Bubble Dynamics and Cavitation Inception Theory

1988 ◽  
Vol 32 (03) ◽  
pp. 155-167
Author(s):  
Blaine R. Parkin ◽  
Brian B. Baker
Keyword(s):  
Initial Conditions ◽  
Second Order ◽  
Order System ◽  
Refined Theory ◽  
Zeroth Order ◽  
Order Problem ◽  
Cavitation Inception ◽  
First Order ◽  
Order Solution ◽  
Forcing Function

In order to provide some theoretical background and to motivate the more refined theory introduced herein, some encouraging known theoretical results on bubble-ring cavitation inception are reviewed. This review is followed by the development of the theory of bubble-ring cavitation cutoff. Its outcome, when compared with experiment, shows the need for a more refined inception theory. The above comparison and the basic ideas behind the cutoff theory's formulation suggest a possible approach for a refinement based on a multiple scales expansion. This seems reasonable because the forcing function pulse in "laboratory time" f, varies slowly compared with the characteristic "bubble time,", which characterizes the response time of a typical microscopic cavitation nucleus. The ratio of these two times gives a small parameter, , appearing in the forcing function, with the result that this problem involves only a soft excitation. Expanding the forced Rayleigh-Plesset equation and its initial conditions to the second order in c, the zeroth-order problem is found to be the well-known autonomous nonlinear equation with nonhomogeneous initial conditions, giving free oscillations of a typical nucleus. The first-order system is a nonautonomous linear system with homogeneous initial conditions which governs the forced bubble growth. The second-order system consists of a linear autonomous differential equation and homogeneous initial conditions. It is needed to establish integrability conditions for the first-order solution. The first-order solution is left for future research and the zeroth-order problem is analyzed in the phase plane. Then a novel approximate integration, = t(u), is given in terms of elliptic integrals and functions. It was not possible to invert this solution and so the inverse u = u() is found numerically. These data are then used to find an analytical approximation for use in future first-order calculations.

scholarly journals From P/E Ratio to Fuzzy Infinite Spreadsheet—Mathematically Rigorous Derivations of the Zeroth and the First Order Solutions of Rate of Return

2022 ◽  
Vol 5 (1) ◽  
pp. p7
Author(s):  
Hugh Ching (USA) ◽  
Chien Yi Lee (China) ◽  
Benjamin Li (Canada)
Keyword(s):  
Exact Solution ◽  
Real Estate ◽  
Initial Conditions ◽  
Investment Analysis ◽  
Rate Of Return ◽  
Exact Equation ◽  
Zeroth Order ◽  
Real Estate Investment ◽  
First Order ◽  
Order Solution

The P/E Ratio (Price/Earning) is one of the most popular concepts in stock analysis, yet its exact interpretation is lacking. Most stock investors know the P/E Ratio as a financial indicator with the useful characteristics of being relatively time-invariant. In this paper, a rigorous mathematical derivation of the P/E Ratio is presented. The derivation shows that, in addition to its assumptions, the P/E Ratio can be considered the zeroth order solution to the rate of return on investment. The commonly used concept of the Capitalization Rate (Cap Rate = Net Income / Price) in real estate investment analysis      can also be similarly derived as the zeroth order solution of the rate of return on real estate investment. This paper also derives the first order solution to the rate of return (Return = Dividend/Price + Growth) with its assumptions. Both the zeroth and the first order solutions are derived from the exact future accounting equation (Cash Return = Sum of Cash Flow + Cash from Resale). The exact equation has been used in the derivation of the exact solution of the rate of return. Empirically, as an illustration of an actual case, the rates of return are 3%, 73%, and 115% for a stock with 70% growth rate for, respectively, the zeroth order, the first order, and the exact solution to the rate of return; the stock doubled its price in 2004. This paper concludes that the zero-th, the first order, and the exact solution of the rate of return all can be derived mathematically from the same exact equation, which, thus, forms a rigorous mathematical foundation for investment analysis, and that the low order solutions have the very practical use in providing the analytically calculated initial conditions for the iterative numerical calculation for the exact solution. The solution of value belongs to recently classified Culture Level Quotient CLQ = 10 and is in the process of being updated by fuzzy logic with its range of tolerance for predicting market crashes to advance to CLQ = 2.

