fluid-mechanics.bigb

= Fluid mechanics
{wiki}

Fluid mechanics studies the motion and forces of liquids and gases.

= Torricelli's law
{parent=Fluid mechanics}
{c}
{wiki=Torricelli%27s_law}

An ideal fluid issuing from a small hole a vertical depth $h$ below a free surface has speed
$$
v=\sqrt{2gh}.
$$

= Mass density
{parent=Fluid mechanics}
{wiki=Density}

Mass density is mass per unit volume.

= Fluid pressure
{parent=Fluid mechanics}
{wiki=Pressure}

Fluid pressure is the isotropic normal compressive stress in a fluid at rest.

= Dynamic viscosity
{parent=Fluid mechanics}
{wiki=Viscosity}

Dynamic viscosity $\mu$ relates Newtonian shear stress to rate of strain.

= Viscosity
{synonym}

= Kinematic viscosity
{parent=Dynamic viscosity}
{wiki=Viscosity\#Kinematic_viscosity}

Kinematic viscosity is dynamic viscosity divided by mass density: $\nu=\mu/\rho$.

= Reynolds number
{parent=Dynamic viscosity}
{c}
{wiki}

The Reynolds number $\operatorname{Re}=UL/\nu$ compares inertial and viscous effects.

= Vorticity
{parent=Fluid mechanics}
{wiki}

Vorticity is the curl of the velocity field, $\omega=\nabla\times u$.

= Circulation
{parent=Vorticity}
{wiki=Circulation_(fluid_dynamics)}

The circulation around a closed curve is $\Gamma=\oint u\mathbin{\cdot}dl$.

= Irrotational flow
{parent=Vorticity}
{wiki}

An irrotational flow has zero vorticity and locally admits a velocity potential.

= Circular vortex-sheet mode
{parent=Fluid mechanics}

For a circular patch in solid-body rotation next to stationary fluid, a sinusoidal boundary displacement couples an interior $r^k$ potential to an exterior $r^{-k}$ potential. The tangential velocity jump makes the interface Kelvin--Helmholtz unstable.

= Internal gravity wave
{parent=Fluid mechanics}
{wiki=Internal_wave}

In a uniformly stratified fluid with buoyancy frequency $N$, a plane wave of horizontal and vertical wavenumbers $k,m$ obeys
$$
\omega^2=\frac{N^2k^2}{k^2+m^2}.
$$

= Buoyancy frequency
{title2=$N$}
{parent=Internal gravity wave}
{wiki=Brunt%E2%80%93V%C3%A4is%C3%A4_frequency}

The buoyancy frequency is the natural angular frequency of small vertical oscillations in a stably stratified fluid.

= Vertical trapping of an internal gravity wave by planar strain
{parent=Internal gravity wave}

For the planar strain $\mathbf U=\gamma(x,0,-z)$, an internal-wave ray has
$$
k=k_0e^{-\gamma t},
\qquad
m=m_0e^{\gamma t}.
$$
If it initially propagates upward with $m_0<0$, its height remains positive at finite time but decays to zero exponentially as $t\to\infty$.

= Mountain-wave cutoff
{parent=Internal gravity wave}

A steady hill pattern seen by a uniform flow of speed $U$ has intrinsic frequency magnitude $Uk$. It radiates a propagating internal wave only for $k\leq N/U$; above this cutoff the vertical wavenumber is imaginary and the disturbance is evanescent.

= Stream function
{parent=Fluid mechanics}
{wiki}

A two-dimensional incompressible velocity field can be written either as $u=\psi_y$, $v=-\psi_x$ or with both signs reversed. The continuity equation then holds identically.

= Streamline classification of a planar linear saddle or centre
{parent=Stream function}

For $(u,v)=(y,ax)$, the stream function $\psi=(y^2-ax^2)/2$ gives hyperbolic streamlines and a saddle when $a>0$, while $a<0$ gives elliptical streamlines and a centre.

= Streamline
{parent=Fluid mechanics}
{wiki}

A streamline is a curve tangent everywhere to the instantaneous velocity field. In steady two-dimensional incompressible flow, streamlines are level sets of the stream function.

= Velocity potential
{parent=Fluid mechanics}
{wiki}

An irrotational velocity field is locally a gradient $u=\nabla\phi$. The scalar $\phi$ is its velocity potential.

= Velocity potentials
{synonym}

= Potential flow around a circular cylinder
{parent=Velocity potential}

Uniform irrotational flow of speed $U$ past a circular cylinder of radius $a$, with zero circulation, has <velocity potential>
$$
\phi=U\left(r+\frac{a^2}{r}\right)\cos\theta.
$$
On the cylinder, the tangential speed is $-2U\sin\theta$, so the <Bernoulli equation> gives
$$
p(a,\theta)=p_\infty+\frac12\rho U^2(1-4\sin^2\theta).
$$

= Radially symmetric incompressible flow in a planar annulus
{parent=Velocity potential}

For liquid occupying $a(t)<r<b(t)$, radial symmetry and <incompressible flow> imply
$$
u_r(r,t)=\frac{a\dot a}{r},
\qquad
\phi(r,t)=a\dot a\log r+C(t).
$$
The <kinematic boundary condition> at the outer interface gives
$$
b^2-a^2=\text{constant},
$$
which expresses conservation of the liquid's area.

