Fluid mechanics

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

An ideal fluid issuing from a small hole a vertical depth $h$ below a free surface has speed

Mass density is mass per unit volume.

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

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

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

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

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

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

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

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.

In a uniformly stratified fluid with buoyancy frequency $N$, a plane wave of horizontal and vertical wavenumbers $k,m$ obeys

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

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

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 $k\le N/U$; above this cutoff the vertical wavenumber is imaginary and the disturbance is evanescent.

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.

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

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.

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

Uniform irrotational flow of speed $U$ past a circular cylinder of radius $a$, with zero circulation, has velocity potentialOn the cylinder, the tangential speed is $-2U\sin \theta$, so the Bernoulli equation gives

For liquid occupying $a(t)<r<b(t)$, radial symmetry and incompressible flow implyThe kinematic boundary condition at the outer interface giveswhich expresses conservation of the liquid's area.

If the pressure at $r=b(t)$ is $p_{\infty }$, the Unsteady Bernoulli equation gives

For $a=a_0(1+ϵ)$, equilibrium outer radius $b_0$, and a bubble gas obeying $p_0\pi a^2=\text{constant}$, the linearization isThe bubble therefore undergoes simple harmonic motion with

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

For a steady incompressible plume with localized vertical body force $b(x)\delta (y)$, decay of velocity and shear at transverse infinity gives

If $b(x)=B{x}^{-1/5}$, balancing the momentum flux and transverse viscosity gives

With $u=-{\psi }_y$, $v={\psi }_x$, $\eta =y/\Delta$, and $\psi =U\Delta f(\eta )$, the plume boundary-layer equation reduces to

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

For time-dependent irrotational flow with $u=\nabla \varphi$, constant mass density, and no body force, the Euler equations for an inviscid fluid integrate toA time-dependent change of gauge in $\varphi$ may set $C(t)=0$.

For $\mathbf{u}=\nabla \varphi +\beta \nabla \alpha$, the densitygives incompressibility and material conservation of $\alpha$ and $\beta$. Differentiating the Clebsch representation then yields the Euler momentum equation with

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

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

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

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 givefor the free surface to reach the inlet.

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

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

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

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

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

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.

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

For a film of thickness $h$ flowing down a slope of angle $\alpha$, an upslope surface stress of magnitude $\tau$ givesThe surface velocity, total flux, and basal shear reverse atrespectively.

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

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 requires fluid velocity to equal solid-boundary velocity.

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

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

The rate-of-strain tensor is the symmetric part of the velocity gradient,

For a planar velocity field in polar coordinates,

For an incompressible Newtonian fluid of constant density and viscosity,

Steady pressure-driven flow through a circular pipe has the parabolic profile

For a fixed domain $\mathcal{D}$, body-force density $f$, and stress tensor $\sigma$, the kinetic energy satisfiesThe last term is the nonnegative rate at which viscosity converts mechanical energy into heat.

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 $\delta \sim a{\mathrm{Re}}^{-1/2}$ 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

For the zero-circulation flow around a circle, $e:e=8U^2{a}^4/r^6$. 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 is the zero-Reynolds-number limit in which viscous and pressure forces balance while fluid inertia is neglected.

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

For two Stokes velocity-stress fields in the same domain,It follows by integrating the divergence of the corresponding cross-work flux and using symmetry of the Newtonian stress.

For an incompressible Newtonian fluid, the nonnegative rate at which viscosity converts mechanical energy into heat is

Taking the curl of the planar Stokes equation makes vorticity harmonic. Since vorticity is minus the Laplacian of the stream function,

For $\psi =r^{2}f(\theta )$, biharmonicity gives
$f^{(4)}+4f^{\prime \prime }=0$. In the wedge $-\alpha <\theta <0$, no slip at the lower wall and no penetration plus tangential stress $S$ at the upper surface giveThe resulting surface speed is

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 isIt satisfies no slip at $r=a$ and approaches the imposed shear in velocity.

The surface shear stress isThe torque exerted by the fluid on the disk per unit axial length is therefore

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

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 $(U\cdot x)x$.

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

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

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

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

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

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

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.

For concentric spheres of radii $a<b$, with the inner sphere rotating at angular velocity $\mathbf{\Omega }$ and the outer sphere fixed, the Stokes velocity isThe pressure is constant, and may be set to zero.

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

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.

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 requiresThe last condition includes both $\rho |U|H^2/(\mu L)\ll 1$ and $\rho VH/\mu \ll 1$.

