scholarly journals A Comprehensive Study of Mass Accretion and Atmospheric Effect of Raindrop

Author(s):  
Sneha Dey ◽  
◽  
Dr. A. Ghorai ◽  

The mass accretion of a raindrop in different layers of the atmosphere is not dealt with so far. A comprehensive brief study of the motion of raindrops through the atmosphere (i) without mass accretion, (ii) with mass accretion and (iii) finally pressure variation in the atmosphere with altitude using Bernoulli’s equation is illustrated. Acquirement of mass from moist air is mass accretion and mass accretion during the motion of raindrop through resistive medium holds an arbitrary power-law equation. Bernoulli’s equation when applied to it, the generalized first-order differential equation is reduced to a polynomial equation. Results show a single intersecting point of approximate terminal velocity 1 m/s and mass 10-06 mg as illustrated. Terminal velocity is achieved within 25 sec. There is the approximate exponential growth of terminal velocity. An increase in momentum is due to mass accretion during motion. Various conditions of no mass accretion and mass accretion show the same result while for atmospheric effect using Bernoulli’s equation the first-order differential equation reduces to a polynomial equation.

Author(s):  
Sneha Dey ◽  
◽  
A. Ghorai ◽  

Exploration of dynamics of raindrops is one of the simple yet most complicated mechanical problems. Mass accretion from moist air during the motion of raindrop through resistive medium holds an arbitrary power law equation. Its integral part is the change of shape, terminal motions and terminal solutions, etc. Classical Newtonian formalism is used to formulate a mathematical model of generalized first order differential equation. We have discussed about the terminal velocity of raindrop and its variation with the extensive use of python program and library. It is found that terminal velocity 𝐯𝐓𝐜𝛂𝛃 is achieved within 20 seconds where 𝛂=, 𝛃=(𝟎,𝟏) and 𝐧=𝟎,𝟏,𝟐,𝟑,𝟒,…. Its variations due to mass accretion roughly follows the earlier predicted range 𝐠/𝟕 to 𝐠/𝟑.


2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Yun Xin ◽  
Xiaoxiao Cui ◽  
Jie Liu

Abstract The main purpose of this paper is to obtain an exact expression of the positive periodic solution for a first-order differential equation with attractive and repulsive singularities. Moreover, we prove the existence of at least one positive periodic solution for this equation with an indefinite singularity by applications of topological degree theorem, and give the upper and lower bounds of the positive periodic solution.


2021 ◽  
pp. 1-19
Author(s):  
Calogero Vetro ◽  
Dariusz Wardowski

We discuss a third-order differential equation, involving a general form of nonlinearity. We obtain results describing how suitable coefficient functions determine the asymptotic and (non-)oscillatory behavior of solutions. We use comparison technique with first-order differential equations together with the Kusano–Naito’s and Philos’ approaches.


2000 ◽  
Vol 15 (28) ◽  
pp. 4477-4498 ◽  
Author(s):  
P. M. LLATAS ◽  
A. V. RAMALLO ◽  
J. M. SÁNCHEZ DE SANTOS

We analyze the world volume solitons of a D3-brane probe in the background of parallel (p, q) five-branes. The D3-brane is embedded along the directions transverse to the five-branes of the background. By using the S duality invariance of the D3-brane, we find a first-order differential equation whose solutions saturate an energy bound. The SO(3) invariant solutions of this equation are found analytically. They represent world volume solitons which can be interpreted as formed by parallel (-q, p) strings emanating from the D3-brane world volume. It is shown that these configurations are 1/4 supersymmetric and provide a world volume realization of the Hanany–Witten effect.


1963 ◽  
Vol 3 (2) ◽  
pp. 202-206 ◽  
Author(s):  
J. C. Butcher

Huta [1], [2] has given two processes for solving a first order differential equation to sixth order accuracy. His methods are each eight stage Runge-Kutta processes and differ mainly in that the later process has simpler coefficients occurring in it.


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