The Significance of the Net Transfer of Viscous Stress Energy and the Local Production of Kinetic Energy in Stationary Soil Water Flow

Abstract

Abstract For stationary saturated flow of water through a homogeneous porous medium non-equilibrium thermodynamics provides a scheme in which measurable fluxes and forces are connected to each other by linear homogeneous relationships. This set of relationships infers the existence of coupling phenomena and constitutes the main virtue of this branch of science, viz. the then proposed equality of “twin” cross-coefficients. Especially for systems with intricate geometry this way of treating cross-phenomena seems to be promising. To come to a correct and useful set of fluxes and forces one may treat a small subsystem within the liquid phase and then integrate over a certain volume of the porous system. This local treatment brings up the two terms of concern, − ∇ · τ · ν and − d 1/2ρν 2 / dt , in the dissipation function. For stationary flow in a medium with a homogeneous phase distribution these two terms can be shown to disappear upon integration, according to statistical reasonings as used by Mandl [3]. This then leads straightforward to the proposal of a linear relationship between the filter flux and the gradient of the hydraulic head. The existence and the magnitude of the two terms of concern in the dissipation function before integration predicts the lack of a linear relationship between the 1Scal flow velocity and the local gradient of the hydraulic head. For non-homogeneous, non-saturated or non-stationary flow, the two terms will not disappear upon integration. A first estimate of the order of magnitude of the two terms relative to the other terms in the dissipation function is made for isothermal stationary saturated flow in a non-homogeneous porous medium by calculations based on the flow pattern between non-parallel plane walls. These calculations show that for systems with a change in thickness of water layers from 100 μ to 50 μ over a distance of 10 cm the ratio of the integrated value of − ∇ · τ · ν to the integrated value of — ν · ∇ p (which is the main term of the dissipation function) is only 10 −1 . The ratio of the integrated value of − d 1/2ρν 2 / dt to the main term is of the same order of magnitude, assuming a filter flux of 10 −3 cm/sec to be maintained at a moisture content of 0.50 at the point where the water layer thickness is 100 μ.