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Types of flow, streamline, pathline, streakline

Short answer

Steady flow: properties constant at a point; uniform: velocity constant along flow. Streamline: curve tangent to velocity vectors; pathline: particle trajectory; streakline: particles from a point. Streamline equation: dx/u = dy/v = dz/w. In steady flow, streamlines, pathlines and streaklines coincide.

(a) Differentiate between steady and unsteady flow, uniform and non-uniform flow, and one-, two- and three-dimensional flow with examples. (b) Define streamline, pathline and streakline. Derive the differential equation of a streamline and state when the three coincide.

(a) Classification of flow

TypeMeaningExample
SteadyProperties at a point do not change with time: ∂/∂t=0\partial/\partial t = 0Constant discharge through a pipe
UnsteadyProperties change with timeWater hammer, emptying a tank
UniformVelocity does not change with position along the flow: ∂V⃗/∂s=0\partial \vec V/\partial s = 0Flow in a pipe of constant diameter
Non-uniformVelocity changes from point to pointFlow in a tapering pipe
1DVelocity depends on one space coordinate onlyAverage velocity in a pipe
2DVelocity depends on two coordinates, no variation in the thirdFlow over a long wide dam spillway
3DVelocity depends on all three coordinatesFlow around a finite wing, river bend

(b) Streamline, pathline, streakline

  • Streamline: an imaginary curve drawn so that the tangent at every point gives the direction of the velocity vector at that instant. No flow crosses a streamline.
  • Pathline: the actual path traced by a single fluid particle over time (Lagrangian concept).
  • Streakline: the locus of all particles that have passed through a fixed point earlier, e.g. a line of dye injected steadily at a point.

Equation of streamline

Let ds⃗=dx i^+dy j^+dz k^d\vec s = dx\,\hat i + dy\,\hat j + dz\,\hat k be an element along the streamline. It is parallel to V⃗=ui^+vj^+wk^\vec V = u\hat i + v\hat j + w\hat k, so ds⃗×V⃗=0d\vec s \times \vec V = 0. Hence

dxu=dyv=dzw\frac{dx}{u} = \frac{dy}{v} = \frac{dz}{w}

For 2D flow, dy/dx=v/udy/dx = v/u, which is integrated to get the streamline family.

When they coincide

In steady flow the velocity field does not change with time, so streamlines, pathlines and streaklines are identical. In unsteady flow they differ.

✓ Answer checked by Claude Opus 5.5 (high effort)

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