continuum-mechanics.bigb

= Continuum mechanics
{wiki}

= Material derivative
{title2=$D/Dt$}
{parent=Continuum mechanics}
{wiki}

The material derivative follows a moving continuum parcel:
$$
\frac D{Dt}=\frac{\partial}{\partial t}+\mathbf u\cdot\nabla.
$$

= Eulerian time average
{parent=Material derivative}
{c}

An Eulerian time average holds spatial position fixed while averaging a time-dependent field:
$$
\langle u\rangle(x)=\frac1T\int_0^Tu(x,t)\,dt.
$$

= Lagrangian trajectory
{parent=Material derivative}
{c}

A Lagrangian trajectory follows one marked fluid particle and solves
$$
\dot X(t)=u(X(t),t).
$$

= Stokes drift
{parent=Lagrangian trajectory}
{c}
{wiki=Stokes_drift}

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.

= Rayleigh-Taylor instability
{c}
{parent=Continuum mechanics}
{wiki}

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

= Boundary layer
{parent=Continuum mechanics}
{wiki}

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

= Boundary-layer scaling
{parent=Boundary layer}

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

= Two-dimensional boundary-layer equations
{parent=Boundary layer}

For steady incompressible flow with imposed outer pressure gradient,
$$
u_x+v_y=0,
\qquad
uu_x+vu_y=-\frac1\rho p_x+\nu u_{yy}.
$$
The normal momentum equation makes pressure approximately uniform across the layer.

= Boundary layer over a linearly stretching sheet
{parent=Two-dimensional boundary-layer equations}

For a sheet moving with $u(x,0)=\alpha x$ through otherwise stationary fluid, the layer thickness is $\delta=\sqrt{\nu/\alpha}$. With
$$
\eta=y/\delta,\qquad
\psi=\alpha x\delta f(\eta),
$$
the boundary-layer equations reduce to
$$
f'''+ff''-(f')^2=0,
\qquad
f(0)=0,\quad f'(0)=1,\quad f'(\infty)=0.
$$
The exact solution is $f=1-e^{-\eta}$.

= Boundary-layer separation
{parent=Boundary layer}
{wiki=Flow_separation}

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.

= Falkner-Skan equation
{c}
{parent=Continuum mechanics}
{wiki=Falkner–Skan_boundary_layer}

For $U=Ax^m$, $\psi=\sqrt{\nu xU}f(\eta)$ gives
$$
f'''+\frac{m+1}{2}ff''+m(1-f'^2)=0,
$$
with $f(0)=f'(0)=0$ and $f'(\infty)=1$.

= Acoustic energy flux
{parent=Continuum mechanics}
{wiki=Sound_intensity}

For harmonic acoustic amplitudes, $\langle\mathbf I\rangle=\frac12\operatorname{Re}(p\mathbf u^*)$.

= Transmitted velocity potential from a piston through an acoustic interface
{parent=Acoustic energy flux}

For a finite duct segment of phase length $\lambda$ joined to a semi-infinite medium, continuity of pressure and velocity gives the outgoing potential amplitude
$$
T=\epsilon c_-
\frac{i(\rho_+/\rho_-)\sin\lambda-(c_-/c_+)\cos\lambda}
{(\rho_+/\rho_-)^2\sin^2\lambda+(c_-/c_+)^2\cos^2\lambda}.
$$
At half-integer $\lambda/\pi$, a large impedance mismatch can strongly enhance transmitted flux for a prescribed piston displacement.

= Burgers vortex
{c}
{parent=Continuum mechanics}
{wiki}

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

= Elasticity
{parent=Continuum mechanics}
{wiki=Elasticity_(physics)}

Elasticity studies reversible deformation and the stresses produced by it.

= Linear elasticity
{parent=Elasticity}
{wiki}

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

= Nondegenerate one-dimensional elastic material
{parent=Linear elasticity}

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.

= Displacement field
{parent=Linear elasticity}
{wiki=Displacement_field_(mechanics)}

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

= Displacement gradient tensor
{parent=Displacement field}
{wiki=Deformation_(engineering)\#Displacement_gradient_tensor}

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

= Lamé parameter
{parent=Linear elasticity}
{c}
{wiki=Lam%C3%A9_parameters}

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

= Lamé parameters
{synonym}

= Isotropic linear-elastic energy density
{parent=Linear elasticity}

For the two-dimensional convention $\nabla\mathbf u=(\partial_i u_j)$, the energy density
$$
W=\frac\mu2\nabla\mathbf u:\nabla\mathbf u^T
+\frac{\lambda+\mu}{2}(\nabla\cdot\mathbf u)^2
$$
produces the static isotropic elastic equations by variation.

