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In continuum mechanics, the definition of trajectory or pathline is the locus of the positions occupied by a given particle in space throughout time.

And streamlines are a family of curves which for every instant in time are the velocity field envelopes.

For a stationary velocity field, trajectories and streamlines coincide..

How is this possible? Even if the velocity didn't depend on time, and was only changing with the change of position, it doesn't make sense for the streamline and trajectory to coincide because they hold different physical meaning.

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  • $\begingroup$ It helps to look at the equations for them. Please give us a decent starting place when asking questions by posting formulas in the question. $\endgroup$ – Phil Sweet Mar 5 at 23:45
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How is this possible? Even if the velocity didn't depend on time, and was only changing with the change of position, it doesn't make sense for the streamline and trajectory to coincide because they hold different physical meaning.

The only thing that represents a single physical thing is a trajectory. Pathlines, streaklines, and streamlines can all be thought of as aggregations of trajectories that were assembled using different rules. They are traces in space, and somebody decided to make life easy and define them in a way that they are all congruent for a constant velocity field.

A trajectory is a point in space and time. If you start with a velocity field, you need an initial point and time to get started. The trajectory point is then found by integrating to the final time.

$$P\small{(x_0\,,y_0\,,t_0\,,t_1)}\normalsize{ =} \int_{t_0}^{t_1} \mathbf{V}(\small{x(t)\,,y(t)\,,t}\,\normalsize{)\,dt}$$

If $\mathbf{V}$ is a stationary field, several things are simpler. $\mathbf{V}$ doesn't have a time parameter, and the bounds of the integral can be shifted by any constant we like.

$$P\small{(x_0\,,y_0\,,t_1)}\normalsize{ =} \int_{0}^{t_1} \mathbf{V}(\small{x(t)\,,y(t)}\,\normalsize{)\,dt}$$

A pathline is a collection of trajectory points that all have the same starting point $P$, with $t_1$ being anything.

A streamline is basically the same thing, but we travel across the field infinitely fast. No time elapses. But this makes collecting data points from the integral difficult. So some clever person realized that the same curve can be gotten by parameterizing the curve differently. Instead of using time, we can use distance as the variable of integration. And instead of integrating the velocity, we integrate the normalized velocity.

So $s$ is the new variable of integration, and it is related to $t$ by $ds=|\mathbf{V}\small{(x\,,y)}\normalsize{|\,dt}$

And a streampoint becomes $\displaystyle P\small{(x_0\,,y_0\,,s_1)}\normalsize{ =} \int_{0}^{s_1} \mathbf{v}(\small{x(s)\,,y(s)}\,\normalsize{)\,ds}$ where $\displaystyle\mathbf{v}=\frac{\mathbf{V}}{|\mathbf{V}|}$

So what we are doing is reparameterizing the integral to get rid of an infinity problem and provide nice bounds on the definite integral. This parameterization is different at each point in the field, but it all comes back together when you integrate over the field. Different parameterizations, same shaped curve.

And that's okay, because with pathlines, streakline, and streamlines, we don't keep an index of the parameter values that produced a given point the way we do with a trajectory. A trajectory should retain this timestamp info.

Now if you want to test your calculus chops, assume $\mathbf{V}$ isn't stationary, and $ds=|\mathbf{V}\small{(x\,,y\,,t)}\normalsize{|\,dt}$ and figure out what the real restrictions on $\displaystyle\mathbf{v}=\frac{\mathbf{V}}{|\mathbf{V}|}$ are such that streamlines and pathlines are the same. A constant $\mathbf{V}$ is overly strict, there are certain $\mathbf{V}(t)$ for which this is true as well.

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