scholarly journals A Mathematical Model for Transport in Poroelastic Materials with Variable Volume:Derivation, Lie Symmetry Analysis, and Examples

Symmetry ◽  
2020 ◽  
Vol 12 (3) ◽  
pp. 396
Author(s):  
Roman Cherniha ◽  
Joanna Stachowska-Pietka ◽  
Jacek Waniewski
Keyword(s):  
Solute Transport ◽  
Lie Symmetry ◽  
Transport Processes ◽  
Experimental Investigations ◽  
One Dimensional ◽  
Poroelastic Media ◽  
Poroelastic Materials ◽  
Dimensional Version ◽  
The One ◽  
Symmetry Method

Fluid and solute transport in poroelastic media is studied. Mathematical modeling of such transport is a complicated problem because of the volume change of the specimen due to swelling or shrinking and the transport processes are nonlinearly linked. The tensorial character of the variables adds also substantial complication in both theoretical and experimental investigations. The one-dimensional version of the theory is less complex and may serve as an approximation in some problems, and therefore, a one-dimensional (in space) model of fluid and solute transport through a poroelastic medium with variable volume is developed and analyzed. In order to obtain analytical results, the Lie symmetry method is applied. It is shown that the governing equations of the model admit a non-trivial Lie symmetry, which is used for construction of exact solutions. Some examples of the solutions are discussed in detail.

scholarly journals Constrained lattice-field hierarchies and Toda system with Block symmetry

2016 ◽  
Vol 13 (05) ◽  
pp. 1650061 ◽  
Author(s):  
Chuanzhong Li
Keyword(s):  
Difference Equations ◽  
Integrable Systems ◽  
Integrable Hierarchies ◽  
Lie Symmetry ◽  
Toda System ◽  
One Dimensional ◽  
Toda Hierarchy ◽  
Lattice Field ◽  
The One ◽  
Ghost Symmetry

In this paper, we construct the additional [Formula: see text]-symmetry and ghost symmetry of two-lattice field integrable hierarchies. Using the symmetry constraint, we construct constrained two-lattice integrable systems which contain several new integrable difference equations. Under a further reduction, the constrained two-lattice integrable systems can be combined into one single integrable system, namely the well-known one-dimensional original Toda hierarchy. We prove that the one-dimensional original Toda hierarchy has a nice Block Lie symmetry.

scholarly journals On Hydrogen Entanglements and Gravitation: a Lie Symmetry Approach

2021 ◽  
Vol 15 ◽  
pp. 31-37
Author(s):  
JM Manale
Keyword(s):  
Gravitational Constant ◽  
Superposition Principle ◽  
Lie Symmetry ◽  
Quantum Superposition ◽  
One Dimensional ◽  
Symmetry Approach ◽  
Subatomic Particles ◽  
Popular View ◽  
The One ◽  
Lie Symmetry Approach

We depart from the popular view on how gravitation is generated. Ours is entanglement based. For an atom, hydrogen in this case, all subatomic particles, within and outside the nuclei, participate in this eternal dance. To test the idea, we use it to determine a formula for G, the universal gravitational constant. In the process, we note that the one-dimensional Schrodinger equation is not solved, as claimed. For example, existing solutions are silent on the quantum superposition principle. This we address through our modified symmetry group theoretical methods.

A Viscoelastic Model for the Dynamic Behavior of Saturated Poroelastic Soils

1992 ◽  
Vol 59 (1) ◽  
pp. 128-135 ◽  
Author(s):  
J. P. Bardet
Keyword(s):  
Elastic Moduli ◽  
Viscoelastic Model ◽  
Degree Of Saturation ◽  
Biot Theory ◽  
One Dimensional ◽  
Poroelastic Materials ◽  
Proposed Model ◽  
Wide Range ◽  
Dynamic Loadings ◽  
The One

A viscoelastic model is proposed to describe the dynamic response of the saturated poroelastic materials that obey the Biot theory (1956). The viscoelastic model is defined from the velocity and attenuation of dilatational and distortional waves in poroelastic materials. Its material properties are defined in terms of the elastic moduli, porosity, specific gravity, degree of saturation, and permeability of the soils. The proposed model is tested by comparing its response with the one of poroelastic materials in the case of axial and lateral harmonic loadings of one-dimensional columns. The viscoelastic model is simpler to use than poroelastic materials but yields similar results for a wide range of soils and dynamic loadings.

scholarly journals On the thermoelasticity with two porosities: asymptotic behaviour

2018 ◽  
Vol 24 (9) ◽  
pp. 2713-2725 ◽  
Author(s):  
N. Bazarra ◽  
J.R. Fernández ◽  
M.C. Leseduarte ◽  
A. Magaña ◽  
R. Quintanilla
Keyword(s):  
Asymptotic Stability ◽  
Exponential Decay ◽  
Existence And Uniqueness ◽  
Semigroup Theory ◽  
Sufficient Conditions ◽  
Uniqueness Result ◽  
Porous Structures ◽  
One Dimensional ◽  
Dimensional Version ◽  
The One

In this paper we consider the one-dimensional version of thermoelasticity with two porous structures and porous dissipation on one or both of them. We first give an existence and uniqueness result by means of semigroup theory. Exponential decay of the solutions is obtained when porous dissipation is assumed for each porous structure. Later, we consider dissipation only on one of the porous structures and we prove that, under appropriate conditions on the coefficients, there exists undamped solutions. Therefore, asymptotic stability cannot be expected in general. However, we are able to give suitable sufficient conditions for the constitutive coefficients to guarantee the exponential decay of the solutions.

