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The Bernoulli Integral for a Certain Class of Non‐Stationary Viscous Vortical Flows of Incompressible Fluid
Author(s) -
Stepanyants Y.A.,
Yakubovich E.I.
Publication year - 2015
Publication title -
studies in applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.164
H-Index - 46
eISSN - 1467-9590
pISSN - 0022-2526
DOI - 10.1111/sapm.12087
Subject(s) - inviscid flow , bernoulli's principle , vorticity , compressibility , mathematics , stream function , fluid parcel , vortex , vorticity equation , fluid dynamics , flow (mathematics) , vector field , incompressible flow , classical mechanics , scalar (mathematics) , conservative vector field , mathematical analysis , physics , mechanics , geometry , thermodynamics
It has been shown in our paper [1] that there is a wide class of 3D motions of incompressible viscous fluid which can be described by one scalar function dabbed the quasi‐potential . This class of fluid flows is characterized by three‐component velocity field having two‐component vorticity field; both these fields can depend of all three spatial variables and time, in general. Governing equations for the quasi‐potential have been derived and simple illustrative example of 3D flow has been presented. Here, we derive the Bernoulli integral for that class of flows and compare it against the known Bernoulli integrals for the potential flows or 2D stationary vortical flows of inviscid fluid. We show that the Bernoulli integral for this class of fluid motion possesses unusual features: it is valid for the vortical nonstationary motions of a viscous incompressible fluid. We present a new very nontrivial analytical example of 3D flow with two‐component vorticity which hardly can be obtained by any of known methods. In the last section, we suggest a generalization of the developed concept which allows one to describe a certain class of 3D flows with the 3D vorticity.