Scaling‐equivalent rotating flows,ZAMM - Journal of Applied Mathematics and Mechanics

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Scaling‐equivalent rotating flows
ZAMM - Journal of Applied Mathematics and Mechanics ( IF 2.3 ) Pub Date : 2020-09-17 , DOI: 10.1002/zamm.201900229
Eugen Magyari ₁

Affiliation

Three outstanding rotating disk flows described by an exact solution of the Navier–Stokes equations are revisited in this paper. The purpose is to find out to what extent the corresponding boundary value problems can be mapped on each other by scaling transformations. The three addressed, and seemingly basically different axisymmetric flows are (A) the flow induced by a rough rotating disk, (B) the flow induced by a simultaneously rotating and radially stretching disk, and (C) the classical von Kármán swirl. The main results of the paper can be summarized as follows. (i) The continuous set of solutions of the problem (A) corresponding to

(λ = 0, η > 0)

is scaling‐equivalent to the solution of von Kármán, (C), (ii) Every given solution of the problem (A) with

(λ > 0, η \geq 0)

generates by a suitable scaling transformation a unique solution of (B) with

c > c_{c r i t} = 2.3848

, (iii) every given solution of the problem (B) with a

c > c_{c r i t}

can generate a continuous set of solutions of the problem (A), (iv) in the stretching dominated subcritical range

c < c_{c r i t}

of (B), no scaling‐equivalent solutions of the three problems exist. All the above results are also valid for the magnetohydrodynamic (MHD) extensions of the problems (A), (B), and (C). For these cases an approximate solution could be found in a closed analytical form which furnishes for large values of the magnetic parameter M highly accurate results.

中文翻译：

标度等效旋转流

本文将重述由Navier–Stokes方程的精确解所描述的三种出色的旋转盘流。目的是找出通过缩放变换可以将对应的边界值问题相互映射到什么程度。三种已解决且看似基本不同的轴对称流是（A）由粗糙旋转圆盘引起的流，（B）由同时旋转并径向拉伸的圆盘引起的流，以及（C）经典的vonKármán旋流。本文的主要结果可归纳如下。（i）对应于（A）的问题的连续解集