Lie symmetry analysis and some exact solutions for modified Zakharov–Kuznetsov equation

2018 ◽  
Vol 32 (31) ◽  
pp. 1850383 ◽  
Author(s):  
Xuan Zhou ◽  
Wenrui Shan ◽  
Zhilei Niu ◽  
Pengcheng Xiao ◽  
Ying Wang
Keyword(s):  
Exact Solutions ◽  
Detailed Analysis ◽  
Symmetry Group ◽  
Nonlinear Waves ◽  
Lie Symmetry ◽  
Lie Symmetry Analysis ◽  
One Dimensional ◽  
Infinitesimal Generators ◽  
Lie Symmetry Method ◽  
Symmetry Method

In this study, the Lie symmetry method is used to perform detailed analysis on the modified Zakharov–Kuznetsov equation. We have obtained the infinitesimal generators, commutator table of Lie algebra and symmetry group. In addition to that, optimal system of one-dimensional subalgebras up to conjugacy is derived and used to construct distinct exact solutions. These solutions describe the dynamics of nonlinear waves in isothermal multicomponent magnetized plasmas.

scholarly journals HEAT TRANSPORT IN A THREE-DIMENSIONAL SLAB GEOMETRY AND THE TEMPERATURE PROFILE OF INGEN-HAUSZ'S EXPERIMENT

2013 ◽  
Vol 27 (13) ◽  
pp. 1350057 ◽  
Author(s):  
SHILADITYA ACHARYA ◽  
KRISHNENDU MUKHERJEE
Keyword(s):  
Temperature Profile ◽  
Three Dimensional ◽  
Initial State ◽  
Slab Geometry ◽  
Similar Nature ◽  
One Dimensional ◽  
Different Temperatures ◽  
Dimensional Version ◽  
The One ◽  
The Continuum

We study the transport of heat in a three-dimensional, harmonic crystal of slab geometry whose boundaries and the intermediate surfaces are connected to stochastic, white noise heat baths at different temperatures. Heat baths at the intermediate surfaces are required to fix the initial state of the slab in respect of its surroundings. We allow the flow of energy fluxes between the intermediate surfaces and the attached baths and impose conditions that relate the widths of Gaussian noises of the intermediate baths. The radiated heat obeys Newton's law of cooling when intermediate baths collectively constitute the environment surrounding the slab. We show that Fourier's law holds in the continuum limit. We obtain an exponentially falling temperature profile from high to low temperature end of the slab and this very nature of the profile was already confirmed by Ingen-Hausz's experiment. Temperature profile of similar nature is also obtained in the one-dimensional version of this model.

Theoretical and experimental investigations of the reflexion of normal shock waves with vibrational relaxation

1967 ◽  
Vol 30 (1) ◽  
pp. 51-64 ◽  
Author(s):  
N. H. Johannesen ◽  
G. A. Bird ◽  
H. K. Zienkiewicz
Keyword(s):  
Shock Wave ◽  
Vibrational Relaxation ◽  
Experimental Investigations ◽  
Rayleigh Line ◽  
Dimensional Problem ◽  
Density Measurements ◽  
Wave Characteristic ◽  
One Dimensional ◽  
The One ◽  
Theoretical Results

The one-dimensional problem of shock-wave reflexion with relaxation is treated numerically by combining the shock-wave, characteristic, and Rayleigh-line equations. The theoretical results are compared with pressure and density measurements in CO2, and the agreement is found to be excellent.

On a Square Packing Problem

2002 ◽  
Vol 11 (2) ◽  
pp. 113-127
Author(s):  
S. BOUCHERON ◽  
W. FERNANDEZ de la VEGA
Keyword(s):  
Packing Problem ◽  
Concentration Of Measure ◽  
Change Of Measure ◽  
Tail Probabilities ◽  
One Dimensional ◽  
Large Fluctuations ◽  
Dimensional Version ◽  
Square Packing ◽  
The One ◽  
Average Size

An instance of the square packing problem consists of n squares with independently, uniformly distributed side-lengths and independently, uniformly distributed locations on the unit d-dimensional torus. A packing is a maximum family of pairwise disjoint squares. The one-dimensional version of the problem is the classical random interval packing problem. This paper deals with the asymptotic behaviour of packings as n tends to infinity while d = 2. Coffman, Lueker, Spencer and Winkler recently proved that the average size of packing is Θ(nd/(d+1)). Using partitioning techniques, sub-additivity and concentration of measure arguments, we show first that, after normalization by n2/3, the size of two-dimensional square packings tends in probability toward a genuine limit γ. Straightforward concentration arguments show that large fluctuations of order n2/3 should have probability vanishing exponentially fast with n2/3. Even though γ remains unknown, using a change of measure argument we show that this upper bound on tail probabilities is qualitatively correct.

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