Fluid Dynamics
Fluid dynamics is the branch of continuum mechanics that studies the motion of liquids and gases. From blood flow to weather, from pipe networks to galactic winds, the same equations — Navier–Stokes in their full generality — describe astonishingly diverse phenomena.
A substance that deforms continuously under shear stress, however small. Both liquids and gases qualify; the principal difference is compressibility, gases being far more compressible than liquids.
Eulerian description and the material derivative
In the Eulerian description, we attach the coordinate system to space and follow how velocity v(r, t), density ρ and pressure p change at each fixed point. For any quantity f, the rate of change following a fluid particle is
Df/Dt = ∂f/∂t + (v·∇)f
(the material or substantial derivative).
Continuity equation
Conservation of mass for a control volume gives the continuity equation:
∂ρ/∂t + ∇·(ρv) = 0
For an incompressible fluid (constant ρ), this reduces to
∇·v = 0 (incompressibility / solenoidal velocity)
Euler's equation
For an ideal (inviscid) fluid, Newton's second law applied to a fluid element gives Euler's equation:
ρ Dv/Dt = −∇p + ρg
(g is gravitational acceleration). Integrating along a streamline for steady, incompressible, inviscid flow yields Bernoulli's equation:
p + ½ρv² + ρgh = constant along a streamline
This single equation — relating pressure, kinetic energy and potential energy per unit volume — explains airfoil lift, Venturi flow meters, Pitot tubes and the curl of a spinning cricket ball.
- Streamline: line everywhere tangent to v at a given instant.
- Pathline: actual trajectory of a fluid particle.
- Streakline: locus of all particles that previously passed through a fixed point.
- In steady flow these three coincide.
- Vorticity ω = ∇×v; Kelvin's circulation theorem: in inviscid barotropic flow, circulation around a material loop is conserved.
Navier–Stokes equation
Adding viscosity (Newtonian, isotropic) gives the Navier–Stokes equation:
ρ Dv/Dt = −∇p + μ∇²v + ρg
where μ is the dynamic viscosity (Pa·s). For incompressible flow this together with ∇·v = 0 forms a closed system. Solutions are notoriously difficult; existence and smoothness in 3-D is one of the Millennium Prize Problems.
Reynolds number
The dimensionless ratio of inertial to viscous forces is:
Re = ρVL/μ = VL/ν, where ν = μ/ρ is kinematic viscosity
Re classifies flow regimes:
| Re | Regime |
|---|---|
| ≪ 1 | Creeping (Stokes) flow |
| ~10² | Laminar |
| ~10³–10⁴ | Transition |
| ≫ 10⁴ | Turbulent |
For pipe flow, transition typically occurs around Re ≈ 2300.
Bernoulli applications
| Device | Principle |
|---|---|
| Pitot tube | Stagnation point: p_stag = p_static + ½ρv² gives airspeed |
| Venturi meter | Narrow throat → lower pressure → measures flow rate |
| Airfoil lift | Faster flow over curved upper surface → lower pressure (with circulation, Kutta condition) |
| Magnus effect | Spinning ball generates asymmetric flow and sideways force |
Viscous flow examples
Poiseuille flow (laminar pipe flow)
Steady incompressible flow through a horizontal pipe of radius R driven by pressure gradient Δp/L:
v(r) = (Δp/(4μL))(R² − r²)
The volumetric flow rate is
Q = (πR⁴/8μL) Δp
(Hagen–Poiseuille). Note the R⁴ dependence — halving the radius reduces flow 16-fold; basis of blood-pressure physics.
Couette flow
Steady shear between parallel plates of separation h, upper plate moving at U: linear velocity profile v(y) = Uy/h, shear stress τ = μU/h.
Boundary layer
Near a solid surface in high-Re flow, viscosity remains important in a thin boundary layer (Prandtl, 1904), beyond which the flow is essentially inviscid. Boundary layer thickness δ scales as δ/L ~ Re^(−1/2) for laminar flow. Separation of the boundary layer at adverse pressure gradients causes form drag — important in aircraft, ships and cars.
For dimensional-analysis questions, list the variables, count their independent dimensions, and apply the Buckingham π theorem: a relation among n variables involving k independent dimensions can be expressed in terms of (n − k) dimensionless groups. Classic example: drag on a sphere depends on (D, V, ρ, μ) → 4 − 3 = 1 group, Re; hence C_d = f(Re).
Compressible flow
When velocities approach the speed of sound a = √(γRT/M), density changes become important. The Mach number M = V/a partitions flows:
- M < 0.3: incompressible approximation valid
- 0.3 < M < 1: subsonic compressible
- M = 1: sonic
- M > 1: supersonic; shock waves and oblique shocks appear
For isentropic flow the area-Mach relation predicts a converging-diverging (de Laval) nozzle for supersonic acceleration — central to rocket engines.
Dimensionless numbers
Beyond Re, several other groups appear:
| Number | Definition | Importance |
|---|---|---|
| Mach | V/a | Compressibility |
| Froude | V/√(gL) | Free-surface flows |
| Weber | ρV²L/σ | Surface tension |
| Prandtl | ν/α | Heat/momentum diffusion ratio |
| Nusselt | hL/k | Convective heat transfer |
These groups underpin the scaling of laboratory experiments to real-world scale — naval architects, aerodynamicists and chemical engineers all rely on them.