Fluid Kinematics & Continuity Equation

Fluid kinematics (~1–2 marks in GATE ME) describes motion without forces, and the continuity equation is one of the most-used relations in the whole subject. Stream function and velocity potential are frequent conceptual marks.

Flow description & continuity

Continuity (mass conservation): for incompressible steady flow, A₁V₁ = A₂V₂ (volume flow rate Q constant). Differential form: ∂ρ/∂t + ∇·(ρV) = 0; for incompressible flow ∇·V = 0 (∂u/∂x + ∂v/∂y + ∂w/∂z = 0).

Flow types: steady/unsteady (∂/∂t), uniform/non-uniform (∂/∂s), laminar/turbulent, compressible/incompressible, rotational/irrotational (vorticity zero?). Acceleration has local (∂V/∂t) and convective (V·∇V) parts — a steady flow can still accelerate convectively (nozzle).

Stream function & velocity potential

• Stream function ψ exists for any 2D incompressible flow: u = ∂ψ/∂y, v = −∂ψ/∂x. ψ = constant along a streamline; automatically satisfies continuity.
• Velocity potential φ exists only for irrotational flow: u = ∂φ/∂x, v = ∂φ/∂y (some texts use −). For flow that is both incompressible and irrotational, ψ and φ both satisfy Laplace's equation ∇²=0, and streamlines (ψ) are orthogonal to equipotentials (φ).

Vorticity = ∇×V = 2ω (twice the angular velocity); irrotational means vorticity = 0.

Exam Tricks & Tips

• 🎯 A₁V₁ = A₂V₂: velocity rises where area falls — a nozzle speeds flow up, a diffuser slows it.
• 🎯 Stream function ψ exists for ANY incompressible flow; velocity potential φ only for IRROTATIONAL flow — a favourite MCQ distinction.
• 🎯 Existence of ψ automatically satisfies continuity; existence of φ implies irrotationality — check which the question is probing.
• 🎯 Streamlines (ψ = const) and equipotential lines (φ = const) are orthogonal for potential flow.
• 🎯 Convective acceleration exists even in steady flow — the flow accelerates through a nozzle though ∂V/∂t = 0.
• ❌ Common mistake: assuming a velocity potential exists for every flow — it exists only when the flow is irrotational (zero vorticity); a rotational flow has ψ but no φ.

Expected exam pattern

A 1-mark continuity or "does ψ/φ exist / is the flow irrotational" MCQ, and a 2-mark problem finding velocity components from ψ or φ, or checking continuity/irrotationality of a given velocity field.

Quick recap

Continuity: A₁V₁ = A₂V₂; incompressible ∇·V = 0. ψ (stream function): any 2D incompressible flow, ψ = const on streamlines. φ (velocity potential): only irrotational flow. Both satisfy Laplace for potential flow; streamlines ⊥ equipotentials. Vorticity = ∇×V = 2ω.

Cover the answer, recall, then check. 11 cards on fluid kinematics for GATE ME.

Q1. State the continuity equation for incompressible steady pipe flow.
A1. A₁V₁ = A₂V₂ — the volume flow rate Q is constant along the pipe.

Q2. Differential continuity for incompressible flow?
A2. ∇·V = ∂u/∂x + ∂v/∂y + ∂w/∂z = 0. (General: ∂ρ/∂t + ∇·(ρV) = 0.)

Q3. Define the stream function and its key property.
A3. u = ∂ψ/∂y, v = −∂ψ/∂x. ψ = constant along a streamline, and it automatically satisfies continuity.

Q4. For which flows does a stream function exist?
A4. Any two-dimensional incompressible flow (rotational or irrotational).

Q5. Define velocity potential and its existence condition.
A5. u = ∂φ/∂x, v = ∂φ/∂y. It exists ONLY for irrotational flow (zero vorticity).

Q6. What equation do ψ and φ satisfy for incompressible irrotational flow?
A6. Laplace's equation, ∇²ψ = 0 and ∇²φ = 0.

Q7. Geometric relation between streamlines and equipotential lines?
A7. They are mutually orthogonal (perpendicular) in potential flow.

Q8. Define vorticity and irrotational flow.
A8. Vorticity = ∇×V = 2ω (twice the local angular velocity). Irrotational flow has zero vorticity.

Q9. What are the two parts of fluid acceleration?
A9. Local (∂V/∂t, time change) and convective (V·∇V, spatial change). Steady flow keeps the convective part.

Q10. Can steady flow accelerate?
A10. Yes — through convective acceleration (e.g. a nozzle), even though the local term ∂V/∂t is zero.

Q11. A 2D flow has ψ but no φ. What does that tell you?
A11. It is incompressible (ψ exists) but rotational (φ absent → non-zero vorticity).

