scholarly journals Some Workout Problems on Motion of Raindrop

2021 ◽  
Vol 9 (2) ◽  
Author(s):  
Sneha Dey ◽  
Amitava Ghorai
Keyword(s):  
Mathematical Modeling ◽  
Close Agreement ◽  
Classical Problem ◽  
Terminal Velocity ◽  
Air Resistance ◽  
Rain Drop

The motion of rain drop through atmosphere is an interesting classical problem because of the fact that air resistance and moisture accretion are integral part of it. Mathematical modeling of it using Newtonian formalism is considered here and discussions are made for no mass accretion and air resistance proportional to nth power of velocity. We use python program and library extensively to find the terminal velocity of rain drop. Graphs show close agreement and velocity power up to n=3 is good.

Particle motion with air resistance

2012 ◽  
Vol 8 (1) ◽  
pp. 1-15
Author(s):  
Gy. Sitkei
Keyword(s):  
Basic Equation ◽  
Initial Conditions ◽  
Equation Of Motion ◽  
Terminal Velocity ◽  
Dimensionless Form ◽  
Wood Industry ◽  
Acceptable Error ◽  
General Validity ◽  
Practical Applications ◽  
Air Resistance

Motion of particles with air resistance (e.g. horizontal and inclined throwing) plays an important role in many technological processes in agriculture, wood industry and several other fields. Although, the basic equation of motion of this problem is well known, however, the solutions for practical applications are not sufficient. In this article working diagrams were developed for quick estimation of the throwing distance and the terminal velocity. Approximate solution procedures are presented in closed form with acceptable error. The working diagrams provide with arbitrary initial conditions in dimensionless form of general validity.

Research on the Flight Status of Badminton Based on the Method of Mechanics Analysis

2012 ◽  
Vol 507 ◽  
pp. 246-251
Author(s):  
Xiao Liang Xie ◽  
Kun Jiang
Keyword(s):  
Research Method ◽  
Terminal Velocity ◽  
Measured Data ◽  
Motion Equation ◽  
Movement Trajectory ◽  
Flight Trajectory ◽  
Motion Equations ◽  
Air Resistance ◽  
Mechanics Analysis ◽  
The Relationship

The main purpose of this research is to establish and validate the flight movement equation of a badminton, and found the relationship between the velocity of badminton and the air resistance. The research method is based on the motion laws of aerodynamics,which establish a motion equation of flight trajectory under the influence of the gravity and air resistance by applying the theory of aerodynamics. The results show that the motion equations of flight trajectory of a badminton can be determined through the structure of terminal velocity,and the measured data is quite good fitting with the prediction of badminton movement trajectory. Findings also show, the resistance is proportional to the square of the badminton's speed. In addition, the angle and intensity of the wind may affect the trajectory.

A Combined Wind Profiler and Polarimetric Weather Radar Method for the Investigation of Precipitation and Vertical Velocities

2009 ◽  
Vol 26 (9) ◽  
pp. 1940-1955 ◽  
Author(s):  
Michihiro S. Teshiba ◽  
Phillip B. Chilson ◽  
Alexander V. Ryzhkov ◽  
Terry J. Schuur ◽  
Robert D. Palmer
Keyword(s):  
Weather Radar ◽  
Terminal Velocity ◽  
Doppler Velocity ◽  
Drop Diameter ◽  
Wind Profiler ◽  
Rain Drop ◽  
Air Motion ◽  
Polarimetric Weather Radar ◽  
Uhf Wind Profiler ◽  
Differential Reflectivity

Abstract A method is presented by which combined S-band polarimetric weather radar and UHF wind profiler observations of precipitation can be used to extract the properties of liquid phase hydrometeors and the vertical velocity of the air through which they are falling. Doppler spectra, which contain the air motion and/or fall speed of hydrometeors, are estimated using the vertically pointing wind profiler. Complementary to these observations, spectra of rain drop size distribution (DSD) are simulated by several parameters as related to the DSD, which are estimated through the two polarimetric parameters of radar reflectivity (ZH) and differential reflectivity (ZDR) from the scanning weather radar. These DSDs are then mapped into equivalent Doppler spectra (fall speeds) using an assumed relationship between the equivolume drop diameter and the drop’s terminal velocity. The method is applied to a set of observations collected on 11 March 2007 in central Oklahoma. In areas of stratiform precipitation, where the vertical wind motion is expected to be small, it was found that the fall speeds obtained from the spectra of the rain DSD agree well with those of the Doppler velocity estimated with the profiler. For those cases when the shapes of the Doppler spectra are found to be similar in shape but shifted in velocity, the velocity offset is attributed to vertical air motion. In convective rainfall, the Doppler spectra of the rain DSD and the Doppler velocity can exhibit significant differences owing to vertical air motions together with atmospheric turbulence. Overall, it was found that the height dependencies of Doppler spectra measured by the profiler combined with vertical profiles of Z, ZDR, and the cross correlation (ρHV) as well as the estimated spectra of raindrop physical terminal fall speeds from the polarimetric radar provide unique insight into the microphysics of precipitation. Vertical air motions (updrafts/downdrafts) can be estimated using such combined measurements.

