Closed vortex lines in fluid and gas

Continuous fluid and gas flows with closed vortex tubes are investigated. The circulation along the vortex line of the ratio of the density of the resultant of all forces (applied to the fluid or gas) to the density of the fluid or gas is considered. It coincides with the circulation (along the same vortex line) of the partial derivative of the velocity vector with respect to time and, therefore, for stationary flows, it is equal to zero on any closed vortex line. For non-stationary flows, vortex tubes are considered, which remain closed for at least a certain time interval. A previously unknown regularity has been discovered, consisting in the fact that at, each fixed moment of time, such circulation is the same for all closed vortex lines that make up the vortex tube. This regularity is true for compressible and incompressible, viscous (various rheologies) and non-viscous fluids in a field of potential and non-potential external mass forces. Since this regularity is not embedded in modern numerical algorithms, it can be used to verify the numerical calculations of unsteady flows with closed vortex tubes by checking the equality of circulations on different closed vortex lines (in a tube).
The expression for the distribution density of the resultant of all forces applied to fluid or gas may contain higher-order derivatives. At the same time, the expression for the partial derivative of the velocity vector with respect to time and the expression for the vector of vorticity (which is necessary for constructing the vortex line) contain only the first derivatives; which makes it possible to use new regularity for verifying the calculations made by methods of high and low orders simaltaniously.