the maximum value near the drain electrode. Drain current is given by two components – diffusion and drift ones. Diffusion current component is independent on the x-coordinate and it is equal to the drift current component for the low electric field. The model explaining the experimentally observed capture time constant dependence on the lateral electric field and the trap position is given. From the dependence of the capture time constant τc on the drain current could be calculated longitudinal coordinate of the trap position.

White-light interferometry on rough surfaces—measurement uncertainty caused by noise

White-light interferometry on rough surfaces is an optical method for the measurement of the geometrical form of objects. The longitudinal coordinate of the measured surface is obtained from the measured interferogram by means of an evaluation method. However, the longitudinal coordinate cannot be determined completely accurately because the interferogram is affected by noise. We calculate the lower limit of the longitudinal measurement uncertainty caused by noise by use of the Cramer-Rao inequality. Additionally, we calculate the lower limit of the longitudinal measurement uncertainty caused by shot noise only.

Generation and Characterization of Electron Bunches with Ramped Current Profiles in a Dual-Frequency Superconducting Linear Accelerator

We report on the successful experimental generation of electron bunches with ramped current profiles. The technique relies on impressing nonlinear correlations in the longitudinal phase space using a superconducing radio frequency linear accelerator operating at two frequencies and a current-enhancing dispersive section. The produced ~700-MeV bunches have peak currents of the order of a kilo-Ampère. Data taken for various accelerator settings demonstrate the versatility of the method and, in particular, its ability to produce current profiles that have a quasilinear dependency on the longitudinal (temporal) coordinate. The measured bunch parameters are shown, via numerical simulations, to produce gigavolt-per-meter peak accelerating electric fields with transformer ratios larger than 2 in dielectric-lined waveguides.

Heat and mass diffusion fluxes on a permeable wall with foreign-gas blowing

In the present paper, we consider the variation of heat and mass diffusion fluxes on a permeable plate with blowing of a foreign gas into the boundary layer, the fluxes being considered as functions of the permeability parameter, varied through variation of blowing intensity, free-stream velocity, or longitudinal coordinate. It is shown that at a fixed distance from the leading edge of the plate one can, varying the value of blowing intensity while preserving the uniformity of the blowing over the plate length, obtain a non-monotonic variation of the wall heat and mass diffusion fluxes. In contrast to the heat and mass diffusion fluxes, the shear stress always monotonically decreases with increasing the blowing intensity. Similar to the shear stress, on increase of permeability parameter achieved through changing either the free-stream velocity or the longitudinal coordinate the heat flux and the mass diffusion flux both show a monotonic reduction. Using the integral relations of boundary-layer theory, we have derived simple analytical expressions allowing determination of the maximum values of the heat and mass diffusion fluxes in laminar and turbulent flow regimes. The obtained analytical relations were verified by performed numerical simulations.

Tunable quasioptical power dividers operating in the millimeter wavelength range

Circuits and principle of operation of quasioptical power dividers based on two-row periodic structures, formed by grids of metal bars and a dielectric waveguide placed along their longitudinal axis, are described. The experimental studies of power divider prototypes in a frequency range of 60–80 GHz have shown a possibility of regulating the emitted power level in the main divider arm by changing longitudinal and angular coordinates of the two-row periodic structure, which can be used for designing quasioptical attenuators.

Intrinsic Feature Extraction on Hippocampal Surfaces and Its Applications

This paper proposes a novel approach for extracting two intrinsic feature curves on hippocampal (HC) surfaces. The hippocampus is a key target of study in medical imaging, as it degenerates in conditions such as epilepsy and Alzheimer's disease (AD), but its structure is complex. To facilitate HC morphometry, we generate two intrinsic feature curves that describe their global geometries. For example, the separation of them captures thickness changes in HC surfaces, which can be used to effectively measure HC atrophy found in patients with AD. They also separate HC surfaces into upper and lower surface patches where intrinsic shape analysis using conformal modules can be carried out. Based on these curves, we further propose a parameterization of HC surfaces called the eigen-harmonic parameterization (EHP). EHP maps each HC surface onto a parameter domain and imposes longitudinal and azimuthal coordinates on each surface, which follow the gradient and level sets of its first nontrivial Laplace--Beltrami ei...