For a film of thickness $h(x,t)$ flowing down a plane inclined by $\alpha$, lubrication theory gives the fluxwhen surface tension is neglected. Mass conservation gives the thin-film equation $h_t+q_x=0$.

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 $\sqrt{a{h}_0}$.

Between surfaces of velocities $U_0,U_1$ separated by $h$, the local lubrication profile is

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

The flux of the local Couette-Poiseuille profile is

Incompressible mass conservation in a moving gap givesCombined with the local Couette-Poiseuille flux, this is the Reynolds lubrication equation.

For lower speed $U$, stationary upper tangential speed, upper normal speed $-V$, and equal endpoint pressures, define $⟨f⟩=L^{-1}{\int }_0^{L}f(x)dx$. Thenwhere

The horizontal force on the lower wall vanishes when

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

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.

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

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.

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 has gradient equal to fluid density times gravity.

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

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

For radial flow outside a sphere, incompressible flow makesindependent of radius. Its velocity potential is $\varphi =-Q/(4\pi r)$.

A vacuum cavity of radius $a(t)$ in an infinite inviscid incompressible fluid of density $\rho$ and far-field pressure $p_0$ obeysIf $a(0)=a_0$ and $\dot{a}(0)=0$, its collapsing branch satisfies

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

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

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

For small kinematic viscosity $\nu$, the mean viscous dissipation per unit horizontal area is $2\rho \nu g{k}^2|A{|}^2$. Since the mean wave energy is $\rho g|A{|}^2/2$, the amplitude obeys

For small-amplitude potential flow beneath a mean surface $z=0$, the linearized kinematic and dynamic conditions are

In a rectangular box with Neumann side walls, a horizontal modehas 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 $\Omega =\sqrt{gk}$.

Pressure forcing of one rectangular free-surface mode reduces its amplitude toAt $\omega =\sqrt{gk}$, the inviscid undamped response contains $t\sin (\omega t)$ and grows without bound in the linear model.

For planar incompressible flow, a streamfunction satisfies

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

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

For a free surface $z=\eta (x,y,t)$ and velocity $(u,v,w)$, material conservation of $z-\eta$ giveson the surface.

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

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 equateson the two sides, up to a removable function of time.

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

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

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.

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

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 exponentOne sign has positive growth rate, so every nonzero wavenumber is unstable in the inviscid zero-thickness model.

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

For one-dimensional homentropic ideal-gas flow,is constant along the characteristic $dx/dt=u\pm c$.

In a right-moving simple wave entering gas initially at rest with sound speed $c_0$, the other Riemann invariant is constant, soThe isentropic relationconverts the velocity disturbance into its nonlinear pressure disturbance.

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 $\partial x(\tau ,t)/\partial \tau =0$.

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

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.

In the rest frame of a one-dimensional shock, states with velocities $w_0,w_1$ satisfyand

If a shock travels into stationary gas at speed $V$ while the downstream piston and gas travel at $V/3$, then for $\gamma =3$,

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

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

For irrotational linear acoustic flow, $\mathbf{u}=\nabla \varphi$, $p^{\prime }=-{\rho }_0{\varphi }_t$, and $\varphi$ obeys the wave equation.

For $R(t)=a+\mathrm{Re}(ϵe^{i\omega t})$, put $k=\omega /c_0$. To first order in $ϵ/a$, the exterior outgoing velocity-potential amplitude isIt satisfies ${\hat{u}}_r(a)={\hat{\varphi }}^{\prime }(a)=i\omega ϵ$.

The mean power radiated by the pulsating sphere is

At $r=a$,For $\omega a/c_0\gg 1$, kinetic and compressional energies are equal and the field is locally radiative. For $\omega a/c_0\ll 1$, reactive kinetic energy dominates and the mean radiated flux is small.

For $R<r<2R$, a rigid inner sphere and harmonic pressureproduce a standing radial field. With $k=\omega /c_0$ and $\alpha =kR$,The pressure and radial velocity are in temporal quadrature, so the period-averaged acoustic intensity vanishes.

The ideal pressure-forced response becomes resonant whenequivalently $\tan \alpha =-\alpha$. Damping regularizes the divergent linear response and introduces a nonzero phase lag and mean supplied power.

Linear acoustic fields obeywhere

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

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

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

A disturbance decaying over depth $1/k$ accelerates fluid mass of order ${\rho }_0/k$ per unit area, giving a frequency-dependent added inertia.

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

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