Rivulet Instability with Effect of Coriolis Force

2006 ◽  
Vol 22 (3) ◽  
pp. 221-227 ◽  
Author(s):  
H.-C. Cho ◽  
F.-C. Chou
Keyword(s):  
Coriolis Force ◽  
Critical Radius ◽  
Deflection Angle ◽  
Characteristic Length ◽  
Bond Number ◽  
Viscous Force ◽  
Fingering Instability ◽  
High Bond ◽  
The Deflection Angle ◽  
Visualization Experiments

AbstractThe effect of Coriolis force on the rivulet (fingering) instability, the onset of rivulet phenomena during spin coating, is investigated by flow visualization experiments incorporating with dimensional analysis. This study demonstrates that the Coriolis force will affect significantly the critical radius of rivulet instability and the deflection angle of instability rivulet. For the cases of low Bond number, the effect of Coriolis force is a stabilizing factor, and the dimensionless critical radius increases slightly with increasing rotational Reynolds number Reω. In the case of high Bond number, the effect of Coriolis force becomes a destabilizing factor while Reω < 1, and a characteristic length is found by balancing the viscous force with the surface tension. For Reω > 1, the radial Corilois force, which is always pointing inward, plays a stabilizing role with magnitude Reω2.

scholarly journals Quantitative evaluation indexes of the spatial steering effect of directional perforation hydraulic fractures

2021 ◽  
pp. 014459872110102
Author(s):  
Lu Weiyong ◽  
He Changchun
Keyword(s):  
Quantitative Evaluation ◽  
Hydraulic Fracture ◽  
Deflection Angle ◽  
Hydraulic Fractures ◽  
Principal Stress Direction ◽  
Perforation Hole ◽  
Evaluation Indexes ◽  
Initiation Pressure ◽  
The Deflection Angle ◽  
Bending Fracture

To better evaluate the spatial steering effect of directional perforation hydraulic fractures, evaluation indexes for the spatial steering effect are first proposed in this paper. Then, these indexes are used to quantitatively evaluate existing physical experimental results. Finally, with the help of RFPA2D-Flow software, the influence of perforation length and azimuth on the spatial steering process of hydraulic fracture are quantitatively analysed using four evaluation indexes. It is shown by the results that the spatial deflection trajectory, deflection distance, deflection angle and initiation pressure of hydraulic fractures can be used as quantitative evaluation indexes for the spatial steering effect of hydraulic fractures. The deflection paths of directional perforation hydraulic fractures are basically the same. They all gradually deflect to the maximum horizontal principal stress direction from the perforation hole and finally represent a double-wing bending fracture. The deflection distance, deflection angle and initiation pressure of hydraulic fractures increase gradually with increasing perforation azimuth, and the sensitivity of the deflection angle to the perforation azimuth of hydraulic fractures also increases. With increasing perforation length, the deflection distance of hydraulic fractures increases gradually. However, the deflection angle and initiation pressure decrease gradually, as does the sensitivity.

scholarly journals OPERATOR-PRODUCT EXPANSION AND ANALYTICITY

1999 ◽  
Vol 14 (30) ◽  
pp. 4819-4840
Author(s):  
JAN FISCHER ◽  
IVO VRKOČ
Keyword(s):  
Operator Product Expansion ◽  
Deflection Angle ◽  
Vacuum Expectation ◽  
Vacuum Expectation Value ◽  
Operator Product ◽  
Coupling Constant ◽  
Expectation Value ◽  
Euclidean Region ◽  
The Deflection Angle ◽  
Class Of Functions

We discuss the current use of the operator-product expansion in QCD calculations. Treating the OPE as an expansion in inverse powers of an energy-squared variable (with possible exponential terms added), approximating the vacuum expectation value of the operator product by several terms and assuming a bound on the remainder along the Euclidean region, we observe how the bound varies with increasing deflection from the Euclidean ray down to the cut (Minkowski region). We argue that the assumption that the remainder is constant for all angles in the cut complex plane down to the Minkowski region is not justified. Making specific assumptions on the properties of the expanded function, we obtain bounds on the remainder in explicit form and show that they are very sensitive both to the deflection angle and to the class of functions considered. The results obtained are discussed in connection with calculations of the coupling constant αs from the τ decay.

