Optimal shape of slowly rising gas bubbles

2005 ◽  
Vol 83 (7) ◽  
pp. 761-766
Author(s):  
Alexei M Frolov
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Incompressible Fluid ◽  
Three Dimensional ◽  
Gas Bubbles ◽  
Ideal Incompressible Fluid ◽  
Optimal Shape ◽  
Dimensional Problem ◽  
Optimal Form ◽  
One Dimensional

The variational optimal shape of slowly rising gas bubbles in an ideal incompressible fluid is determined. It is shown that the original three-dimensional problem can be reduced to a relatively simple one-dimensional (i.e., ordinary) differential equation. The solution of this equation allows one to obtain the variational optimal form of slowly rising gas bubbles. PACS No.: 47.55.Dz

scholarly journals Extreme Analysis of a Random Ordinary Differential Equation

2014 ◽  
Vol 51 (4) ◽  
pp. 1021-1036 ◽  
Author(s):  
Jingchen Liu ◽  
Xiang Zhou
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Elliptic Equation ◽  
Gaussian Process ◽  
Stochastic System ◽  
Random Coefficient ◽  
Asymptotic Approximations ◽  
Tail Probabilities ◽  
One Dimensional ◽  
Spatially Varying

In this paper we consider a one dimensional stochastic system described by an elliptic equation. A spatially varying random coefficient is introduced to account for uncertainty or imprecise measurements. We model the logarithm of this coefficient by a Gaussian process and provide asymptotic approximations of the tail probabilities of the derivative of the solution.

Nonlinear stationary magnetoconvection in a rotating fluid

1997 ◽  
Vol 58 (3) ◽  
pp. 395-408 ◽  
Author(s):  
S. G. TAGARE
Keyword(s):  
Magnetic Field ◽  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Pitchfork Bifurcation ◽  
Time Dependent ◽  
Vertical Magnetic Field ◽  
Stationary Convection ◽  
Rotating Fluid ◽  
Order Ordinary Differential Equation ◽  
One Dimensional

We investigate finite-amplitude magnetoconvection in a rotating fluid in the presence of a vertical magnetic field when the axis of rotation is parallel to a vertical magnetic field. We derive a nonlinear, time-dependent, one-dimensional Landau–Ginzburg equation near the onset of stationary convection at supercritical pitchfork bifurcation whenformula hereand a nonlinear time-dependent second-order ordinary differential equation when Ta=T*a (from below). Ta=T*a corresponds to codimension-two bifurcation (or secondary bifurcation), where the threshold for stationary convection at the pitchfork bifurcation coincides with the threshold for oscillatory convection at the Hopf bifurcation. We obtain steady-state solutions of the one-dimensional Landau–Ginzburg equation, and discuss the solution of the nonlinear time-dependent second-order ordinary differential equation.

Diffraction of Electrons by Two and Three Mutually Orthogonal Standing Light Waves

1975 ◽  
Vol 53 (2) ◽  
pp. 157-164 ◽  
Author(s):  
F. Ehlotzky
Keyword(s):  
Electron Scattering ◽  
Light Wave ◽  
Three Dimensional ◽  
Three Dimensions ◽  
Dimensional Problem ◽  
One Dimensional ◽  
Standing Light Wave ◽  
Light Waves ◽  
The One ◽  
Rectangular Lattices

The one-dimensional problem of electron scattering by a standing light wave, known as the Kapitza–Dirac effect, is shown to be easily extendable to two and three dimensions, thus showing all characteristics of diffraction of electrons by simple two- and three-dimensional rectangular lattices.

scholarly journals Efficient C2 Continuous Surface Creation Technique Based on Ordinary Differential Equation

Symmetry ◽  
2019 ◽  
Vol 12 (1) ◽  
pp. 38
Author(s):  
Shaojun Bian ◽  
Greg Maguire ◽  
Willem Kokke ◽  
Lihua You ◽  
Jian J. Zhang
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Shape Control ◽  
Three Dimensional ◽  
3D Models ◽  
Partial Derivatives ◽  
Surface Patches ◽  
C2 Continuity ◽  
Continuous Surface ◽  
Boundary Curves

In order to reduce the data size and simplify the process of creating characters’ 3D models, a new and interactive ordinary differential equation (ODE)-based C2 continuous surface creation algorithm is introduced in this paper. With this approach, the creation of a three-dimensional surface is transformed into generating two boundary curves plus four control curves and solving a vector-valued sixth order ordinary differential equation subjected to boundary constraints consisting of boundary curves, and first and second partial derivatives at the boundary curves. Unlike the existing patch modeling approaches which require tedious and time-consuming manual operations to stitch two separate patches together to achieve continuity between two stitched patches, the proposed technique maintains the C2 continuity between adjacent surface patches naturally, which avoids manual stitching operations. Besides, compared with polygon surface modeling, our ODE C2 surface creation method can significantly reduce and compress the data size, deform the surface easily by simply changing the first and second partial derivatives, and shape control parameters instead of manipulating loads of polygon points.

ON THE STRUCTURE OF THE SOLUTIONS OF A TWO-PARAMETER FAMILY OF THREE-DIMENSIONAL ORDINARY DIFFERENTIAL EQUATIONS

2003 ◽  
Vol 13 (05) ◽  
pp. 1287-1298 ◽  
Author(s):  
SERKAN T. IMPRAM ◽  
RUSSELL JOHNSON ◽  
RAFFAELLA PAVANI
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Dynamical Behavior ◽  
Three Dimensional ◽  
Wide Range ◽  
Two Parameters ◽  
Two Parameter ◽  
Autonomous Ordinary Differential Equation

We analyze the global structure of the solutions of a three-dimensional, autonomous ordinary differential equation which depends on two parameters. We use graphical, heuristic, and rigorous arguments to show that as the parameters vary, a wide range of dynamical behavior is displayed.

Difference Solution to Chloride Diffusion in Concrete

2011 ◽  
Vol 99-100 ◽  
pp. 754-757
Author(s):  
Ai Ping Yu ◽  
Hai Bo Lu ◽  
Yan Lin Zhao ◽  
Ke Yu Wei
Keyword(s):  
Differential Equation ◽  
Discrete Model ◽  
Three Dimensional ◽  
Chloride Diffusion ◽  
Second Law ◽  
Continuous Variables ◽  
Convergence Conditions ◽  
One Dimensional ◽  
The Difference ◽  
2D And 3D

Fick’s second law and its analytical solution were usually used for analysis of chloride diffusion in concrete. But discretization of the continuous variables is more appropriate in modeling and discrete model is more suitable for engineering applications. In this paper, Difference Equation equivalent to partial differential equation was established with the difference method, and three-dimensional differential equation of Fick's second law was resolved. Based on this study，the convergence conditions of difference Equations for one-dimensional, 2D and 3D diffusion was given.

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