Kinematics of flow, streamlines and continuity equation

Why it matters

Before you can apply forces or energy to a flowing fluid you must describe its motion: where it goes, how fast, and whether mass is conserved. Kinematics gives that description, and the continuity equation is the first equation written in every pipe, nozzle, heat-exchanger or reactor flow problem. Getting velocities right from flow rates and areas is the step on which every later pressure-drop and pump calculation depends.

Key ideas

Two descriptions of motion.

• Lagrangian: follow individual fluid particles and track their position with time (like tracking one tracer bead).
• Eulerian: fix points in space and record the velocity field V(x, y, z, t) there. Almost all engineering fluid mechanics uses the Eulerian view.

Classifying flows.

• Steady: properties at a fixed point do not change with time (∂/∂t = 0). Unsteady: they do (draining tank, valve closure).
• Uniform: velocity does not change along the flow direction at a given instant (constant-area pipe). Non-uniform: it does (nozzle, diffuser).
• One-, two-, three-dimensional: number of space coordinates the velocity depends on. Pipe flow with average velocity is treated as one-dimensional.
• Laminar / turbulent, rotational / irrotational, compressible / incompressible.

Flow lines.

• Streamline: a line everywhere tangent to the instantaneous velocity vector. No flow crosses a streamline, and streamlines cannot cross (except at stagnation points), because a point has only one velocity.
• Pathline: the actual trajectory of one particle over time.
• Streakline: the locus of all particles that have passed through one fixed point (what dye injected at a point shows).
• Streamtube: a tube formed by streamlines through a closed curve; fluid enters and leaves only at its ends.
• In steady flow, streamlines, pathlines and streaklines coincide. In unsteady flow they generally differ.

Acceleration of a fluid particle. Even in steady flow a particle accelerates if it moves into a region of different velocity (a nozzle). The total (material) acceleration has a local part ∂V/∂t, present only in unsteady flow, and a convective part (V·∇)V, present only in non-uniform flow.

Continuity (conservation of mass). For a control volume, rate of accumulation of mass = mass in − mass out.

• Steady flow in a streamtube or pipe: ṁ = ρAV is constant.
• Incompressible flow: Q = AV is constant, so velocity is inversely proportional to area (to d² for circular pipes).
• Differential form: ∂ρ/∂t + ∇·(ρV) = 0. For incompressible flow (steady or unsteady) this reduces to ∇·V = 0.
• Branched pipes: the sum of inflows equals the sum of outflows at a junction.

Stream function and velocity potential (2-D).

• Stream function ψ exists for any 2-D incompressible flow and satisfies continuity automatically. Lines of constant ψ are streamlines, and the difference ψ₂ − ψ₁ equals the volumetric flow per unit depth between them.
• Velocity potential φ exists only for irrotational flow. Lines of constant φ are perpendicular to streamlines.
• Rotation: vorticity ω_z = ∂v/∂x − ∂u/∂y; the flow is irrotational if ω = 0.

Formulas

• ṁ = ρ·A·V — mass flow rate (kg/s); ρ (kg/m³), A (m²), V average velocity (m/s).
• Q = A·V, A₁·V₁ = A₂·V₂ — incompressible, steady; Q (m³/s).
• V₂ = V₁·(d₁/d₂)² — circular pipes, d diameter (m).
• ∂ρ/∂t + ∇·(ρV) = 0 — general continuity equation.
• ∂u/∂x + ∂v/∂y + ∂w/∂z = 0 — incompressible flow; u, v, w velocity components (m/s).
• a_x = ∂u/∂t + u·∂u/∂x + v·∂u/∂y + w·∂u/∂z — x-acceleration (m/s²); similar for a_y, a_z.
• u = ∂ψ/∂y, v = −∂ψ/∂x — stream function ψ (m²/s).
• u = −∂φ/∂x, v = −∂φ/∂y — velocity potential φ (m²/s); some texts use the opposite sign, so follow your textbook.
• ω_z = ∂v/∂x − ∂u/∂y — vorticity (s⁻¹).

Worked examples

Example 1 (standard). Water flows at 2 m/s in a 0.30 m pipe that splits into two branches of 0.20 m and 0.15 m diameter. The velocity in the 0.20 m branch is 2.5 m/s. Find the velocity in the 0.15 m branch.

1. Q = A·V. Q = (π/4)(0.30)² × 2 = 0.1414 m³/s.
2. Q₁ = (π/4)(0.20)² × 2.5 = 0.0785 m³/s.
3. Continuity at the junction: Q₂ = Q − Q₁ = 0.0628 m³/s.
4. V₂ = Q₂/A₂ = 0.0628 / [(π/4)(0.15)²] = 0.0628 / 0.01767 = 3.56 m/s. V₂ ≈ 3.56 m/s.

Example 2 (GATE level). A 2-D flow has u = 2x and v = −2y (m/s, with x and y in m). (a) Is it a possible incompressible flow? (b) Find the acceleration at (1, 2). (c) Find the stream function and the flow per unit depth between the streamlines through (1, 1) and (2, 2).

1. Continuity: ∂u/∂x + ∂v/∂y = 2 − 2 = 0. Yes, incompressible flow is possible.
2. Steady, so a_x = u·∂u/∂x + v·∂u/∂y = (2x)(2) + (−2y)(0) = 4x. At (1, 2): a_x = 4 m/s².
3. a_y = u·∂v/∂x + v·∂v/∂y = 0 + (−2y)(−2) = 4y. At (1, 2): a_y = 8 m/s².
4. |a| = √(4² + 8²) = 8.94 m/s².
5. From u = ∂ψ/∂y = 2x, ψ = 2xy + f(x). Then v = −∂ψ/∂x = −2y − f′(x) = −2y, so f is a constant: ψ = 2xy.
6. ψ(1, 1) = 2 and ψ(2, 2) = 8, so the flow between the streamlines is 8 − 2 = 6 m²/s, i.e. 6 m³/s per metre depth.
7. Vorticity ω_z = ∂v/∂x − ∂u/∂y = 0, so the flow is also irrotational. (b) |a| ≈ 8.94 m/s²; (c) ψ = 2xy, q = 6 m³/s per metre depth.

Common mistakes

• Using area ratio d₁/d₂ instead of (d₁/d₂)²: halving the diameter quadruples the velocity.
• Saying steady flow has zero acceleration; convective acceleration exists wherever the velocity changes along the path.
• Applying Q = AV across a compressible gas flow with large density change; use ṁ = ρAV.
• Confusing streamlines with pathlines in unsteady flow.
• Assuming a velocity potential exists for any flow; it needs irrotational flow, while ψ needs only 2-D incompressible flow.

For GATE CH

Expect continuity numericals for reducers and branches, checks of whether a given velocity field satisfies continuity, finding a missing velocity component, computing local and convective acceleration at a point, and stream-function questions (flow between two streamlines). Practise partial differentiation of simple polynomial velocity fields quickly and accurately.

Quick check

1. The diameter of a pipe is halved. By what factor does the velocity change?
2. For u = x² + y in 2-D incompressible flow, find v (taking v = 0 at y = 0).
3. In what kind of flow do streamlines and pathlines coincide?
4. Which acceleration component is zero in steady flow?

Answers: 1. Increases four times. 2. v = −2xy. 3. Steady flow. 4. The local (temporal) acceleration ∂V/∂t.