The initial value problem of a rising bubble in a two‐dimensional vertical channel

1992 ◽  
Vol 4 (5) ◽  
pp. 913-920 ◽  
Author(s):  
Yumin Yang
Keyword(s):  
Initial Value Problem ◽  
Vertical Channel ◽  
Two Dimensional ◽  
Initial Value ◽  
Rising Bubble

scholarly journals The influence of circulation on the stability of vortices to mode-one disturbances

1995 ◽  
Vol 451 (1943) ◽  
pp. 747-755 ◽  
Keyword(s):  
Velocity Field ◽  
Angular Velocity ◽  
Initial Value Problem ◽  
Large Time ◽  
Asymptotic Methods ◽  
Two Dimensional ◽  
Initial Value ◽  
The Stability ◽  
Ultimate Fate ◽  
Solution Behaviour

The initial value problem for the two-dimensional inviscid vorticity equation, linearized about an azimuthal basic velocity field with monotonic angular velocity, is solved exactly for mode-one disturbances. The solution behaviour is investigated for large time using asymptotic methods. The circulation of the basic state is found to govern the ultimate fate of the disturbance: for basic state vorticity distributions with non-zero circulation, the perturbation tends to the steady solution first mentioned in Michalke & Timme (1967), while for zero circulation, the perturbation grows without bound. The latter case has potentially important implications for the stability of isolated eddies in geophysics.

A Numerical Simulation on Effects of Distance Between Bubble and the Wall on Behavior of Rising Bubble

2013 ◽  
Author(s):  
Ying Huang ◽  
Puzhen Gao
Keyword(s):  
Numerical Simulation ◽  
Vertical Channel ◽  
Narrow Channel ◽  
Horizontal Position ◽  
Two Dimensional ◽  
Volume Of Fluid Method ◽  
Air Bubble ◽  
Rising Bubble ◽  
Water Based ◽  
Different Diameters

A numerical investigation of two-dimensional air bubble behaviors under the effect of gravity in still water based on the VOF (Volume-Of-Fluid) method is carried out. Initially, the surface tension effects on the behavior of the bubble is analyzed, which contains the simulation of the ascending motion of a single air bubble in liquid and the study of the interaction between bubbles in terms of coalescence. Additionally, the differences of single bubble’s rising motion in an infinite surroundings and in a vertical narrow channel are analyzed. The coalescence of bubbles is also studied. The motion of bubbles with different diameters in a vertical channel is simulated. It is found that the bubbles’ behavior depends on the distance between the bubble and the wall. Finally, numerical simulation of the motion of several bubbles of the same size, at the same initial horizontal position and with uniform distribution is carried out. The result reveals that the bubbles at different distances from the wall have different velocities, after a while, the bubbles distribution presents as “U”.

HOPF SOLUTIONS TO A FLUID-ELASTIC INTERACTION MODEL

2008 ◽  
Vol 18 (02) ◽  
pp. 215-269 ◽  
Author(s):  
M. GUIDORZI ◽  
M. PADULA ◽  
P. I. PLOTNIKOV
Keyword(s):  
Global Existence ◽  
Existence Theorem ◽  
Initial Value Problem ◽  
Interaction Model ◽  
Elastic Interaction ◽  
Initial Condition ◽  
Two Dimensional ◽  
Initial Value ◽  
Global Existence Theorem ◽  
Model Equations

In this paper, we give a global existence theorem of weak solutions to model equations governing interaction fluid structure in a two-dimensional layer, cf. Refs. 8 and 14. To our knowledge this is the first existence theorem of global in time solutions for such model. The interest of our result is double because, first, we change the original initial value problem by deleting one initial condition, second, we construct a solution through the classical Galerkin method for which several computing codes have been constructed.

