# Does a non-changing fluid streamline imply steady state

If a fluid streamline does not change in time, could still the velocity vector change its modulus (but not its direction) in time? The equations of a streamline are

$\frac{dx_i(s)}{ds}=u_i$, thus the modulus of $\mathbf{u}$ would be

$|\mathbf{u}|=\sqrt{\sum_i \left(\frac{dx_i(s)}{ds} \right)^2 }$

and then if the streamline is the same the modulus will stay the same. Does this sound correct?

## 1 Answer

Consider inviscid fluid in a pipe, with a piston at one end of the pipe. When the piston moves with a certain velocity, the fluid in front of it moves as a single block with the same velocity (assuming incompressible flow). In this flow the streamlines are all straight and parallel to the axis of the pipe. Now suppose that the piston is accelerating. The streamlines still remain the same: straight and parallel to pipe's axis. However the modulus of velocity (i.e. the speed) of the flow increases with time.

• Will this scenario change when the viscosity is no longer zero. Also, what if the pipe is curved? – user7641 Apr 20 '17 at 15:01
• You may consider fully developed steady viscous flow in a pipe, straight or curved. The streamlines are parallel to pipe's axis and again the same conclusion holds. – Deep Apr 21 '17 at 4:57
• Thank you for commenting... you said 'steady'. Would this mean $|\mathbf{u}|=\mbox{const}$. If the modulus is not constant in this case, could you provide a reference so I can read up more? – user7641 Apr 21 '17 at 14:04
• It seems the pipe is crucial to achieve what you said – user7641 Apr 21 '17 at 14:57
• @user7641 I am sorry, I should not have said steady. To increase speed you must have acceleration, but not so much acceleration that the flow becomes unstable and streamlines do not remain parallel to pipe's axis. You can achieve the same effect with flow between parallel plates, in which one plate accelerates w.r.t the other. I do not know of any reference, this is something you can think up yourself. You should read general books on fluid mechanics, for eg. Fluid Mechanics by F. M. White. – Deep Apr 22 '17 at 3:51