Exact solutions to generalized plane Beltrami–Trkal and Ballabh flows
- Authors: Prosviryakov E.Y.1
-
Affiliations:
- Institute of Engineering Science, Urals Branch, Russian Academy of Sciences
- Issue: Vol 24, No 2 (2020)
- Pages: 319-330
- Section: Articles
- URL: https://journals.eco-vector.com/1991-8615/article/view/41991
- DOI: https://doi.org/10.14498/vsgtu1766
- ID: 41991
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Abstract
Nonstationary plane flows of a viscous incompressible fluid in a potential field of external forces are considered. An elliptic partial differential equation is obtained, with each solution being a vortex flow stream function described by an exact solution to the Navier–Stokes equations. The obtained solutions generalize the Beltrami–Trkal and Ballabh flows. Examples of such new solutions are given. They are intended to verify numerical algorithms and computer programs.
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\centerline{\bf\large Introduction} \smallskip Starting from the studies by Gromeka and Lamb~[1, 2] proposing a new method of writing the Euler equations, a method for integrating the fluid motion equations began to be developed. The essence of this method is the rearrangement of the initial equations to the form convenient for integration. As applied to the Navier–Stokes equations, this line of research is discussed in studies where new forms of writing the equations make it possible to obtain previously unknown invariants and hidden symmetries of the constitutive equations~[3–10]. One of the methods for representing the motion equation (the Aristov–Pukhnachev method~[6, 8, 9]) has been introduced to computational fluid dynamics~[8, 9]. The numerical solutions of the Navier–Stokes axisymmetric equations were tested by conventional procedures~[8, 9], which took no account of the invariant Helmholtz theorems and their extensions~[10]. Few nontrivial exact solutions to the Navier–Stokes equations have been known so far~[11–25]. The very notion of exact solution is unsettled and expanding~[11]. It seems obvious that the exact solutions to the Navier–Stokes equations, which offer new problem statements in terms of different areas of mathematics, mechanics, and physics~[11–25], are of the greatest interest. The main difficulty in the analytical and numerical integration of the fluid motion equations lies in the absence of a clear relation of pressure to the velocity vector components. The evolutionary equation relating pressure to the velocity components has yet to be known~[26]. Let us now illustrate the ensuing difficulties by plane flow. If an arbitrary function is given on a plane and viewed as a stream function, the velocity components calculated via the partial derivatives of this function will satisfy the continuity equation. We then can substitute these components into the Navier–Stokes equation, thus arriving at an equation for determining the gradient of pressure×
About the authors
Eugenii Yurevich Prosviryakov
Institute of Engineering Science, Urals Branch, Russian Academy of Sciences
Email: evgen_pros@mail.ru
Doctor of physico-mathematical sciences, no status
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