Proceed at your own pace, because of terminal velocity

2020 ◽  
pp. 109-112
Author(s):  
Susan D'Agostino
Keyword(s):  
Terminal Velocity ◽  
Mathematics Students ◽  
Short Introduction ◽  
Air Resistance ◽  
The Moment

“Proceed at your own pace, because of terminal velocity” offers a short introduction to the mathematics behind terminal velocity—the speed at which the force of air resistance equals the force of gravity on that object. The discussion is illustrated with hand-drawn sketches. Mathematics students and enthusiasts are encouraged to find their own metaphorical terminal velocity in mathematical and life pursuits—the moment at which the force of resistance they encounter matches the force they are able to expend. At the chapter’s end, readers may check their understanding by working on a problem. A solution is provided.

scholarly journals MATHEMATICAL MODELING OF STRENGTH-DEFORMED CONDITION OF PIPE CONCRETE CONSTRUCTIONS AT SUSTAINABLE TEMPERATURE

Fire Safety ◽  
2020 ◽  
Vol 35 ◽  
pp. 63-68
Author(s):  
M. Semerak ◽  
N. Ferents ◽  
D. Kharyshyn ◽  
S. Vovk
Keyword(s):  
Mathematical Modeling ◽  
Mutual Influence ◽  
Classical Problem ◽  
Structural Type ◽  
Force Response ◽  
Thermophysical Characteristics ◽  
Tensile Stresses ◽  
Maximum Value ◽  
Layer Structures ◽  
Steel Sleeve

The mathematical modeling of the thermo-stressed state of pipe-concrete structures under conditions of stationary thermal conductivity is carried out. During the study, the most common structural type of TBC was selected and mathematical models of the stress-strain state of TBA under heating conditions were taken into account, taking into account their geometric dimensions and thermophysical characteristics of metal and concrete. The main feature of the TBC study is that the heat exchange in the structure, as well as the corresponding force response, are investigated independently of each other, whereas the problem of fire resistance should be posed as a classical problem of elasticity, taking into account the mutual influence of temperature and mechanical stresses. In two-layer structures (concrete - metal), the largest radial stresses that occur in concrete work in tension. Tensile stresses occur in the thickness of the outer tube. The stresses occurring on the inner surface of the steel sleeve coincide with the stresses in the concrete. When approaching the outer surface, they decrease and on the surface r = R2 equal to zero. In the case where the coefficients of linear extension αt (i = 1, 2) are equal to each other, the maximum tensile stresses are reduced within 0≤r≤R1 .; if νi (i = 1, 2) within 0≤ r≤R2, the stresses will also decrease. Axial stresses work on compression. They reach maximum value in the outer shell. For equal values αt (1) = αt (2), the magnitude of the stresses does not change, and at ν1 = ν2 the stresses in the metal will decrease. Ring stresses in the region 0≤ r≤R1 are tensile stresses and in the region R1≤ r≤R2 are compression stresses, and the compression stresses are greater than the tensile stresses in concrete. For αt (1) = αt (2), the stresses in the concrete decrease and for ν1 = ν2.

scholarly journals Improving the rainfall rate estimation in the midstream of the Heihe River Basin using rain drop size distribution

2009 ◽  
Vol 6 (5) ◽  
pp. 6107-6134 ◽  
Author(s):  
G. Zhao ◽  
R. Chu ◽  
X. Li ◽  
T. Zhang ◽  
J. Shen ◽  
...  
Keyword(s):  
Size Distribution ◽  
Rain Rate ◽  
Terminal Velocity ◽  
Tibet Plateau ◽  
Heihe River ◽  
Drop Diameter ◽  
Observation Data ◽  
Raindrop Size Distribution ◽  
Rain Drop ◽  
Raindrop Size

