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首页> 外文期刊>Zeitschrift fur Angewandte Mathematik und Mechanik >Analytical determination of seeping soil slopes of a constant exit gradient
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Analytical determination of seeping soil slopes of a constant exit gradient

机译：恒定出口坡度渗流土坡的解析确定

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Soil slopes satisfying the condition of a constant exit gradient (constant Darcian velocity) and the seepage face (isobaricity) condition are found by complex analysis. For an empty drainage trench, soil is infinitely deep and the flow domain is bounded by two branches of a phreatic surface and the trench contour. The problem is solved by conformal mapping of the Zhukovskii domain (half-plane) onto the hodograph domain (lune), The ultimately stable seepage face and the inflow rate are determined as functions of a specified exit gradient. With decreasing of the gradient the trench flattens. If the gradient is 1, the depth-width aspect ratio of the trench reaches 0.21. Similar conformal mapping of a lune in the hodograph plane on a half-strip in the complex potential plane is used to solve the problem of a phreatic surface flow from a soil channel, whose equipotential contour satisfies the condition of constant "entrance" gradient. For a dam slope, the flow domain is underlain by an impermeable bottom and the hodograph is a circular triangle with a unit-gradient circle as the slope image. The Polubarinova-Kochina method of analytical differential equations is modified to reconstruct the complex coordinate and the complex potential as a function of an auxiliary variable. The resulting slope of unit exit gradient has a depth-width ratio of 1.22. [References: 41]
机译：通过复杂分析发现满足恒定出口梯度（恒定达西速度）和渗流面（等压）条件的土壤坡度。对于空的排水沟，土壤无限深，流域由潜水面和沟渠轮廓的两个分支限制。通过将Zhukovskii域（半平面）保形映射到Hodograph域（管形）上，可以解决该问题。最终稳定的渗流面和流入速率取决于特定出口梯度的函数。随着梯度的减小，沟槽变平。如果梯度为1，则沟槽的深度-宽度纵横比达到0.21。在复数势平面中的半条带上，全息图平面中月牙的类似共形映射用于解决来自土壤通道的潜水表面流的问题，该土壤流的等势轮廓满足恒定“入口”梯度的条件。对于大坝坡度，流域被不可渗透的底部所覆盖，并且全息图是带有单位渐变圆作为坡度图像的圆形三角形。修改了Polubarinova-Kochina解析微分方程方法，以根据辅助变量重建复数坐标和复数势。单位出口梯度的最终斜率的深宽比为1.22。 [参考：41]

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