Classical Mechanics · Upper Division

Fluid Mechanics

Fluids are continuous media described by field equations — the Navier-Stokes equations — that govern everything from blood flow to ocean currents to atmospheric weather, and remain one of the great unsolved problems of mathematics.

Prerequisites

Newton's laws (Ch. 3) \cdot Vectors and calculus (Ch. 21-22) \cdot Partial differential equations

Learning Goals

Distinguish Eulerian and Lagrangian descriptions and write the material derivative D/Dt in terms of partial derivatives.
Derive th $e continuity equation for a compressible fluid and state the incompressibility condition \nabla$
Apply Bernoulli's equation to find flow velocities and pressure differences in steady, inviscid, incompressible flow.
Define the Reynolds number and use it to classify flow regimes from creeping (Stokes) to fully turbulent.
$Explain Prandtl's boundary layer concept and the \delta \propto \sqrt(\nux/U) scaling for a flat plate$ .

FM.1 Kinematics and the Continuity Equation

A fluid is described by the velocity field v(r, t) — the velocity of the fluid element at position r at time t. Two frames of description:

Eulerian: fixed position r, varying t. What velocity do we observe at a fixed point in space? The relevant time derivative is ∂/∂t (local).

Lagrangian: follow a fluid parcel. The material derivative(rate of change following the fluid) is:

\frac{D}{Dt} = \frac{\partial}{\partial t} + \mathbf{v}\cdot\nabla \qquad \text{(material/substantial derivative)}

(FM.1)

Mass conservation gives the continuity equation:

\frac{\partial \rho}{\partial t} + \nabla\cdot(\rho\mathbf{v}) = 0 \qquad \text{(continuity, compressible)}

(FM.2)

For an incompressible fluid (ρ = const): ∇·v = 0 — the flow is divergence-free. This holds for liquids and subsonic gas flows.

FM.2 Euler and Navier-Stokes Equations

Newton's second law for a fluid element, including pressure and viscosity:

\rho\left(\frac{\partial \mathbf{v}}{\partial t}+(\mathbf{v}\cdot\nabla)\mathbf{v}\right) = -\nabla P + \eta\nabla^2\mathbf{v} + \rho\mathbf{g} \qquad \text{(Navier-Stokes)}

(FM.3)

For inviscid flow (η = 0): Euler's equation. The nonlinear term (v·∇)v is the inertial term — it causes turbulence, vortex stretching, and makes Navier-Stokes brutally difficult. Whether smooth solutions always exist in 3D is one of the Millennium Prize Problems (unsolved, $1M prize).

Definition FM.1 — Reynolds Number

The Reynolds number Re = \rhovL/\eta = vL/\nu (\nu = \eta/\rho is kinematic viscosity) measures the ratio

of inertial to viscous forces:

Re = (inertial force) / (viscous force) = \rhov^{2}L^{2} / (\etavL) = \rhovL/\eta

Re ≪ 1: viscous dominates (Stokes flow — creeping, reversible, s

perm swimming). Re ≫ 1: inertial dominates (turbulent, mixing, aircraft). Transition: Re \approx

FM.3 Bernoulli's Equation

For inviscid, steady, incompressible flow along a streamline:

P + \frac{1}{2}\rho v^2 + \rho gh = \mathrm{constant} \qquad \text{(Bernoulli's equation)}

(FM.4)

This is energy conservation for fluid elements. Bernoulli's principle — faster flow, lower pressure — explains airfoil lift (wing), carburetor operation, and the Venturi effect. It is derived from Euler's equation by integration along a streamline.

Example FM.1 — Torricelli's Law

A large tank has a small hole at depth h below the surface. Find the exit velocity.

