A particular analytical solution of a steady-state flow of a groundwater mound

Hydromechanical calculation of symmetrical potential flow with a free surface and sewer (drainage)

“…For special cases (for example, b = ϕ A = 0.3, 0.5, and 0.8 at H = 1 and q = 1), the data computed by the suggested calculated dependences coincide with their exact values including these for sewer widths l = 0.79, 0.75, and 0.84 [3].…”

“…It is known [1][2][3][4] that the Zhukowski complex is determined by the dependence (2) where W = ϕ A + iψ is the complex potential with coor dinates of head function ϕ and stream function ψ accepted in the right half plane in the form of the region delineated by the profile of another "inverse ellipse,"…”

“…Here, θ 1 and θ 2 are the coordinates of the region of the Zhukowski complex: θ 2 is the piezometric height (equal pressure lines, or isobars), and θ 1 is the conju gated pressure height (lines orthogonal to them) [1][2][3][4]; b is the half width of the impermeable base; and ϕ A is the head function at central point A (Fig. 1).…”

Solvability of free boundary problems for steady groundwater flow

“…Boundary conditions (2.4) and (2.5) on the fixed boundary ∂B could be interpreted in the sense of traces if the distributions of subsets ∂B N and ∂B D on ∂B were sufficiently regular. To avoid any regularity assumptions with respect to these distributions let us introduce a subspace W 1,2 0 (B, ∂B D ) ⊂ W 1,2 (B) as the closure of the set of smooth functions C ∞ 0 B\∂B D with compact support in B\∂B D . Then condition (2.5) implies that p(•) − P (•) ∈ W 1,2 0 (B, ∂B D ) for some given function P ∈ W 1,2 (B).…”

“…Most of them are obtained by means of the conformal mapping method developed by G. Kirchhoff and N. Zhukovsky (see, for instance, [19] or [6], Chapter 7). This method is still in use in engineering hydrology [2,11]. Any solution of this kind can be regarded as a particular solvability result.…”