Applications of the Separation of Variables Method and Duhamel’s Principle to Instantaneously Released Point-Source Solute Model in Water Environmental Flow

• Published in: Sustainability Vol. 16; no. 16; p. 6912
• Main Authors: Gao, Ran, Gao, Juncai, Chu, Linlin
• Format: Journal Article
• Language: English
• Published: Basel MDPI AG 12.08.2024
• Subjects: Air pollution; Boundary conditions; Environmental impact; Flow velocity; Ordinary differential equations; Partial differential equations; Variables; Water; Water pollution
• ISSN: 2071-1050; 2071-1050
• DOI: 10.3390/su16166912

The transport–diffusion problem of point-source solutes in water environmental flows is an important issue in environmental fluid mechanics, with significant theoretical and practical implications for sustainable development and the ecological management and environmental protection of water. This study presents a model for instantaneously released multi-point-source solutes, utilizing the separation of variables method and Duhamel’s principle to solve classical mathematical physics equations. The zeroth-order and first-order concentration moment equations, which are crucial for predicting the cross-sectional average concentration of instantaneously released point-source solutes, are systematically addressed. The accuracy of the analytical results is confirmed by comparing them with the relevant literature. Furthermore, a general discussion is provided based on the study’s findings (including an ideal physical model of Couette flow), and an analytical solution (a recursive relationship) for higher-order concentration moments is deduced. Finally, this study quantitatively discusses downstream environmental ecological effects by examining the movement of released point-source solute centroids in the river, illustrating that the time needed for the released point-source solute to have an environmental–ecological impact downstream of the river is dependent on the initial release location. Under the specified engineering parameters, for the release location at the bottom boundary point of the channel (z0 = 0 m), the midpoint (z0 = 5 m), and the water-surface point (z0 = 10 m), the time for additional displacement of released solute centroid to reach the asymptotic value in three cases is 4.0 h, 1.0 h, and 4.5 h; the asymptotic values are approximately −0.087 km, 0.012 km, and 0.055 km, respectively. These results not only correspond with the conclusions of previous research but also provide a more extensive range of numerical results. This study establishes the groundwork for theoretical research on more complex water environmental flow models and provides a theoretical basis for engineering computations aimed at contributing to the environmental management of rivers and lakes.