Fluid mechanics in one page

Material derivative, continuity, Euler, Navier–Stokes, Bernoulli, Reynolds, Stokes flow, boundary layers.

A fluid is a continuum that cannot resist a sustained shear elastically: any applied shear produces a flow. The governing equations of motion follow from applying Newton’s second law to a small fluid element (mechanics chapter) together with a constitutive law specifying how the element responds to stress. The Sound and Cavitation books and the cochlear-mechanics lessons of Hearing all rest on this small clutch of equations.

Eulerian vs. Lagrangian — and the material derivative

A scalar field $\phi(\mathbf{r}, t)$ on a flowing fluid can be tracked two ways. The Eulerian view sits at a fixed point in space and records what passes by; the Lagrangian view follows a particular fluid parcel along its trajectory. For a parcel moving with the local fluid velocity $\mathbf{u}(\mathbf{r}, t)$ , the time-derivatives in the two frames are related by the material derivative:

\frac{D\phi}{Dt} \;\equiv\; \frac{\partial \phi}{\partial t} \;+\; \mathbf{u}\cdot\nabla\phi.

The first term is the change at a fixed point; the second is the change the parcel experiences by being carried into a region where $\phi$ has a different value.

Eulerian observation point x = 2.0

The field T(x, y, t) is a wave advecting at speed U: T = T₀ + A sin(k(x − Ut)). A fixed-frame observer sees T oscillating in time — that's ∂T/∂t. A parcel drifting *with* the flow sees no change — DT/Dt = 0, because the parcel moves with the wave. The material derivative D/Dt = ∂/∂t + u·∇ is the time derivative *that matters for Newton's second law on a fluid element*.

The field is a wave moving with the flow, $T = T_0 + A\sin(k(x - Ut))$ . The Eulerian observer at fixed $x_E$ sees $T$ oscillating in time — $\partial T/\partial t \ne 0$ . The Lagrangian parcel drifts with the wave at speed $U$ , so it sees $T$ never change — $DT/Dt = 0$ . Newton’s second law applied to a fluid element involves $D\mathbf{u}/Dt$ , not $\partial \mathbf{u}/\partial t$ — this is the source of the convective nonlinearity in fluid dynamics.

Conservation of mass — continuity

Mass cannot disappear; if the mass density of a fluid changes at a point, mass must be flowing in or out. The differential statement is the continuity equation:

\frac{\partial \rho}{\partial t} \;+\; \nabla \cdot (\rho \mathbf{u}) \;=\; 0.

inflow v_in

0.50

outflow v_out

0.50

v_in = v_out → density parked. This is the *incompressible* case ∇·v = 0.

The Sound book’s continuity lesson develops this slowly with the slab-balance argument that makes the divergence term concrete. For an incompressible flow $D\rho/Dt = 0$ which, together with the above, reduces to the divergence-free condition

\nabla\cdot\mathbf{u} \;=\; 0.

Most of the flows in this bookshelf — perilymph in the cochlea, water in a cavitating volume, low-Mach-number atmospheric flows — are well-approximated as incompressible.

Euler’s equation — Newton’s second law in an inviscid fluid

Newton’s second law applied to a small fluid element: net force per unit volume equals $\rho\,D\mathbf{u}/Dt$ . In an inviscid fluid the only surface force on the element comes from pressure on its faces, summing to $-\nabla p$ per unit volume.

∂p/∂x-2.00 Pa/m

p_L − p_R0.800 Pa

a = −∇p/ρ (ρ = 1)2.000 m/s²

Pressure gradient ∂p/∂x = -2.00 Pa/m Slab thickness dx = 0.40 m

Pressure pushes a slab from both sides. When the pressures are equal the slab is in static equilibrium; when they differ, the imbalance is a force per unit area = (p_L − p_R). For a slab of thickness dx that's −(∂p/∂x)·dx — equivalently, −∇p per unit volume. Newton's second law on this volume element is exactly Euler's equation: ρ·Du/Dt = −∇p.

The pressure-difference reasoning is direct: on the left face $+p_L A$ pushes right; on the right face $-p_R A$ pushes left; the net is $(p_L - p_R) A = -(\partial p/\partial x)\,dx\,A = -(\partial p/\partial x)\,dV$ . Putting this into Newton’s second law gives Euler’s equation:

\rho \frac{D\mathbf{u}}{Dt} \;=\; -\nabla p \;+\; \rho \mathbf{g}.

p_left

0.50

p_right

0.50

∂p/∂x = 0 → no net force → slab coasts at constant velocity.

