Kinematics: streamlines, stream function and velocity potential

Why it matters

Kinematics describes how fluid moves without asking why: velocity fields, accelerations, rotation and flow patterns. It is how you check that a proposed velocity field is physically possible, how CFD results are read (streamlines, vorticity contours) and how classical potential-flow solutions for flow around bodies, nozzles and inlets are built.

Key ideas

Two descriptions. The Lagrangian view follows individual particles; the Eulerian view watches fixed points in space and gives the velocity field V(x, y, z, t) = u·i + v·j + w·k. Fluid mechanics normally uses the Eulerian view.

Classifying flows.

• Steady: nothing changes with time at a fixed point (∂/∂t = 0). Unsteady: properties change with time.
• Uniform: velocity does not change along the flow direction at an instant. Non-uniform: it does (for example, in a converging nozzle).
• One-, two- or three-dimensional, depending on how many space coordinates the velocity depends on.
• Laminar or turbulent; compressible or incompressible; rotational or irrotational.

Flow lines.

• Streamline: a line everywhere tangent to the velocity at an instant. No fluid crosses a streamline. Two streamlines cannot cross except at a stagnation point or singularity, since a point cannot have two velocity directions.
• Pathline: the actual trajectory of one particle over time.
• Streakline: the line joining all particles that have passed through one fixed point (dye injected from a nozzle).
• In steady flow all three coincide. In unsteady flow they generally differ.
• A stream tube is a bundle of streamlines; it behaves like a pipe with no flow across its walls.

Acceleration. A particle's velocity can change because the field changes with time (local acceleration) and because the particle moves to a place where velocity is different (convective acceleration). Uniform steady flow has neither; steady flow through a nozzle has only convective acceleration.

Continuity (differential form). Mass conservation at a point gives ∂ρ/∂t + ∇·(ρV) = 0. For incompressible flow this reduces to ∇·V = 0. Any proposed velocity field must satisfy it to be a "possible flow".

Rotation, vorticity and circulation. The angular velocity of a fluid element about z is ω_z = ½(∂v/∂x − ∂u/∂y). Vorticity is twice the rotation, ζ = ∇ × V. Circulation Γ is the line integral of velocity around a closed curve and equals the vorticity flux through it (Stokes' theorem). Flow is irrotational if ζ = 0 everywhere. Uniform flow away from walls, and a free vortex (except at its centre), are irrotational; boundary layers, wakes and a forced vortex (rigid-body rotation) are rotational.

Stream function ψ (2D incompressible flow). Defined so that u = ∂ψ/∂y and v = −∂ψ/∂x. It satisfies continuity automatically. Lines of constant ψ are streamlines, and the difference ψ₂ − ψ₁ is the volume flow rate per unit depth between two streamlines. ψ exists for any 2D incompressible flow, rotational or not. If the flow is also irrotational, ∇²ψ = 0.

Velocity potential φ (irrotational flow). Defined so that V = ∇φ, i.e. u = ∂φ/∂x and v = ∂φ/∂y (some textbooks use a minus sign; be consistent with your book). φ exists only if the flow is irrotational, in 2D or 3D. If the flow is also incompressible, ∇²φ = 0 (Laplace's equation). Lines of constant φ are equipotential lines.

Flow net. For 2D irrotational incompressible flow, both ψ and φ exist, satisfy Laplace's equation and obey the Cauchy–Riemann relations ∂φ/∂x = ∂ψ/∂y and ∂φ/∂y = −∂ψ/∂x. Streamlines and equipotential lines therefore cross at right angles, forming a flow net. Where streamlines crowd together, velocity is high. Because Laplace's equation is linear, simple flows (uniform flow, source, sink, vortex, doublet) can be superposed to model flow past cylinders and bodies.

Formulas

dx / u = dy / v = dz / w

• Equation of a streamline (u, v, w in m/s).

a_x = ∂u/∂t + u·∂u/∂x + v·∂u/∂y + w·∂u/∂z (similarly a_y, a_z)

• a_x = acceleration (m/s²): first term local, the rest convective.

∂u/∂x + ∂v/∂y + ∂w/∂z = 0

• Continuity for incompressible flow (each term in s⁻¹).

ω_z = ½ · (∂v/∂x − ∂u/∂y), ζ_z = 2·ω_z

• ω_z = rotation (rad/s), ζ_z = vorticity (s⁻¹). Irrotational if zero.

u = ∂ψ/∂y, v = −∂ψ/∂x, q = ψ₂ − ψ₁

• ψ = stream function (m²/s), q = flow rate per unit depth between two streamlines (m²/s). 2D incompressible flow.

u = ∂φ/∂x, v = ∂φ/∂y

• φ = velocity potential (m²/s). Irrotational flow.

