Browsing by Author "Dilip Kumar Jaiswal"

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Analytical solution for transport of pollutant from time-dependent locations along groundwater
(Elsevier B.V., 2022) Dilip Kumar Jaiswal; Naveen Kumar; Raja Ram Yadav
The present work derives analytical solutions of advection–dispersion equation (ADE) with temporal coefficients, and a pollutant's point source moving linearly along the axis of a one-dimensional semi-infinite domain. The source is considered a varying and a uniform pulse source, respectively. The dispersion of pollutant originating from a varying pulse source may be supposed to occur along groundwater flow domain, and that from a uniform pulse source in an open medium like air or along a river flow. The location of the input concentration that is the pollutant's concentration emanating from the source in an open medium or that reaching the groundwater domain being infiltrated from its source on the ground, is considered moving linearly along the flow direction. The motion of the source is described through an asymptotically increasing temporal function. The illustration of the analytical solution clearly reflects this feature. It also renders that the concentration pattern of the proposed solution is proximal to that of an existing solution obtained with the stationary source. The pertinent existing solutions may also be derived from the proposed solutions. The proposed solutions are found approximate but it is also found that the error of approximation of one of them is too small to have any effect on the concentration pattern. To get these solutions, firstly, the moving source is reduced into a stationary source at the origin, then the governing equations including the ADE, are made free from the three temporal functions, one occurring in the time-dependent position of the source, and the other two as the coefficients of the ADE. In this process, three new position variables, and a new time variable are introduced using as many coordinate transformations. Then the Laplace Integral Transformation Technique (LITT) is used to get the final solutions. The solution in Laplacian domain with uniform pulse source is obtained as a special case of that with the varying pulse source. © 2022 Elsevier B.V.
Analytical solutions for temporally and spatially dependent solute dispersion of pulse type input concentration in one-dimensional semi-infinite media
(2009) Dilip Kumar Jaiswal; Atul Kumar; Naveen Kumar; R.R. Yadav
A linear advection-diffusion equation with variable coefficients in a one-dimensional semi-infinite medium is solved analytically using a Laplace transformation technique, for two dispersion problems: temporally dependent dispersion along a uniform flow and spatially dependent dispersion along a non-uniform flow. Uniform and varying pulse type input conditions are considered. The variable coefficients in the advection-diffusion equation are reduced into constant coefficients with the help of two transformations which introduce new space and time variables, respectively. It is observed that the temporal dependence of increasing nature causes faster solute transport through the medium than that of decreasing nature. Similarly the effect of inhomogeneity of the medium on the solute transport is studied with the help of a function linearly interpolated in a finite space domain. © 2009 International Association for Hydraulic Engineering and Research, Asia Pacific Division.
Analytical solutions of one-dimensional advection- diffusion equation with variable coefficients in a finite domain
(2009) Atul Kumar; Dilip Kumar Jaiswal; Naveen Kumar
Analytical solutions are obtained for one-dimensional advection-diffusion equation with variable coefficients in a longitudinal finite initially solute free domain, for two dispersion problems. In the first one, temporally dependent solute dispersion along uniform flow in homogeneous domain is studied. In the second problem the velocity is considered spatially dependent due to the inhomogeneity of the domain and the dispersion is considered proportional to the square of the velocity. The velocity is linearly interpolated to represent small increase in it along the finite domain. This analytical solution is compared with the numerical solution in case the dispersion is proportional to the same linearly interpolated velocity. The input condition is considered continuous of uniform and of increasing nature both. The analytical solutions are obtained by using Laplace transformation technique. In that process new independent space and time variables have been introduced. The effects of the dependency of dispersion with time and the inhomogeneity of the domain on the solute transport are studied separately with the help of graphs. © Printed in India.
Analytical solutions to one-dimensional advection-diffusion equation with variable coefficients in semi-infinite media
(2010) Atul Kumar; Dilip Kumar Jaiswal; Naveen Kumar
In the present study one-dimensional advection-diffusion equation with variable coefficients is solved for three dispersion problems: (i) solute dispersion along steady flow through an inhomogeneous medium, (ii) temporally dependent solute dispersion along uniform flow through homogeneous medium and (iii) solute dispersion along temporally dependent flow through inhomogeneous medium. Continuous point sources of uniform and increasing nature are considered in an initially solute free semi-infinite medium. Analytical solutions are obtained using Laplace transformation technique. The inhomogeneity of the medium is expressed by spatially dependent flow. Its velocity is defined by a function interpolated linearly in a finite domain in which concentration values are to be evaluated. The dispersion is considered proportional to square of the spatially dependent velocity. The solutions of the third problem may help understand the concentration dispersion pattern along a sinusoidally varying unsteady flow through an inhomogeneous medium. New independent variables are introduced through separate transformations, in terms of which the advection-diffusion equation in each problem is reduced into the one with the constant coefficients. The effects of spatial and temporal dependence on the concentration dispersion are studied with the help of respective parameters and are shown graphically. © 2009 Elsevier B.V. All rights reserved.
Discussion of "analytical Solutions for Advection-Dispersion Equations with Time-Dependent Coefficients" by Baoqing Deng, Fei Long, and Jing Gao
(American Society of Civil Engineers (ASCE), 2020) Dilip Kumar Jaiswal; Atul Kumar; Naveen Kumar
[No abstract available]
Fuzzy model: Time dependent dispersion in rivers
(2009) Vinay Singh; Atul Kumar; Dilip Kumar Jaiswal
In this paper, two dimensional dispersion equation is considered with time dependent along uniform flow. The solution of time dependent dispersion equation is converted in fuzzy environment. The fuzzy arithmetic used to simulate the fuzzy relation in modeling river water qual-ity. The parameters of two dimensional dispersion equation of water quality model are assumed as trapezoidal fuzzy numbers. From fuzzy model the concentrations can be obtained by corresponding to the specified α-cut technique and arithmetic operations of trapezoidal fuzzy numbers. Solution of Fuzzy model is compared with determinate solution of dispersion equation. Copyright © 2009 by IICAI.
One-dimensional solute dispersion along unsteady flow through a heterogeneous medium, dispersion being proportional to the square of velocity; [Dispersion de soluté à une dimension dans un écoulement non permanent à travers un milieu hétérogène, la dispersion étant proportionnelle au carré de la vitesse]
(2012) Atul Kumar; Dilip Kumar Jaiswal; Naveen Kumar
One-dimensional solute transport, originating from a continuous uniform point source, is studied along unsteady longitudinal flow through a heterogeneous medium of semi-infinite extent. Velocity is considered as directly proportional to the linear spatially-dependent function that defines the heterogeneity. It is also assumed temporally dependent. It is expressed in both the independent variables in degenerate form. The dispersion parameter is considered to be proportional to square of the velocity. Certain new independent variables are introduced through separate transformations to reduce the variable coefficients of the advection-diffusion equation to constant coefficients. The Laplace Transformation Technique (LTT) is used to obtain the desired solution. The effects of heterogeneity and unsteadiness on the solute transport are investigated. © 2012 Copyright 2012 IAHS Press.