= Navier-Cauchy equation
{parent=Linear elasticity}
{c}
{wiki=Navier%E2%80%93Cauchy_equations}

In a homogeneous isotropic elastic body without body forces, static equilibrium of the displacement field is
$$
\mu\nabla^2\mathbf u
+(\lambda+\mu)\nabla(\nabla\cdot\mathbf u)=0.
$$

= Uniform extension of a one-dimensional elastic body
{parent=Navier-Cauchy equation}

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

= Elastic wave in an isotropic solid
{parent=Linear elasticity}

For density $\rho$ and <Lamé parameters> $\lambda,\mu$, the displacement satisfies
$$
\rho\mathbf u_{tt}
=(\lambda+\mu)\nabla(\nabla\cdot\mathbf u)+\mu\nabla^2\mathbf u.
$$
Its longitudinal and transverse wave speeds are
$$
c_P=\sqrt{\frac{\lambda+2\mu}{\rho}},
\qquad
c_S=\sqrt{\frac{\mu}{\rho}}.
$$

= Mode conversion at a planar elastic interface
{parent=Elastic wave in an isotropic solid}

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.

= Snell law for elastic and acoustic waves
{parent=Mode conversion at a planar elastic interface}
{c}

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
$$
\frac{\sin\theta_j}{c_j}=\text{constant}.
$$

= Acoustic impedance
{title2=$Z=\rho c$}
{parent=Elastic wave in an isotropic solid}
{wiki}

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, $Z\cos\theta$.

= Displacement reflection from an interface between two inviscid elastic liquids
{parent=Acoustic impedance}

For a P-wave incident at angle $\theta$ from a liquid $(\rho,c,\lambda)$ onto $(\rho',c',\lambda')$, with transmitted angle $\theta'$ obeying <Snell law for elastic and acoustic waves>,
$$
R=\frac{\lambda'\sin2\theta-\lambda\sin2\theta'}
{\lambda'\sin2\theta+\lambda\sin2\theta'}.
$$
No reflection is equivalent to $\rho'c'\cos\theta=\rho c\cos\theta'$.

= Elastic-wave energy flux
{parent=Elasticity}

For displacement velocity $\dot u$ and stress $\sigma$, the instantaneous elastic-energy flux is
$$
P_i=-\sigma_{ij}\dot u_j.
$$
For complex harmonic amplitudes, its average over one temporal period is
$$
\langle P_i\rangle=-\frac12\operatorname{Re}(\widehat\sigma_{ij}\widehat{\dot u}_j^*).
$$

= Reflection of an SV-wave from a rigid plane
{parent=Elastic-wave energy flux}
{c}

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.

= Evanescent reflected P-wave at a rigid plane
{parent=Reflection of an SV-wave from a rigid plane}
{c}

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

= Unit-modulus SV reflection with evanescent P conversion
{parent=Evanescent reflected P-wave at a rigid plane}
{c}

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.

= Rigid boundary condition for an elastic wave
{parent=Reflection of an SV-wave from a rigid plane}

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

= Wavevector
{title2=$\mathbf k$}
{parent=Elastic-wave energy flux}
{wiki}

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

= Hamiltonian ray-tracing equations
{parent=Wavevector}
{wiki=Ray_tracing_(physics)}

For a slowly varying local dispersion relation $\omega=\Omega(\mathbf k;\mathbf x,t)$, wave-packet rays obey
$$
\dot x_i=\frac{\partial\Omega}{\partial k_i},
\qquad
\dot k_i=-\frac{\partial\Omega}{\partial x_i},
\qquad
\dot\omega=\frac{\partial\Omega}{\partial t}.
$$

= WKB method
{c}
{parent=Hamiltonian ray-tracing equations}
{wiki=WKB_approximation}

The WKB method seeks a rapidly oscillating field in the form
$$
\phi(x,t)=A(x,t;\varepsilon)e^{i\theta(x,t)/\varepsilon},
\qquad 0<\varepsilon\ll1,
$$
and determines the phase and slowly varying amplitude order by order in $\varepsilon$.

= Eikonal equation
{parent=WKB method}
{wiki}

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

= Total derivative along a ray
{parent=Hamiltonian ray-tracing equations}

Along a ray $\mathbf x(t)$, the derivative of a field $f(\mathbf x,t)$ is
$$
\frac{df}{dt}
=\frac{\partial f}{\partial t}
+\dot{\mathbf x}\cdot\nabla f.
$$

= Evanescent wave
{parent=Elastic-wave energy flux}
{wiki}

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.