Radiant Heat Transfer From Isothermal Dispersions With Isotropic Scattering

1967 ◽  
Vol 89 (4) ◽  
pp. 300-308 ◽  
Author(s):  
R. H. Edwards ◽  
R. P. Bobco
Keyword(s):  
Heat Transfer ◽  
Approximate Solutions ◽  
Radiant Heat ◽  
Second Order ◽  
Isotropic Scattering ◽  
Radiant Heat Transfer ◽  
Approximate Methods ◽  
First Order ◽  
Order Solution ◽  
Radiant Transfer

Two approximate methods are presented for making radiant heat-transfer computations from gray, isothermal dispersions which absorb, emit, and scatter isotropically. The integrodifferential equation of radiant transfer is solved using moment techniques to obtain a first-order solution. A second-order solution is found by iteration. The approximate solutions are compared to exact solutions found in the literature of astrophysics for the case of a plane-parallel geometry. The exact and approximate solutions are both expressed in terms of directional and hemispherical emissivities at a boundary. The comparison for a slab, which is neither optically thin nor thick (τ = 1), indicates that the second-order solution is accurate to within 10 percent for both directional and hemispherical properties. These results suggest that relatively simple techniques may be used to make design computations for more complex geometries and boundary conditions.

Mean-Square Response of a Second-Order System to Nonstationary, Random Excitation

1970 ◽  
Vol 37 (3) ◽  
pp. 612-616 ◽  
Author(s):  
L. L. Bucciarelli ◽  
C. Kuo
Keyword(s):  
Narrow Band ◽  
Random Excitation ◽  
Second Order ◽  
Envelope Function ◽  
Order System ◽  
Mean Square ◽  
Forcing Function ◽  
Mean Square Response ◽  
Noise Function ◽  
The Mean

The mean-square response of a lightly damped, second-order system to a type of non-stationary random excitation is determined. The forcing function on the system is taken in the form of a product of a well-defined, slowly varying envelope function and a noise function. The latter is assumed to be white or correlated as a narrow band process. Taking advantage of the slow variation of the envelope function and the small damping of the system, relatively simple integrals are obtained which approximate the mean-square response. Upper bounds on the mean-square response are also obtained.

First and Second Order Wave-Current Interactions for Floating Bodies

2012 ◽  
Author(s):  
Charles Monroy ◽  
Yann Giorgiutti ◽  
Xiao-Bo Chen
Keyword(s):  
Field Method ◽  
Boundary Integral ◽  
Second Order ◽  
Boundary Integral Method ◽  
Diffraction Radiation ◽  
Floating Bodies ◽  
Order Problem ◽  
First Order ◽  
Wave Drift Damping ◽  
Order Quantities

The influence of current in sea-keeping problems is felt not only for first order quantities such as wave run-ups in front of the structure, but also mainly for second order quantities. In particular, the wave drift damping (which is expressed as the derivative of drift force with respect to the current) is of special interest for mooring systems. The interaction effects of a double-body steady flow on wave diffraction-radiation is studied through a decomposition of the time-harmonic potential into linear and interaction components. A boundary integral method is used to solve the first order problem. Ultimately, a far-field method is proposed to get access to second order drift forces.

scholarly journals Study on the Effect of Fractional Derivative on the Hyperspectral Data of Soil Organic Matter Content in Arid Region

2019 ◽  
Vol 2019 ◽  
pp. 1-11 ◽  
Author(s):  
Cheng-Biao Fu ◽  
Hei-Gang Xiong ◽  
An-Hong Tian
Keyword(s):  
Organic Matter ◽  
Soil Organic Matter ◽  
Fractional Derivative ◽  
Arid Region ◽  
Arid Zone ◽  
Organic Matter Content ◽  
Matter Content ◽  
Second Order ◽  
Zeroth Order ◽  
First Order