MHD effects on unstedy dynamic, thermal and diffusion boundary layer flow over a horizontal circular cylinder

This paper is devoted to the analysis of unsteady two-dimensional dynamic, thermal and diffusion magnetohydrodynamic laminar boundary layer flow over a horizontal circular cylinder of incompressible and electrical conductivity fluid, in a porous medium, in the presence of a heat source or sink, and chemical reactions. The present magnetic field is homogenous and perpendicular to the body surface. It is assumed that the induction of the outer magnetic field is the function of the longitudinal coordinate and time. Fluid electrical conductivity is constant. The outer electric field is neglected and the magnetic Reynolds number is significantly lower than one i. e. the considered the problem is in induction-less approximation. Free stream velocity, temperature and concentration on the body are arbitrary differentiable functions. The developed governing boundary layer equations and associated boundary conditions are converted into a nondimensional form using a suitable similarity transformation and similarity parameters. The system of dimensionless equations is solved using the finite difference method and iteration method. Numerical results are obtained and presented for incompressible fluid for different numbers, such as Sc, Pr, Ec and magnetic number, and the parameter of the porous medium, temperature parameters, thermal parameter, diffusion parameters and chemical reaction parameter. The solutions for the flow, temperature and diffusion transfer and other integral characteristics, boundary layer, are evaluated numerically for different values of the magnetic field. Transient effects of velocity, temperature and diffusion are analyzed. A part of obtained results is given in the form of figures and corresponding conclusions.

Torsional vibrations of piezoceramic hollow cylinders with circular polarization

A problem of natural torsional vibrations of hollow piezoceramic cylinders is considered. The piezoceramic is polarized in the circular direction. The lateral surfaces of the cylinder are covered with shortcircuited thin electrodes. The method of solution of the problem is based on the combination of the spline collocation methods along the longitudinal coordinate and the stepwise search method along the radial coordinate. We present results of a numerical analysis of a cylinder made from PZT 4 ceramic in a wide range of change in the geometrical characteristics of the cylinder.

Self-similar turbulent boundary layer in pressure gradient. Four flow regimes

Self-similar flows in a turbulent boundary layer when the free-stream velocity is specified as a power, with the exponent m, function of longitudinal coordinate are investigated. The self-similar formulation not only simplifies solving of the problem by reducing the equations of motion to ordinary differential equations but also provides a mean for formulating closure conditions. It is shown that for the class of flows under consideration that depend on three governing parameters the dimensionless mixing length is a function of the normalized distance from the wall and the exponent m in the outer region and a universal function of local Reynolds number in the wall region, the latter corollary being true even when the skin friction vanishes. In calculations this function is set to be independent of pressure gradient, which gives the results very close to experimental data.There exist four different self-similar flow regimes. Each regime is related to its similarity parameter, one of which is the well-known Clauser equilibrium parameter and the other three are established for the first time. In case of adverse pressure gradient when the exponent m lies within certain limits, which depend on Reynolds number, the problem has two solutions with different values of the boundary layer thickness and the skin friction, which points out the possibility of hysteresis in near-separating flow. Separation occurs not at the minimal value of m that corresponds to the strongest adverse pressure gradient but at m = −0.216 − 0.4Rep−1/3 + O(Rep−2/3), where Rep is the Reynolds number based on the longitudinal pressure gradient. The results of the theory are in good agreement with experimental data.