= Pressure in radially symmetric annular potential flow
{parent=Radially symmetric incompressible flow in a planar annulus}

If the pressure at $r=b(t)$ is $p_\infty$, the <unsteady Bernoulli equation> gives
$$
p(a,t)-p_\infty
=\rho\left[\frac d{dt}(a\dot a)\log\frac ba
-\frac{(a\dot a)^2}{2}\left(\frac1{a^2}-\frac1{b^2}\right)\right].
$$

= Small oscillation of a planar gas bubble in an annular liquid
{parent=Pressure in radially symmetric annular potential flow}

For $a=a_0(1+\epsilon)$, equilibrium outer radius $b_0$, and a bubble gas obeying $p_0\pi a^2=\text{constant}$, the <linearization> is
$$
\rho a_0^2\log\frac{b_0}{a_0}\,\ddot\epsilon
=-2p_\infty\epsilon.
$$
The bubble therefore undergoes <simple harmonic motion> with
$$
\omega^2=\frac{2p_\infty}{\rho a_0^2\log(b_0/a_0)}.
$$

= Laminar plume
{parent=Fluid mechanics}
{wiki=Plume_(fluid_dynamics)}

A laminar plume is a narrow buoyancy-driven flow in which viscous diffusion balances inertia across the plume.

= Integral momentum-flux balance for a two-dimensional plume
{parent=Laminar plume}

For a steady incompressible plume with localized vertical body force $b(x)\delta(y)$, decay of velocity and shear at transverse infinity gives
$$
\frac{d}{dx}\int_{-\infty}^{\infty}\rho u^2\,dy=b(x).
$$

= Similarity scaling of a two-dimensional laminar plume
{parent=Laminar plume}

If $b(x)=Bx^{-1/5}$, balancing the momentum flux and transverse viscosity gives
$$
\Delta\sim\left(\frac{\mu^2}{\rho B}\right)^{1/3}x^{2/5},
\qquad
U\sim\left(\frac{B^2}{\rho\mu}\right)^{1/3}x^{1/5}.
$$

= Similarity equation for a two-dimensional laminar plume
{parent=Similarity scaling of a two-dimensional laminar plume}

With $u=-\psi_y$, $v=\psi_x$, $\eta=y/\Delta$, and $\psi=U\Delta f(\eta)$, the plume boundary-layer equation reduces to
$$
f'''+\frac15(f')^2-\frac35ff''=\delta(\eta).
$$

= Euler equations for an inviscid fluid
{parent=Fluid mechanics}
{c}
{wiki}

Euler’s equations express momentum conservation in a fluid without viscosity.

= Unsteady Bernoulli equation
{parent=Euler equations for an inviscid fluid}
{c}
{wiki=Bernoulli%27s_principle}

For time-dependent <irrotational flow> with $u=\nabla\phi$, constant <mass density>, and no body force, the <Euler equations for an inviscid fluid> integrate to
$$
\partial_t\phi+\frac12|\nabla\phi|^2+\frac p\rho=C(t).
$$
A time-dependent change of gauge in $\phi$ may set $C(t)=0$.

= Clebsch-potential variational derivation of incompressible Euler flow
{parent=Euler equations for an inviscid fluid}
{c}

For $\mathbf u=\nabla\phi+\beta\nabla\alpha$, the density
$$
\mathcal L=-\beta\alpha_t-\frac12|\mathbf u|^2
$$
gives incompressibility and material conservation of $\alpha$ and $\beta$. Differentiating the Clebsch representation then yields the Euler momentum equation with
$$
p=-\frac12|\mathbf u|^2-\phi_t-\beta\alpha_t.
$$

= Bernoulli equation
{parent=Euler equations for an inviscid fluid}
{c}
{wiki}

For steady inviscid flow, pressure, kinetic energy density, and conservative potential are constant along a streamline.

= Bernoulli function
{parent=Bernoulli equation}
{c}
{wiki}

The Bernoulli function combines pressure, kinetic energy density, and conservative potential.

= Bernoulli equation over topography
{parent=Bernoulli equation}
{c}

For steady potential flow over small hills, linearized Bernoulli pressure combines a hydrostatic term $-\rho g\eta$ with a dynamic term $-\rho U\phi_x$.

= Draining time of a uniform tank through a siphon
{parent=Bernoulli equation}

If a tank and siphon tube have areas $A$ and $a$, the tube outlet is $H$ below an inlet initially submerged by $h_0$, and both free boundaries are at atmospheric pressure, continuity and Bernoulli's equation give
$$
t=\sqrt{2}\left(\frac{A^2}{a^2}-1\right)^{1/2}
\frac{\sqrt{H+h_0}-\sqrt H}{\sqrt g}
$$
for the free surface to reach the inlet.

= Integral momentum equation
{parent=Euler equations for an inviscid fluid}
{wiki}

The integral momentum equation balances momentum flux, pressure, body force, and forces on a control volume.

= Force on a pipe junction
{parent=Integral momentum equation}
{wiki}

The force on a pipe junction follows by balancing inlet and outlet momentum fluxes and pressure forces.

= Conservation of mass in a pipe
{parent=Fluid mechanics}
{wiki}

For steady incompressible pipe flow, volume flux equals cross-sectional area times mean speed and is conserved through a junction.

= Viscous fluid flow
{parent=Fluid mechanics}
{wiki}

Viscous fluid flow includes shear stresses proportional to velocity gradients for a Newtonian fluid.

= Viscous diffusion time
{title2=$L^2/\nu$}
{parent=Viscous fluid flow}

Momentum diffuses across a distance $L$ in the characteristic time $L^2/\nu$, where $\nu$ is the <kinematic viscosity>.

