Kinematics: streamlines, stream function and velocity potential

Why it matters

Kinematics describes how fluid moves — velocity, acceleration, rotation and flow lines — before you ask what forces cause it. Streamline pictures tell you where an intake or diffuser will separate. The stream function and velocity potential give compact descriptions of 2-D flows around bodies, and they appear in GATE as quick differentiation problems.

Key ideas

• Lagrangian vs Eulerian. Lagrangian follows individual particles; Eulerian watches fixed points in space. Fluid mechanics uses the Eulerian velocity field V(x, y, z, t) = u i + v j + w k.
• Flow classification.
  • Steady: nothing changes with time at a fixed point (∂/∂t = 0). Unsteady otherwise.
  • Uniform: velocity does not change with position along the flow at a given instant.
  • One-, two- or three-dimensional, by how many space coordinates the velocity depends on.
• Flow lines.
  • Streamline: a curve tangent to the velocity vector everywhere at one instant: dx/u = dy/v = dz/w. No flow crosses a streamline, and a solid wall is itself a streamline.
  • Pathline: the actual trajectory of one particle over time.
  • Streakline: the locus of all particles that have passed through one fixed point (dye or smoke filament).
  • In steady flow all three coincide. Streamlines cannot cross except at a stagnation point, where velocity is zero.
  • Stream tube: a bundle of streamlines through a closed curve; its walls carry no flow.
• Acceleration. A particle accelerates for two reasons: the field changes with time (local acceleration) and the particle moves to a place where velocity is different (convective acceleration). Steady flow through a nozzle has zero local but non-zero convective acceleration.
• Continuity in differential form. For incompressible flow, ∂u/∂x + ∂v/∂y + ∂w/∂z = 0. Any proposed velocity field must satisfy this to be physically possible.
• Rotation and vorticity. A fluid element can translate, rotate and deform. Rotation about z is ω_z = ½(∂v/∂x − ∂u/∂y); vorticity is twice the rotation. Flow with zero vorticity everywhere is irrotational. Viscous boundary layers are rotational; the flow outside them is often close to irrotational.
• Circulation Γ is the line integral of velocity around a closed curve and equals the total vorticity enclosed (Stokes' theorem). It is the basis of lift (Kutta–Joukowski).
• Stream function ψ exists for any 2-D incompressible flow, rotational or not, because it is defined so continuity is automatically satisfied. Lines of constant ψ are streamlines, and the difference in ψ between two streamlines equals the volume flow rate between them per unit depth.
• Velocity potential φ exists only for irrotational flow (in 2-D or 3-D). Its gradient is the velocity. Lines of constant φ (equipotential lines) are perpendicular to streamlines wherever the velocity is non-zero.
• If a flow is both incompressible and irrotational, both φ and ψ satisfy Laplace's equation. The grid of streamlines and equipotentials is a flow net.
• Sign convention. This lesson uses u = ∂ψ/∂y, v = −∂ψ/∂x and u = ∂φ/∂x, v = ∂φ/∂y. Some textbooks put a minus sign on φ (V = −∇φ). Always check which convention a question uses.

Formulas

dx/u = dy/v = dz/w Equation of a streamline; u, v, w velocity components (m/s).

a_x = ∂u/∂t + u·∂u/∂x + v·∂u/∂y + w·∂u/∂z Local plus convective acceleration in x (m/s²); a_y and a_z similarly.

∂u/∂x + ∂v/∂y + ∂w/∂z = 0 Continuity for incompressible flow (s⁻¹ terms).

ω_z = ½·(∂v/∂x − ∂u/∂y) ζ_z = 2·ω_z Rotation ω_z (rad/s) and vorticity ζ_z (s⁻¹) about z; irrotational if zero.

u = ∂ψ/∂y, v = −∂ψ/∂x Stream function ψ (m²/s for 2-D flow). Exists for 2-D incompressible flow.

q = ψ₂ − ψ₁ Volume flow rate per unit depth between two streamlines (m²/s, i.e. m³/s per metre).

u = ∂φ/∂x, v = ∂φ/∂y Velocity potential φ (m²/s). Exists only for irrotational flow.

