Subjects: Physics >> General Physics: Statistical and Quantum Mechanics, Quantum Information, etc. submitted time 2017-11-24
Abstract： This study considers the periodic orbital period of an n-body system from the perspective of dimension analysis. According to characteristics of the n-body system with point masses $(m_1,m_2,...,m_n)$, the gravitational field parameter, $\alpha \sim Gm_im_j$, the n-body system reduction mass $M_n$, and the area, $A_n$, of the periodic orbit are selected as the basic parameters, while the period, $T_n$, and the system energy, $|E_n|$, are expressed as the three basic parameters. By using Buckingham $\pi$-theorem of dimensional analysis, these two relations can be reduced to a dimensionless form, which can surprisingly produce only one dimensionless $\pi$, respectively. Because there is only one $\pi$, therefore the $\pi$ must be a constant. Since the two-body system is a special case of the n-body, we can uniquely determine the two constants by using the two-body Kepler's third law. Thus, the n-body system Kepler's third law is deduced and is given by $T_n|E_n|^{3/2}=\frac{\pi}{\sqrt{2}} G\left(\frac{\sum_{i=1}^n\sum_{j=i+1}^n(m_im_j)^3}{\sum_{k=1}^n m_k}\right)^{1/2}$. A numerical validation and comparison study was hence conducted.
Peer Review Status:
Awaiting Review
Subjects: Physics >> Electromagnetism, Optics, Acoustics, Heat Transfer, Classical Mechanics, and Fluid Dynamics submitted time 2017-09-30
Abstract： This paper uses dimensional analysis to define the general expression of the pair $(N,\ell)$, and identifies the dominant combination-parameters of the capillary wrinkling problem, while it also determines the dominant parameters of different problems relating to its use. The dimensional analysis results reveal that, in general, there are no universal scaling laws for capillary wrinkling. Only for a small/moderate deformation, it was found that the wrinkling number $N$ is mainly controlled by the ratio of bending stiffness and surface tension, while the wrinkling length $\ell$ is controlled by the ratio of in-plane stiffness and surface tension. Having linear physical relationship in the case of the small deformation, simpler scaling laws are proposed for the pair $(N,\ell)$. The universality of the scaling laws, which are verified by the dimensional analysis, will give us more confidence. As a natural extension, we gave the pair $(N,\ell)$ a thin film case made of axisymmetric anisotropic materials. By using Tanner's scaling laws, we obtained dynamical scaling laws for a drop radius and the pair $(N,\ell)$, which shows that the pair $(N,\ell)$ will fade away with time. Finally, we obtained the pair $(N,\ell)$ within the gravity regime.
Peer Review Status:Awaiting Review
Subjects: Physics >> General Physics: Statistical and Quantum Mechanics, Quantum Information, etc. submitted time 2017-09-13
Abstract： Capillary rise is one of the most well-known and vivid illustrations of capillarity, however, there is no complete solution has been obtained except some segmental solutions for asymptotic regimes. In this paper, we use singularity-free Bush's equation and successfully obtain its series and perturbation solutions. The solutions reveal that the capillary rise dynamics is mainly controlled by the Bond number and the Galileo number, of which the Bond number is a key parameter on the existence of the solution. Due to the poor rate of convergence of the series solution, we propose an approximate analytic solution. The proposed analytic solution has been verified numerically.
Peer Review Status:Awaiting Review
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