There may be a convoluted way to do this. Results may not be perfect but it could be a possible solution. Just hear me out.
1) Query the lines and determine the direction in which the "it" flows.
2) If flows "Left" put a Green square via a graphic layer on the node/point where the end of the line segment is.
3) Repeat this process for each direction and apply a different colour for each direction. West = Green, East = Blue, North = Red, South = Pink.
4) When you load your application this colour schematic will be apply throughout the entire map on the fly.
5) If you are live synchronizing, then call this method to "colour the flow".
I use this similar technique for colouring my points, my points have a value, I query each point in the layer, get the value, then assign a colour using a graphic layer.
If you figure out how to use PointPaintOperation please let me know, but for now I just use a Graphic Layer.
Other than that I don't think there is a way to determine flow natively with ArcGIS Mobile.