Dimensionless Modulus Research Articles

Inspiraling compact objects with generic deformations

Self-gravitating bodies can have an arbitrarily complex shape, which implies a much richer multipolar structure than that of a black hole in General Relativity. With this motivation, we study the corrections to the dynamics of a binary system due to generic, nonaxisymmetric mass quadrupole moments to leading post-Newtonian (PN) order. Utilizing the method of osculating orbits and a multiple scale analysis, we find analytic solutions to the precession and orbital dynamics of a (generically eccentric) binary in terms of the dimensionless modulus parameters $\epsilon_{m}$, corresponding to axial $m=1$ and polar $m=2$ corrections from oblateness/prolateness. The solutions to the precession dynamics are exact for $0 \le \epsilon_{2} < 1$, and perturbative in $\epsilon_{1} \ll 1$. We further compute the leading order corrections to the gravitational wave amplitude and phase for a quasi-circular binary due to mass quadrupole effects. Making use of the stationary phase approximation and shifted uniform asymptotics (SUA), the corrections to the phase enter at relative 2PN order, while the amplitude modulations enter at -0.5PN order with a SUA amplitude correction at 3.25PN order, relative 2PN order to the leading order SUA correction. By investigating the dephasing due to generic quadrupole moments, we find that a phase difference $\gtrsim 0.1$~radians is achievable for $\epsilon_{m} \gtrsim 10^{-3}$, which suggests that constraints with current and future ground-based gravitational wave detectors are possible. Our results can be implemented in parameter estimation studies to constrain generic multipolar deformations of the Kerr geometry and of neutron stars.

Low-frequency pseudo-Rayleigh and pseudo-Scholte waves at an interface of liquid/soft porous sediment with underlying hard porous sediment substrate

Summary The paper focuses on the propagation of low-frequency pseudo-Rayleigh and pseudo-Scholte waves at the liquid/soft porous sediment interface with an underlying hard porous sediment half-space. The overlying liquid is assumed to be ideal compressible medium and the porous sediments are modelled by Biot theory. Based on the boundary conditions, the closed-form dispersion equations of far-field interface waves are deduced using 2-D Helmholtz decomposition theorem and Fourier transform. The velocity and attenuation of pseudo-Rayleigh and pseudo-Scholte waves are determined by Newton iteration in a reasonable rooting interval. The analytical expressions of the displacement field and liquid pressure distribution caused by interface waves are also derived. Then, the dispersion equations for four degenerate systems are derived as special cases by assuming the thickness of the liquid layer or the sandwiched porous soft sediment layer to be zero or infinite. Lastly, numerical examples are used to verify the degeneracy of the system and to analyse the propagation characteristics of pseudo-Rayleigh and pseudo-Scholte waves. They show the dependences of the velocity and displacement field on dimensionless modulus and dimensionless wavelength. When the dimensionless wavelength is small or very large, the phase velocity and displacement field calculated by the present system is the same as the special cases, thus proving the validating of the new system.

Temperature dependent Young’s modulus of ZnO nanowires

A thermal resonant method was developed to accurately determine the temperature-dependent Young’s moduli of nanowires. In this method, the frequency spectra of a [0001]-oriented ZnO nanowire cantilever at elevated temperatures were measured using scanning laser Doppler vibrometry. The temperature-dependent Young’s moduli were derived from the resonant frequencies using Euler–Bernoulli beam theory. It was found that the modulus of ZnO nanowires decreased linearly with the increase of temperature from 300 to 650 K, independent of the nanowire diameter ranged from 101 to 350 nm. The temperature coefficient that defines the linear relationship between the dimensionless modulus and temperature is which agrees with that of being calculated using molecular dynamics with a partially charged rigid ion model.

