Continuum mechanics

The material derivative follows a moving continuum parcel:

An Eulerian time average holds spatial position fixed while averaging a time-dependent field:

A Lagrangian trajectory follows one marked fluid particle and solves

Stokes drift is the difference between the mean velocity of fluid particles and the Eulerian mean velocity at fixed positions. It appears at second order because particles sample the oscillatory velocity at displaced positions.

The Rayleigh-Taylor instability occurs when a denser fluid lies above a lighter fluid in a gravitational field, causing interface perturbations to grow.

A boundary layer is a thin region next to a surface in which viscosity remains significant even when inertia dominates the outer flow.

Balancing $U^2/x$ with $\nu U/{\delta }^2$ gives $\delta \sim \sqrt{\nu x/U}$.

For steady incompressible flow with imposed outer pressure gradient,The normal momentum equation makes pressure approximately uniform across the layer.

For a sheet moving with $u(x,0)=\alpha x$ through otherwise stationary fluid, the layer thickness is $\delta =\sqrt{\nu /\alpha }$. Withthe boundary-layer equations reduce toThe exact solution is $f=1-e^{-\eta }$.

Boundary-layer separation occurs where near-wall flow detaches from a surface, commonly after an adverse pressure gradient drives the wall shear to zero and then reverses it.

For $U=A{x}^m$, $\psi =\sqrt{\nu xU}f(\eta )$ giveswith $f(0)=f^{\prime }(0)=0$ and $f^{\prime }(\infty )=1$.

For harmonic acoustic amplitudes, $⟨\mathbf{I}⟩=\frac{1}{2}\mathrm{Re}(p{\mathbf{u}}^{\ast })$.

For a finite duct segment of phase length $\lambda$ joined to a semi-infinite medium, continuity of pressure and velocity gives the outgoing potential amplitudeAt half-integer $\lambda /\pi$, a large impedance mismatch can strongly enhance transmitted flux for a prescribed piston displacement.

A Burgers vortex balances axial vortex stretching against radial viscous diffusion, producing a Gaussian vorticity profile.

Elasticity studies reversible deformation and the stresses produced by it.

Linear elasticity approximates strain and stress to first order in a small displacement field.

For the one-dimensional reduction of isotropic linear elasticity, nondegeneracy means that the effective longitudinal modulus $\lambda +2\mu$ is nonzero. The static Navier-Cauchy equation then forces the displacement to be affine.

A displacement field $\mathbf{u}(\mathbf{x})$ gives the change in position of each material point from a reference configuration.

The displacement gradient has components $(\nabla \mathbf{u}{)}_{ij}={\partial }_i{u}_j$ and records local changes of the displacement field.

The two Lamé parameters $\lambda$ and $\mu$ determine the linear isotropic relation between stress and strain; $\mu$ is the shear modulus.

For the two-dimensional convention $\nabla \mathbf{u}=({\partial }_i{u}_j)$, the energy densityproduces the static isotropic elastic equations by variation.

In a homogeneous isotropic elastic body without body forces, static equilibrium of the displacement field is

For a one-dimensional body with endpoints displaced by zero and $\Delta$, the static Navier-Cauchy equation reduces to $u^{\prime \prime }=0$ and gives the uniform strain solution $u(x)=\Delta x/L$.

For density $\rho$ and Lamé parameters $\lambda ,\mu$, the displacement satisfiesIts longitudinal and transverse wave speeds are

An obliquely incident in-plane P- or SV-wave generally produces reflected and transmitted P- and SV-waves. Continuity of displacement and traction supplies four scalar conditions for their four amplitudes, while frequency and tangential wavenumber are shared.

Phase matching at a planar interface preserves frequency and tangential wavenumber. Thus waves of speeds $c_j$ and angles ${\theta }_j$ from the normal satisfy

The acoustic impedance of a fluid is $Z=\rho c$. At an oblique planar interface, zero reflection requires a matching condition involving the normal impedances $Z/\cos \theta$ or, equivalently for displacement amplitudes, $Z\cos \theta$.

For a P-wave incident at angle $\theta$ from a liquid $(\rho ,c,\lambda )$ onto $({\rho }^{\prime },c^{\prime },{\lambda }^{\prime })$, with transmitted angle ${\theta }^{\prime }$ obeying Snell law for elastic and acoustic waves,No reflection is equivalent to ${\rho }^{\prime }{c}^{\prime }\cos \theta =\rho c\cos {\theta }^{\prime }$.

For displacement velocity $\dot{u}$ and stress $\sigma$, the instantaneous elastic-energy flux isFor complex harmonic amplitudes, its average over one temporal period is

An incident SV-wave generally reflects as both an SV-wave and a P-wave. A rigid boundary determines their amplitudes by requiring both displacement components to vanish, while phase matching preserves frequency and tangential wavenumber.

For incident SV angle $\theta$, tangential phase matching gives $\sin \varphi =(c_P/c_S)\sin \theta$ for the reflected P-wave. It is evanescent when $\sin \theta >c_S/c_P$.

When the converted P-wave is evanescent, its amplitude decays away from the boundary and the reflected SV amplitude has modulus one. The P field stores reactive energy near the boundary but carries zero mean normal power.

At a perfectly rigid plane, every component of the displacement field vanishes.

A wavevector points in the direction of phase propagation; its magnitude is the wavenumber and its scalar product with position gives the spatial phase.

For a slowly varying local dispersion relation $\omega =\Omega (\mathbf{k};\mathbf{x},t)$, wave-packet rays obey

The WKB method seeks a rapidly oscillating field in the formand determines the phase and slowly varying amplitude order by order in $\epsilon$.

The leading WKB phase obeys an eikonal equation. For a local dispersion relation $\omega =\Omega (k;x,t)$ and $k=\nabla \theta$, it is

Along a ray $\mathbf{x}(t)$, the derivative of a field $f(\mathbf{x},t)$ is

An evanescent wave has an imaginary component of its wavevector and therefore decays exponentially in that direction instead of transporting energy away as a propagating wave.