Navier-Stokes equations and exact solutions

Why it matters

The Navier–Stokes equations are the complete momentum balance for a Newtonian fluid; every CFD package solves them. A handful of flows — between plates, in pipes, in thin falling films — can be solved exactly, and these exact solutions are the backbone of lubrication, falling-film absorbers and evaporators, coating flows, viscometers and laminar heat- and mass-transfer analysis. They also show how to reduce a general equation to a problem you can integrate by hand.

Key ideas

What the equations say. For a fluid element: mass × acceleration = pressure force + viscous force + body force. Together with continuity they give four equations for u, v, w and p (for incompressible flow).

Assumptions behind the usual (incompressible) form.

• Continuum fluid.
• Newtonian: viscous stress is linear in strain rate (Stokes' constitutive relations).
• Constant density and constant viscosity. The general compressible form also exists; it adds a ∇(∇·V) term and is used for gas dynamics. Temperature-dependent properties need the energy equation as well.

Meaning of each term (x-direction): ρ(∂u/∂t + u·∂u/∂x + v·∂u/∂y + w·∂u/∂z) is inertia (local + convective acceleration); −∂p/∂x is the pressure force; μ∇²u is the net viscous force; ρg_x is gravity. Dropping the viscous term gives Euler's equation; dropping the inertia term gives creeping (Stokes) flow, valid for Re ≪ 1.

Why exact solutions are rare. The convective terms are non-linear, so solutions cannot be superposed. Exact solutions exist where the geometry makes the convective terms vanish identically: parallel flows, in which only one velocity component exists and it varies only across the flow (fully developed).

Recipe for an exact solution.

1. Choose coordinates aligned with the flow (Cartesian for plates and films, cylindrical for pipes and annuli).
2. State assumptions: steady, fully developed, one velocity component, no variation in the third direction.
3. Use continuity to show that the convective terms vanish.
4. Reduce the momentum equation to an ordinary differential equation and integrate twice.
5. Apply boundary conditions: no slip at a solid wall (fluid velocity equals wall velocity); zero shear (du/dy = 0) at a free surface in contact with a gas; symmetry (du/dy = 0) on a centreline.

Classical exact solutions.

• Plane Couette flow: fluid between a fixed and a moving plate, no pressure gradient: linear profile u = U·y/h.
• Plane Poiseuille flow: fixed plates, pressure-driven: parabolic profile, u_max = 1.5 × average.
• Couette–Poiseuille: moving plate plus pressure gradient; the profiles superpose because the reduced equation is linear.
• Hagen–Poiseuille: circular pipe, parabolic in r, u_max = 2 × average.
• Falling film on an inclined or vertical plate: gravity-driven, half-parabola with the maximum at the free surface — the basis of Nusselt's film theory for wetted-wall columns and condensation.
• Others: flow in an annulus, flow between rotating cylinders (Couette viscometer), Stokes' first problem (suddenly started plate).

Formulas

• ρ·(∂u/∂t + u·∂u/∂x + v·∂u/∂y + w·∂u/∂z) = −∂p/∂x + μ·(∂²u/∂x² + ∂²u/∂y² + ∂²u/∂z²) + ρ·g_x — x-momentum, incompressible Newtonian; u, v, w (m/s), p (Pa), μ (Pa·s), ρ (kg/m³).
• ∂u/∂x + ∂v/∂y + ∂w/∂z = 0 — continuity.
• u = U·y/h — plane Couette flow; U plate speed (m/s), h gap (m).
• u = (1/2μ)·(−dp/dx)·(h² − y²) — plane Poiseuille flow between plates at y = ±h (half-gap h).
• u_max = (−dp/dx)·h²/(2μ), u_avg = (2/3)·u_max, q = (−dp/dx)·(2h)³/(12μ) — q flow per unit width (m²/s).
• τ_w = (−dp/dx)·h — wall shear stress between plates (Pa).
• u = (ρ·g·sinθ/μ)·(δ·y − y²/2) — falling film on a plane inclined at θ to the horizontal; y from the wall, δ film thickness (m).
• u_max = ρ·g·sinθ·δ²/(2μ), u_avg = ρ·g·sinθ·δ²/(3μ), q = ρ·g·sinθ·δ³/(3μ) — film results; vertical wall θ = 90°.
• Re_film = 4·ρ·q/μ — film Reynolds number; smooth laminar only up to about 30, wavy-laminar to about 1000–1800 (varies by source), then turbulent.

