A stagnation streamline is a streamline that ends on the surface of an object, resulting in a stagnation point. Consider now the stagnation streamline for a symmetric object, which is along the centerline as shown below. Fluid will keep on entering from the left, so I assume that at some point the fluid has to escape to either side of the object (up or down). But at the centerline we know that $v=0$. So how is this streamline physical if it does not satisfy continuity?
My initial guess is that the streamline is infinitely thin and therefore nothing is building up, however I would appreciate any additional explanation or links to resources :)