SIMPLE ALGORITHM'S FOR THE CALCULATION OF HEAT TRANSPORT PROBLEM IN PLATE

2001 ◽  
Vol 6 (1) ◽  
pp. 85-96
Author(s):  
H. Kalis ◽  
I. Kangro
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Heat Transport ◽  
Initial Value Problem ◽  
Transport Problem ◽  
Two Dimensional ◽  
Initial Value ◽  
Equation Solution ◽  
Averaging Methods ◽  
First Order

The approximations of some heat transport problem in a thin plate are based on the finite volume and conservative averaging methods [1,2]. These procedures allow one to reduce the two dimensional heat transport problem described by a partial differential equation to an initial‐value problem for a system of two ordinary differential equations (ODEs) of the first order or to an initial‐value problem for one ordinary differential equations of the first order with one algebraic equation. Solution of the corresponding problems is obtained by using MAPLE routines “gear”, “mgear” and “lsode”.

Global well-posedness and linearization of a semilinear 2D wave equation with exponential type nonlinearity

2010 ◽  
Vol 17 (3) ◽  
pp. 543-562 ◽  
Author(s):  
Olfa Mahouachi ◽  
Tarek Saanouni
Keyword(s):  
Wave Equation ◽  
Exponential Type ◽  
Initial Value Problem ◽  
Energy Estimate ◽  
Linear Wave ◽  
Well Posedness ◽  
Two Dimensional ◽  
Initial Value ◽  
Linear Wave Equation ◽  
Funct Anal

Abstract We consider the initial value problem for a two-dimensional semi-linear wave equation with exponential type nonlinearity. We obtain global well-posedness in the energy space. We also establish the linearization of bounded energy solutions in the spirit of Gérard [J. Funct. Anal. 141: 60–98, 1996]. The proof uses Moser–Trudinger type inequalities and the energy estimate.

scholarly journals Spatially Quasi-Periodic Water Waves of Infinite Depth

2021 ◽  
Vol 31 (3) ◽  
Author(s):  
Jon Wilkening ◽  
Xinyu Zhao
Keyword(s):  
Initial Value Problem ◽  
Water Waves ◽  
Numerical Study ◽  
Equations Of Motion ◽  
Small Scale ◽  
Normal Velocity ◽  
Natural Setting ◽  
Two Dimensional ◽  
Periodic Functions ◽  
Initial Value

AbstractWe formulate the two-dimensional gravity-capillary water wave equations in a spatially quasi-periodic setting and present a numerical study of solutions of the initial value problem. We propose a Fourier pseudo-spectral discretization of the equations of motion in which one-dimensional quasi-periodic functions are represented by two-dimensional periodic functions on a torus. We adopt a conformal mapping formulation and employ a quasi-periodic version of the Hilbert transform to determine the normal velocity of the free surface. Two methods of time-stepping the initial value problem are proposed, an explicit Runge–Kutta (ERK) method and an exponential time-differencing (ETD) scheme. The ETD approach makes use of the small-scale decomposition to eliminate stiffness due to surface tension. We perform a convergence study to compare the accuracy and efficiency of the methods on a traveling wave test problem. We also present an example of a periodic wave profile containing vertical tangent lines that is set in motion with a quasi-periodic velocity potential. As time evolves, each wave peak evolves differently, and only some of them overturn. Beyond water waves, we argue that spatial quasi-periodicity is a natural setting to study the dynamics of linear and nonlinear waves, offering a third option to the usual modeling assumption that solutions either evolve on a periodic domain or decay at infinity.

Excitation of surface waves in magnetohydrodynamics

1990 ◽  
Vol 43 (2) ◽  
pp. 183-188 ◽  
Author(s):  
Bhimsen K. Shivamoggi
Keyword(s):  
Magnetic Field ◽  
Gravitational Field ◽  
Asymptotic Behaviour ◽  
Pressure Distribution ◽  
Surface Waves ◽  
Initial Value Problem ◽  
Large Time ◽  
Far Field ◽  
Two Dimensional ◽  
Initial Value

A study is made of the transient development of two-dimensional linearized surface waves generated by a localized steady pressure distribution on the interface between a uniformly streaming, semi-infinite, infinitely conducting plasma subjected to a gravitational field and the confining vacuum magnetic field. The linearized equations associated with an initial-value problem are used to obtain the large-time asymptotic behaviour of the disturbance in the far field.

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