(See the Sound book’s Euler-equation lesson for the slow build-up.) Together with continuity and an equation of state — adiabatic, for the Sound book — this forms a closed system for $(\rho, \mathbf{u}, p)$ .

Navier–Stokes — what viscosity adds

Real fluids resist shear, not just compression. For a Newtonian fluid — water, air, every fluid in this bookshelf at moderate strain rates — the viscous stress is linear in the velocity-gradient tensor. For incompressible flow, the momentum equation becomes

\rho \frac{D\mathbf{u}}{Dt} \;=\; -\nabla p \;+\; \mu\, \nabla^2 \mathbf{u} \;+\; \rho \mathbf{g}.

The viscous term $\mu \nabla^2 \mathbf{u}$ redistributes momentum across velocity gradients exactly the way Fick’s law redistributes mass across concentration gradients (viscosity & diffusion chapter). The kinematic viscosity $\nu = \mu/\rho$ has units of $\text{m}^2/\text{s}$ — a diffusivity of momentum.

Bernoulli — an integrated form of Euler

For a steady, inviscid, incompressible flow along a streamline, Euler’s equation can be integrated once:

▶ From Euler to Bernoulli, along a streamline

For steady flow, $\partial \mathbf{u}/\partial t = 0$ , so

\rho\,(\mathbf{u}\cdot\nabla)\mathbf{u} \;=\; -\nabla p \;+\; \rho \mathbf{g}.

Use the vector identity $(\mathbf{u}\cdot\nabla)\mathbf{u} = \nabla(\tfrac12 |\mathbf{u}|^2) - \mathbf{u} \times (\nabla \times \mathbf{u})$ . Along a streamline the curl term is perpendicular to $\mathbf{u}$ and drops out when dotted with the tangent direction. With $\mathbf{g} = -g\hat{\mathbf{z}}$ and $\rho$ constant,

\nabla\left(\tfrac12 \rho |\mathbf{u}|^2 + p + \rho g z\right) \cdot d\mathbf{s} \;=\; 0,

so along the streamline

\tfrac12 \rho |\mathbf{u}|^2 \;+\; p \;+\; \rho g z \;=\; \text{const}.

This is Bernoulli’s equation. It is exact for steady, inviscid, incompressible flow along a streamline. Importantly, the constant can differ between streamlines — Bernoulli holds along, not across.

u(x)2.500

p(x)-1.625

½ρu²3.125

p + ½ρu²1.500

Probe location x = 0.00 Volume flow Q = 1.00 Throat opening = 40% of pipe area

Slide the probe across the Venturi. Where the pipe narrows the velocity rises (continuity: Q = A u = const) and the pressure drops (Bernoulli: p + ½ρu² = const along a streamline). The black total-energy curve is *flat* — the trade-off of pressure for kinetic energy along the streamline is exact for steady, inviscid, incompressible flow.

Slide the probe along the Venturi: where the pipe narrows the speed rises (continuity), the pressure drops (Bernoulli), and the total energy is rigorously constant along the streamline. This is the cavitation-inception argument of Cavitation Ch 2.4, the Pitot tube, the Venturi meter, and every aerodynamic-lift argument that does not invoke circulation explicitly.

The Reynolds number — the parameter of incompressible flow

From the Navier–Stokes equation, rescaling lengths by a characteristic $L$ , velocities by $U$ , time by $L/U$ , and pressure by $\rho U^2$ gives

\frac{D\tilde{\mathbf{u}}}{D\tilde{t}} \;=\; -\tilde{\nabla}\tilde{p} \;+\; \frac{1}{\mathrm{Re}}\, \tilde{\nabla}^2 \tilde{\mathbf{u}}, \qquad \mathrm{Re} \;\equiv\; \frac{\rho U L}{\mu} \;=\; \frac{U L}{\nu}.

The Reynolds number is the only dimensionless parameter of incompressible Newtonian flow. It expresses the ratio of inertial to viscous forces and dictates the flow regime.

Alternating vortices shed periodically; pressure on the cylinder oscillates.

Re100

C_d1.40

Strouhal St0.165

log₁₀ Re = 2.00 (Re = 100)

The diagram shows flow past a smooth cylinder across nine decades of Reynolds number. Five qualitatively distinct regimes appear: creeping (Stokes), attached, steady symmetric eddies, von Kármán vortex street, and post-drag-crisis turbulent boundary layer. The equation governing all five is the same single Navier–Stokes equation; all the behaviour comes from the relative weight of two terms, parametrised by Re.

