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THE GOVERNING PARTIAL DIFFERENTIAL EQUATION FOR ONE-DIMENSIONAL FLOW OF WATER IN A DEFORMING BODY OF SOIL (MATERIAL COORDINATES, JACOBIAN)
Abstract
A partial differential equation in the spatial (Eulerian) coordinate system, which is a fixed, stationary frame of reference, that describes the one-dimensional, isothermal, and isosaline flux of water through a one-dimensionally non-homogeneous, deforming body of soil that consists of solids, water and gas, is derived. The derivation is also based on analysis in the material (Lagrangian) coordinate system, which is a moving and deforming frame of reference. The method of transformation from one system to the other for an one-dimensional application is fully discussed. Proof is offered that for a given location the ratio of the volume fraction of the soil solids at a reference time to the volume fraction of the soil solids at a given time is equal to the Jacobian at the given location at that given time. A general partial integrodifferential equation for the one-dimensional velocity of the soil solids is derived. The correct use of Darcy's law as it applies to the flow of water relative to the solids is explained and demonstrated. Some simple approaches for the solution of the governing partial differential equation are discussed.
Subject Area
Agronomy
Recommended Citation
BAUMER, OTTO WEICHSEL, "THE GOVERNING PARTIAL DIFFERENTIAL EQUATION FOR ONE-DIMENSIONAL FLOW OF WATER IN A DEFORMING BODY OF SOIL (MATERIAL COORDINATES, JACOBIAN)" (1986). ETD collection for University of Nebraska-Lincoln. AAI8706222.
https://digitalcommons.unl.edu/dissertations/AAI8706222