fluid dynamics: sketching streamlines of velocity field when there is only one non-zero velocity component

Question

I have been asked to sketch the streamlines in the $x_1$$x_2$-plane for the two-dimensional field: $$v=(x_1x_2,0,0)$$ All the examples I have seen of this kind of question use the $v_1$ and $v_2$ components to create a ratio and use this to work out the type of graph needed to be plotted. However, in this example, there is only one component that is non-zero and so I don't know how to go about working out the sketch. I've worked out that the flow is not irrotational, since its vorticity is not zero and also that it is not solenoidal as its divergence is not zero either. Is that right? Where do I go from here? Any help would be greatly appreciated.

Answer 1

The field is always directed along the first coordinate axis. Therefore, its streamlines are lines parallel to that axis. Very easy to sketch. If you wish to connect this with the slope-from-the-ratio method, consider that the ratio $0/(x_1x_2)$ is zero. So, the slope of streamlines is everywhere zero.

Sure, the points with $x_1x_2=0$ are exceptional: they are stationary points of the field. You may want to mark them somehow to indicate this fact.