Exact Solutions for Two-Dimensional Time-Dependent Flow and Deformation Within a Poroelastic Medium

1999 ◽  
Vol 66 (2) ◽  
pp. 536-540 ◽  
Author(s):  
S. I. Barry ◽  
G. N. Mercer
Keyword(s):  
Exact Solutions ◽  
Applied Pressure ◽  
Arbitrary Point ◽  
Analytic Solutions ◽  
Time Dependent ◽  
Finite Domain ◽  
Two Dimensional ◽  
Poroelastic Medium ◽  
Exact Analytic Solutions ◽  
Dependent Flow

Exact analytic solutions are derived for the time-dependent deformation of a poroelastic medium within a two-dimensional finite domain. Solutions are given with a specific set of boundary conditions for the case of a source of fluid at an arbitrary point and for an applied pressure on the boundary. These solutions are ideal for testing numerical schemes for poroelastic flow and deformations due to their relative simplicity.

Cusp development in free boundaries, and two-dimensional slow viscous flows

1995 ◽  
Vol 6 (5) ◽  
pp. 441-454 ◽  
Author(s):  
S. D. Howison ◽  
S. Richardson
Keyword(s):  
Surface Tension ◽  
Viscous Flows ◽  
Analytic Solutions ◽  
Time Dependent ◽  
Two Dimensional ◽  
Free Boundaries ◽  
Mathematical Solution ◽  
Simply Connected Domain ◽  
Exact Analytic Solutions ◽  
Simply Connected

We consider a family of problems involving two-dimensional Stokes flows with a time dependent free boundary for which exact analytic solutions can be found; the fluid initially occupies some bounded, simply-connected domain and is withdrawn from a fixed point within that domain. If we suppose there to be no surface tension acting, we find that cusps develop in the free surface before all the fluid has been extracted, and the mathematical solution ceases to be physically relevant after these have appeared. However, if we include a non-zero surface tension in the theory, no matter how small this may be, the cusp development is inhibited and the solution allows all the fluid to be removed.

scholarly journals New Soliton Solutions for the Higher-Dimensional Non-Local Ito Equation

2021 ◽  
Vol 10 (1) ◽  
pp. 374-384
Author(s):  
Mustafa Inc ◽  
E. A. Az-Zo’bi ◽  
Adil Jhangeer ◽  
Hadi Rezazadeh ◽  
Muhammad Nasir Ali ◽  
...  
Keyword(s):  
Exact Solutions ◽  
Solution Process ◽  
Soliton Solutions ◽  
Analytic Solutions ◽  
Bright Solitons ◽  
Exact Analytic Solutions ◽  
Higher Dimensional ◽  
Non Local ◽  
Free Parameters ◽  
Ito Equation

Abstract In this article, (2+1)-dimensional Ito equation that models waves motion on shallow water surfaces is analyzed for exact analytic solutions. Two reliable techniques involving the simplest equation and modified simplest equation algorithms are utilized to find exact solutions of the considered equation involving bright solitons, singular periodic solitons, and singular bright solitons. These solutions are also described graphically while taking suitable values of free parameters. The applied algorithms are effective and convenient in handling the solution process for Ito equation that appears in many phenomena.

scholarly journals Two- and three-dimensional numerical simulations of the transition to oscillatory convection in low-Prandtl-number fluids

1998 ◽  
Vol 374 ◽  
pp. 145-171 ◽  
Author(s):  
DANIEL HENRY ◽  
MARC BUFFAT
Keyword(s):  
Prandtl Number ◽  
Relative Concentration ◽  
Three Dimensional ◽  
Power Spectra ◽  
Hadley Circulation ◽  
Time Dependent ◽  
Two Dimensional ◽  
General Behaviour ◽  
Shallow Cavities ◽  
Dependent Flow

The convective flows which arise in shallow cavities filled with low-Prandtl-number fluids when subjected to a horizontal temperature gradient are studied numerically with a finite element method. Attention is focused on a rigid cavity with dimensions 4×2×1, for which experimental data are available. The three-dimensional results indicate that, after a relative concentration of the initial Hadley circulation, a transition to time-dependent flows occurs in the form of a roll oscillation with a purely dynamical origin. This transition corresponds to a Hopf bifurcation with a breaking of symmetry that gives some specific properties to the time evolution of the flow: these properties are shown to be the result of the general behaviour of the dynamical systems. Calculations performed in the case of mercury compare well with the experiments with similar power spectra of the temperature, and this validates the analysis of the nature of the global flow performed in the limiting case Pr=0. All these results are discussed with respect to the linear and nonlinear analyses and to other computational experiments. Numerical results obtained in the corresponding two-dimensional situation give a different transition to the time-dependent flow: it is shown that in the three-dimensional cavity this type of two-dimensional transition is less probable than the observed transition with breaking of symmetry.