= Couette flow
{parent=Viscous fluid flow}
{c}
{wiki}

Couette flow is viscous shear flow driven by relative tangential motion of parallel boundaries. In the steady no-pressure-gradient case its velocity profile is linear.

= Falling film flow
{parent=Viscous fluid flow}
{wiki}

A steady film down a wall has a parabolic velocity profile set by gravity, ambient pressure gradient, and surface shear.

= Inclined viscous film with opposing surface shear
{parent=Falling film flow}

For a film of thickness $h$ flowing down a slope of angle $\alpha$, an upslope surface stress of magnitude $\tau$ gives
$$
u(z)=\frac{\rho g\sin\alpha}{\mu}
\left(hz-\frac{z^2}{2}\right)-\frac{\tau z}{\mu}.
$$
The surface velocity, total flux, and basal shear reverse at
$$
\tau=\frac12\rho gh\sin\alpha,
\qquad
\frac23\rho gh\sin\alpha,
\qquad
\rho gh\sin\alpha,
$$
respectively.

= Stress boundary condition
{parent=Viscous fluid flow}
{wiki}

At a fluid interface, normal and tangential tractions satisfy the imposed stress balance.

= Stress-free boundary condition
{parent=Stress boundary condition}
{wiki=Free_surface#Free_surface_in_fluid_mechanics}

A clean fluid interface with negligible exterior viscosity has zero tangential traction. Its normal traction is determined by exterior pressure and surface tension.

= No-slip boundary condition
{parent=Viscous fluid flow}
{wiki}

No slip requires fluid velocity to equal solid-boundary velocity.

= Newtonian fluid stress tensor
{parent=Viscous fluid flow}
{c}
{wiki=Newtonian_fluid}

For an incompressible Newtonian fluid, $\sigma=-pI+\mu(\nabla u+(\nabla u)^T)$.

= Shear stress
{title2=$\tau$}
{parent=Newtonian fluid stress tensor}
{wiki}

Shear stress is the tangential component of traction on a surface. For leading unidirectional flow $u(y)$ in a Newtonian fluid, $\tau=\mu\,du/dy$.

= Rate-of-strain tensor
{parent=Newtonian fluid stress tensor}
{wiki=Strain-rate_tensor}

The rate-of-strain tensor is the symmetric part of the velocity gradient,
$$
e=\frac12\left(\nabla u+(\nabla u)^T\right).
$$

= Polar shear stress in a Newtonian fluid
{parent=Newtonian fluid stress tensor}

For a planar velocity field in polar coordinates,
$$
\sigma_{r\theta}
=\mu\left[
r\frac{\partial}{\partial r}\left(\frac{u_\theta}{r}\right)
+\frac1r\frac{\partial u_r}{\partial\theta}
\right].
$$

= Navier-Stokes equation
{parent=Viscous fluid flow}
{c}
{wiki=Navier%E2%80%93Stokes_equations}

For an incompressible Newtonian fluid of constant density and viscosity,
$$
\rho\left(\partial_tu+u\mathbin{\cdot}\nabla u\right)
=-\nabla p+\mu\nabla^2u+\rho f,
\qquad \nabla\mathbin{\cdot}u=0.
$$

= Hagen-Poiseuille flow
{c}
{parent=Navier-Stokes equation}
{wiki=Hagen%E2%80%93Poiseuille_equation}

Steady pressure-driven flow through a circular pipe has the parabolic profile
$$
w(r)=-\frac1{4\mu}\frac{dp}{dz}(R^2-r^2).
$$

= Kinetic-energy balance for an incompressible Newtonian fluid
{parent=Viscous fluid flow}

For a fixed domain $\mathcal D$, body-force density $f$, and stress tensor $\sigma$, the kinetic energy satisfies
$$
\frac d{dt}\int_{\mathcal D}\frac12\rho|u|^2dV
+\int_{\partial\mathcal D}\frac12\rho|u|^2u\cdot n\,dS
=\int_{\mathcal D}u\cdot f\,dV
+\int_{\partial\mathcal D}u\cdot\sigma\cdot n\,dS
-2\mu\int_{\mathcal D}e:e\,dV.
$$
The last term is the nonnegative rate at which viscosity converts mechanical energy into heat.

= Dissipation estimate for high-Reynolds-number bubble drag
{parent=Kinetic-energy balance for an incompressible Newtonian fluid}

For a clean bubble of size $a$ and speed $U$ at large <Reynolds number>, the outer <potential flow> violates the stress-free condition by a strain of order $U/a$. A boundary layer of thickness $\delta\sim a\operatorname{Re}^{-1/2}$ needs a velocity correction only of order $U\delta/a$, so its dissipation is smaller than the outer-flow dissipation by order $\delta/a$. The leading drag power is therefore obtained from
$$
DU\sim 2\mu\int_{\rm outer}e:e\,dV.
$$

= Drag on a two-dimensional circular bubble from outer-flow dissipation
{title2=$D=8\pi\mu U$}
{parent=Dissipation estimate for high-Reynolds-number bubble drag}

For the zero-circulation flow around a circle, $e:e=8U^2a^4/r^6$. Integration over the exterior plane gives dissipation $8\pi\mu U^2$ per unit axial length and hence drag $D=8\pi\mu U$ per unit length.

= Stokes flow
{parent=Viscous fluid flow}
{c}
{wiki}

Stokes flow is the zero-Reynolds-number limit in which viscous and pressure forces balance while fluid inertia is neglected.

