scholarly journals Analytic solution to an interfacial flow with kinetic undercooling in a time-dependent gap Hele-Shaw cell

2017 ◽  
Vol 22 (11) ◽  
pp. 0-0
Author(s):  
Xuming Xie ◽  
Keyword(s):  
Analytic Solution ◽  
Time Dependent ◽  
Interfacial Flow ◽  
Kinetic Undercooling ◽  
Hele Shaw Cell

QUANTUM DYNAMICS OF WAVE PACKET IN HARMONIC POTENTIAL

2005 ◽  
Vol 19 (24) ◽  
pp. 3745-3754
Author(s):  
ZHAN-NING HU ◽  
CHANG SUB KIM
Keyword(s):  
Schrödinger Equation ◽  
Wave Packet ◽  
Quantum Dynamics ◽  
Schrodinger Equation ◽  
Analytic Solution ◽  
Nonlinear Differential Equations ◽  
Time Dependent ◽  
Dimensional System ◽  
Two Dimensional ◽  
Vibrational States

In this paper, the analytic solution of the time-dependent Schrödinger equation is obtained for the wave packet in two-dimensional oscillator potential. The quantum dynamics of the wave packet is investigated based on this analytic solution. To our knowledge, this is the first time we solve, analytically and exactly this kind of time-dependent Schrödinger equation in a two-dimensional system, in which the Gaussian parameters satisfy the coupled nonlinear differential equations. The coherent states and their rotations of the system are discussed in detail. We find also that this analytic solution includes four kinds of modes of the evolutions for the wave packets: rigid, rotational, vibrational states and a combination of the rotation and vibration without spreading.

scholarly journals Numerical investigation of controlling interfacial instabilities in non-standard Hele-Shaw configurations

2019 ◽  
Vol 877 ◽  
pp. 1063-1097 ◽  
Author(s):  
Liam C. Morrow ◽  
Timothy J. Moroney ◽  
Scott W. McCue
Keyword(s):  
Viscous Fluid ◽  
Linear Prediction ◽  
Applied Mathematics ◽  
Viscous Fingering ◽  
Time Dependent ◽  
Linear Stability Theory ◽  
Inviscid Fluid ◽  
Interfacial Instabilities ◽  
Hele Shaw Cell ◽  
Injection Rates

Viscous fingering experiments in Hele-Shaw cells lead to striking pattern formations which have been the subject of intense focus among the physics and applied mathematics community for many years. In recent times, much attention has been devoted to devising strategies for controlling such patterns and reducing the growth of the interfacial fingers. We continue this research by reporting on numerical simulations, based on the level set method, of a generalised Hele-Shaw model for which the geometry of the Hele-Shaw cell is altered. First, we investigate how imposing constant and time-dependent injection rates in a Hele-Shaw cell that is either standard, tapered or rotating can be used to reduce the development of viscous fingering when an inviscid fluid is injected into a viscous fluid over a finite time period. We perform a series of numerical experiments comparing the effectiveness of each strategy to determine how these non-standard Hele-Shaw configurations influence the morphological features of the inviscid–viscous fluid interface. Surprisingly, a converging or diverging taper of the plates leads to reduced metrics of viscous fingering at the final time when compared to the standard parallel configuration, especially with carefully chosen injection rates; for the rotating plate case, the effect is even more dramatic, with sufficiently large rotation rates completely stabilising the interface. Next, we illustrate how the number of non-splitting fingers can be controlled by injecting the inviscid fluid at a time-dependent rate while increasing the gap between the plates. Our simulations compare well with previous experimental results for various injection rates and geometric configurations. We demonstrate how the number of non-splitting fingers agrees with that predicted from linear stability theory up to some finger number; for larger values of our control parameter, the fully nonlinear dynamics of the problem leads to slightly fewer fingers than this linear prediction.

scholarly journals Rigorous results in existence and selection of Saffman–Taylor fingers by kinetic undercooling

2018 ◽  
Vol 30 (1) ◽  
pp. 63-116
Author(s):  
XUMING XIE
Keyword(s):  
Surface Tension ◽  
Rigorous Results ◽  
Engineering Mathematics ◽  
Stokes Lines ◽  
Kinetic Undercooling ◽  
Finger Width ◽  
Linear Differential ◽  
Finger Tip ◽  
Selection Of ◽  
Hele Shaw Cell

