[Physics] Why will fluid particles always move along streamlines in a steady flow

flowfluid dynamics

It is stated at numerous places that in a steady flow, all particles move along the streamlines, i.e. streamlines and pathlines coincide.

For example –

  1. WIkipedia states-

In steady flow (when the velocity vector-field does not change with time), the streamlines, pathlines, and streaklines coincide. This is because when a particle on a streamline reaches a point, $a_{0}$, further on that streamline the equations governing the flow will send it in a certain direction $\vec {x}$. As the equations that govern the flow remain the same when another particle reaches $a_{0}$ it will also go in the direction $\vec {x}$. If the flow is not steady then when the next particle reaches position $a_{0}$ the flow would have changed and the particle will go in a different direction.

  1. NPTEL–

In a steady flow, all the four basic line patterns are identical. Since, the velocity at each point in the flow field remains constant with time, consequently streamline shapes do not vary. It implies that the particle located on a given streamline will always move along the same streamline. Further, the consecutive particles passing through a fixed point in space will be on the same streamline. Hence, all the lines are identical in a steady flow. They do not coincide for unsteady flows.

  1. And this answer on physics.SE-

On a steady flow, streamlines correspond to the trajectory of "fluids particles" or parcels. (Not to be confused with the one of the real "particles" that are the molecules.)

While all these sources provide justification for this claim, I do not follow any of them for the case when the streamlines are curves other than straight lines.

My doubt is that, if a particle is at any point on a non-straight streamline, it shall have a velocity tangential to the curve. Now, there is 'centripedal force' on the 'particle'. Within the continuum hypothesis, should't Pascal's law implies that hydrostatic pressure cancels out in all directions. The only force shall be the driving force causing the flow (the difference in dynamic pressure).

Thus –

  1. The situation should be analogous to a 'magnetic monopole' and magnetic field lines. (It is of course not exactly same as the tangent there gives the direction of acceleration not velocity)

  2. The particle is moving tangentially to the streamline, so, at the next instant it should leave the streamline while still following its general direction. Now, it is on some other streamline adjacent to the first, so the same thing should happen again. So, the particle should slowly 'drift away' from the streamline.

My question is –

What is wrong with my reasoning above. Is there some force other than the simple dynamic pressure difference keeping the particle along the streamline? What exactly is this force and how does it work? Why precisely would the particle move along a streamline in 'any' steady flow?

Best Answer

Pascal's law says that pressure is isotropic and that, under hydrostatic conditions, this means that it is the same at all horizontal locations within within the fluid. However, if the fluid is flowing, the pressure varies with spatial position to balance any centripetal forces. This is captured in Euler's equation of fluid motion.

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