Multiple Expansion Method Research Articles

Maker Chemistry: Exploring a New Multiple Expansion Method through Sensor and Software to Determine Vapor Pressure

Maker Chemistry: Exploring a New Multiple Expansion Method through Sensor and Software to Determine Vapor Pressure

Gaseous PVT Property Measurements of cis-1,3,3,3-Tetrafluoropropene

PVT properties in the vapor phase of cis-1,3,3,3-tetrafluoropropene (R1234ze(Z)) were measured by a multiple expansion method in the temperature range from 353 to 413 K and at pressures up to 2.7 MPa. Thirty data along four isotherms are obtained in the temperatures between 353 and 413 K. The vapor pressures at the temperatures were also measured by adding a sample of R1234ze(Z) to a sample cell at the vapor–liquid equilibrium conditions. The uncertainties in temperature and pressure measurements are estimated to be within 6 mK and 0.3 kPa, respectively. The expanded uncertainty in density measurement is estimated within no greater than 0.12% (k = 2). The obtained PVT properties and vapor pressures are compared with the existing equation of state.

Homogenization of linear hyperbolic stochastic partial differential equation with rapidly oscillating coefficients: The two scale convergence method

In this paper we establish new homogenization results for stochastic linear hyperbolic equations with periodically oscillating coefficients. We first use the multiple expansion method to drive the homogenized problem. Next we use the two scale convergence method and Prokhorov's and Skorokhod's probabilistic compactness results. We prove that the sequence of solutions of the original problem converges in suitable topologies to the solution of a homogenized stochastic hyperbolic problem with constant coefficients. We also prove a corrector result. Homogenization is a mathematical theory aimed at understanding the behavior of processes that take place in heterogeneous media with highly oscillating heterogeneities at the microscopic level using prop- erties of the homogeneous media obtained by homogenizing these materials. These heterogeneous ma- terials consist of finely mixed different components like soil, paper, concrete for building, fibreglass, materials used in the manufacturing of high tech equipments such as planes, rockets and so on. This signifies that almost everything around us in real life is a heterogeneous material. The physical problems described on heterogeneous materials such as heat, mechanical constraints, flow of fluids in these me- dia leads to the study of PDEs with highly oscillating coefficients depending on macroscopic scales or boundary value problems for PDEs in domain with fine grained boundaries. The main obstacle in solv- ing these problems arises either from the character of the domain or the presence of high oscillations in the coefficients of the governing equation. To this end, it is expensive to compute solutions to these type of problems. Numerical methods have proved inefficient in solving such problems due to the fact that even the most advanced parallel computers are unable to simulate schemes related to the physically interesting such problems.

Bonding modes of some labile Si compounds studied by charge density analysis

Some organic molecules containing Si atom(s) are very labile, even if the corresponding carbon analogs are very stable. To gain information on bonding modes of such compounds, we analyzed valence density distribution, which play critical roles in chemistry of molecule, by applying multipole expansion method. Very recently, an imine coordinated silacyclopropan-1-one, 1, has synthesized by Baceiredo, Kato and co-workers.[1] To clarify the bonding mode of 1, the electron density distributions of 1 and its precursor have analyzed by a multiple expansion method using single crystal X-ray diffraction data. As shown in static model density maps, bonding electrons of Si-C bonds distribute on the outside of the silacyclopropane ring (Si1-C1-C2 ring) (Fig. 1a) with largely extent, in compared with that of the precursor, indicating an in-plane pi-interaction on the Si1-C1 and Si1-C2 bonds. On the other hand, the C1-C2 bonding electrons distribute on the bond, and the bond critical point (BCP) is located on the inside of the three membered ring. In addition, the C1-C2 bonding electrons elongates inside the ring toward the Si1 atom, indicating electron donation from sigma(C1-C2)-bond to the Si1 (Fig. 1b). Consequently, these maps propose greater contribution of canonical structures in Fig. 1c.

Nonlinear Analysis of Rayleigh–Taylor Instability of Cylindrical Flow With Heat and Mass Transfer

We study the nonlinear Rayleigh–Taylor instability of the interface between two viscous fluids, when the phases are enclosed between two horizontal cylindrical surfaces coaxial with the interface, and when there is mass and heat transfer across the interface. The fluids are considered to be viscous and incompressible with different kinematic viscosities. The method of multiple expansions has been used for the investigation. In the nonlinear theory, it is shown that the evolution of the amplitude is governed by a Ginzburg–Landau equation. The various stability criteria are discussed both analytically and numerically and stability diagrams are obtained. It has been observed that the heat and mass transfer has stabilizing effect on the stability of the system in the nonlinear analysis.

