PDE: Parabolic form

Author(s):  
Bijan Kumar Bagchi
Keyword(s):  
2021 ◽  
Vol 11 ◽  
pp. 184798042110011
Author(s):  
Mahmoud M Selim ◽  
Taher A Nofal

In this work, an attempt is done to apply the Kirchhoff plate theory to find out the vibrational analyses of a nanoplate incorporating surface irregularity effects. The effects of surface irregularity on natural frequency of vibration of nanomaterials, especially for nanoplates, have not been investigated before, and most of the previous research have been carried for regular nanoplates. Therefore, it must be emphasized that the vibrations of irregular nanoplate are novel and applicable for the nanodevices, in which nanoplates act as the main structure of the nanocomposite. The surface irregularity is assumed in the parabolic form at the surface of the nanoplate. A novel equation of motion and frequency equation is derived. The obtained results provide a better representation of the vibration behavior of irregular nanoplates. It has been observed that the presence of surface irregularity affects considerably on the natural frequency of vibrational nanoplates. In addition, it has been seen that the natural frequency of nanoplate decreases with the increase of surface irregularity parameter. Finally, it can be concluded that the present results may serve as useful references for the application and design of nano-oscillators and nanodevices, in which nanoplates act as the most prevalent nanocomposites structural element.


2012 ◽  
Vol 189 ◽  
pp. 448-452
Author(s):  
Yan Jun Chen ◽  
Guo Qing Wu ◽  
Yang Cao ◽  
Dian Gui Huang ◽  
Qin Wang ◽  
...  

Numerical studies are conducted to research the performance of a kind of lift-drag type vertical axis wind turbine (VAWT) affected by solidity with the CFD method. Moving mesh technique is used to construct the model. The Spalart-Allmaras one equation turbulent model and the implicit coupled algorithm based on pressure are selected to solve the transient equations. In this research, how the tip speed ratio and the solidity of blade affect the power coefficient (Cp) of the small H-VAWT is analyzed. The results indicate that Cp curves exhibit approximate parabolic form with its maximum in the middle range of tip speed ratio. The two-blade wind turbine has the lowest Cp while the three-blade one is more powerful and the four-blade one brings the highest power. With the certain number of blades, there is a best chord length, and too long or too short chord length may reduce the Cp.


2016 ◽  
Vol 851 ◽  
pp. 377-382
Author(s):  
Raisa Bolotnova ◽  
Andrey Topolnikov ◽  
Valeria Korobchinskaya

To describe the explosive flow in thin tubes the model of vapor-liquid mixture with heat and mass transfer in two-dimensional axisymmetric formulation is employed. The phase transformation is significantly amplified with increasing of initial saturation temperature. The radial expansion of the jet outflow occurs due to the intensification of vaporization from cylindrical to conical form and parabolic form for supercritical initial state. Another problem applied to the outflow of a detonation wave in liquid filled with chemically active gas bubbles from the thin cylindrical tube. Modeling shows the important role of the opening angle of the outflow jet, which can either support the detonation or put it down.


1.It may be taken as an axiom of electrical ignition that the closer its conditions resemble those in the explosion wave front the more readily will it occur. These conditions are high temperature and pressure, and in the case of hydrocarbons combustion to carbon monoxide. They are also characteristic of condenser discharge sparks, for in the first place the surface of a platinum pole to which condenser discharge has been made becomes pitted to a remarkable extent, greater than when large currents are broken by separation of the poles. The sparks have therefore a high temperature. That they give rise to high gas pressure is clear from the intensity of the sound of a single spark discharge, and finally it will be shown that combustion to carbon monoxide rather than to carbon dioxide is peculiar in certain cases to ignition by capacity sparks. In addition to these the sparks are of very short duration, are oscillatory in character, and start with ionisation or breakdown of the gas between the poles. The belief that all visible sparks will ignite explosive mixtures no doubt arose from observations of the activity of condenser discharge in this respect, but while in certain cases, especially in the ignition of hydrogen, the least is in every case a well marked limit to their igniting power, and as the percentage of gas limits of inflammability are approached they require to be large. 2. The Paraffins—Ethane, Propane, and Butane .—The gases used in the present work were from the same stocks as those used for break-spark ignition. The results obtained from them are given in fig. 1. They have two interesting features; their minimum igniting current is the same in every case, in this resembling their ignition by continuous current break-sparks, but they have the parabolic form characteristic of alternating current break-spark ignition. Ethane has, however, a minimum at 7∙7 per cent., the point of combustion to carbon monoxide, the others, as before, midway between this and combustion to carbon dioxide. There is, however, on the higher side of the ethane and propane curves—the supply of butane gave out before this could be examined fully—a step or increase in difficulty of ignition corresponding to mixtures midway between four and five atoms of oxygen to one molecule of ethane, and between six and seven to one of propane. This point was thought to be some failure in the quality of the gas, but when it appeared in both, and to a still greater extent in methane, it was more fully examined. Condenser-spark ignition has therefore some of the features of both continuous and low frequency alternating current break-spark ignition, but it has a characteristic type of its own.


