Unsteady motions with steady streamlines (Q1825073)

We study unsteady motions whose streamlines are steady. The velocity field for such motions is given by \(v(x,t)=v(x,t)s(x)\), \(v(x,t)>0\), where x is the spatial point, t is the time, and s is the steady unit vector tangent to the streamline. Evidently \(\partial s/\partial t=0\). For the given velocity field v(x,t) to be endowed with steady streamlines it is necessary and sufficient that (*) \(v\times \partial v/\partial t=0\) or equivalently (**) \(\partial v(x,t)/\partial t=\lambda (x,t)v(x,t)\). By an application of the Helmholtz-Zorwawski theorem it is shown that (*) or (**) is a necessary and sufficient condition for the streamlines to be material lines. We come to perhaps the most important property of motions with steady streamlines. The streamlines and the path lines of the fluid particles coincide if and only if the streamlines are steady. We prove the following statements concerning unsteady motions with steady streamlines. Theorem 1. If \(\text{grad}\lambda(x,t)=0\), where \(\lambda(x,t)\) is determined by (***) \(\partial v(x,t)/\partial t=\lambda (x,t)v(x,t)\), the motion \(v(x,t)\) is a d'Alembert motion. Conversely if \(v(x,t)\) is a d'Alembert motion, then \(\text{grad}\lambda(x,t)=0.\) Theorem 2. An isochoric irrotational motion with steady streamlines must be a d'Alembert motion. Theorem 3. A rotational screw motion with steady streamlines, not necessarily an isochoric motion, must be a d'Alembert motion. Theorem 4. A rotational motion, not necessarily an isochoric motion, with steady streamlines and vortex lines (such motions include screw motions), must be a d'Alembert motion or a complex lamella motion, or both. If the motion is not a d'Alembert motion, then at any fixed time \(t=t_ 0\) the vortex lines must lie on the surfaces of the family \(\lambda(x,t_ 0)=c\) (the family being determined by continuous variation of c), here \(\lambda\) is given by (***). If, further, this motion is isochoric, the streamlines also lie on these surfaces, and the streamlines and vortex lines determine a family of orthogonal curves on each surface. Theorem 5 is essentially a specialization of the last statement of Theorem 4. Theorem 5. Let \(v(x,t)\) be an isochoric rotational circulation-preserving motion with steady streamlines and vortex lines which is not a d'Alembert motion. The motion is complex-lamellar. Furthermore, at any fixed time \(t=t_ 0\) the streamlines and vortex lines are respectively geodesics and geodesic parallels on the surfaces \(\lambda(x,t_ 0)=c\), where \(\lambda\) is given by (***). The vorticity magnitude \(\omega_ b\) is given by \({\dot\omega}{}_ b=\partial \omega_ b/\partial t+v\cdot \text{grad} \omega_ b=\theta v\omega_ b\), where \(\theta\) is the geodesic curvature of the vortex lines on the surfaces.