= Hydrodynamic resistance matrix
{parent=Stokes flow}

At fixed orientation, linearity of Stokes flow relates a rigid body's translational velocity $\mathbf U$ to the hydrodynamic force $\mathbf F$ by
$$
\mathbf F=-\mathbf R\mathbf U.
$$
The Lorentz reciprocal theorem makes $\mathbf R$ symmetric, and positive viscous dissipation makes it positive definite.

= Lorentz reciprocal theorem for Stokes flow
{c}
{parent=Stokes flow}
{wiki=Lorentz_reciprocal_theorem}

For two Stokes velocity-stress fields in the same domain,
$$
\int_{\partial\mathcal D}\mathbf u^{(1)}\cdot\boldsymbol\sigma^{(2)}\mathbf n\,dS
=
\int_{\partial\mathcal D}\mathbf u^{(2)}\cdot\boldsymbol\sigma^{(1)}\mathbf n\,dS.
$$
It follows by integrating the divergence of the corresponding cross-work flux and using symmetry of the Newtonian stress.

= Viscous dissipation
{parent=Stokes flow}
{wiki=Viscosity#Viscous_dissipation}

For an incompressible Newtonian fluid, the nonnegative rate at which viscosity converts mechanical energy into heat is
$$
2\mu\int_{\mathcal D}\mathbf e:\mathbf e\,dV,
\qquad
\mathbf e=\frac12(\nabla\mathbf u+\nabla\mathbf u^T).
$$

= Biharmonic stream function for planar Stokes flow
{parent=Stokes flow}
{c}

Taking the curl of the planar Stokes equation makes vorticity harmonic. Since vorticity is minus the Laplacian of the stream function,
$$
\nabla^4\psi=0.
$$

= Similarity solution for tangentially forced Stokes wedge
{parent=Biharmonic stream function for planar Stokes flow}

For $\psi=r^2f(\theta)$, biharmonicity gives
$f^{(4)}+4f''=0$. In the wedge $-\alpha<\theta<0$, no slip at the lower wall and no penetration plus tangential stress $S$ at the upper surface give
$$
f(-\alpha)=f'(-\alpha)=f(0)=0,
\qquad \mu f''(0)=S.
$$
The resulting surface speed is
$$
U(r)=\frac{Sr}{\mu}
\frac{1-\cos2\alpha-\alpha\sin2\alpha}
{\sin2\alpha-2\alpha\cos2\alpha}.
$$

= Biharmonic stream function for a fixed disk in planar shear
{parent=Biharmonic stream function for planar Stokes flow}
{c}

For a stationary disk of radius $a$ in the far-field shear $u_\infty=\gamma y e_x$, using $u_r=\psi_\theta/r$ and $u_\theta=-\psi_r$, the exterior solution is
$$
\psi=\frac\gamma4\left[
r^2-a^2-2a^2\log\frac ra
-\left(r^2-2a^2+\frac{a^4}{r^2}\right)\cos2\theta
\right].
$$
It satisfies no slip at $r=a$ and approaches the imposed shear in velocity.

= Hydrodynamic torque on a fixed disk in planar shear
{parent=Biharmonic stream function for a fixed disk in planar shear}

The surface shear stress is
$$
\sigma_{r\theta}(a,\theta)
=-\mu\gamma+2\mu\gamma\cos2\theta.
$$
The torque exerted by the fluid on the disk per unit axial length is therefore
$$
\mathcal T_z
=a^2\int_0^{2\pi}\sigma_{r\theta}(a,\theta)\,d\theta
=-2\pi\mu\gamma a^2.
$$

= Harmonic pressure and vorticity in Stokes flow
{parent=Stokes flow}
{c}

Divergence and curl of $-\nabla p+\mu\nabla^2u=0$ with $\nabla\cdot u=0$ give $\nabla^2p=0$ and $\nabla^2\omega=0$.

= Velocity gradient of translating-sphere Stokes flow
{parent=Stokes flow}

Writing $u=A(r)U+B(r)(U\cdot x)x$ reduces its gradient to radial derivatives of $A,B$ plus the product rule for $(U\cdot x)x$.

= Vorticity of translating-sphere Stokes flow
{parent=Stokes flow}

For the classical translating-sphere solution, $\omega=(3a/(2r^3))U\times x$.

= Harmonicity of derivatives of the Newtonian potential
{parent=Stokes flow}

Since $\nabla^2(1/r)=0$ away from the origin, every constant-coefficient derivative of $1/r$ is harmonic there as well.

= Direct incompressibility check for translating-sphere flow
{parent=Stokes flow}

For $u=A(r)U+B(r)(U\cdot x)x$, its divergence is $(U\cdot x)(A'/r+rB'+4B)$, which vanishes for the translating-sphere coefficients.

= Surface traction in translating-sphere Stokes flow
{parent=Stokes flow}

On the sphere, pressure and the normal part of the viscous stress cancel, leaving uniform traction $-3\mu U/(2a)$.

= Stokes drag law
{parent=Surface traction in translating-sphere Stokes flow}
{c}
{wiki=Stokes%27_law}

Integrating the translating-sphere traction gives the drag force $F=-6\pi\mu aU$.

= Kinematic reversibility of Stokes flow
{parent=Stokes flow}
{wiki=Stokes_flow\#Time_reversibility}

Linearity of the Stokes equations implies that reversing all imposed forces and boundary velocities reverses the entire velocity field and retraces particle paths.