Approximate analytical solution of the concentration of phenol and oxygen and rate of phenol degradation in fluidized bed bioreactor

A theoretical model for fluidized bioreactor (FBR) is presented. The model is based on system of reaction-diffusion equation containing a non-linear term related to specific growth rate of biomass. Homotopy analysis method (HAM) is successfully employed in the development of non-linear steady state phenol degradation. In this paper simple and approximate polynomial expression of concentration of phenol and oxygen and rate of phenol degradation are derived for all possible values of parameters ϕO (dimensionless modulus for oxygen), ϕS (dimensionless modulus for phenol), Ki* (dimensionless inhibition constant for phenol) and KO*, KS* (dimensionless monod constant for oxygen and phenol). The limiting case results are compared with our common results and are found to be in good agreement.

Minimum spouting velocity under vacuum and high temperature in conical spouted beds

AbstractA study has been carried out on spouting performance in conical spouted beds at high temperatures and under vacuum by using materials of different textures, densities and particle diameters. The applicability of existing correlations for determining the minimum spouting velocity has been checked and, based on a statistical analysis to assess the influence of the dimensionless modulus, a new one has been proposed for the prediction of the minimum spouting velocity under vacuum and at high temperatures. This correlation provides good predictions in the pyrolysis of tyres under vacuum.

Comparison of the Effectiveness Factors for a Reaction at a Pore Wall Calculated on the Assumption of the Langmuir-Hinshelwood Mechanism and According to a Power Equation

Langmuir-Hinshelwood and power equations were used for the description of a chemical reaction on a pore wall in the transition kinetic region, where the reaction rate is determined both by diffusion into the pore and by the chemical reaction at the interface. The exponent in the power equation corresponding to the chosen constants in the Langmuir-Hinshelwood equation was calculated by the least squares method. The concentration distribution in the pore and the effectiveness factor as function of the dimensionless modulus h∈(0,4> were calculated for both kinetic equations. The deviations of the effectiveness factor found for the corresponding power equation with respect to the Langmuir-Hinshelwood equation were smaller than ± 5% for a porous catalyst in the form of an infinite plate or a sphere, assuming that the true reaction order was 1 or 2 and that the exponent in the power equation was higher than 0.125.

Effect of surface conduction on charge relaxation in partially filled vessels

The decay of electrostatic charge within a conducting vessel partially filled with charged liquid is slowed by the temporary accumulation of a portion of the initial charge at the free surface of the liquid. This effect will prolong the period of exposure to vapor ignition hazards due to electrostatic discharge, especially when the fill level approaches 100%. However, if surface conduction is taken into account, surface charge accumulation tends to be offset by opposite sign charge flowing in from the conducting side walls. In this paper, the existing model for charge relaxation in partially filled conducting cylindrical vessels is modified to account for surface conduction. The principal result is that the extrinsic relaxation time constants associated with the modal solution are reduced. A dimensionless modulus is identified that compares the surface and volume conduction mechanisms and indicates that the effect of surface conduction is inversely proportional to the absolute vessel diameter. When the vessel diameter is small or when the surface conductivity is sufficiently high, the relaxation time for the electric field in the vapor space is reduced below the intrinsic charge relaxation time of the liquid. The practical implication of these results is that surface conduction can easily dominate in establishing the rate of decay of the electric field in the vapor space.

Fluid Flow Model in a Vertical Multiwafer Stacked-in-Tube CVD Reactor

Mass transfer in a vertical multiwafer stacked-in-tube CVD reactor was analyzed by simple fluid-flow models, and the applicable range of the models was determined. Carbon plate was burnt under the condition of relatively low burning rate in the CVD reactor. The local burning rate of the carbon plate was analyzed by a one-dimensional diffusion model in the wafer part and a cell-in-series model based upon the wafer distance in the annular part or cell-in-series model in the space between the wafers. The applicable range of the simple model was specified by a dimensionless modulus, J (=rw√k/HD)

Modeling Stress Relaxation Response of Wheat En Masse Using the Triaxial Test

ABSTRACT Time dependent load response data were obtained from triaxial tests to characterize the stress relaxation behavior of wheat en masse. Stress relaxation data were measured for two axial strains, two confining pressures, two moisture contents, and two initial bulk densities. The generalized Maxwell model was used to simulate the stress relaxation response of wheat. The test data were transformed into dimensionless quantities using initial or maximum elastic modulus E^^x ^^^ ^^^^ relaxation time T50. The data converged for t < T50 using dimensionless modulus <t^ = E/En^a^ ^s dependent variable and dimensionless time 0 = t/T^Q as independent variable. Two boundary condition variables — confining pressure and axial strain, significantly affected the initial elastic modulus; whereas, three variables — confining pressure, initial bulk density and moisture content, significantly affected the half relaxation time.