The Riemann–Volterra time-domain technique for waveguides: A case study for elliptic geometry

The method of incomplete separation of variables is applied for solving the wave propagation problems in which the source distribution and the emanated wave are constrained by an elliptic cylinder. Solutions are obtained in the form of expansions in terms of the Mathieu modes, whose completeness makes possible to solve the problem for arbitrary source distribution and initial values of the wavefunction and its time derivative defined within the cylinder. Transient modal amplitudes are found using the Riemann (Riemann–Volterra) method. An important feature of this approach is the straightforward definition of the essentially bounded effective integration areas on the plane spanned by the longitudinal and time coordinates, taking into account the spatiotemporal constraints imposed on the source. For source turned on in a fixed instant, the method is capable to model wave propagation inside the semi-infinite and finite elliptic cylinders provided that the Dirichlet or Neumann boundary conditions are specified on the limiting cross-section(s). Recent techniques of transverse–longitudinal wave decomposition open the prospect of adapting the method to more general cylindrical configurations and to other cases, in which the incomplete separation of variables results in partial differential equations of a known Riemann function (such as the Euler–Poisson–Darboux equation).

Film boiling on a hot body moving in a fluid

We investigate the transient film boiling in the vicinity of a stagnation point on the frontal surface of a very hot blunt body which moves with a constant velocity in an incompressible viscous fluid in the presence of a vapor layer near the body surface. Within the unsteady two-phase boundary layer approximation, the equations of motion of the liquid and vapor phases are formulatedwith account of the conservation of mass, momentum, and energy on the a priori unknown phase interface. In the vicinity of the stagnation point on the body surface, the solution of the boundary layer equations is sought in the form of series in the longitudinal coordinate. For the leading terms of the series, a parabolic system of partial differential equations is obtained, which is solved numerically. The similarity parameters controlling the film boiling process are determined. On the basis of parametric numerical calculations, the dynamics of the vapor layer are investigated for the case of a plane hot body moving in water with the room pressure and temperature. In the space of governing parameters, the limits of the existence of steady and unsteady film boiling regimes are found.

Self‐similar turbulent boundary layer in pressure gradient

AbstractSelf‐similar flows in a turbulent boundary layer when the free‐stream velocity is specified as a power function of longitudinal coordinate are investigated. The self‐similar formulation not only simplifies solving of the problem by reducing the equations of motion to ordinary differential equations but also provides a mean for formulating closure conditions. It is shown that for the class of flows under consideration that depend on three governing parameters the dimensionless mixing length is a function of the normalised distance from the wall and the exponent in the law specifying the free‐stream velocity distribution in the outer region and a universal function of local Reynolds number in the wall region, the latter corollary being true even when the skin friction vanishes. In calculations this function is set to be independent of pressure gradient, which gives the results very close to experimental data.There exist four different self‐similar flow regimes. Each regime is related to its similarity parameter, one of which is the well‐known Clauser equilibrium parameter and the other three are established for the first time. In case of adverse pressure gradient when the exponent lies within certain limits, which depend on Reynolds number, the problem has two solutions with different values of the boundary layer thickness and skin friction, which points out the possibility of hysteresis in near‐separating flow. Separation occurs not at the minimum value of the exponent that corresponds to the strongest adverse pressure gradient but at a higher one whose dependence on Reynolds number is calculated in the paper. The results of the theory are in good agreement with experimental data. (© 2011 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Perturbation solution for the viscoelastic flow around a rigid sphere under pure uniaxial elongation

We study the steady, three-dimensional creeping and viscoelastic flow around a rigid sphere subject to steady uniaxial extensional flow imposed at infinity. The viscoelastic response of the ambient fluid to the flow deformation is modeled using the second-order-fluid model, the Upper Convected Maxwell, the exponential affine Phan-Thien and Tanner and the Giesekus constitutive equations. A spherical coordinate system with origin at the center of the sphere is used to describe the flow field and the solution of the governing equations is expanded as a series in the Deborah number. The resulting sequence of differential equations is solved analytically up to second order and numerically up to fourth order in Deborah number by employing fully spectral representations for all the primary variables. In particular, Chebyshev polynomials are utilized in the radial coordinate and the Double Fourier Series in the longitudinal and latitudinal coordinates. The numerical results up to second-order agree within machine accuracy with the available analytical solutions clearly indicating the correctness and accuracy of the numerical method used here.