= Reflection argument for zero Stokes migration
{parent=Kinematic reversibility of Stokes flow}

If spatial reflection leaves a Stokes configuration and forcing equivalent to flow reversal while preserving one candidate velocity component, uniqueness forces that component to vanish.

= Rotational Stokes flow between concentric spheres
{c}
{parent=Stokes flow}

For concentric spheres of radii $a<b$, with the inner sphere rotating at angular velocity $\boldsymbol\Omega$ and the outer sphere fixed, the Stokes velocity is
$$
\mathbf u(\mathbf x)
=\frac{a^3}{b^3-a^3}
\left(\frac{b^3}{r^3}-1\right)
\boldsymbol\Omega\times\mathbf x.
$$
The pressure is constant, and may be set to zero.

= Torque in rotational Stokes flow between concentric spheres
{c}
{parent=Rotational Stokes flow between concentric spheres}

The torque transmitted across any concentric sphere is
$$
\mathbf G
=\frac{8\pi\mu a^3b^3}{b^3-a^3}\boldsymbol\Omega.
$$
It approaches $8\pi\mu a^3\boldsymbol\Omega$ when $a\ll b$, and for a thin gap $h=b-a\ll a$ it approaches
$$
\frac{8\pi\mu a^4}{3h}\boldsymbol\Omega.
$$

= Lubrication theory
{parent=Viscous fluid flow}
{wiki}

Lubrication theory describes viscous flow through a gap whose thickness is much smaller than its streamwise length; pressure is nearly uniform across the gap and controls a locally parabolic velocity profile.

= Lubrication-limit scaling for a moving thin gap
{parent=Lubrication theory}

For gap scale $H$, length $L$, lower-wall tangential speed $U$, upper-wall normal speed $V$, and induced horizontal scale $W=|U|+VL/H$, the lubrication limit requires
$$
\frac HL\ll1,
\qquad |h'|\ll1,
\qquad
\frac{\rho W H^2}{\mu L}\ll1.
$$
The last condition includes both $\rho|U|H^2/(\mu L)\ll1$ and $\rho VH/\mu\ll1$.

= Gravity-driven thin film on an incline
{parent=Lubrication theory}

For a film of thickness $h(x,t)$ flowing down a plane inclined by $\alpha$, lubrication theory gives the flux
$$
q=\frac{\rho gh^3}{3\mu}
\left(\sin\alpha-\cos\alpha\,h_x\right)
$$
when surface tension is neglected. Mass conservation gives the thin-film equation $h_t+q_x=0$.

= Parabolic lubrication gap
{parent=Lubrication theory}

Near the closest approach of a circle of radius $a$ to a wall, the gap is $h(x)=h_0+x^2/(2a)+O(x^4/a^3)$ and has streamwise scale $\sqrt{ah_0}$.

= Couette-Poiseuille flow in a thin gap
{parent=Lubrication theory}
{c}

Between surfaces of velocities $U_0,U_1$ separated by $h$, the local lubrication profile is
$$u=U_0+(U_1-U_0)y/h+(p_x/2\mu)y(y-h).$$

= Couette-Poiseuille flow with a stress-free stationary wall
{parent=Couette-Poiseuille flow in a thin gap}
{c}

For $U_0=0$, $U_1=U$, and constant $G=p_x$, zero shear at $y=0$ requires
$$
G=\frac{2\mu U}{h^2}.
$$
The velocity becomes $u=Uy^2/h^2$, the flux per unit width is $Uh/3$, and the fluid exerts shear $-2\mu U/h$ on the moving upper wall.

= Moving-boundary lubrication flux
{parent=Couette-Poiseuille flow in a thin gap}

The flux of the local Couette-Poiseuille profile is
$$Q=h(U_0+U_1)/2-h^3p_x/(12\mu).$$

= Reynolds lubrication equation
{c}
{parent=Moving-boundary lubrication flux}
{wiki=Reynolds_equation}

Incompressible mass conservation in a moving gap gives
$$
\partial_t h+\partial_xQ=0.
$$
Combined with the local Couette-Poiseuille flux, this is the Reynolds lubrication equation.

= Pressure gradient in a translating and squeezing finite gap
{parent=Reynolds lubrication equation}

For lower speed $U$, stationary upper tangential speed, upper normal speed $-V$, and equal endpoint pressures, define $\langle f\rangle=L^{-1}\int_0^Lf(x)\,dx$. Then
$$
p_x=\frac{6\mu U}{h^2}-\frac{12\mu(Vx+C)}{h^3},
$$
where
$$
C=\frac U2\frac{\langle h^{-2}\rangle}{\langle h^{-3}\rangle}
-V\frac{\langle xh^{-3}\rangle}{\langle h^{-3}\rangle}.
$$

= Zero-shear lower wall in a translating and squeezing finite gap
{parent=Pressure gradient in a translating and squeezing finite gap}

The horizontal force on the lower wall vanishes when
$$
U=6V\frac{
\langle xh^{-2}\rangle
-\langle h^{-2}\rangle\langle xh^{-3}\rangle/
\langle h^{-3}\rangle}
{4\langle h^{-1}\rangle
-3\langle h^{-2}\rangle^2/\langle h^{-3}\rangle}.
$$

= Annular lubrication drag on a settling cylinder
{parent=Moving-boundary lubrication flux}

For a cylinder of radius $a$ in a coaxial container with gap $h\ll a$, global displacement forces reverse flow through the gap and creates
$$
\Delta p\simeq\frac{6\mu aLU}{h^3}.
$$
The pressure drag exceeds side shear by order $a/h$. Balancing it with excess weight gives $U\simeq\Delta\rho,gh^3/(6\mu a)$.