Worked examples

Example 1 (standard). Oil (μ = 0.3 Pa·s) flows between two fixed parallel plates 10 mm apart under a pressure gradient of −2000 Pa/m. Find u_max, the flow per metre width and the wall shear stress.

1. Half-gap h = 0.005 m.
2. u_max = (−dp/dx)·h²/(2μ) = 2000 × 2.5 × 10⁻⁵ / 0.6 = 0.0833 m/s.
3. q = (−dp/dx)·(2h)³/(12μ) = 2000 × (0.01)³ / 3.6 = 5.56 × 10⁻⁴ m²/s. Check: u_avg × gap = (2/3 × 0.0833) × 0.01 = 5.56 × 10⁻⁴ ✓.
4. τ_w = (−dp/dx)·h = 2000 × 0.005 = 10 Pa. u_max ≈ 0.083 m/s, q ≈ 5.56 × 10⁻⁴ m³/s per metre width, τ_w = 10 Pa.

Example 2 (GATE level). Water (ρ = 1000 kg/m³, μ = 1 × 10⁻³ Pa·s) runs down a vertical wall as a laminar film 0.3 mm thick. Derive and evaluate u_max, u_avg, the flow per metre of wall width and the film Reynolds number.

1. Assumptions: steady, fully developed, u = u(y) only, no pressure gradient along the film (free surface at atmospheric pressure).
2. x-momentum reduces to μ·d²u/dy² = −ρg.
3. Integrate: du/dy = −(ρg/μ)y + C₁. Zero shear at the free surface y = δ gives C₁ = ρgδ/μ.
4. Integrate again with u = 0 at y = 0: u = (ρg/μ)(δy − y²/2).
5. u_max (at y = δ) = ρgδ²/(2μ) = 1000 × 9.81 × (3 × 10⁻⁴)² / (2 × 10⁻³) = 0.441 m/s.
6. u_avg = (2/3)u_max = 0.294 m/s; q = u_avg·δ = ρgδ³/(3μ) = 8.83 × 10⁻⁵ m²/s.
7. Re_film = 4ρq/μ = 4 × 1000 × 8.83 × 10⁻⁵ / 10⁻³ = 353 — in the wavy-laminar range, so the smooth-film result is an approximation. u_max ≈ 0.441 m/s, u_avg ≈ 0.294 m/s, q ≈ 8.83 × 10⁻⁵ m³/s per metre, Re_film ≈ 353.

Common mistakes

• Using the full gap where the half-gap h appears (factor of 4 in u_max).
• Applying no-slip at a free surface; a gas–liquid interface has zero shear, not zero velocity.
• Forgetting that −dp/dx is positive for flow in +x.
• Writing u_max = 2u_avg for plates (that is the pipe result; plates give 1.5).
• Assuming exact solutions apply at high Re; they require laminar, fully developed flow.

For GATE CH

Expect derivation-based numericals for flow between plates, Couette–Poiseuille combinations, falling films on inclined plates, flow in an annulus, and questions identifying which NS terms vanish under given assumptions. Practise integrating d²u/dy² = constant and applying no-slip and zero-shear conditions cleanly.

Quick check

1. Which term of the NS equations makes them non-linear?
2. What boundary condition applies at the free surface of a falling film?
3. Ratio u_max/u_avg for flow between fixed parallel plates?
4. If film thickness doubles, by what factor does the flow per unit width change?

Answers: 1. The convective acceleration term (V·∇)V. 2. Zero shear stress, du/dy = 0. 3. 1.5. 4. 8 times (q ∝ δ³).