The precise calculations of the constant terms in the equations of motions of planets and photons of general relativity

2021 ◽  
Vol 34 (2) ◽  
pp. 183-192
Author(s):  
Mei Xiaochun
Keyword(s):  
General Relativity ◽  
Gravitational Field ◽  
Deflection Angle ◽  
Constant Term ◽  
Equation Of Motion ◽  
Time Dependent ◽  
Newtonian Gravity ◽  
Newtonian Theory ◽  
The Deflection Angle ◽  
Equations Of Motions

In general relativity, the values of constant terms in the equations of motions of planets and light have not been seriously discussed. Based on the Schwarzschild metric and the geodesic equations of the Riemann geometry, it is proved in this paper that the constant term in the time-dependent equation of motion of planet in general relativity must be equal to zero. Otherwise, when the correction term of general relativity is ignored, the resulting Newtonian gravity formula would change its basic form. Due to the absence of this constant term, the equation of motion cannot describe the elliptical and the hyperbolic orbital motions of celestial bodies in the solar gravitational field. It can only describe the parabolic orbital motion (with minor corrections). Therefore, it becomes meaningless to use general relativity calculating the precession of Mercury's perihelion. It is also proved that the time-dependent orbital equation of light in general relativity is contradictory to the time-independent equation of light. Using the time-independent orbital equation to do calculation, the deflection angle of light in the solar gravitational field is <mml:math display="inline"> <mml:mrow> <mml:mn>1.7</mml:mn> <mml:msup> <mml:mn>5</mml:mn> <mml:mo>″</mml:mo> </mml:msup> </mml:mrow> </mml:math> . But using the time-dependent equation to do calculation, the deflection angle of light is only a small correction of the prediction value <mml:math display="inline"> <mml:mrow> <mml:mn>0.87</mml:mn> <mml:msup> <mml:mn>5</mml:mn> <mml:mo>″</mml:mo> </mml:msup> </mml:mrow> </mml:math> of the Newtonian gravity theory with a magnitude order of <mml:math display="inline"> <mml:mrow> <mml:msup> <mml:mrow> <mml:mn>10</mml:mn> </mml:mrow> <mml:mrow> <mml:mo>−</mml:mo> <mml:mn>5</mml:mn> </mml:mrow> </mml:msup> </mml:mrow> </mml:math> . The reason causing this inconsistency was the Einstein's assumption that the motion of light satisfied the condition <mml:math display="inline"> <mml:mrow> <mml:mi>d</mml:mi> <mml:mi>s</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> in gravitational field. It leads to the absence of constant term in the time-independent equation of motion of light and destroys the uniqueness of geodesic in curved space-time. Meanwhile, light is subjected to repulsive forces in the gravitational field, rather than attractive forces. The direction of deflection of light is opposite, inconsistent with the predictions of present general relativity and the Newtonian theory of gravity. Observing on the earth surface, the wavelength of light emitted by the sun is violet shifted. This prediction is obviously not true. Practical observation is red shift. Finally, the practical significance of the calculation of the Mercury perihelion's precession and the existing problems of the light's deflection experiments of general relativity are briefly discussed. The conclusion of this paper is that general relativity cannot have consistence with the Newtonian theory of gravity for the descriptions of motions of planets and light in the solar system. The theory itself is not self-consistent too.

Gauging the earth

2013 ◽  
Vol 97 (540) ◽  
pp. 413-420
Author(s):  
John D. Mahony
Keyword(s):  
Approximate Formula ◽  
Deflection Angle ◽  
School Science ◽  
Science Department ◽  
Measurement Apparatus ◽  
The Earth ◽  
Interesting Discussion ◽  
Simple Measurement ◽  
The Deflection Angle

In an earlier note by W. J. A. Colman [1], the reader was treated to an interesting discussion concerning a BBC programme about the earth's radius in the context of endeavours to determine it by an 11th century Persian mathematician, Al-Biruni. At the same time a modem but approximate formula that might not have been available to the ancients was proposed for the radius and, following that account, it is the purpose here to explore just how accurately one might determine the earth's radius using this formula together with a simple measurement apparatus (not a sophisticated astrolabe) that might be constructed from materials found in the garage of any DIY handyman, or indeed, in the laboratory of any school science department. A schematic for the configuration of an observer P at a height h above the earth (of radius R) where the deflection angle to the horizon is denoted by θ, is shown in Figure 1. Also shown in the figure is a ‘sighting tube’ of length L about which more will be said later.