∂²φ/∂x² + ∂²φ/∂y² = 0, ∂²ψ/∂x² + ∂²ψ/∂y² = 0

• Laplace's equations: φ for incompressible irrotational flow; ψ for irrotational 2D flow.

Worked examples

Example 1 — checking a velocity field (standard). A 2D field is u = x²y, v = −xy² (m/s, with x and y in m). Check that it is a possible incompressible flow, and find the acceleration and the rotation at (1, 2).

1. Continuity: ∂u/∂x + ∂v/∂y = 2xy − 2xy = 0, so the flow is possible.
2. a_x = u·∂u/∂x + v·∂u/∂y = (x²y)(2xy) + (−xy²)(x²) = x³y². At (1, 2): a_x = 1 × 4 = 4 m/s².
3. a_y = u·∂v/∂x + v·∂v/∂y = (x²y)(−y²) + (−xy²)(−2xy) = x²y³. At (1, 2): a_y = 1 × 8 = 8 m/s².
4. Magnitude = √(4² + 8²) = 8.94 m/s².
5. ω_z = ½(∂v/∂x − ∂u/∂y) = ½(−y² − x²). At (1, 2): ω_z = ½(−4 − 1) = −2.5 rad/s. Answer: possible flow; a ≈ 8.94 m/s²; ω_z = −2.5 rad/s (rotational, clockwise).

Example 2 — stream function to potential and flow rate (GATE level). For ψ = 3x²y − y³ (m²/s), find the velocity components, check irrotationality, find φ, the flow rate between the streamlines through (1, 1) and (2, 3), and the speed at (2, 3).

1. u = ∂ψ/∂y = 3x² − 3y²; v = −∂ψ/∂x = −6xy.
2. Rotation check: ∂v/∂x − ∂u/∂y = −6y − (−6y) = 0, so the flow is irrotational and φ exists.
3. ∂φ/∂x = u = 3x² − 3y², so φ = x³ − 3xy² + f(y). Then ∂φ/∂y = −6xy + f′(y) must equal v = −6xy, so f′ = 0 and φ = x³ − 3xy² (plus a constant).
4. ψ(1, 1) = 3 − 1 = 2; ψ(2, 3) = 3 × 4 × 3 − 27 = 9. Flow rate q = 9 − 2 = 7 m²/s per metre depth.
5. At (2, 3): u = 12 − 27 = −15 m/s, v = −36 m/s, so |V| = √(225 + 1296) = 39 m/s. Answer: φ = x³ − 3xy²; q = 7 m²/s per m depth; speed 39 m/s at (2, 3).

Common mistakes

• Swapping the signs in the stream-function relations. With u = ∂ψ/∂y, v must be −∂ψ/∂x.
• Assuming φ exists for every flow. It exists only for irrotational flow; ψ exists for any 2D incompressible flow.
• Calling a flow irrotational because the streamlines are circular. A free vortex is irrotational, a forced vortex is rotational.
• Forgetting the convective terms when finding acceleration in a steady but non-uniform flow.
• Stating that streamlines, pathlines and streaklines are always the same. That holds only in steady flow.
• Using ψ₂ − ψ₁ as a flow rate in m³/s; in 2D it is per unit depth (m²/s).

For GATE ME

Expect one-line derivations: find v from u using continuity, find acceleration at a point, check whether a given φ or ψ represents a possible or irrotational flow, find ψ from φ (or the reverse), get the flow rate between two streamlines, and identify vorticity or circulation for vortex flows. MCQs test the definitions of streamlines, pathlines and streaklines and when ψ and φ exist. Practise partial differentiation quickly and keep one sign convention for φ.

Quick check

1. In which kind of flow do streamlines, pathlines and streaklines coincide?
2. Does a stream function exist for a 2D incompressible rotational flow?
3. For ψ = 4xy (m²/s), what are u and v?
4. Is u = 2x, v = −2y an irrotational flow?
5. What does the difference in ψ between two streamlines represent?

Answers: 1. steady flow; 2. yes, ψ exists for any 2D incompressible flow; 3. u = 4x, v = −4y; 4. yes, ∂v/∂x − ∂u/∂y = 0; 5. the volume flow rate per unit depth between them.