Key examples — fluid mechanics

Where the chapter’s machinery shows up across the bookshelf.

Example 1: Bernoulli at the threshold of cavitation

The pressure at the throat of a constriction drops by $\tfrac12\rho U_\text{throat}^2 - \tfrac12\rho U_\infty^2$ relative to the far field. When this drop is large enough to push the local pressure below the liquid’s vapour pressure, a vapour cavity nucleates — cavitation. The cavitation number $\mathrm{Ca} = (p_\infty - p_v) / (\tfrac12 \rho U_\infty^2)$ measures how close the flow is to this threshold; cavitation occurs roughly when $\mathrm{Ca} < -C_{p,\min}$ where $C_{p,\min}$ is the most-negative pressure coefficient anywhere on the body. See Cavitation Ch 2.4.

Example 2: lubrication theory and the cochlear traveling wave

The basilar membrane sits in a thin fluid-filled gap of the cochlea (~1 mm wide, ~100 μm scale height). For long-wavelength pressure variations along the cochlea, the flow in the gap is approximately parallel to the membrane: $u_x \gg u_y$, $\partial/\partial y \gg \partial/\partial x$. This is the lubrication-theory limit. Navier–Stokes reduces to

which is exactly the long-wave hydrodynamic equation coupling membrane displacement and pressure that the Hearing book’s traveling-wave chapter develops. The whole tonotopic frequency-place map of the cochlea is a consequence of this lubrication-theory reduction applied to a membrane with a stiffness gradient.

Example 3: the Rayleigh–Plesset velocity field

For a spherical bubble of radius $R(t)$ in an incompressible liquid, mass conservation $\nabla\cdot\mathbf{u} = 0$ in spherical coordinates gives $\partial(r^2 u_r)/\partial r = 0$, i.e. $r^2 u_r = R^2 \dot R$ (matching the boundary condition $u_r(R) = \dot R$). Integrating the radial Euler equation from $R$ to $\infty$ gives the Rayleigh–Plesset equation. This is one of the cleanest deployments of continuity + Euler in spherical geometry. See Cavitation Ch 3.1.

Example 4: very-low-Re Stokes drag on a stereocilium

A single stereocilium of a hair-bundle has radius $\sim 100\,\text{nm}$ and moves at frequencies up to 20 kHz with displacements of a few nm. The Reynolds number is $\mathrm{Re} \sim 10^{-5}$ — deep in Stokes flow. The drag force per cilium is $\gamma_\text{drag} \dot{x}$ with $\gamma_\text{drag} \sim 6\pi\mu a$, contributing to the damping of the basilar-membrane response. The hair-cell amplifier (Hearing Ch 4.5) is precisely the cell’s strategy for cancelling this viscous damping with an active feedback force.

Example 5: the four routes to the acoustic wave equation

The Sound book derives the acoustic wave equation four different ways: from fluid mechanics (continuity + Euler + adiabatic EOS), from a lattice of oscillators, from kinetic theory, and from Hamilton’s principle. The fluid-mechanics route (Sound Ch 4.5) is the one this chapter directly supports. Linearising continuity and Euler around a uniform rest state, with $p = p_0 + p'$, $\rho = \rho_0 + \rho'$, $\mathbf{u} = \mathbf{u}'$, gives

the acoustic wave equation, with $c^2 = \gamma p_0/\rho_0$ from the adiabatic equation of state.

Cross-book backlinks

• Sound Ch 4.2 — conservation of mass: the continuity equation, slowly.
• Sound Ch 4.3 — Euler’s equation: F = ma on a slab.
• Sound Ch 4.5 — fluid-mechanics route to the wave equation: linearising continuity + Euler.
• Sound Ch 9 — moving media: the material-derivative version of acoustic propagation.
• Hearing Ch 4.3 — the cochlear traveling wave: lubrication-theory long-wave equation.
• Cavitation Ch 3.1 — Rayleigh–Plesset derivation: continuity + Euler in spherical symmetry.
• Cavitation Ch 2.4 — nucleation in flow: Bernoulli at the cavitation threshold.