scholarly journals The Accelerated Motion of Droplets and Bubbles

1956 ◽  
Vol 9 (1) ◽  
pp. 19 ◽  
Author(s):  
T Pearcey ◽  
GW Hill
Keyword(s):  
Differential Equation ◽  
Drag Force ◽  
Equation Of Motion ◽  
Viscous Medium ◽  
Rate Of Change ◽  
Uniform Motion ◽  
Small Particles ◽  
Accelerated Motion ◽  
Scale Transformations ◽  
Accelerated Particles

The drag force acting upon small particles such as droplets and bubbles, moving through a viscous medium, depends upon the rate of change of the state of motion of the medium and upon the diffusion of vorticity from the surface' of the particle. For accelerated particles the drag force changes with time and depends upon all previous accelerations. The equation of motion of a particle takes the form of an integro-differential equation, which has been solved numerically in the case of deceleration to rest from uniform motion with no impressed body forces. In any experiments designed according to the principles of dynamical similarity, the ratios of viscosities and densities of the medium and the particle must be maintained constant in the scale transformations involved.

Dynamic Response of Risers Conveying Fluid

2000 ◽  
Vol 122 (3) ◽  
pp. 188-193 ◽  
Author(s):  
Patrick Bar-Avi
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Drag Force ◽  
Fluid Velocity ◽  
Equation Of Motion ◽  
Finite Difference Approximation ◽  
Difference Approximation ◽  
Partial Differential ◽  
Highly Nonlinear ◽  
Conveying Fluid

Riser conveying fluid is one category in a wide research area known as moving continua dynamic that was the subject to many research studies in the past 50 yr. The structure investigated in this paper is assumed to be a flexible pipe having a geometry of a simple catenary with a buoyancy force at the top. The riser is modeled as a beamlike continuous system subjected to wave, current, and wind forces. The derivation of the partial differential equation of motion included the following aspects: nonlinearities due to geometry, i.e., large displacements as well as wave drag force, fluid internal and external pressure, and internal fluid velocity and acceleration. The wave forces are approximated via Morison’s equation, where the drag force is proportional to the square of the relative velocity between the riser and the waves. All forces are calculated at the instantaneous position of the riser, causing the equation to be highly nonlinear. The nonlinear partial differential equation of motion is then solved numerically. Finite difference approximation code that was imbedded in ACSL software is used. The equation of motion is solved to evaluate the riser’s response to different environmental conditions and other physical parameters such as internal pressure and fluid velocity and acceleration. [S0892-7219(00)00203-X]

The Numerical Methods of Peridynamics Theory Used in Failure Analysis of Materials

2014 ◽  
Vol 1030-1032 ◽  
pp. 223-227
Author(s):  
Lin Fan ◽  
Song Rong Qian ◽  
Teng Fei Ma
Keyword(s):  
Differential Equation ◽  
Integral Equation ◽  
Numerical Integration ◽  
Integration Method ◽  
Equation Of Motion ◽  
Numerical Integration Method ◽  
Volume Integral ◽  
Volume Integral Equation ◽  
Method Of Molecular Dynamics ◽  
Classic Theory

In order to analysis the force situation of the material which is discontinuity,we can used the new theory called peridynamics to slove it.Peridynamics theory is a new method of molecular dynamics that develops very quickly.Peridynamics theory used the volume integral equation to constructed the model,used the volume integral equation to calculated the PD force in the horizon.So It doesn’t need to assumed the material’s continuity which must assumed that use partial differential equation to formulates the equation of motion. Destruction and the expend of crack which have been included in the peridynamics’ equation of motion.Do not need other additional conditions.In this paper,we introduce the peridynamics theory modeling method and introduce the relations between peridynamics and classic theory of mechanics.We also introduce the numerical integration method of peridynamics.Finally implementation the numerical integration in prototype microelastic brittle material.Through these work to show the advantage of peridynamics to analysis the force situation of the material.

scholarly journals Modeling the dynamical sinking of biogenic particles in oceanic flow

2017 ◽  
Vol 24 (2) ◽  
pp. 293-305 ◽  
Author(s):  
Pedro Monroy ◽  
Emilio Hernández-García ◽  
Vincent Rossi ◽  
Cristóbal López
Keyword(s):  
Three Dimensional ◽  
Finite Size ◽  
Equation Of Motion ◽  
Particle Dynamics ◽  
Particle Sizes ◽  
Small Particles ◽  
Physical Processes ◽  
Mesoscale Simulation ◽  
Sinking Particles ◽  
Theoretical Results

