Fluid kinematics and the continuity equation

Why it matters

Before you can ask what forces act on a fluid, you need to describe how it moves: how fast, in which direction, and how the speed changes from point to point. Sizing a coolant line, predicting the speed of a hydraulic cylinder, or choosing a nozzle for a spray system all start with the continuity equation, which is simply conservation of mass written for a flowing fluid.

Key ideas

Kinematics versus dynamics. Fluid kinematics describes motion (velocity, acceleration, flow patterns) without asking which forces cause it. Dynamics (Bernoulli, momentum equation) comes next and uses the kinematic description.

Lagrangian and Eulerian descriptions. The Lagrangian view follows individual fluid particles. The Eulerian view, used in almost all engineering work, fixes points in space and gives the velocity field V(x, y, z, t) = u i + v j + w k at those points.

Types of flow.

• Steady: properties at a fixed point do not change with time (∂/∂t = 0). Unsteady: they do (for example, flow while a valve is closing).
• Uniform: velocity does not change along the flow direction at an instant. Non-uniform: it does (a tapering nozzle).
• One-, two- or three-dimensional, depending on how many space coordinates the velocity depends on. Pipe flow is usually analysed as one-dimensional using the mean velocity.
• Rotational or irrotational: whether fluid particles spin about their own axes (vorticity non-zero or zero).
• Compressible or incompressible: whether density changes appreciably.

Flow lines.

• Streamline: a line everywhere tangent to the velocity vector at an instant. No fluid crosses a streamline, and two streamlines cannot cross (that would mean two velocities at one point).
• Pathline: the actual track of one particle over time.
• Streakline: the locus of all particles that have passed through one fixed point (what you see when dye is injected continuously).
• 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 through its walls.

Acceleration of a fluid particle. Even in steady flow a particle accelerates if it moves into a region of different velocity. The total (material) acceleration has a local part (∂V/∂t, zero in steady flow) and a convective part (u ∂V/∂x + v ∂V/∂y + w ∂V/∂z, non-zero in non-uniform flow). Flow through a nozzle at constant rate is steady but has convective acceleration.

Continuity equation. Mass is neither created nor destroyed. For steady flow through any stream tube or pipe, the mass flow rate is the same at every section: ρ₁A₁V₁ = ρ₂A₂V₂. If density is constant (liquids, and low-speed gas flow), this reduces to A₁V₁ = A₂V₂ = Q, the volume flow rate. For incompressible flow in a rigid pipe this holds instantaneously even when the flow is unsteady. At a junction, the flow in equals the sum of the flows out.

Differential form. For incompressible flow at a point: ∂u/∂x + ∂v/∂y + ∂w/∂z = 0. A proposed velocity field that does not satisfy this cannot describe an incompressible flow.

Stream function and velocity potential (2-D). For 2-D incompressible flow a stream function ψ exists with u = ∂ψ/∂y and v = −∂ψ/∂x; lines of constant ψ are streamlines, and the difference ψ₂ − ψ₁ is the volume flow per unit depth between two streamlines. If the flow is also irrotational, a velocity potential φ exists with u = ∂φ/∂x, v = ∂φ/∂y; lines of constant φ cut streamlines at right angles.

Formulas

Q = A·V

• Q: volume flow rate (m³/s); A: flow area (m²); V: mean velocity (m/s).

ṁ = ρ·A·V

• ṁ: mass flow rate (kg/s); ρ: density (kg/m³).

ρ₁·A₁·V₁ = ρ₂·A₂·V₂

• Steady flow, any fluid.

A₁·V₁ = A₂·V₂, so for circular pipes V₂ = V₁·(d₁/d₂)²

• Incompressible flow; d: diameters (m).

∂u/∂x + ∂v/∂y + ∂w/∂z = 0

• Incompressible continuity at a point; u, v, w velocity components (m/s).

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

• Total acceleration (m/s²) = local + convective.

