Can all 2D solenoidal fluid flows be given in terms of a stream function? Is it true that for any 2D solenoidal fluid flow $\mathbf{u}$ ( i.e one with zero divergence, $\mathbf{\nabla\cdot u  = 0}$ ), that $\mathbf{u}$ may be obtained from a stream function?
I had always thought a stream function could only be used for steady flows.
 A: The question that I asked was not that well posed. Stream functions are used for two dimensional fluid velocity fields, where their is a zero component for the fluid velocity in the third spatial dimension.
According to a possibly less than accessible reference, that I have found, ANY two dimensional solenoidal flow field $\mathbf{u}$ can be derived from  a scalar stream function $\psi$.
For a two dimensional solenoidal flow $\mathbf{u}$ which hence obeys , 
\begin{align}
\mathbf{\nabla\cdot u}&= 0\\
\text{or      }  \frac {\partial  u_1 } {\partial  x   } +  \frac{\partial  u_2   } { \partial  y  }&=0
\end{align}
we can have $\mathbf{u}$ given in terms of a stream function $\psi$, as
\begin{align}
u_1&= \frac {\partial  \psi } {\partial  y   }     \\
u_2&=  -\frac{\partial \psi   } { \partial  x  }
\end{align}
For the following quote, see Reference 1 ( a margin note on pg.15 )

it can be shown that the equation $\mathbf{\nabla\cdot u= 0}$
  guarantees the existence of such a $\psi$

Reference 1: Mathematical Methods and Fluid Mechanics, Unit 5 Kinematics of fluids, The Open University, Walton Hall, Milton Keynes, (1984). Reprinted 1992.