Abstract. During the intensive observation period of the Watershed Allied Telemetry Experimental Research (WATER), a total of 1074 raindrop size distribution were measured by the Parsivel disdrometer, a latest state of the art optical laser instrument. Because of the limited observation data in Qinghai-Tibet Plateau, the modeling behavior was not well-done. We used raindrop size distributions to improve the rain rate estimator of meteorological radar, in order to obtain many accurate rain rate data in this area. We got the relationship between the terminal velocity of the rain drop and the diameter (mm) of a rain drop: v(D)=4.67 D0.53. Then four types of estimators for X-band polarimetric radar are examined. The simulation results show that the classical estimator R(Z) is most sensitive to variations in DSD and the estimator R (KDP, Z, ZDR) is the best estimator for estimating the rain rate. The lowest sensitivity of the rain rate estimator R (KDP, Z, ZDP) to variations in DSD can be explained by the following facts. The difference in the forward-scattering amplitudes at horizontal and vertical polarizations, which contributes KDP, is proportional to the 3rd power of the drop diameter. On the other hand, the exponent of the backscatter cross section, which contributes to Z, is proportional to the 6th power of the drop diameter. Because the rain rate R is proportional to the 3.57th power of the drop diameter, KDP is less sensitive to DSD variations than Z.

Development and validation of mathematical modeling for terminal velocity of cantaloupe

2019 ◽  
Vol 42 (3) ◽  
pp. e13000 ◽  
Author(s):  
Mehdi Moradi ◽  
Amin Mousavi Khaneghah ◽  
Majid Parvaresh ◽  
Hossein Balanian
Keyword(s):  
Mathematical Modeling ◽  
Terminal Velocity ◽  
Development And Validation

Near-surface turbulence and buoyancy induced by heavy rainfall

2017 ◽  
Vol 830 ◽  
pp. 602-630 ◽  
Author(s):  
E. L. Harrison ◽  
F. Veron
Keyword(s):  
Energy Dissipation ◽  
Kinetic Energy ◽  
Turbulent Kinetic Energy ◽  
Energy Flux ◽  
Terminal Velocity ◽  
Near Surface ◽  
Wave Channel ◽  
Surface Turbulence ◽  
Rain Drop ◽  
The Impact

We present results from experiments designed to measure near-surface turbulence generated by rainfall. Laboratory experiments were performed using artificial rain falling at near-terminal velocity in a wind–wave channel filled with synthetic seawater. In this first series of experiments, no wind was generated and the receiving seawater was initially at rest. Rainfall rates from 40 to $190~\text{mm}~\text{h}^{-1}$ were investigated. Subsurface turbulent velocities of the order of $O(10^{-2})~\text{m}~\text{s}^{-1}$ are generated near the interface below the depth of the cavities generated by the rain drop impacts. The turbulence appears independent of rainfall rates. At depth larger than the size of the cavities, the turbulent velocity fluctuations decay as $z^{-3/2}$. Turbulent length scales also appear to scale with the size of the impact cavities. In these seawater experiments, a freshwater lens is established at the water surface due to the rain. At the highest rain rate studied, the resulting buoyancy flux appears to lead to a shallower subsurface mixed layer and a slight decrease of the turbulent kinetic energy dissipation. Finally, direct measurements and inertial estimates of the turbulent kinetic energy dissipation show that approximately 0.1–0.3 % of the kinetic energy flux from the rain is dissipated in the form of turbulence. This is consistent with existing freshwater measurements and suggests that high levels of dissipation occur at depths and scales smaller than those resolved here and/or that other phenomena dissipate a considerable amount of the total kinetic energy flux provided by rainfall.

The quantitative comparison of experimental and simulated diffraction contrast images

1995 ◽  
Vol 53 ◽  
pp. 102-103
Author(s):  
Stuart McKernan
Keyword(s):  
Image Processing ◽  
Personal Computer ◽  
Close Agreement ◽  
Quantitative Comparison ◽  
Image Simulation ◽  
Computing Power ◽  
Diffraction Contrast ◽  
Simulated Images ◽  
Made In

For many years the concept of quantitative diffraction contrast experiments might have consisted of the determination of dislocation Burgers vectors using a g.b = 0 criterion from several different 2-beam images. Since the advent of the personal computer revolution, the available computing power for performing image-processing and image-simulation calculations is enormous and ubiquitous. Several programs now exist to perform simulations of diffraction contrast images using various approximations. The most common approximations are the use of only 2-beams or a single systematic row to calculate the image contrast, or calculating the image using a column approximation. The increasing amount of literature showing comparisons of experimental and simulated images shows that it is possible to obtain very close agreement between the two images; although the choice of parameters used, and the assumptions made, in performing the calculation must be properly dealt with. The simulation of the images of defects in materials has, in many cases, therefore become a tractable problem.

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