ω_z = ½·(∂v/∂x − ∂u/∂y)

• Rotation about z (rad/s); zero for irrotational flow. Vorticity = 2ω.

u = ∂ψ/∂y, v = −∂ψ/∂x; q = ψ₂ − ψ₁

• ψ: stream function (m²/s); q: flow per unit depth between two streamlines (m²/s).

Worked examples

Example 1 (standard): reducer in a water line. Given: water flows through a 150 mm pipe at 2 m/s, which reduces to 100 mm. ρ = 1000 kg/m³. Find Q, ṁ and the velocity in the smaller pipe.

1. A₁ = π·d₁²/4 = π × 0.15² / 4 = 0.017 671 m²
2. Q = A₁·V₁ = 0.017 671 × 2 = 0.035 34 m³/s (35.3 L/s)
3. ṁ = ρ·Q = 1000 × 0.035 34 = 35.34 kg/s
4. V₂ = V₁·(d₁/d₂)² = 2 × (150/100)² = 4.5 m/s Answer: Q ≈ 0.0353 m³/s, ṁ ≈ 35.3 kg/s, V₂ = 4.5 m/s

Example 2 (GATE level): checking a velocity field and finding acceleration. Given: a steady 2-D field u = 3x, v = −3y (m/s, with x and y in m). (a) Is it a possible incompressible flow? (b) Find the acceleration at (2, 1) m. (c) Find the flow per unit depth between the streamlines through (1, 1) and (2, 1).

1. Continuity: ∂u/∂x + ∂v/∂y = 3 + (−3) = 0, so the flow is possible.
2. Steady, so no local acceleration. a_x = u·∂u/∂x + v·∂u/∂y = (3x)(3) + (−3y)(0) = 9x = 18 m/s² at x = 2.
3. a_y = u·∂v/∂x + v·∂v/∂y = (3x)(0) + (−3y)(−3) = 9y = 9 m/s² at y = 1.
4. |a| = √(18² + 9²) = 20.12 m/s²
5. Stream function: u = ∂ψ/∂y = 3x gives ψ = 3xy + f(x); v = −∂ψ/∂x = −3y gives f′(x) = 0. So ψ = 3xy.
6. ψ(1,1) = 3, ψ(2,1) = 6, so q = 6 − 3 = 3 m²/s per metre depth. Answer: possible flow; |a| ≈ 20.1 m/s²; q = 3 m³/s per metre depth

Note that the flow is steady, yet the particle accelerates — purely convective acceleration.

Example 3 (quick): cylinder speed. A pump delivers 20 L/min to a 50 mm bore cylinder: V = Q/A = (20/60 000) / (π × 0.05²/4) = 0.170 m/s.

Common mistakes

• Using diameter instead of area: velocity scales with (d₁/d₂)², not d₁/d₂.
• Forgetting to convert L/min or mm to SI before calculating.
• Saying steady flow has no acceleration; it can have convective acceleration.
• Assuming streamlines equal pathlines in unsteady flow.
• Getting the sign wrong in v = −∂ψ/∂x, or confusing ψ (streamlines) with φ (equipotential lines).
• Applying A₁V₁ = A₂V₂ to a gas with large density changes; use ρAV instead.

For GATE PI

Expect quick numericals on pipe reducers, branching pipes and cylinder speeds using Q = AV, and 2-D velocity-field questions: check continuity, compute convective acceleration at a point, find the rotation or vorticity, or obtain the stream function and the flow between two streamlines. Conceptual one-markers test the definitions of steady, uniform, streamline, pathline and streakline. Practise partial derivatives of simple polynomial fields quickly and carefully.

Quick check

1. In which type of flow do streamlines, pathlines and streaklines coincide?
2. Water flows at 1 m/s in a 200 mm pipe that reduces to 100 mm. What is the velocity in the smaller pipe?
3. Is u = 2x, v = 2y a possible incompressible 2-D flow?
4. Can a steady flow have non-zero acceleration?

Answers: 1. Steady flow 2. 4 m/s 3. No, ∂u/∂x + ∂v/∂y = 4 ≠ 0 4. Yes, convective acceleration in non-uniform flow