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?

  • $\begingroup$ Doesn't Pascal's law apply specifically to hydrostatic pressure? $\endgroup$ – probably_someone Mar 7 at 12:28
  • $\begingroup$ @probably_someone I have only mentioned it for the hydrostatic component. I mean to say that there can be no local hydrostatic force causing the effect. $\endgroup$ – Devashish Kaushik Mar 7 at 12:33
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    $\begingroup$ Is there a reason to believe that only hydrostatic forces are acting here? $\endgroup$ – probably_someone Mar 7 at 12:38
  • $\begingroup$ @probably_someone Pressure represents the isotropic part of the stress tensor, and always satisfies Pascal's law. For an inviscid fluid, the entire stress tensor is isotropic (and equal to the pressure). $\endgroup$ – Chet Miller Mar 7 at 12:39
  • $\begingroup$ Fluid mechanics is/can be constructed from statistical N-body theory via the usage of the moments of the Boltzmann equation. Therefore the fluid streamlines always represent particle trajectories for particles perfectly coupled to the flow in steady-state. The particles feel all forces that the flow feels, as they ARE the flow. The main difference between a flow system and a uncoupled bunch of particles is pressure. $\endgroup$ – AtmosphericPrisonEscape Mar 7 at 16:53

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.

  • $\begingroup$ OK, so does that mean that I should be able to use Euler's equation to show that the pressure will vary in such a way so as to cause forces which make the particles move along the streamline (provided there is no time dependence)? I don't see how your observation implies that pathlines and streamlines should coincide for steady flows. Could you clarify? $\endgroup$ – Devashish Kaushik Mar 7 at 12:50
  • $\begingroup$ In an incompressible 2D flow, the velocity components are related to the stream function by $v_x=\frac{\partial \psi}{\partial y}$ and $v_y=-\frac{\partial \psi}{\partial x}$. If the spatial distribution of the stream function were changing with time, the velocity could not be constant at each point. $\endgroup$ – Chet Miller Mar 7 at 12:57

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