1971 ◽  
Vol 25 (2) ◽  
pp. 185-193
Author(s):  
William G. Doty
Keyword(s):  

The way God reigns has analogues in terribly ordinary human events. This is the mysterious element which can never be fully applied and remains always in parabolic form.


Author(s):  
X-B Chen

The singular and highly oscillatory properties of unsteady ship waves are studied by considering potential flows generated by a point source pulsating and advancing at a uniform forward speed located close to or at the free surface. The wave component of the free-surface potential defined by Noblesse and Chen by a single integral along the dispersion curves defined by the dispersion relation is analysed by developing asymptotic expansions of the open dispersion curves at large wave numbers. The asymptotic analysis of the wave component contributed by the leading asymptotic term of a parabolic form shows that unsteady ship waves are highly oscillatory with infinitely increasing amplitude and infinitely decreasing wavelength, when a field point approaches the track of the source point at the free surface. The highly oscillatory property and complex singular behaviour of unsteady ship waves are further expressed in an original and analytically closed form.


1980 ◽  
Vol 10 (4) ◽  
pp. 491-497 ◽  
Author(s):  
Andrew N. Rencz ◽  
Allan N. Auclair

Parabolic and logarithmic regressions were used to relate tree diameter to the biomass of root, root crown, bole, branch, needle, cone, and epiphytic lichens for a sample of 15 Piceamariana (Mill) trees occupying lichen woodland in the subarctic of eastern Canada. In 22 of 27 regressions both models yielded r2 values >0.82. Biomass estimates of total tree and individual component dry weights resulted in estimates with less than 6.5% difference between biomass estimates by logarithmic versus parabolic equations. For this data set the logarithmic model appeared more appropriate than the parabolic form. Validity of the regressions was judged on r2, analysis of variance, and examination of residuals. Equations generated in this study were considered to be inapplicable to P. mariana growing in closed forest. Problems in extrapolation were discussed.


2013 ◽  
Vol 307 ◽  
pp. 364-367 ◽  
Author(s):  
Ali Ozturk ◽  
Müfit Gülgeç

This paper presents analytical solutions of the thermal stresses in a functionally graded solid cylinder with fixed ends in elastic region. These thermal stresses are due to the uniform heat generation inside the cylinder. Material properties of the functionally graded (FG) cylinder vary radially according to a parabolic form. The material properties are assumed to be independent of the temperature which are yield strength, elasticity modulus, thermal conduction coefficient, thermal expansion coefficient and Poisson’s ratio. The solutions for the thermal stresses are valid for both homogeneous and functionally graded materials.


Author(s):  
John G. Michopoulos ◽  
Andrew Birnbaum ◽  
Athanasios P. Iliopoulos

Despite significant efforts examining the suitability of the proper form of the heat transfer partial differential equation (PDE) as a function of the time scale of interest (e.g. seconds, picoseconds, femtoseconds, etc.), very little work has been done to investigate the millisecond-microsecond regime. This paper examines the differences between the parabolic and one of the hyber-bolic forms of the heat conduction PDE that govern the thermal energy conservation on these intermediate timescales. Emphasis is given to the types of problems where relatively fast heat flux deposition is realized. Specifically, the classical parabolic form is contrasted against the lesser known Cattaneo-Vernotte hyperbolic form. A comparative study of the behavior of these forms over various pulsed conditions are applied at the center of a rectangular plate. Further emphasis is given to the variability of the solutions subject to constant or temperature-dependent thermal properties. Additionally, two materials, Al-6061 and refractory Nb1Zr, with widely varying thermal properties, were investigated.


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