An Analytical Solution of Two-Dimensional Flow and Deformation Coupling Due to a Point Source Within a Finite Poroelastic Media

2011 ◽  
Vol 78 (6) ◽  
Author(s):  
Peichao Li ◽  
Detang Lu
Keyword(s):  
Analytical Solution ◽  
Boundary Conditions ◽  
Point Source ◽  
Transient Behavior ◽  
Finite Domain ◽  
Two Dimensional ◽  
Pressure Derivative ◽  
Poroelastic Media ◽  
Two Dimensional Flow ◽  
Dependent Flow

An analytical solution is derived for the time-dependent flow and deformation coupling of a saturated isotropic homogeneous incompressible poroelastic media within a two-dimensional (2D) finite domain due to a point source at some arbitrary position. In this study, the pore pressure field is assumed to conform to the second type of boundary conditions. Boundary conditions of the displacement field are chosen with care to match the appropriate finite sine and cosine transforms and simplify the resulting solution. It is found that the analytical solution is always independent of the Poisson’s ratio. The detailed solutions are given for the case of a periodic point source with zero pressure derivatives on the boundaries and for an imposed pressure derivative on the lower edge in the absence of a source. The presented analytical solutions are highly applicable for calibrating numerical codes, and meanwhile they can be used to further investigate the transient behavior of flow and deformation coupling induced by fluid withdrawal within a 2D finite poroelastic media.

EXACT SOLUTIONS OF A MANY-ANYON PROBLEM

1992 ◽  
Vol 07 (32) ◽  
pp. 7931-7942 ◽  
Author(s):  
STEFAN V. MASHKEVICH
Keyword(s):  
Exact Solutions ◽  
Stationary State ◽  
Arbitrary Number ◽  
Statistical Parameter ◽  
Analytic Solutions ◽  
Wave Functions ◽  
Successive Approximations ◽  
Method Of Successive Approximations ◽  
Exact Analytic Solutions ◽  
State Functions

A scheme for obtaining exact analytic solutions of the problem of an arbitrary number of anyons in a harmonic well is developed. Its essence consists in establishing a set of wave functions with the demanded interchange properties followed by finding stationary state functions within this set by the method of successive approximations. The energy of the corresponding states depends linearly on the statistical parameter. The discussion of the obtained states is carried out.

scholarly journals Exact Steady Solutions for the Two Dimensional Broadwell Model

2019 ◽  
pp. 1-14
Author(s):  
Amah Séna D'Almeida ◽  
Kokou Anani Agosseme
Keyword(s):  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Boundary Value ◽  
Analytic Solutions ◽  
Two Dimensional ◽  
Exact Analytic Solutions ◽  
Accommodation Coefficients ◽  
Steady Solutions

Existence and boundedness of the solutions of the boundary value problem for the four velocity two dimensional Broadwell model for bounded boundary conditions is proved and exact analytic solutions are built. An application to the determination of the accommodation coefficients on the boundaries of a flow in a box is performed.

scholarly journals Lie Symmetry Reductions and Exact Solutions to the Rosenau Equation

2014 ◽  
Vol 2014 ◽  
pp. 1-6
Author(s):  
Ben Gao ◽  
Hongxia Tian
Keyword(s):  
Exact Solutions ◽  
Analytic Solutions ◽  
Lie Symmetry ◽  
Series Method ◽  
Lie Symmetry Analysis ◽  
Power Series Method ◽  
Exact Analytic Solutions ◽  
Rosenau Equation ◽  
Physical Phenomena ◽  
Similarity Reductions

The Lie symmetry analysis is performed on the Rosenau equation which arises in modeling many physical phenomena. The similarity reductions and exact solutions are presented. Then the exact analytic solutions are considered by the power series method.

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