Why will fluid particles always move along streamlines in a steady flow?

Question

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.

2. 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.

3. 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.)

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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.

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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?

Answer 1

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.

Answer 2

The issue is really what is meant by "particle-line". Fundamentally, the molecules in the fluid move randomly whenever there is non-zero pressure, since pressure is proportional to the random motion of the molecules, $P \sim \sum_{i=1}^3 \langle (v^i - \langle v^i\rangle)^2 \rangle $. However, you can ask about the motion of a "virtual particle" that has the velocity $\langle v^i \rangle$ at any given point, and at any given time instant. One can then integrate the motion of this virtual particle and that is what would be called a "particle-line".

But consider 2 equal streams of microscopic particles with exactly opposite velocities. In this case the average velocity of the ensemble will be zero, $\langle v \rangle = 0$, (while there will be anisotropic pressure). So there needs to be no real particle moving on a "particle-line". So the "particle" on the particle-line fulfills different equations of motion than any real particle. One particular modification is an artificial acceleration by the (isotropic) pressure gradient $- \nabla (P/\rho)$ (see Euler fluid equations). The term does not appear because of any real force between the particles, it appears only thanks to the averaging over the ensemble and there are examples where it is purely kinematic.

So let us now compare a streamline and a particle-line. A streamline is computed by freezing the flow in a single instant of time, and integrating all curves with the frozen velocity $\langle v^i \rangle$. On the other hand, the particle-line is computed by integrating the velocity $\langle v^i \rangle$ without freezing the flow.

If you want to understand this picture from the dynamical (or rather "virtual-dynamical") point of view in a time-dependent flow, the difference is that the streamline encounters a different artificial acceleration $-\nabla (P/\rho)$ than the particle-line. This is because $P, \rho$ and their gradients have changed "before the particle gets there" from the same point as the streamline.