= Pressure recovery condition in lubrication flow
{parent=Lubrication theory}

If pressure approaches the same ambient value at both ends of a lubrication region, then $\int_{-\infty}^{\infty}p_x\,dx=0$, which determines the conserved flux.

= Viscous shear torque
{parent=Viscous fluid flow}

Tangential viscous traction produces torque equal to its moment integrated over the solid surface.

= Torque-free cylinder in a lubrication gap
{parent=Viscous shear torque}

For a circular cylinder translating parallel to a nearby wall at zero Reynolds number, leading lubrication shear and pressure recovery make the zero-torque condition force zero angular velocity.

= Body force
{parent=Fluid mechanics}
{wiki}

A body force acts throughout a fluid volume. Gravity and electromagnetic forces are standard examples; a body force per unit mass $f$ contributes $\rho f$ to the momentum equation.

= Hydrostatic pressure
{parent=Fluid mechanics}
{wiki}

Hydrostatic pressure has gradient equal to fluid density times gravity.

= Potential flow
{parent=Fluid mechanics}
{wiki}

Potential flow is irrotational flow represented as the gradient of a scalar potential.

= Potential flow around a translating sphere
{parent=Potential flow}

For a sphere of radius $a$ translating at speed $U$ along the polar axis through fluid at rest at infinity, the laboratory-frame potential is
$$
\phi=-\frac{Ua^3}{2r^2}\cos\theta.
$$
It gives $u_r=Ua^3\cos\theta/r^3$ and $u_\theta=Ua^3\sin\theta/(2r^3)$.

= Spherically symmetric incompressible radial flow
{parent=Potential flow}

For radial flow outside a sphere, <incompressible flow> makes
$$
Q(t)=4\pi r^2u_r(r,t)
$$
independent of radius. Its velocity potential is $\phi=-Q/(4\pi r)$.

= Rayleigh collapse of a spherical cavity
{parent=Spherically symmetric incompressible radial flow}
{c}
{wiki=Rayleigh%E2%80%93Plesset_equation}

A vacuum cavity of radius $a(t)$ in an infinite inviscid incompressible fluid of density $\rho$ and far-field pressure $p_0$ obeys
$$
a\ddot a+\frac32\dot a^2=-\frac{p_0}{\rho}.
$$
If $a(0)=a_0$ and $\dot a(0)=0$, its collapsing branch satisfies
$$
\dot a=-\sqrt{\frac{2p_0}{3\rho}
\left(\frac{a_0^3}{a^3}-1\right)}.
$$

= Squeezing wedge flow
{parent=Potential flow}
{wiki}

Inviscid flow between closing hinged plates is a quadratic potential flow that expels fluid radially.

= Surface gravity wave
{parent=Potential flow}
{wiki=Gravity_wave}

A surface gravity wave is a free-surface oscillation restored by gravity. In inviscid deep water its angular frequency and horizontal wavenumber satisfy $\omega^2=gk$.

= Deep-water gravity wave
{parent=Surface gravity wave}
{wiki=Airy_wave_theory}

A small-amplitude deep-water wave with surface elevation $\eta=Ae^{i(kx-\omega t)}$ has dispersion relation $\omega^2=gk$ and a velocity potential that decays as $e^{kz}$ below the surface.

= Viscous decay of a deep-water gravity wave
{parent=Deep-water gravity wave}

For small kinematic viscosity $\nu$, the mean viscous dissipation per unit horizontal area is $2\rho\nu gk^2|A|^2$. Since the mean wave energy is $\rho g|A|^2/2$, the amplitude obeys
$$
\frac{d|A|}{dt}=-2\nu k^2|A|,
\qquad
|A(t)|=|A(0)|e^{-2\nu k^2t}.
$$

= Linearized free-surface boundary conditions
{parent=Deep-water gravity wave}

For small-amplitude potential flow beneath a mean surface $z=0$, the linearized kinematic and dynamic conditions are
$$
\eta_t=\phi_z,
\qquad
\phi_t+g\eta=-\frac{p}{\rho}
\qquad(z=0).
$$

= Rectangular standing surface-gravity mode
{parent=Linearized free-surface boundary conditions}

In a rectangular box with Neumann side walls, a horizontal mode
$$
\cos\frac{m\pi x}{L_x}\cos\frac{n\pi y}{L_y}
$$
has horizontal wavenumber $k=[(m\pi/L_x)^2+(n\pi/L_y)^2]^{1/2}$, depth dependence $e^{kz}$ in infinitely deep fluid, and natural frequency $\Omega=\sqrt{gk}$.

= Resonance of a pressure-forced rectangular surface-gravity mode
{parent=Rectangular standing surface-gravity mode}

Pressure forcing of one rectangular free-surface mode reduces its amplitude to
$$
H''+gkH=-\frac{kp_0}{\rho}\cos(\omega t).
$$
At $\omega=\sqrt{gk}$, the inviscid undamped response contains $t\sin(\omega t)$ and grows without bound in the linear model.

