A numerical model of a nonlinear and degenerate diffusion equation on connected graphs, arising in a diversity of industrial applications, is presented. This paper shows that concurrently using a staggered finite volume spatial discretization with an analytical solution-based flux evaluation method and an operator-splitting technique computes stable and physicallyconsistent numerical solutions to the equation. A demonstrative application example of the numerical model to moisture dynamics in a hypothetical non-woven fibrous strip network under an evaporative environment is presented in order to show its versatility. Mathematical issues to be addressed in a future for better comprehending the model are finally discussed.
Rainwater use and rooftop vegetation can be considered as one of the primal means for mitigating environmental problems encountered in urban areas, such as urban flooding and heat island. However, urban areas are too crowded for installing these facilities. In this paper, Rooftop vegetation with rainwater storage is created that is more flexibly than existing ones because of their less restriction on some constraints. It consists of two containers: an upper container for vegetation and a lower one for storing rainwater. Stored rainwater is supplied to the upper one by bottom irrigation system which consists of two kinds of fibrous sheets: one is horizontally placed on the upper container bottom for distributing the water into the soils and the other is connected vertically under the former for transporting the stored water to upper one. Some experiments are conducted for examining the ability of regulating soil moisture. Experimental results indicate that moisture movements to the upper in vertical fibrous decrease with increase in the distance from rainwater surface to the bottom of upper container, which affects moisture movements in horizontal sheet in quantity and the speed. Therefore, it is suggested that the present bottom irrigation can regulate soil moisture.
Comprehension of moisture dynamics in fibrous sheets is indispensable in a wide variety of industrial areas. This paper proposes a practical mathematical model, which is referred to as the 2-D extended porous medium equation (PME), to physically describe moisture dynamics in fibrous sheets under evaporative environment. A numerical method, which is referred to as the 2-D dual-finite volume method (DFVM), to approximate its solutions in a stable manner is also presented so that the moisture dynamics is reasonably simulated. The 2-D DFVM, which can optionally be equipped with isotone numerical fluxes, is examined with test cases to show its satisfactory accuracy and versatility. The parameters and coefficients involved in the mathematical model for a non-woven fibrous sheet are identified with laboratory experiments. Numerical simulation of moisture dynamics in the horizontally or vertically placed sheet is performed as a demonstrative application example of the present model and the numerical method.
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