Abstract. We study the problem of sinking particles in a realistic oceanic flow, with major energetic structures in the mesoscale, focussing on the range of particle sizes and densities appropriate for marine biogenic particles. Our aim is to evaluate the relevance of theoretical results of finite size particle dynamics in their applications in the oceanographic context. By using a simplified equation of motion of small particles in a mesoscale simulation of the oceanic velocity field, we estimate the influence of physical processes such as the Coriolis force and the inertia of the particles, and we conclude that they represent negligible corrections to the most important terms, which are passive motion with the velocity of the flow, and a constant added vertical velocity due to gravity. Even if within this approximation three-dimensional clustering of particles can not occur, two-dimensional cuts or projections of the evolving three-dimensional density can display inhomogeneities similar to the ones observed in sinking ocean particles.

scholarly journals Transient Analysis of Thermal Treatment Processes Carburizing and Nitriding of Steel Components by Numerical Modeling and Simulation

2019 ◽  
Vol 9 (2) ◽  
pp. 2584-2593
Keyword(s):  
Differential Equation ◽  
Numerical Modeling ◽  
Ambient Temperature ◽  
Modeling And Simulation ◽  
Geometric Dimension ◽  
Rate Of Change ◽  
Critical Parameters ◽  
Numerical Modeling And Simulation ◽  
Linear Differential ◽  
Energy Balance Approach

The purpose of carburizing, nitriding and carbonitriding is to increase the strength of components. Elements such as carbon, nitrogen and carbon-nitride are diffused into the components at high temperature convective environment. The amount of diffusion is to be regulated by controlling the temperature and time of diffusion. Time and temperature of process govern diffusion rate and strength of the component. Numerical modeling is applied by energy balance approach i.e., equating rate of change of energy is equal to energy transferred by conduction, convection and radiation. By non dimensionalising relations for the mentioned critical parameters were obtained. The phenomenon of convection, radiation and conduction are taken together for the purpose of numerical modeling. Variation of temperature and depth of diffusion of component for the taken components i.e., sphere and cube was plotted in transient state. For both numerical analysis and simulation the boundary conditions i.e., for carburization the ambient temperature is 9500C with carbon monoxide as the carburizing agent and for nitriding the ambient temperature is 5300C with nitrogen as nitriding agent and the component taken is of steel which is initially at room temperature were taken. Results obtained from numerical modeling and simulation were compared with each other and observed that in both analyses the variation of temperature with time and depth of diffusion is almost linear. Final differential equation obtained in numerical modeling is a single order non linear differential equation which is solved in MATLAB using finite difference approach. Data obtained from MATLAB were plotted for variation of surface temperature and geometric dimension with respect to time.

scholarly journals ON OSCILLATIONS DESCRIBING A GENERALIZED DIFFERENTIAL RELAY EQUATION

2020 ◽  
pp. 53-60
Author(s):  
Vasiliy Olshansky ◽  
Stanislav Olshansky ◽  
Oleksіі Tokarchuk
Keyword(s):  
Differential Equation ◽  
Initial Conditions ◽  
Oscillatory System ◽  
Equation Of Motion ◽  
Resistance Force ◽  
Oscillatory Process ◽  
Initial Deviation ◽  
Approximate Analytical Solutions ◽  
Positive Exponent ◽  
Generalized Differential

The motion of an oscillatory system with one degree of freedom, described by the generalized Rayleigh differential equation, is considered. The generalization is achieved by replacing the cubic term, which expresses the dissipative strength of the equation of motion, by a power term with an arbitrary positive exponent. To study the oscillatory process involved the method of energy balance. Using it, an approximate differential equation of the envelope of the graph of the oscillatory process is compiled and its analytical solution is constructed from which it follows that quasilinear frictional self-oscillations are possible only when the exponent is greater than unity. The value of the amplitude of the self-oscillations in the steady state also depends on the value of the indicator. A compact formula for calculating this amplitude is derived. In the general case, the calculation involves the use of a gamma function table. In the case when the exponent is three, the amplitude turned out to be the same as in the asymptotic solution of the Rayleigh equation that Stoker constructed. The amplitude is independent of the initial conditions. Self-oscillations are impossible if the exponent is less than or equal to unity, since depending on the initial deviation of the system, oscillations either sway (instability of the movement is manifested) or the range decreases to zero with a limited number of cycles, which is usually observed with free oscillations of the oscillator with dry friction. These properties of the oscillatory system are also confirmed by numerical computer integration of the differential equation of motion for specific initial data. In the Maple environment, the oscillator trajectories are constructed for various values of the nonlinearity index in the expression of the viscous resistance force and a corresponding comparative analysis is carried out, which confirms the adequacy of approximate analytical solutions.