Discussion on the application of fractional derivative algorithm in monitoring organic matter content in field soil is scarce. This study is aimed at improving the accuracy of soil organic matter (SOM) content estimation in arid region, and the undesirable model precision caused by the missing information associated with the larger discrepancy between conventional integer-order, i.e., first order and second order, derivative, and raw spectral data. We utilized fractional derivative (of zeroth order to second order in 0.2-order interval) processing on the field spectral reflectance (R) of the salinized soil sample from Fukang, Xinjiang, and its square root-transformed (R), log-transformed (lgR), inverse-transformed (1/R), and inverse log-transformed (1/lgR) values. The correlation coefficient of each fractional derivative of transformed value with SOM content was calculated. The simulation showed the derivative reflectance value approximates zero. When increasing from zeroth order to first order, the derivative curve gradually aligns to the first-order curve, and the destination alignment was also seen while increasing from first order to second order. The significance test of 0.05 showed initial increase and later decay of bands in the five spectral transformations as the order increases. For specific bands, the derivative algorithm clearly justifies the correlation between soil spectra and organic matter content, and all of the absolute highest correlation coefficient values were obtained at fractional orders. When compared with integer-order derivative, fractional derivative is significantly better in improving correlation, showing overall superiority. The result supports the application of fractional derivative in the hyperspectral remote monitor of SOM in arid zone, which may in turn realize the timely and accurate SOM monitor in arid zone, and provides the basis for ecological restoration.

New fixed-time sliding mode control for a mismatched second-order system

2020 ◽  
pp. 014233122095230
Author(s):  
Guo Jianguo ◽  
Yang Shengjiang
Keyword(s):  
Sliding Mode Control ◽  
Sliding Mode ◽  
Initial Conditions ◽  
Nonlinear Function ◽  
Fixed Time ◽  
Second Order ◽  
Order System ◽  
Time Stability ◽  
Mode Control ◽  
Mismatched Uncertainties

A fixed-time sliding mode control (FTSMC) method is proposed for a second-order system with mismatched uncertainties in this paper. A new sliding mode, which is insensitive to the mismatched disturbance, is present to eliminate the effect of mismatched uncertainties by adopting the differentiable nonlinear function, and to obtain the fixed time stability independent of initial conditions by using the fraction-order function. In order to improve the performance of control system, the extended disturbance-observer-based fixed-time sliding mode control (EDO-FTSMC) approach is investigated to obtain the fixed-time stability subject to the mismatched uncertainties. Finally, the performance of the proposed control method is illustrated to compare other control approaches with numerical simulation results and application examples.

Second-Order Diffraction Forces on an Array of Vertical Cylinders in Bichromatic Bidirectional Waves

1995 ◽  
Vol 117 (1) ◽  
pp. 12-18 ◽  
Author(s):  
J. H. Vazquez ◽  
A. N. Williams
Keyword(s):  
Cross Section ◽  
Finite Depth ◽  
Hydrodynamic Forces ◽  
Arbitrary Cross Section ◽  
Second Order ◽  
Order Problem ◽  
First Order ◽  
Wave Directionality ◽  
Vertical Cylinders ◽  
Green Function Approach

A complete second-order solution is presented for the hydrodynamic forces due to the action of bichromatic, bidirectional waves on an array of bottom-mounted, surface-piercing cylinders of arbitrary cross section in water of uniform finite depth. Based on the constant structural cross section, the first-order problem is solved utilizing a two-dimensional Green function approach, while an assisting radiation potential approach is used to obtain the hydrodynamic loads due to the second-order potential. Results are presented which illustrate the influence of wave directionality on the second-order sum and difference frequency hydrodynamic forces on a two-cylinder array. It is found that wave directionality may have a significant influence on the second-order hydrodynamic forces on these arrays and that the assumption of unidirectional waves does not always lead to conservative estimates of the second-order loading.

scholarly journals Lie transforms and motion of a charged particle in a periodic magnetic field

1984 ◽  
Vol 7 (1) ◽  
pp. 159-169
Author(s):  
Sikha Bhattacharyya ◽  
R. K. Roy Choudhury
Keyword(s):  
Magnetic Field ◽  
Charged Particle ◽  
Averaging Method ◽  
Second Order ◽  
Lie Series ◽  
Periodic Magnetic Field ◽  
First Order ◽  
Order Solution ◽  
Lie Transforms ◽  
Spatially Periodic

We use the Lie series averaging method to obtain a complete second order solution for motion of a charged particle in a spatially periodic magnetic field. A comparison is made with the first order solution obtained previously by Coffey.

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