The selection of Saffman–Taylor fingers by surface tension has been extensively investigated. In this paper, we are concerned with the existence and selection of steadily translating symmetric finger solutions in a Hele–Shaw cell by small but non-zero kinetic undercooling (ε2). We rigorously conclude that for relative finger width λ near one half, symmetric finger solutions exist in the asymptotic limit of undercooling ε2 → 0 if the Stokes multiplier for a relatively simple non-linear differential equation is zero. This Stokes multiplier S depends on the parameter $\alpha \equiv \frac{2 \lambda -1}{(1-\lambda)}\epsilon^{-\frac{4}{3}}$ and earlier calculations have shown this to be zero for a discrete set of values of α. While this result is similar to that obtained previously for Saffman–Taylor fingers by surface tension, the analysis for the problem with kinetic undercooling exhibits a number of subtleties as pointed out by Chapman and King (2003, The selection of Saffman–Taylor fingers by kinetic undercooling, Journal of Engineering Mathematics, 46, 1–32). The main subtlety is the behaviour of the Stokes lines at the finger tip, where the analysis is complicated by non-analyticity of coefficients in the governing equation.

Aspects of the Shi–Seinfeld–Okuyama theory of transient nucleation

1995 ◽  
Vol 10 (7) ◽  
pp. 1674-1679 ◽  
Author(s):  
G. Sundar ◽  
J.J. Hoyt
Keyword(s):  
High Temperature ◽  
Singular Perturbation ◽  
Incubation Time ◽  
Cluster Size ◽  
Analytic Solution ◽  
Time Dependent ◽  
Transient Kinetics ◽  
High Temperature Anneal ◽  
Transient Nucleation

The analytic solution to the time-dependent nucleation problem by Shi-Seinfeld-Okuyama (SSO) [Phys. Rev. A 41, 2101 (1990)] is reviewed. The singular perturbation solution employed by SSO is extended to examine the effect of initial quench position on the incubation time. Two cases are discussed. The first investigates the role of excess vacancies from the high temperature quench on the transient kinetics. The second case examines the change in the incubation time due to the effects of a preexisting subcritical cluster size distribution which forms during the high temperature anneal.

QUANTUM TREATMENT OF A TIME DEPENDENT SINGLE-TRAPPED ION INTERACTING WITH A BIMODAL CAVITY FIELD

2003 ◽  
Vol 17 (31n32) ◽  
pp. 5925-5941 ◽  
Author(s):  
MAHMOUD ABDEL-ATY ◽  
A.-S. F. OBADA ◽  
M. SEBAWE ABDALLA
Keyword(s):  
Cross Correlation ◽  
Analytic Solution ◽  
Coupling Parameter ◽  
Equations Of Motion ◽  
Time Dependent ◽  
Field Interaction ◽  
Trapped Ion ◽  
Heisenberg Equations ◽  
Quantum Treatment ◽  
Heisenberg Equations Of Motion

In the present communication we consider a time dependent ion-field interaction. Here we discuss the interaction between a single trapped ion and two fields taking into account the coupling parameter to be time dependent and allowing for amplitude modulation of the laser field radiating the trapped ion. At exact resonances the analytic solution for the Heisenberg equations of motion is obtained. We examine the effect of the velocity and the acceleration on the Rabi oscillations by studying the second order correlation function. The phenomenon of squeezing for single and two fields cases is considered. The cross correlation between the fields is discussed.

Recurring Molecular Alignment Induced by Pulsed Nonresonant Laser Fields

2001 ◽  
Vol 66 (7) ◽  
pp. 991-1004 ◽  
Author(s):  
Long Cai ◽  
Břetislav Friedrich
Keyword(s):  
Laser Pulse ◽  
Laser Field ◽  
Analytic Solution ◽  
Short Pulse ◽  
Molecular Alignment ◽  
Time Dependent ◽  
Molecular Axis ◽  
Linear Molecule ◽  
Numerical Computations ◽  
Laser Fields

We examine the rotational wavepackets created by the nonadiabatic interaction of a linear molecule with a pulsed nonresonant laser field. We map out the recurrences of the wavepackets and of the concomitant alignment as a function of the duration and intensity of the laser pulse. We derive an analytic solution to the time-dependent Schrödinger equation in the short-pulse limit and find it to agree quantitatively with our numerical computations. This indicates that the recurrences are favored under an impulsive transfer of action from the radiative field to the molecule. The recurring wavepackets afford field-free alignment of the molecular axis.

Analytic solution for two bubbles in a Hele-Shaw cell

2000 ◽  
Vol 62 (3) ◽  
pp. R3047-R3050 ◽  
Author(s):  
Giovani L. Vasconcelos
Keyword(s):  
Analytic Solution ◽  
Two Bubbles ◽  
Hele Shaw Cell

Numerical study of miscible fingering in a time-dependent gap Hele-Shaw cell

2005 ◽  
Vol 71 (5) ◽  
Author(s):  
Ching-Yao Chen ◽  
Chen-Hau Chen ◽  
José A. Miranda
Keyword(s):  
Numerical Study ◽  
Time Dependent ◽  
Hele Shaw Cell
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