A Semi-Discrete Approach for the Numerical Simulation of Freestanding Blocks

2016 ◽  
pp. 416-439
Author(s):  
Fernando Peña
Keyword(s):  
Differential Equation ◽  
Numerical Simulation ◽  
Numerical Modeling ◽  
Discrete Model ◽  
Mathematical Formulation ◽  
Equation Of Motion ◽  
Rigid Bodies ◽  
The Body ◽  
Discrete Approach ◽  
Rocking Motion

This chapter addresses the numerical modeling of freestanding rigid blocks by means of a semi-discrete approach. The pure rocking motion of single rigid bodies can be easily studied with the differential equation of motion, which can be solved by numerical integration or by linearization. However, when we deal with sliding and jumping motion of rigid bodies, the mathematical formulation becomes quite complex. In order to overcome this complexity, a Semi-Discrete Model (SMD) is proposed for the study of rocking motion of rigid bodies, in which the rigid body is considered as a mass element supported by springs and dashpots, in the spirit of deformable contacts between rigid blocks. The SMD can detect separation and sliding of the body; however, initial base contacts do not change, keeping a relative continuity between the body and its base. Extensive numerical simulations have been carried out in order to validate the proposed approach.

Casimir Effect on Voltage Response of Superharmonic Resonance of NEMS Resonators

2017 ◽  
Author(s):  
Martin Botello ◽  
Christian Reyes ◽  
Julio Beatriz ◽  
Dumitru I. Caruntu
Keyword(s):  
Differential Equation ◽  
Multiple Scales ◽  
Nonlinear Partial Differential Equation ◽  
Equation Of Motion ◽  
Second Order ◽  
Mode Shapes ◽  
Voltage Response ◽  
Casimir Forces ◽  
Superharmonic Resonance ◽  
Reference Length

This paper investigates the voltage response of superharmonic resonance of the second order of electrostatically actuated nano-electro-mechanical system (NEMS) resonator sensor. The structure of the NEMS device is a resonator cantilever over a ground plate under Alternating Current (AC) voltage. Superharmonic resonance of second order occurs when the AC voltage is operating in a frequency near-quarter the natural frequency of the resonator. The forces acting on the system are electrostatic, damping and Casimir. To induce a bifurcation phenomenon in superharmonic resonance, the AC voltage is in the category of hard excitation. The gap distance between the cantilever resonator and base plate is in the range of 20 nm to 1 μm for Casimir forces to be present. The differential equation of motion is converted to dimensionless by choosing the gap as reference length for deflections, the length of the resonator for the axial coordinate, and reference time based on the characteristics of the structure. The Method of Multiple Scales (MMS) and Reduced Order Model (ROM) are used to model the characteristic of the system. MMS transforms the nonlinear partial differential equation of motion into two simpler problems, namely zero-order and first-order. ROM, based on the Galerkin procedure, uses the undamped linear mode shapes of the undamped cantilever beam as the basis functions. The influences of parameters (i.e. Casimir, damping, fringe, and detuning parameter) were also investigated.

scholarly journals Alternative integration procedure for scale-invariant ordinary differential equations

1979 ◽  
Vol 2 (1) ◽  
pp. 143-145 ◽  
Author(s):  
Gerald Rosen
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
General Solution ◽  
Differential Equations ◽  
Parametric Form ◽  
Scale Invariant ◽  
Invariant Variable ◽  
Parameter Group ◽  
Independent Variables ◽  
Scale Transformations

For an ordinary differential equation invariant under a one-parameter group of scale transformationsx→λx,y→λαy,y′→λα−1y′,y″→λα−2y″, etc., it is shown by example that an explicit analytical general solution may be obtainable in parametric form in terms of the scale-invariant variableξ=∫xy−1/αdx. This alternative integration may go through, as it does for the example equationy″=kxy−2y′, in cases for which the customary dependent and independent variables(x−αy)and(ℓnx)do not yield an analytically integrable transformed equation.

Sign in / Sign up

Export Citation Format

Share Document