= Streamfunction in polar coordinates
{parent=Fluid mechanics}
{wiki}

For planar incompressible flow, a streamfunction satisfies
$$
u_r=\frac1r\frac{\partial\psi}{\partial\theta},
\qquad
u_\theta=-\frac{\partial\psi}{\partial r}.
$$

= Incompressible flow
{parent=Fluid mechanics}
{wiki}

= Streakline
{parent=Fluid mechanics}
{wiki}

A streakline at a given time is the locus of all fluid particles that previously passed through one fixed release point.

= Kinematic boundary condition
{parent=Fluid mechanics}
{wiki}

A material fluid boundary has no relative normal flow; at a fixed impermeable graph $y=\eta(x)$ this is $v=u\eta_x$.

= Kinematic boundary condition for a free-surface graph
{parent=Kinematic boundary condition}

For a free surface $z=\eta(x,y,t)$ and velocity $(u,v,w)$, material conservation of $z-\eta$ gives
$$
w=\eta_t+u\eta_x+v\eta_y=\frac{D\eta}{Dt}
$$
on the surface.

= Linearized boundary condition
{parent=Kinematic boundary condition}

For a small boundary displacement and weak disturbance, evaluate boundary data on the undisturbed surface and discard products of small quantities.

= Dynamic boundary condition for an inviscid interface
{parent=Fluid mechanics}

At an interface between two inviscid fluids without surface tension, the pressure is continuous. For potential flows of equal density, the <unsteady Bernoulli equation> therefore equates
$$
\partial_t\phi+\frac12|\nabla\phi|^2
$$
on the two sides, up to a removable function of time.

= Pressure continuity
{parent=Dynamic boundary condition for an inviscid interface}

Without surface tension or another singular normal stress, the mechanical pressure has the same limiting value on both sides of a fluid interface.

= Wake
{disambiguate=fluid dynamics}
{parent=Fluid mechanics}
{wiki=Wake_(physics)}

A fluid wake is the region downstream of a body or velocity defect in which the flow differs from the surrounding stream.

= Fluid wake
{synonym}

= Shear layer
{parent=Fluid mechanics}
{wiki=Shear_layer}

A shear layer is a thin region across which the tangential <velocity> changes rapidly. In an inviscid idealization it can collapse to a <vortex sheet>.

= Vortex sheet
{parent=Shear layer}
{wiki}

A vortex sheet is a surface across which tangential velocity is discontinuous while normal velocity remains continuous.

= Kelvin-Helmholtz instability
{parent=Vortex sheet}
{c}
{wiki=Kelvin%E2%80%93Helmholtz_instability}

For a planar vortex sheet separating equal-density streams of velocities $U_1$ and $U_2$, a mode of nonzero <wavenumber> $k$ has complex growth exponent
$$
\sigma=-ik\frac{U_1+U_2}{2}
\mathbin{\pm}\frac{|k|\,|U_1-U_2|}{2}.
$$
One sign has positive <growth rate>, so every nonzero wavenumber is unstable in the inviscid zero-thickness model.

= Compressible flow
{parent=Fluid mechanics}
{wiki=Compressible_flow}

Compressible flow allows density to change materially and is required for finite-amplitude pressure waves and shocks.

= Riemann invariant
{parent=Compressible flow}
{c}
{wiki=Riemann_invariant}

For one-dimensional homentropic ideal-gas flow,
$$
R_\pm=u\pm\frac{2(c-c_0)}{\gamma-1}
$$
is constant along the characteristic $dx/dt=u\pm c$.

= Pressure in a perfect-gas simple wave
{parent=Riemann invariant}

In a right-moving simple wave entering gas initially at rest with sound speed $c_0$, the other Riemann invariant is constant, so
$$
c=c_0+\frac{\gamma-1}{2}u.
$$
The isentropic relation
$$
\frac p{p_0}=\left(\frac c{c_0}\right)^{2\gamma/(\gamma-1)}
$$
converts the velocity disturbance into its nonlinear pressure disturbance.

= Shock formation by characteristic intersection
{parent=Riemann invariant}

A smooth compressive simple wave forms a shock when characteristics first intersect. If characteristics are parametrized by their emission time $\tau$, the first shock occurs at the minimum positive time for which $\partial x(\tau,t)/\partial\tau=0$.

= Normal shock wave
{parent=Compressible flow}
{wiki=Normal_shock_tables}

A normal shock is a discontinuity perpendicular to the flow direction across which mass, momentum, and total energy fluxes are conserved.

= Shock frame
{parent=Normal shock wave}

The shock frame is an inertial reference frame in which the shock is stationary. A travelling shock then becomes a steady flow through a fixed discontinuity.

= Rankine-Hugoniot conditions for a perfect gas
{parent=Normal shock wave}
{c}
{wiki=Rankine%E2%80%93Hugoniot_conditions}

In the rest frame of a one-dimensional shock, states with velocities $w_0,w_1$ satisfy
$$
\rho_0w_0=\rho_1w_1,
$$
$$
p_0+\rho_0w_0^2=p_1+\rho_1w_1^2,
$$
and
$$
\frac{\gamma p_0}{(\gamma-1)\rho_0}+\frac12w_0^2
=\frac{\gamma p_1}{(\gamma-1)\rho_1}+\frac12w_1^2.
$$

= Piston-driven shock with specific-heat ratio three
{parent=Rankine-Hugoniot conditions for a perfect gas}

If a shock travels into stationary gas at speed $V$ while the downstream piston and gas travel at $V/3$, then for $\gamma=3$,
$$
\frac{\rho_1}{\rho_0}=\frac32,
\qquad
\frac{p_1}{p_0}=4,
\qquad
V=3\sqrt{\frac{p_0}{\rho_0}}.
$$

= Weak shock
{parent=Normal shock wave}
{wiki=Shock_wave#Weak_shocks}

A weak shock has a small relative pressure jump. Its density jump agrees with the reversible adiabatic prediction through second order in the pressure jump, while entropy production first appears at third order.