An Investigation on the Profile of Microlens by the Thermal Reflow Process Due to Surface Tension and Gravity

2008 ◽  
Author(s):  
Jhy-Cherng Tsai ◽  
Yong-Sung Hsu
Keyword(s):  
Surface Tension ◽  
Critical Radius ◽  
Critical Size ◽  
Liquid Droplet ◽  
Bond Number ◽  
Diameter Ratio ◽  
Experimental Result ◽  
Critical Number ◽  
Reflow Process ◽  
Thermal Reflow

Microlens and its mold fabricated by thermal reflow using photoresist have been widely used for forming patterns in different scales. When the photoresist solidifies from melting condition, for example by the reflow process, its profile is formed based on the balance between surface tension and gravity. This research is aimed to investigate the influence of surface tension and gravity on the profile of microlens in thermal reflow process. Theoretical analysis based on the interaction between surface tension and gravity of liquid droplet is first investigated. The result showed that the height to diameter ratio (h/D), or the sag ratio, of the liquid droplet is affected by the Bond number (Bo), a number defined as the ratio of gravity to surface tension. The sag ratio is not sensitive to Bo when Bo is small but the ratio decreases as Bo increases if Bo is over the critical number. Based on the analysis, the critical number for the AZ4620 photoresist on a silicon substrate is 1, corresponding to the critical radius of droplet R = 2,500μm. When the size of the droplet is less then the critical size, the profile is mainly controlled by the surface tension and thus the sag ratio is about the same regardless the size. The profile, in contrast, is highly affected by the gravity if the size of the droplet is larger then the critical size. The sag ratio decreases exponentially with respect to Bo in this case. Experiments are also designed and conducted to verify the analysis. Experimental result showed that the sag ratio of the photoresist reduces to 0.065 from 0.095 when Bo increases from 0.0048 to 0.192. The results showed that the trend is consistent to the theoretical model.

scholarly journals SPECIAL VEHICLE MOTION STABILIZATION SYSTEM

2021 ◽  
pp. 37-41
Author(s):  
M. V. Lyashenko ◽  
V. V. Shekhovtsov ◽  
P. V. Potapov ◽  
A. A. Shvedunenko
Keyword(s):  
Pid Controller ◽  
Deflection Angle ◽  
Angular Displacement ◽  
Dynamic Equations ◽  
Engine Torque ◽  
Special Vehicle ◽  
Vehicle Motion ◽  
Motion Stabilization ◽  
The Deflection Angle ◽  
The Mathematical Model

The system of special vehicle (SV) motion stabilization during moving on a straight surface is modeled on the base of dynamic equations of the mathematical model. The movement is stabilized by using a PID controller, the angular displacement of the mass is selected to ensure a given speed of movement, and the deflection angle is stabilized by controlling the engine torque.

scholarly journals Values estimation of deflection angle of a floating sensor for various objects (to the resistance to measuring the water surface rate)

2019 ◽  
Vol 5 (2) ◽  
pp. 181-196
Author(s):  
K. Presnyakov ◽  
G. Kerimkulova ◽  
G. Askalieva
Keyword(s):  
Water Surface ◽  
Deflection Angle ◽  
Surface Velocity ◽  
Informative Parameter ◽  
The Deflection Angle

A device is proposed for measuring the surface velocity of water, the use of which is based on the use of a new informative parameter — the deflection angle of the movable float sensor (together with the guide slide) from the vertical in the direction of the dynamic axis of the flow, allowing you to simply and reliably measure the surface velocity of water.

scholarly journals Weak Deflection Angle of some Classes of non-linear Electrodynamics Black Holes via Gauss-Bonnet Theorem

2020 ◽  
Author(s):  
Hasan El Moumni ◽  
Karima Masmar ◽  
Ali Övgün
Keyword(s):  
Black Holes ◽  
Black Hole ◽  
Gravitational Lensing ◽  
Deflection Angle ◽  
Weak Field ◽  
Leading Order ◽  
Plasma Medium ◽  
Non Linear ◽  
The Deflection Angle ◽  
Bonnet Theorem