Aeroelastic tailoring of aft-swept high-aspect-ratio composite wings

The integrating matrix method is used to study the aeroelastic performance of aft-swept high-aspect-ratio wings. Aeroelastic stability boundaries are shown as a function of fiber angle and dimensionless modulus parameters for straight wings of aspect ratio 14. Certain laminates show divergence for wings with low sweep angles, but for most laminates and wing sweeps, flutter was the instability. The bending-torsion coupling that is beneficial for forward-swept wings is shown to be of no advantage for aft-swept wings, for which torsional stiffness is much more significant. Both symmetric and nonsymmetric ply orientations were studied, and it is demonstrated that there need not be a severe penalty for using general laminates.

Effect of internal diffusion on catalytic reactions

The effectiveness factors η have been calculated for a reversible, isothermal, catalytic reaction proceeding without a change in the number of moles, for the case when the reaction rate is expressed by means of rate Eq. (3). The effectiveness factor depends on the dimensionless modulus M L [Eq. (20)] and dimensionless parameter B. If equilibrium composition is attained at the centre plane of the slab of catalyst, the effectiveness factor η can also be calculated analytically from Eqs. (32) and (33). A comparison with the analogous irreversible reaction reveals that the retardation effect of internal diffusion is always stronger for a reversible reaction.

Pressure Drop and Flow Characteristics of Short Capillary Tubes at Low Reynolds Numbers

Abstract The pressure drop and flow characteristics of short capillary tubes have been investigated experimentally for length-to-diameter ratios varying from 0.45 to 18 at diameter Reynolds numbers ranging from 8 to 1500. In the range of the dimensionless modulus (Lμ)/(VD2ρ) from 4 × 10−3 to 3 to 10−1, the experimental data agree within 15 per cent with a mathematical theory by Langhaar (1). At a value of (Lμ)/(VD2ρ) of about 0.3 the experimental data approach the Poiseuille laminar-flow theory (2). For very short tubes (L/D &amp;lt; 0.5) the experimental results deviate from Langhaar’s theory at values of Lμ/VD2ρ less than 4 × 10−3, and at Lμ/VD2ρ equal to 5 × 10−4, the pressure drop is twice as large as that predicted by Langhaar’s theory (1). The experimental results for tubes having very short aspect ratios are in agreement with data obtained by Zucrow (12) with short square-edged jets. It was found that the flow rate Q˙ through a short capillary tube can be related empirically to the over-all pressure drop Δp¯ raised to a power N. The exponent N is a function of the length-to-diameter ratio L/D varying from 0.5 at L/D equal to 0.45 to 0.91 at L/D of 18. The trend of the curve suggests an asymptotic approach to unity, the exponent for Poiseuille-type flow. The results of this study have application to: (a) Simulating flow through screens, doors, cracks, and fissures in small-scale model testing of buildings in atmospheric wind tunnels. (b) Automatic control devices where capillary tubes are used as hydraulic resistances in a larger line and in nozzle-flapper combinations. (c) Heat pumps and air-conditioning equipment where short capillary tubes are used as two-way control valves. (d) Flow through compact heat exchangers and porous materials.

Heat Transmission in High Speed Gas Flow (1st and 2nd Reports)

From the dimensional analysis for high velocity gas flow in a tube, our recommendable dimensionless modulus, Jλ (Tw-Tg) /3 600 μV2, is introduced, which is named Dissipation Ratio and expretssd by Di. In order to recognize the effect of Di, we performed an experiment, in which air velocity flowing through an annular cross section was 70∼200 m/s, the rate of heat transfered into air from the inner surface 300∼20000 kcal/m2·h and the length of tube 55∼280 mm. From our result, the relation between heat flux and dissipation ratio is obtained and our experimental formulae is expressed by Nu, Re, L/D and Di.