An analytic solution for three-dimensional axisymmetric equilibrium crystal shapes

2009 ◽  
Vol 60 (8) ◽  
pp. 631-634 ◽  
Author(s):  
Ping Du ◽  
Harris Wong
Keyword(s):  
Analytic Solution ◽  
Three Dimensional ◽  
Crystal Shapes ◽  
Equilibrium Crystal Shapes ◽  
Axisymmetric Equilibrium

scholarly journals Step-step interactions and universal exponents studied via three-dimensional equilibrium crystal shapes

2002 ◽  
Vol 4 ◽  
pp. 60-60 ◽  
Author(s):  
M Nowicki ◽  
C Bombis ◽  
A Emundts ◽  
H P Bonzel ◽  
P Wynblatt
Keyword(s):  
Three Dimensional ◽  
Crystal Shapes ◽  
Equilibrium Crystal Shapes ◽  
Step Step

scholarly journals An approximate analytic solution of a particular boundary value problem

2001 ◽  
Vol 27 (8) ◽  
pp. 513-520
Author(s):  
Ugur Tanriver ◽  
Aravinda Kar
Keyword(s):  
Boundary Value Problem ◽  
Heat Conduction Equation ◽  
Analytic Solution ◽  
Boundary Value ◽  
Three Dimensional ◽  
Welding Process ◽  
Approximate Analytic Solution ◽  
Analytic Expression ◽  
Steady State Heat ◽  
Laser Welding Process

This note is concerned with the three-dimensional quasi-steady-state heat conduction equation subject to certain boundary conditions in the wholex′y′-plane and finite inz′-direction. This type of boundary value problem arises in laser welding process. The solution to this problem can be represented by an integral using Fourier analysis. This integral is approximated to obtain a simple analytic expression for the temperature distribution.

scholarly journals Rigorous probabilistic analysis of equilibrium crystal shapes

2000 ◽  
Vol 41 (3) ◽  
pp. 1033-1098 ◽  
Author(s):  
T. Bodineau ◽  
D. Ioffe ◽  
Y. Velenik
Keyword(s):  
Probabilistic Analysis ◽  
Crystal Shapes ◽  
Equilibrium Crystal Shapes

scholarly journals Re-Evaluating the Classical Falling Body Problem

Mathematics ◽  
2020 ◽  
Vol 8 (4) ◽  
pp. 553 ◽  
Author(s):  
Essam R. El-Zahar ◽  
Abdelhalim Ebaid ◽  
Abdulrahman F. Aljohani ◽  
José Tenreiro Machado ◽  
Dumitru Baleanu
Keyword(s):  
Differential Equations ◽  
Body Problem ◽  
Inertial Frame ◽  
Analytic Solution ◽  
Equations Of Motion ◽  
Three Dimensional ◽  
Three Dimensions ◽  
Rotating Frame ◽  
Dimensional Model ◽  
Three Dimensional Model

This paper re-analyzes the falling body problem in three dimensions, taking into account the effect of the Earth’s rotation (ER). Accordingly, the analytic solution of the three-dimensional model is obtained. Since the ER is quite slow, the three coupled differential equations of motion are usually approximated by neglecting all high order terms. Furthermore, the theoretical aspects describing the nature of the falling point in the rotating frame and the original inertial frame are proved. The theoretical and numerical results are illustrated and discussed.

Self-excited oscillations in three-dimensional collapsible tubes: simulating their onset and large-amplitude oscillations

2010 ◽  
Vol 652 ◽  
pp. 405-426 ◽  
Author(s):  
MATTHIAS HEIL ◽  
JONATHAN BOYLE
Keyword(s):  
Large Amplitude ◽  
Critical Reynolds Number ◽  
Equilibrium Configuration ◽  
Three Dimensional ◽  
Collapsible Tubes ◽  
High Frequency Oscillations ◽  
Equilibrium Configurations ◽  
Axisymmetric Equilibrium ◽  
Outflow Boundary ◽  
Good Agreement

We employ numerical simulations to explore the development of flow-induced self-excited oscillations in three-dimensional collapsible tubes which are subject to boundary conditions (flow rate prescribed at the outflow boundary) that encourage the development of high-frequency oscillations via an instability mechanism originally proposed by Jensen & Heil (J. Fluid Mech., vol. 481, 2003, p. 235). The simulations show that self-excited oscillations tend to arise preferentially from steady equilibrium configurations in which the tube is buckled non-axisymmetrically. We follow the growing oscillations into the large-amplitude regime and show that short tubes tend to approach an approximately axisymmetric equilibrium configuration in which the oscillations decay, whereas sufficiently long tubes develop sustained large-amplitude limit-cycle oscillations. The period of the oscillations and the critical Reynolds number beyond which their amplitude grows are found to be in good agreement with theoretical scaling estimates.

scholarly journals Manufactured solutions and the verification of three-dimensional Stokes ice-sheet models

2013 ◽  
Vol 7 (1) ◽  
pp. 19-29 ◽  
Author(s):  
W. Leng ◽  
L. Ju ◽  
M. Gunzburger ◽  
S. Price
Keyword(s):  
Computational Models ◽  
Analytic Solution ◽  
Initial Conditions ◽  
Model Simulation ◽  
Three Dimensional ◽  
Ice Sheet ◽  
Solution Technique ◽  
Order Partial Differential Equation ◽  
Manufactured Solutions ◽  
Manufactured Solution

Abstract. The manufactured solution technique is used for the verification of computational models in many fields. In this paper, we construct manufactured solutions for the three-dimensional, isothermal, nonlinear Stokes model for flows in glaciers and ice sheets. The solution construction procedure starts with kinematic boundary conditions and is mainly based on the solution of a first-order partial differential equation for the ice velocity that satisfies the incompressibility condition. The manufactured solutions depend on the geometry of the ice sheet, basal sliding parameters, and ice softness. Initial conditions are taken from the periodic geometry of a standard problem of the ISMIP-HOM benchmark tests. The upper surface is altered through the manufactured solution procedure to generate an analytic solution for the time-dependent flow problem. We then use this manufactured solution to verify a parallel, high-order accurate, finite element Stokes ice-sheet model. Simulation results from the computational model show good convergence to the manufactured analytic solution.

Sign in / Sign up

Export Citation Format

Share Document