In this paper, we study the gravitational lensing by some black hole classes within the non-linear electrodynamics in weak field limits. First, we calculate an optical geometry of the non-linear electrodynamics black hole then we use the Gauss-Bonnet theorem for finding deflection angle in weak field limits. The effect of non-linear electrodynamics on the deflection angle in leading order terms is studied. Furthermore, we discuss the effects of the plasma medium on the weak deflection angle.

scholarly journals Perturbative deflection angle, gravitational lensing in the strong field limit and the black hole shadow

2021 ◽  
Vol 81 (3) ◽  
Author(s):  
Junji Jia ◽  
Ke Huang
Keyword(s):  
Gravitational Lensing ◽  
Deflection Angle ◽  
Strong Field ◽  
Distance Effect ◽  
Critical Value ◽  
Field Limit ◽  
Perturbative Method ◽  
The Deflection Angle ◽  
The Impact ◽  
Strong Field Limit

AbstractA perturbative method to compute the deflection angle of both timelike and null rays in arbitrary static and spherically symmetric spacetimes in the strong field limit is proposed. The result takes a quasi-series form of $$(1-b_c/b)$$ ( 1 - b c / b ) where b is the impact parameter and $$b_c$$ b c is its critical value, with coefficients of the series explicitly given. This result also naturally takes into account the finite distance effect of both the source and detector, and allows to solve the apparent angles of the relativistic images in a more precise way. From this, the BH angular shadow size is expressed as a simple formula containing metric functions and particle/photon sphere radius. The magnification of the relativistic images were shown to diverge at different values of the source-detector angular coordinate difference, depending on the relation between the source and detector distance from the lens. To verify all these results, we then applied them to the Hayward BH spacetime, concentrating on the effects of its charge parameter l and the asymptotic velocity v of the signal. The BH shadow size were found to decrease slightly as l increases to its critical value, and increase as v decreases from light speed. For the deflection angle and the magnification of the images however, both the increase of l and decrease of v will increase their values.

scholarly journals FLUCTUATIONS OF THE CARGO TRANSPORTED BY LIFTING CRANE: SIMULATION AND ANALYSIS

2019 ◽  
Vol 16 (5) ◽  
pp. 526-533
Author(s):  
M. S. Korytov ◽  
V. S. Shcherbakov ◽  
V. E. Belyakov
Keyword(s):  
Objective Function ◽  
Arbitrary Order ◽  
Deflection Angle ◽  
Suspension Point ◽  
Program Code ◽  
First Derivative ◽  
Urgent Task ◽  
Acceleration And Deceleration ◽  
The Deflection Angle ◽  
Simulation Results

Introduction. Reducing fluctuations in the load transported by hoisting cranes with a flexible rope suspension of the load is an urgent task since it can significantly reduce the time taken to complete the operation of moving the load. A promising direction for reducing load fluctuations is to optimize the trajectory of movement of the load suspension upper point.Materials and methods. The paper discussed the method of mathematical simulation of plane vibrations of a load moved by a crane with a horizontally moving suspension point, using the software of the MATLAB system. For modeling, the authors used the function of the MATLAB ode45 system, intended for the numerical solution of systems of non-stationary differential equations of arbitrary order.The second-order differential equation used to describe the fluctuations of the transported load and its implementation in the form of program code was presented. Moreover, the authors demonstrated the elements of program code for the analysis and visualization of simulation results.Results. The authors obtained and presented the series of graphs in the inclination angle’s changing of the cargo rope, the acceleration of the suspension point and the value of the objective function with the sinusoidal nature of the acceleration. The objective function was the sum of the absolute values of the deflection angle of the rope and the first derivative at the final moment of the suspension point’s movement with acceleration.Discussion and conclusions. As a result, the paper shows that the system with energy dissipation does not reach the zero value of the objective function even by a symmetrical nature of acceleration and deceleration of the suspension point. Therefore, it is necessary to give asymmetry to the acceleration and deceleration periods of the suspension point in order to completely absorb the residual fluctuations of the load.

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