Stokes flow and the time-reversibility theorem

When $\mathrm{Re} \ll 1$ , the inertial term in Navier–Stokes is negligible and the equation reduces to the linear Stokes equation:

\nabla p \;=\; \mu \nabla^2 \mathbf{u}, \qquad \nabla \cdot \mathbf{u} \;=\; 0.

Three consequential properties:

Linearity. Doubling the boundary velocity doubles the entire flow.
Time-reversibility. Reverse the boundary motion and the flow exactly reverses.
Stokes drag. $F = 6\pi \mu a U$ on a rigid no-slip sphere of radius $a$ .

The time-reversibility is the famous scallop theorem: a swimmer using a stroke that is identical to its time-reverse cannot move forward at low Re.

Stroke:

At low Reynolds number, the Stokes equation is *time-reversible*: reverse all the boundary velocities and the entire flow reverses too. A reciprocal stroke (one whose time-reverse traces the same shape, like a scallop opening and closing) therefore generates zero net motion. Bacteria escape this trap by using non-reciprocal motions — flagellar rotation, the breaststroke-like flexible-arm motion of Chlamydomonas — that break the symmetry. Purcell stated this as the "scallop theorem" in his 1977 *Life at Low Reynolds Number*.

The scallop opens and closes — its stroke is identical to its time-reverse, so the net displacement is zero. The corkscrew rotation traces a different shape forward and backward, breaks time-reversal symmetry, and can swim. Bacteria use this trick; so do spermatozoa and ciliated cells.

Boundary layers — viscosity hiding near walls

At high Reynolds number the viscous term is small almost everywhere, but it cannot be neglected in thin boundary layers adjacent to solid surfaces, where the velocity transitions from the bulk-flow value to zero (the no-slip condition). The boundary-layer thickness on a flat plate scales as

\delta(x) \;\sim\; \sqrt{\frac{\nu x}{U}} \;=\; \frac{x}{\sqrt{\mathrm{Re}_x}}.

U1.00

ν0.020

δ(x=1)0.707

Re_x=1 = Ux/ν50

Free-stream U = 1.00 Kinematic viscosity ν = 0.020

The boundary layer is the thin viscous region near the plate where the no-slip condition forces velocity to drop from U to 0. Its thickness grows as δ ~ √(νx/U) — slower with downstream distance, faster with viscosity, thinner at higher speed. Outside the layer the flow is essentially inviscid; inside, viscous stresses cannot be ignored. This split is the foundation of *boundary-layer theory*.

The boundary layer thickens with downstream distance as $\sqrt{x}$ ; outside it the flow is effectively inviscid (governed by Euler alone), and the velocity matches the free-stream $U$ . Inside it, the no-slip condition $u = 0$ at the wall forces a velocity profile that picks up momentum diffusively from the wall.

A related thin-layer approximation — lubrication theory — applies when the flow is confined between nearby surfaces, as in the gap between the basilar membrane and the cochlear bone. The long-wave equation of Hearing Ch 4.3 is precisely a lubrication-theory limit of the full Navier–Stokes equations.

⏳ The history — Euler, Navier, Stokes, and the slow domestication of viscosity

Leonhard Euler in 1755 wrote down what we now call Euler’s equation in a memoir to the Berlin Academy; he had developed the entire formalism of inviscid fluid dynamics by purely deductive reasoning from Newton’s laws applied to fluid elements. For nearly a century Euler’s equation was the fluid equation, and the persistent discrepancies between its predictions and reality — most famously, d’Alembert’s paradox that a body in steady inviscid flow experiences zero drag — were treated as embarrassments rather than evidence of a missing term.

The missing term is viscosity. Claude-Louis Navier (1822) and George Gabriel Stokes (1845) independently added the viscous-stress term, producing the equation we now call Navier–Stokes. The molecular justification — that microscopic momentum transport across velocity gradients gives a stress linear in the velocity gradient — was supplied later by Maxwell and Boltzmann via the kinetic theory of gases.

The mathematical maturity of Navier–Stokes is uneven. Existence and smoothness of three-dimensional solutions is still open — one of the Clay Millennium Prize problems. But for engineering purposes the equations are unambiguously right: they predict every flow regime, every transition, and every drag-coefficient curve in the bookshelf.

For the cross-book applications — Bernoulli at cavitation inception, lubrication-theory cochlea, the Rayleigh–Plesset velocity field, low-Re hair-bundle dynamics — see the key examples sub-page.