Improved method for calculation of a series of parameters in slotted channels with stepped sealing surfaces for laminar medium flow

This paper presents computational relationships derived on the basis of solution of Reynolds equations for a layer of lubricant, which are required for design of the slotted seals of a stepped channel in a laminar-flow regime. Relationships are derived for determination of the pressure curve, and also the flow rate of medium in the stepped channel for steps of various dimensions with respect to the radial and longitudinal coordinates. Qualities of the geometry of the stepped channel are analyzed, and design recommendations are given for selection of the dimensions and shape of the slotted channel. Illustrations of the pattern of the pressure curves, and also leakage are presented for a typical series of effective gaps in the slotted seal. The solutions obtained are applicable for analyses of sealing subassemblies for the flow of a medium in stepped channels. This shape of channels is employed in slotted seals or sliding ring-oil bearings.

Heat transfer in the thermal entrance region for drag reduction surfactant solutions in pipe flow

Experimental results relating to heat transfer for a flow of surfactant solutions in the thermal entrance region of a pipe were discussed. Aqueous solutions of hexadecyltrimethylammonium chloride with addition of sodium salicylate were used as model fluids. It was observed that the effect of buoyancy on the heat transfer process was less pronounced in the entrance region of a pipe than in the scope of a fully developed flow. It was demonstrated that it was possible to correlate experimental results for the flow of drag-reducing surfactant solutions, based on the dependence of Nusselt number on a dimensionless longitudinal coordinate. In addition, simple empirical equations were proposed which enable the local and mean values of the heat transfer coefficient to be calculated.

Propagation of unsteady waves in an elastic layer

We consider a plane problem of propagation of unsteady waves in a plane layer of constant thickness filled with a homogeneous linearly elastic isotropic medium in the absence of mass forces and with zero initial conditions. We assume that, on one of the layer boundaries, the normal stresses are given in the form of the Dirac delta function, the tangential stresses are zero, and the second boundary is rigidly fixed. The problem is solved by using the Laplace transform with respect to time and the Fourier transform with respect to the longitudinal coordinate. The normal displacements at an arbitrary point are obtained in the form of finite sums.

Understanding the Non-Gaussian Nature of Linear Reactive Solute Transport in 1D and 2D

In the present study, we examine non-Gaussian spreading of solutes subject to advection, dispersion and kinetic sorption (adsorption/desorption). We start considering the behavior of a single particle and apply a random walk to describe advection/dispersion plus a Markov chain to describe kinetic sorption. We show in a rigorous way that this model leads to a set of differential equations. For this combination of stochastic processes, such a derivation is new. Then, to illustrate the mechanism that leads to non-Gaussian spreading, we analyze this set of equations at first leaving out the Gaussian dispersion term (microdispersion). The set of equations now transforms to the telegrapher’s equation. Characteristic for this system is a longitudinal spreading that becomes Gaussian only in the longtime limit. We refer to this as kinetics-induced spreading. When the microdispersion process is included back again, the characteristics of the telegraph equations are still present. Now, two spreading phenomena are active, the Gaussian microdispersive spreading plus the kinetics-induced non-Gaussian spreading. In the long run, the latter becomes Gaussian as well. Another non-Gaussian feature shows itself in the 2D situation. Here, the lateral spread and the longitudinal displacement are no longer independent, as should be the case for a 2D Gaussian spreading process. In a displacing plume, this interdependence is displayed as a ‘tailing’ effect. We also analyze marginal and conditional moments, which confirm this result. With respect to effective properties (velocity and dispersion), we conclude that effective parameters can be defined properly only for large times (asymptotic times). In the two-dimensional case, it appears that the transverse spreading depends on the longitudinal coordinate. This results in ‘cigar-shaped’ contours.