= Linear acoustics
{parent=Fluid mechanics}
{wiki}

Linear acoustics describes small pressure, density, and velocity perturbations about a uniform quiescent compressible fluid.

= Homentropic pressure perturbation
{parent=Linear acoustics}

For a homentropic reference state, first-order pressure and density perturbations satisfy $p'=c_0^2\rho'$, where $c_0$ is the sound speed.

= Acoustic velocity potential
{parent=Linear acoustics}
{wiki=Velocity_potential\#Acoustics}

For irrotational linear acoustic flow, $\mathbf u=\nabla\phi$, $p'=-\rho_0\phi_t$, and $\phi$ obeys the wave equation.

= Outgoing acoustic field of a pulsating sphere
{parent=Acoustic velocity potential}

For $R(t)=a+\operatorname{Re}(\epsilon e^{i\omega t})$, put $k=\omega/c_0$. To first order in $\epsilon/a$, the exterior outgoing velocity-potential amplitude is
$$
\widehat\phi(r)
=-\frac{i\omega\epsilon a^2}{1+ika}
\frac{e^{-ik(r-a)}}r.
$$
It satisfies $\widehat u_r(a)=\widehat\phi'(a)=i\omega\epsilon$.

= Mean acoustic power of a pulsating sphere
{parent=Outgoing acoustic field of a pulsating sphere}

The mean power radiated by the pulsating sphere is
$$
\overline P
=2\pi a^2\rho_0\omega^2\epsilon^2c_0
\frac{\omega^2a^2}{c_0^2+\omega^2a^2}.
$$

= Surface acoustic energy-to-flux ratio of a pulsating sphere
{parent=Outgoing acoustic field of a pulsating sphere}

At $r=a$,
$$
\frac{c_0\langle K+W\rangle}{|\langle I_r\rangle|}
=1+\frac{c_0^2}{2\omega^2a^2}.
$$
For $\omega a/c_0\gg1$, kinetic and compressional energies are equal and the field is locally radiative. For $\omega a/c_0\ll1$, reactive kinetic energy dominates and the mean radiated flux is small.

= Pressure-forced standing acoustic wave in a spherical annulus
{parent=Acoustic velocity potential}

For $R<r<2R$, a rigid inner sphere and harmonic pressure
$$
p'(2R,t)=\varepsilon p_0\cos\omega t
$$
produce a standing radial field. With $k=\omega/c_0$ and $\alpha=kR$,
$$
\phi(r,t)
=-\frac{2R\varepsilon p_0}{\rho_0\omega r}
\frac{\cos k(r-R)+(kR)^{-1}\sin k(r-R)}
{\cos\alpha+\alpha^{-1}\sin\alpha}
\sin\omega t.
$$
The pressure and radial velocity are in temporal quadrature, so the period-averaged <acoustic energy conservation>[acoustic intensity] vanishes.

= Radial acoustic resonance in a spherical annulus
{parent=Pressure-forced standing acoustic wave in a spherical annulus}

The ideal pressure-forced response becomes resonant when
$$
\cos\alpha+\frac{\sin\alpha}{\alpha}=0,
$$
equivalently $\tan\alpha=-\alpha$. Damping regularizes the divergent linear response and introduces a nonzero phase lag and mean supplied power.

= Acoustic energy conservation
{parent=Linear acoustics}

Linear acoustic fields obey
$$
\frac{\partial}{\partial t}(K+W)+\nabla\cdot\mathbf I=0,
$$
where
$$
K=\frac12\rho_0|\mathbf u|^2,
\qquad
W=\frac{p'^2}{2\rho_0c_0^2},
\qquad
\mathbf I=p'\mathbf u.
$$

= Evanescent acoustic surface wave
{parent=Linear acoustics}

A surface wave with phase speed below the bulk sound speed decays normally as $e^{-\alpha z}$, where $\alpha=k\sqrt{1-c^2/c_0^2}$.

= Spring-supported acoustic membrane wave
{parent=Evanescent acoustic surface wave}

For a massless spring-supported membrane adjoining a fluid half-space, kinematic and pressure balance give the dispersion relation $As^4+s^2-1=0$ with $s=c/c_0$.

= Acoustic membrane dispersion asymptotics
{parent=Spring-supported acoustic membrane wave}

For $As^4+s^2-1=0$, the physical speed approaches $c_0(1-A/2)$ as $A$ tends to zero and scales as $c_0A^{-1/4}$ as $A$ tends to infinity.

= Added mass of an evanescent fluid layer
{parent=Evanescent acoustic surface wave}

A disturbance decaying over depth $1/k$ accelerates fluid mass of order $\rho_0/k$ per unit area, giving a frequency-dependent added inertia.

= Time average of harmonic power
{parent=Linear acoustics}

For real harmonic fields with complex amplitudes $\widehat p$ and $\widehat u$, their period-averaged product is $\operatorname{Re}(\widehat p\widehat u^*)/2$.

= Reactive acoustic energy flux
{parent=Time average of harmonic power}

When pressure and velocity are in quadrature, their instantaneous energy flux oscillates but its period average vanishes.