Analytical solution for monovalent-divalent ion exchange transport in groundwater

1999 ◽  
Vol 36 (6) ◽  
pp. 1197-1201 ◽  
Author(s):  
Yee-Chung Jin ◽  
Songlin Ye
Keyword(s):  
Ion Exchange ◽  
Analytical Solution ◽  
Groundwater Flow ◽  
Coordinate System ◽  
Solute Transport ◽  
Numerical Solution ◽  
Approximate Analytical Solution ◽  
Initial Conditions ◽  
Direct Integration ◽  
Advection Velocity

An approximate analytical solution is derived for solute transport with monovalent-divalent ion exchange in saturated, steady, groundwater flow. The analytical solution is obtained by simplifying the complicated heterovalent ion exchange. Using a coordinate system that moves at the average solute advection velocity, a solution is obtained by direct integration with given boundary and initial conditions. The results agree well with a numerical solution using the original isotherm. The analytical solution presented is similar to that of monovalent-monovalent exchange, which suggests that a simple ion exchange model may be assumed for approximation.

scholarly journals On the Solution of Burgers’ Equation with the New Fractional Derivative

Open Physics ◽  
2015 ◽  
Vol 13 (1) ◽  
Author(s):  
Ali Kurt ◽  
Yücel Çenesiz ◽  
Orkun Tasbozan
Keyword(s):  
Exact Solution ◽  
Analytical Solution ◽  
Fractional Derivative ◽  
Burgers Equation ◽  
Approximate Analytical Solution ◽  
Initial Conditions ◽  
Conformable Fractional Derivative ◽  
Analysis Method ◽  
Auxiliary Parameter ◽  
Homotopy Analysis

AbstractFirstly in this article, the exact solution of a time fractional Burgers’ equation, where the derivative is conformable fractional derivative, with dirichlet and initial conditions is found byHopf-Cole transform. Thereafter the approximate analytical solution of the time conformable fractional Burger’s equation is determined by using a Homotopy Analysis Method(HAM). This solution involves an auxiliary parameter ~ which we also determine. The numerical solution of Burgers’ equation with the analytical solution obtained by using the Hopf-Cole transform is compared.

scholarly journals Bifurcation analysis of a composite cantilever beam via 1:3 internal resonance

2020 ◽  
Vol 28 (1) ◽  
Author(s):  
M. Sayed ◽  
A. A. Mousa ◽  
D. Y. Alzaharani ◽  
I. H. Mustafa ◽  
S. I. El-Bendary
Keyword(s):  
Steady State ◽  
Analytical Solution ◽  
Numerical Solution ◽  
Bifurcation Analysis ◽  
Multiple Scales ◽  
Approximate Analytical Solution ◽  
Feedback Signal ◽  
Beam Model ◽  
Signal Gain ◽  
Periodic Attractors

Abstract In this paper, we study a multiple scales perturbation and numerical solution for vibrations analysis and control of a system which simulates the vibrations of a nonlinear composite beam model. System of second order differential equations with nonlinearity due to quadratic and cubic terms, excited by parametric and external excitations, are presented. The controller is implemented to control one frequency at primary and parametric resonance where damage in the mechanical system is probable. Active control is applied to the system. The multiple scales perturbation (MSP) method is implemented to obtain an approximate analytical solution. The stability analysis of the system is obtained by frequency response (FR). Bifurcation analysis is conducted using various control parameters such as natural frequency (ω1), detuning parameter (σ1), feedback signal gain (β), control signal gain (γ), and other parameters. The dynamic behavior of the system is predicted within various ranges of bifurcation parameters. All of the stable steady state (point attractor), stable periodic attractors, unstable steady state, and unstable periodic attractors are determined efficiently using bifurcation analysis. The controller’s influence on system behavior is examined numerically. To validate our results, the approximate analytical solution using the MSP method is compared with the numerical solution using the Runge-Kutta (RK) method of order four.

