A streamline at any instant can be defined as an imaginary curve or line in the flow field so that the tangent to the curve at any point represents the direction of the instantaneous velocity at that point. In an unsteady flow where the velocity vector changes with time, the pattern of streamlines also changes from instant to instant.

In a steady flow, the orientation or the pattern of streamlines will be fixed. From the above definition of streamline, it can be written as

$\bar{V} \times d\bar{s}=0$

Description of the terms:

1. d$\bar{s}$ is the length of an infinitesimal line segment along a streamline at a point .

2. $\bar{v}$ is the instantaneous velocity vector.

The above expression therefore represents the differential equation of a streamline. In a Cartesian coordinate-system, representing

the above equation may be simplified as

$\frac{dx}{u}=\frac{dy}{y}=\frac{dz}{w}$

Path Lines

Definition: A path line is the trajectory of a fluid particle of fixed identity as defined by $\bar{S}=S(s_{o}-t)$

$\bar{s}$ is the position vector of a particle (with respect to a fixed point of reference) at a time t.

$\bar{s}_{0}$ is its initial position at a given time t =t0

Path lines

A family of path lines represents the trajectories of different particles, say, P1, P 2, P3, etc. Differences between Path Line and Stream Line

Streak Lines

Definition: A streak line is the locus of the temporary locations of all particles that have passed though a fixed point in the flow field at any instant of time.

Features of a Streak Line: While a path line refers to the identity of a fluid particle, a streak line is specified by a fixed point in the flow field. It is of particular interest in experimental flow visualization.

Example: If dye is injected into a liquid at a fixed point in the flow field, then at a later time t, the dye will indicate the end points of the path lines of particles which have passed through the injection point. The equation of a streak line at time t can be derived by the Lagrangian method.

If a fluid particle $\bar({s_{0}})$ passes through a fixed point $\bar({s_{1}})$ in course of time t, then the Lagrangian method of description gives the equation.

$S(\bar{S_{c},t})=\bar{S_{1}}$ (1)

Solving for ,$S_{0}$

$\bar{S}_{0}=F(\bar{S_{1}, t})$ (2)

If the positions $\bar({s_{0}})$ of the particles which have passed through the fixed point $\bar({s_{1}})$ are determined, then a streak line can be drawn through these points.

Equation: The equation of the streak line at a time t is given by

$\bar{s}=f(\bar{S}_{0},t)$ (3)

Substituting Eq. (1) into Eq. (2) we get the final form of equation of the streak line,

$\bar{s}=f[F(\bar{S}, t),t]$ (4)