Kinematics describes how fluid moves without asking what forces cause it. The continuity equation (mass conservation) is the most-used equation in all of fluid mechanics, and the stream/potential-function machinery underlies ideal-flow analysis. GATE tests these steadily.

Core concept: mass is conserved, so what flows into a region must flow out; the continuity equation enforces this, while stream and potential functions describe the flow field of ideal (incompressible, irrotational) flows.

Deep explanation

Beginner — describing flow

• Lagrangian (follow a particle) vs Eulerian (fixed points in space) descriptions.
• Flow classifications: steady/unsteady, uniform/non-uniform, laminar/turbulent, compressible/incompressible, rotational/irrotational.
• Streamline: tangent to velocity everywhere (no flow crosses it); pathline: actual particle track; streakline: locus of particles that passed a point. In steady flow all three coincide.

Intermediate — the continuity equation

Conservation of mass:
∂ρ/∂t + ∇·(ρV) = 0.
For incompressible flow (ρ constant): ∇·V = 0, i.e. ∂u/∂x + ∂v/∂y + ∂w/∂z = 0. For 1D pipe flow this reduces to the workhorse: A₁V₁ = A₂V₂ (or ρ₁A₁V₁ = ρ₂A₂V₂ compressible) — the mass flow rate ṁ = ρAV is constant.

Advanced — stream function and velocity potential

• Stream function ψ (2D, incompressible): u = ∂ψ/∂y, v = −∂ψ/∂x. It automatically satisfies continuity; lines of constant ψ are streamlines, and the difference Δψ between two streamlines equals the volume flow rate between them.
• Velocity potential φ (irrotational flow): u = ∂φ/∂x, v = ∂φ/∂y. Exists only if ∇×V = 0.
• When both exist (ideal flow), ψ and φ are orthogonal and both satisfy Laplace's equation ∇²ψ = 0, ∇²φ = 0 — the basis of potential-flow solutions.
  Acceleration has local (∂V/∂t) and convective (V·∇V) parts; even in steady flow the convective term (e.g. through a nozzle) is non-zero.

Worked example

Water flows through a pipe that contracts from 100 mm diameter to 50 mm. If the velocity in the larger section is 2 m/s, find the velocity in the smaller section.
Continuity A₁V₁ = A₂V₂. Since A ∝ d², A₁/A₂ = (d₁/d₂)² = (100/50)² = 4.
V₂ = V₁ (A₁/A₂) = 2 × 4 = 8 m/s. (Halving the diameter quarters the area and quadruples the velocity.)

GATE relevance

Continuity (A₁V₁ = A₂V₂), stream-function/velocity-potential relations, and flow classifications (rotational vs irrotational, checking ∇·V = 0) are core Fluid Mechanics questions. Convective acceleration and streamline/pathline distinctions appear as concept items.

Exam tricks & shortcuts

• For area changes, velocity scales as 1/d² — a 2:1 diameter drop gives 4× velocity.
• If a stream function is given, it already satisfies continuity; if a velocity potential exists, the flow is irrotational.
• Check incompressibility by testing ∇·V = 0.
• Mnemonic: "Squeeze the area, speed up the flow (AV = constant)."

Key formulas

• Continuity (incompressible): A₁V₁ = A₂V₂; general ∂ρ/∂t + ∇·(ρV) = 0.
• Discharge: Q = AV.
• Stream function ψ: u = ∂ψ/∂y, v = −∂ψ/∂x (satisfies continuity).
• Velocity potential φ (irrotational): u = ∂φ/∂x; ∇²φ = 0.
• Rotational vs irrotational: vorticity ζ = ∂v/∂x − ∂u/∂y.

Continuity enforces mass conservation; stream function and velocity potential describe 2D flow fields.

Worked Example

Problem: Water flows steadily through a pipe that contracts from a diameter of 100 mm to 50 mm. The velocity in the larger (100 mm) section is 2 m/s. Find the volume flow rate and the velocity in the smaller (50 mm) section.

Solution:
Cross-sectional area of the larger section:
A₁ = (π/4)·d₁² = (π/4)(0.100)² = 7.854 × 10⁻³ m².

Volume flow rate (discharge):
Q = A₁·V₁ = 7.854 × 10⁻³ × 2 = 0.01571 m³/s ≈ 15.71 L/s.

For steady incompressible flow, continuity requires A₁V₁ = A₂V₂. Since area scales with diameter squared:
V₂ = V₁·(d₁/d₂)² = 2 × (100/50)² = 2 × 4 = 8 m/s.

Check via area: A₂ = (π/4)(0.050)² = 1.963 × 10⁻³ m², so V₂ = Q/A₂ = 0.01571/1.963 × 10⁻³ = 8 m/s ✓.

Answer: Q ≈ 15.71 L/s and V₂ = 8 m/s.