Field Observations of Tracer Dispersion Under Natural Flow Conditions in an Unconfined Sandy Aquifer

1979 ◽  
Vol 14 (1) ◽  
pp. 1-18 ◽  
Author(s):  
E.A. Sudicky ◽  
J.A. Cherry
Keyword(s):  
Analytical Solution ◽  
Groundwater Flow ◽  
Solute Transport ◽  
Deterministic Model ◽  
Test Site ◽  
Chloride Salt ◽  
Transport Parameters ◽  
Deterministic Models ◽  
Wide Range ◽  
Sandy Aquifer

Abstract ABSTRACT An exceptionally detailed field determination of the solute transport parameters was performed in an unconfined sandy aquifer near an abandoned landfill at the Canadian Forces Base at Borden, Ontario. The test site is located above the contaminant plume originating from the landfill. The aquifer consists of slightly stratified sands with minor laminations. A chloride salt solution was injected into a two m3 volume of aquifer about one meter below the water-table and then migration of the tracer occurred under the natural hydraulic gradient. The migration of the chloride pulse was monitored in detail using a three-dimensional array of bundle-type multilevel samplers. Hydraulic head measurements in the zone of transport were obtained from a network of miniature piezometers. The test results demonstrated the influence of zones of local aquifer heterogeneity on solute migration rates and the ability of a porous medium to disperse solutes in these zones. Different rates of groundwater flow between a fast and slow transport zone caused the pulse to split into two halves. Each half was found to be Gaussian in shape in accord with the classical theory of solute transport. The measured chloride distributions closely fit an analytical solution of the advection-dispersion equation. Dispersivity values for chloride obtained from the analytical solution increased with mean travel distance in the groundwater flow domain, which suggests that calibration of a deterministic model at one spatial scale may lead to erroneous predictions when applied to a different scale. From this it is concluded that, if deterministic models are to yield useful predictions of contaminant migration, it will be necessary to establish scaling functions from studies of the variability of transport parameters in a wide range of hydrogeological settings.

Reliable analysis for time-fractional nonlinear differential difference equations

2013 ◽  
Vol 3 (4) ◽  
Author(s):  
Mithilesh Singh ◽  
R. Prajapati
Keyword(s):  
Analytical Solution ◽  
Difference Equations ◽  
Present Method ◽  
Approximate Analytical Solution ◽  
Initial Conditions ◽  
Time Derivative ◽  
Explicit Solutions ◽  
Initial Condition ◽  
Hyperbolic Functions ◽  
Fractional Time Derivative

AbstractIn this study, we used HPM to determine the approximate analytical solution of nonlinear differential difference equations of fractional time derivative. By using initial conditions, the explicit solutions of the coupled nonlinear differential difference equations have been derived which demonstrate the effectiveness, potentiality and validity of the method in reality. The present method is very effective and powerful to determine the solution of system of non-linear DDE. The numerical calculations are carried out when the initial condition in the form of hyperbolic functions and the results are shown through the graphs.

A General Correlation for Pool Film Boiling Heat Transfer From a Horizontal Cylinder to Subcooled Liquid: Part 1—A Theoretical Pool Film Boiling Heat Transfer Model Including Radiation Contributions and Its Analytical Solution

1990 ◽  
Vol 112 (2) ◽  
pp. 430-440 ◽  
Author(s):  
A. Sakurai ◽  
M. Shiotsu ◽  
K. Hata
Keyword(s):  
Heat Transfer ◽  
Heat Transfer Coefficient ◽  
Theoretical Model ◽  
Analytical Solution ◽  
Transfer Coefficient ◽  
Numerical Solution ◽  
Boiling Heat Transfer ◽  
Approximate Analytical Solution ◽  
Horizontal Cylinder ◽  
Film Boiling

A rigorous numerical solution of a theoretical model based on laminar boundary layer theory for pool film boiling heat transfer from a horizontal cylinder including the contributions of liquid subcooling and radiation from the cylinder was obtained. The numerical solution predicted accurately the experimental results of pool film boiling heat transfer from a horizontal cylinder in water with high radiation emissivity for a wide range of liquid subcooling in the range of nondimensional cylinder diameters around 1.3, where the numerical solution was applicable to the pool film boiling heat transfer from a cylinder with negligible radiation emissivity. An approximate analytical solution for the theoretical model was also derived. It was given by the sum of the pool film boiling heat transfer coefficient if there were no radiation and the radiation heat transfer coefficient for parallel plates multiplied by a nondimensional radiation parameter similar to the expression for saturated pool film boiling given by Bromley. The approximate analytical solution agreed well with the rigorous numerical solution for various liquids of widely different physical properties under wide ranges of conditions.

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