Streamlines, pathlines, streaklines and the stream function

Why it matters

Smoke in a wind tunnel, dye in a water channel and oil streaks on a wing all show flow lines, but which lines they show depends on how they were produced. Knowing whether you are looking at streamlines, pathlines or streaklines prevents misreading unsteady flows such as vortex shedding. The stream function turns a two-dimensional incompressible flow into a single scalar from which velocities, streamline shapes and volume flow rates follow directly, and it is the basis of potential-flow aerodynamics.

Key ideas

Streamline. A curve that is everywhere tangent to the instantaneous velocity vector. It is a snapshot at one instant. Consequences: there is no flow across a streamline, two streamlines cannot cross (except at stagnation points where V = 0), and a solid wall is itself a streamline. A bundle of streamlines forms a stream tube, which behaves like a pipe with no flow through its walls.

Pathline. The actual trajectory traced by one fluid particle over time (Lagrangian). Found by integrating dr/dt = V(r, t) from the particle's starting point.

Streakline. The locus, at a given instant, of all particles that have passed through one fixed point earlier. Continuous injection of smoke or dye from a nozzle produces a streakline.

Timeline. A line of particles marked at the same instant (e.g. a row of hydrogen bubbles) and then watched as it deforms; it shows velocity profiles.

Steady versus unsteady. In steady flow the velocity field does not change, so a particle keeps moving along the same streamline, and streamlines, pathlines and streaklines through a point all coincide. In unsteady flow they are generally different, which is why smoke pictures of an unsteady wake must be interpreted with care.

Stream function ψ. For two-dimensional incompressible flow, continuity ∂u/∂x + ∂v/∂y = 0 is satisfied identically if u = ∂ψ/∂y and v = −∂ψ/∂x. Properties:

• ψ is constant along a streamline, so lines of constant ψ are the streamlines.
• The volume flow rate per unit depth between two streamlines equals the difference of their ψ values.
• Any differentiable ψ gives an incompressible flow; it need not be irrotational. If the flow is also irrotational, ψ satisfies Laplace's equation ∇²ψ = 0.
• For compressible steady 2-D flow a density-weighted stream function is used instead (ρu = ρ₀∂ψ/∂y, ρv = −ρ₀∂ψ/∂x).

The sign convention matters: with u = ∂ψ/∂y and v = −∂ψ/∂x, ψ increases to the left when you look downstream. Some books use the opposite sign; be consistent with the given definition.

Formulas

dx/u = dy/v = dz/w

• Streamline equation at a fixed instant t (u, v, w in m/s).

dx/dt = u(x,y,z,t), dy/dt = v(x,y,z,t), dz/dt = w(x,y,z,t)

• Pathline equations, integrated from the particle's initial position.

u = ∂ψ/∂y, v = −∂ψ/∂x

• ψ: stream function (m²/s); 2-D incompressible flow in Cartesian coordinates.

u_r = (1/r)·∂ψ/∂θ, u_θ = −∂ψ/∂r

• Polar form (r in m, θ in rad).

q = ψ₂ − ψ₁

• q: volume flow rate per unit depth between the two streamlines (m²/s, i.e. m³/s per metre of depth).

∇²ψ = ∂²ψ/∂x² + ∂²ψ/∂y² = −ω_z

• ω_z: vorticity (s⁻¹). ∇²ψ = 0 only for irrotational flow.

Uniform flow: ψ = U·y = U·r·sinθ.

Worked examples

Example 1 (standard): velocities and flow rate from ψ. Given: ψ = 3xy (m²/s, x and y in m). Find the velocity at (2, 3) and the flow rate between the streamlines through (1, 1) and (2, 3).

1. u = ∂ψ/∂y = 3x = 6 m/s at (2, 3).
2. v = −∂ψ/∂x = −3y = −9 m/s at (2, 3).
3. Speed = √(6² + 9²) = 10.8 m/s.
4. ψ at (1, 1) = 3; ψ at (2, 3) = 18.
5. q = ψ₂ − ψ₁ = 18 − 3 = 15 m²/s. Answer: V = 6i − 9j m/s; q = 15 m³/s per metre depth. Streamlines are the hyperbolas xy = constant (flow into a corner).

Example 2 (GATE level): three different lines in unsteady flow. Given: u = 1 m/s, v = t m/s (uniform in space, changing with time). Find, through the origin, the streamline at t = 2 s, the pathline of the particle at the origin at t = 0, and the streakline at t = 2 s.

1. Streamline at t = 2 s: dy/dx = v/u = 2, so y = 2x (a straight line).
2. Pathline: dx/dt = 1 gives x = t; dy/dt = t gives y = t²/2. Eliminating t: y = x²/2 (a parabola).
3. Streakline: a particle released from the origin at time τ is at x = 2 − τ, y = (2² − τ²)/2 at t = 2. Put s = 2 − τ (so x = s): y = (4 − (2 − s)²)/2 = 2s − s²/2. So y = 2x − x²/2 for 0 ≤ x ≤ 2. All three differ, as expected in unsteady flow. Check at x = 1: streamline y = 2, pathline y = 0.5, streakline y = 1.5.

Example 3: streamline shape. For u = x, v = −y: dx/x = dy/(−y) gives ln x = −ln y + C, so xy = C, the same corner flow as Example 1.

Common mistakes

• Saying streamlines, pathlines and streaklines are always the same. That is true only in steady flow.
• Sign errors in v = −∂ψ/∂x; dropping the minus sign reverses the flow direction in y.
• Using the stream function for 3-D or compressible flow without modification.
• Assuming every stream function describes irrotational flow. Check ∇²ψ = 0 first.
• Forgetting that ψ₂ − ψ₁ is a flow rate per unit depth, not a velocity.
• Using u = ∂ψ/∂x by mistake (that is the velocity potential relation, u = ∂φ/∂x).

For GATE AE

Expect: given ψ, find u, v or the speed at a point; check whether a ψ satisfies continuity or irrotationality; find the flow rate between two streamlines; derive a streamline equation from a velocity field; and conceptual questions on when streamlines, pathlines and streaklines coincide. Practise partial derivatives with the sign convention and the polar form for flows around cylinders.

Quick check

1. ψ = 5xy². What is u at (3, 1)?
2. What is the flow rate per unit depth between ψ = 2 m²/s and ψ = 7 m²/s?
3. In steady flow, do pathlines and streaklines coincide?
4. Can two streamlines cross at a point where the velocity is non-zero?

Answers: 1. u = ∂ψ/∂y = 10xy = 30 m/s 2. 5 m²/s 3. Yes 4. No