Scalar description of three-dimensional vortex flows of incompressible fluid

An essential progress in the investigation of flows of incompressible fluid may be achieved with the help of stream-function. Flow description by means of one scalar stream-function is much simpler than the description based on the three-dimensional vector field. Many interesting and physically important problems were solved by this way. However, the traditional usage of a stream-function is restricted by the assumption of certain symmetry of the flow: the method is applicable only to two-component flows, i.e. when the corresponding velocity field is effectively two-dimensional, e.g., plane flow. This restriction essentially limits the range of applicability of such approach. In this paper we propose another approach, also based on the introduction of only one scalar function. However, we show that with this scalar function a wide class of non-stationary three-dimensional flows can be described. This class of flows includes both potential and vortex flows. In the latter case, the corresponding vorticity field may be two-component, in general. Characteristic features of such flows are described in details. Particular examples of flows are presented in the explicit form