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Integration of a deep fluid equation with a free surface
Theoretical and Mathematical Physics ( IF 1 ) Pub Date : 2020-03-01 , DOI: 10.1134/s0040577920030010
V. E. Zakharov

We show that the Euler equations describing the unsteady potential flow of a two-dimensional deep fluid with a free surface in the absence of gravity and surface tension can be integrated exactly under a special choice of boundary conditions at infinity. We assume that the fluid surface at infinity is unperturbed, while the velocity increase is proportional to distance and inversely proportional to time. This means that the fluid is compressed according to a self-similar law. We consider perturbations of a self-similarly compressible fluid and show that their evolution can be accurately described analytically after a conformal map of the fluid surface to the lower half-plane and the introduction of two arbitrary functions analytic in this half-plane. If one of these functions is equal to zero, then the solution can be written explicitly. In the general case, the solution appears to be a rapidly converging series whose terms can be calculated using recurrence relations.

中文翻译:

深度流体方程与自由表面的积分

我们表明,在没有重力和表面张力的情况下,描述具有自由表面的二维深部流体的不稳定势流的欧拉方程可以在无限远边界条件的特殊选择下精确积分。我们假设无限远的流体表面是未受干扰的,而速度的增加与距离成正比,与时间成反比。这意味着流体根据自相似定律被压缩。我们考虑自相似可压缩流体的扰动,并表明在将流体表面共形映射到下半平面并在该半平面中引入两个任意解析函数后,可以准确地分析描述它们的演化。如果这些函数之一等于零,则可以显式地编写解决方案。
更新日期:2020-03-01
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