Apply Bernoulli:

At surface: P = P_{atm}, v \approx 0 (large tank), height = h. At hole: P = P_{atm}, height = 0

Bernoulli:

P_{atm} + 0 + \rhogh = P_{atm} + \frac{1}{2}\rhov^{2}_exit + 0

Result:

v_{exit} = \sqrt(2gh) \qquad (Torricelli's law, 1643

Same as projectile:This is the velocity a

ball acquires falling freely a height h — as if the water fell freely. Discharge rate: Q =

le \sqrt(2gh

FM.4 Vorticity and Potential Flow

The vorticity ω = ∇ × v measures local rotation of the fluid. For irrotational flow (ω = 0): v = ∇φ for a velocity potential φ. Combined with incompressibility (∇·v = 0):

\nabla^2\phi = 0 \qquad \text{(Laplace's equation for potential flow)}

(FM.5)

Potential flow is solved by the same methods as electrostatics! A cylinder in uniform flow U_∞ has solution φ = U_∞ r(1 + R²/r²) cos θ — D'Alembert's paradox: no drag. Real fluids have viscous boundary layers that separate, creating drag — potential flow misses this entirely. Adding circulation Γ (rotation around the cylinder) gives lift:

L = \rho U_\infty \Gamma \qquad \text{(Kutta-Joukowski theorem)}

(FM.6)

Example FM.2 — Stokes Drag on a Sphere

For very viscous flow (Re ≪ 1) past a sphere of radius R moving at velocity U, the drag force is:

Stokes flow solution:

In the Stokes limit, drop the inertial term in N-S: \eta\nabla^{2}v = \nablaP. Solve with no-slip BC at sp

here surface.

Stokes drag formula:

F_{drag} = 6\pi\etaRU

Terminal velocity:

For a sphere of density \rho_s in fluid \rho_f: weight - buoyancy = drag \to (4/3)\piR^{3}(\rho_s - \rho_f)g

6\pi\etaRU \to U_{terminal} = 2R^{2}(\rho_s-\rho_f)g/(9\eta

Example:

Rain drop R=1mm, \rho_water = 1000, \rho_air = 1.2 kg/m^{3}, \eta_air = 1.8\times10^{-5} Pa\cdots. U = 2(10^{-3})^{2}\times99

9\times9.8/(9\times1.8\times10^{-5}) \approx 12 m/s. (Re = \rhovR/\eta \approx 800 — Stokes is not valid here; the actual term

inal velocity with form drag correction is ~9 m/s.)

Definition FM.2 — Common Traps

Eulerian and Lagrangian are viewpoints: fixed-point derivatives and parcel-following derivatives are not the same.
Incompressible does not mean densityless: $it means density of each parcel is constant, giving \nabla\cdotv = 0$
Bernoulli has assumptions: steady, inviscid, incompressible flow along a streamline.
High Reynolds number does not mean no viscosity: viscosity may dominate thin boundary layers.
Potential flow misses drag: real drag often comes from viscosity, separation, and wake formation.

Exercises — FM.1–FM.4 Fluid Mechanics

A Venturi meter has upstream radius 5 cm and throat radius 2 cm. The pressure difference is 1000 Pa. Find the flow velocity and volume flow rate Q (L/s) for water.

L/s

Straightforward

Describe the dynamics of two parallel line vortices with circulations \pmΓ separated by dist

ance d. What happens if both have the same sign? Opposite signs?

Intermediate

Describe the Prandtl boundary layer. How does the thickness \delta scale with distance x along

a flat plate? When does the boundary layer separate?

Intermediate

4.State Kolmogorov's theory of turbulence and the -5/3 power law for the energy spectrum. What determines the smallest and largest scales? Why is turbulence computationally intractable?

Challenging

Key Takeaways

$Material derivative: D/Dt = \partial/\partialt + v\cdot\nabla (rate of change following fluid parcel$ .
$Continuity: \partial\rho/\partialt + \nabla\cdot(\rhov) = 0. Incompressible: \nabla\cdotv = 0$ .
$Navier-Stokes: \rho Dv/Dt = -\nablaP + \eta\nabla^{2}v + \rhog. Nonlinear, unsolved in 3D$ .
$Reynolds number Re = \rhovL/\eta: Re≪1 laminar (Stokes), Re≫1 turbulent$ .
$Bernoulli (inviscid, steady): P + \frac{1}{2}\rhov^{2} + \rhogh = const along streamline$ .
$Potential flow: \nabla^{2}\phi = 0 = electrostatics. Kutta-Joukowski: L = \rhoU\inftyΓ$ .