∂φ/∂x = ∂ψ/∂y, ∂φ/∂y = −∂ψ/∂x Cauchy–Riemann relations; give orthogonality of φ and ψ lines.

∇²ψ = 0 (irrotational) ∇²φ = 0 (incompressible)

Worked examples

Example 1 (standard) — acceleration, ψ and φ for a corner flow. Given: steady 2-D field u = 3x, v = −3y (m/s, x and y in m). Check continuity, find the acceleration at (1, 2), and find ψ and φ.

1. Continuity: ∂u/∂x + ∂v/∂y = 3 − 3 = 0 → physically possible incompressible flow.
2. a_x = u·∂u/∂x + v·∂u/∂y = 3x × 3 + 0 = 9x = 9 m/s² at x = 1.
3. a_y = u·∂v/∂x + v·∂v/∂y = 0 + (−3y)(−3) = 9y = 18 m/s² at y = 2.
4. |a| = √(9² + 18²) = 20.12 m/s².
5. ψ: ∂ψ/∂y = 3x → ψ = 3xy + f(x); −∂ψ/∂x = −3y − f′(x) = −3y → f = const. So ψ = 3xy.
6. Rotation: ∂v/∂x − ∂u/∂y = 0 − 0 = 0 → irrotational, so φ exists: ∂φ/∂x = 3x → φ = 1.5x² + g(y); ∂φ/∂y = g′(y) = −3y → φ = 1.5(x² − y²). Answer: |a| ≈ 20.1 m/s², ψ = 3xy, φ = 1.5(x² − y²). The flow is steady, yet particles accelerate (convective acceleration).

Example 2 (GATE level) — flow rate from ψ. Given: ψ = x² − y² (m²/s, x and y in m). Find the velocity at (2, 3), check irrotationality, and find the flow rate per metre depth between the streamlines through (1, 1) and (2, 3).

1. u = ∂ψ/∂y = −2y = −6 m/s; v = −∂ψ/∂x = −2x = −4 m/s at (2, 3).
2. |V| = √(36 + 16) = 7.21 m/s.
3. Vorticity: ∂v/∂x − ∂u/∂y = −2 − (−2) = 0 → irrotational.
4. ψ at (1, 1) = 1 − 1 = 0; ψ at (2, 3) = 4 − 9 = −5 m²/s.
5. q = |ψ₂ − ψ₁| = 5 m²/s.
6. Potential: ∂φ/∂x = −2y → φ = −2xy + g(y); ∂φ/∂y = −2x + g′ = −2x → φ = −2xy. Answer: |V| ≈ 7.21 m/s; flow rate = 5 m³/s per metre depth; φ = −2xy.

Common mistakes

• Mixing the sign conventions u = ∂ψ/∂y versus u = −∂ψ/∂y, or V = ∇φ versus V = −∇φ, within one problem.
• Assuming a velocity potential exists for any flow — it needs zero vorticity. A stream function needs only 2-D incompressibility.
• Saying acceleration is zero in steady flow — convective acceleration can be large.
• Forgetting to check continuity before accepting a given velocity field.
• Treating ψ as a velocity; it has units of m²/s and only differences of ψ have physical meaning.
• Calling streamlines and pathlines identical in unsteady flow.

For GATE ME

Questions give a velocity field, ψ or φ and ask for velocity at a point, acceleration (local and convective), whether the flow satisfies continuity or is irrotational, the missing component of a velocity field, or the discharge between two streamlines. Practise partial differentiation quickly and keep one sign convention throughout.

Quick check

1. When do streamlines, pathlines and streaklines coincide?
2. For u = 2x, find the v that satisfies 2-D incompressible continuity with v = 0 on y = 0.
3. ψ = 4xy. What is the flow rate per metre between ψ = 2 and ψ = 10 m²/s?
4. Does a velocity potential exist for u = y, v = 0?
5. What angle do streamlines and equipotential lines make? Answers: 1. In steady flow. 2. v = −2y. 3. 8 m²/s (8 m³/s per metre). 4. No — vorticity ∂v/∂x − ∂u/∂y = −1 ≠ 0. 5. 90°.