SEMI-ANALYTICAL SOLUTION FOR RADIATION OF A HOLLOW SPHERE WITH AN OPENING HOLE IN FINITE DEPTH

An investigation is conducted on radiation of a submerged hollow sphere with an opening hole in finite water depth in this article. Based on the linear theory, the method of multiple expansions is used to obtain the fluid velocity potential in the form of double series of the associated Legendre functions with the unknown coefficients of an infinite set. In terms of the body surface boundary condition and the matching condition between the inner and outer flows at the hole, the complex matrix equations for the coefficients of the series are established. The infinite sets of matrix equations are solved by truncating the series at a finite number. Subsequently, the added mass and radiation damping, associated with the periodic heave motion of a submerged sphere, are evaluated numerically.

Hydrodynamic radii and diffusion coefficients of particle aggregates derived from the bead model

The multiple expansion method was applied for calculating friction tensors and hydrodynamic radii R(H) of rigid molecules of various shape, composed of n(s) equal sized, touching spheres. The maximum value of n(s) studied was 450, which covers most situations met in practice. Calculations were performed for linear chains, half-circles, circles (cyclic molecules) and S-shaped aggregates. It was shown that our results agreed with previous theoretical data obtained for linear chains and cyclic aggregates, for n(s)<100. For larger n(s), studied exclusively in our work, interpolating analytical expressions were formulated for the hydrodynamic radii R(H). These expressions, involving logarithmic function of the aspect ratio parameter (length to width ratio of the macromolecules), are the main finding of our work. Using these expressions, the ratio of the hydrodynamic radius of cyclic-to-linear aggregate q(f) was calculated, which is a parameter of vital significance. It was determined that q(f) attained values close to 0.95 for n(s) approximately 450. This suggests that the previous analytical results derived by Tchen [19], in the slender body limit, who predicted that q(f)-->12/11=1.09, are not applicable for n(s)<450. Using the R(H) values, the average translation diffusion coefficients and the sedimentation coefficients for these aggregate shapes were calculated. It was shown that our theoretical results are in good agreement with experimental data obtained for polyelectrolytes and for DNA fragments of various molecular mass. It was concluded that our results can be effectively used to determine the shape of macromolecules, in particular to discriminate between linear and cyclic DNA configurations.

Nonlinear Instability of Cylindrical Flow with Mass and Heat Transfer

The nonlinear Rayleigh–Taylor stability of the cylindrical interface between the vapor and liquid phases of a fluid is studied when the phases are enclosed between two cylindrical surfaces coaxial with the interface, and when there is mass and heat transfer across the interface. The method of multiple expansion is used for the investigation. The evolution of amplitude is shown be governed by a nonlinear Ginzburg–Landau equation. The various stability criteria are discussed, and the region of stability is displayed graphically.

Nonlinear Rayleigh instability of cylindrical flow with mass and heat transfer

The nonlinear Rayleigh instability of the cylindrical interface between the vapour and liquid phases of a fluid is studied when the phases are enclosed between two cylindrical surfaces coaxial with the interface, and when there is mass and heat transfer across the interface. The method of multiple expansion is used for the investigation. The evolution of amplitude is shown to be governed by a first-order nonlinear differential equation. The stability criterion is discussed, and the region of stability is displayed graphically.

Solitary waves in fluid-filled elastic tubes: weakly dispersive case

In the present work, employing the nonlinear equations of an incompressible, isotropic and elastic thin tube and the approximate equations of an incompressible inviscid fluid, the propagation of weakly nonlinear waves, in such a medium is studied through the use of the modified multiple expansion method. It is shown that the evolution of the lowest order (first-order) term in the perturbation expansion may be described by the Korteweg-de Vries (KdV) equation. The governing equation for the second-order terms and the localized travelling wave solution for these equations are also obtained. The applicability of the present model to flow problems in arteries is discussed.

Some singular solutions of the theory of elasticity

Singular solutions corresponding to “double forces without moment”, an “extension-compression centre”, and “double forces with moment”, obtained previously for an isotropic medium [2, 3] by differentiation of the Kelvin fundamental solution, are constructed for media with an arbitrary elastic anisotropy using the method of multiple expansions [1].

Shunt posts in microstrip transmission lines

Shunt posts in microstrip transmission lines are analyzed by employing the planar waveguide model. A multiple expansion method with greatly enhanced efficiency and accuracy has been achieved with the development of very fast algorithms for the computation of Schloemilch-type series. The method applies to perfectly conducting imperfectly conducting, and composite shunt posts. This approach is compared with the standard point matching technique for perfectly conducting posts. The two methods are found to be in excellent agreement in regions of overlapping applicability. A fairly wide range of computed data is utilized to develop the tools needed to incorporate the post into various microstrip circuit applications. As one application, a procedure is developed for using this post to obtain effective short-circuiting qualities over a broad frequency range. As a second application, the possibility of creating filtering devices is explored using both multiple metallic post structures and single dielectric posts.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>