Navier-Stokes equation

For an incompressible Newtonian fluid of constant density and viscosity,

The Leray-Helmholtz projection $P_{\sigma }$ is the orthogonal projection of square-integrable vector fields onto divergence-free vector fields. Applying it to an incompressible momentum equation removes the pressure gradient.

The Stokes operator is $A=-P_{\sigma }\Delta$ on divergence-free vector fields, where $P_{\sigma }$ is the Leray-Helmholtz projection. On a periodic domain it equals $-\Delta$ on its domain, and its inverse gains two Sobolev derivatives.

The spectral projection $P_N$ retains the first $N$ Stokes eigenfunctions. On its finite-dimensional range, one derivative costs at most ${\lambda }_N^{1/2}$ and all Sobolev norms are equivalent.

For a periodic or boundary-tangent divergence-free vector field $u$, integration by parts givesConsequently $\int (u\cdot \nabla f)f=0$, the cancellation underlying many fluid energy estimates.

For a velocity field that need not be divergence free, the corrected transportsatisfies $⟨\tilde{B}(u,v),w⟩=-⟨\tilde{B}(u,w),v⟩$ and therefore preserves the quadratic energy cancellation.

Replacing the advecting velocity by a fixed finite Stokes projection $P_{N}u$ makes the three-dimensional nonlinearity controlled by ${\lambda }_N^{1/4}|u|\Vert v\Vert \Vert w\Vert$. This yields global existence and uniqueness at the weak energy level.

The steady equation $\nu Au+\tilde{B}(u,u)=f$ retains the energy cancellation without imposing incompressibility. Galerkin bounds, compactness, and elliptic bootstrapping produce an $H^2$ periodic solution for square-integrable forcing.

Rayleigh-Bénard convection is buoyancy-driven flow caused by heating a fluid from below. In an infinite-Prandtl-number reduction, velocity is determined instantaneously by a forced Stokes equation, while temperature satisfies an advection-diffusion equation.

For a fixed-temperature layer with stress-free boundary conditions, a vertical Fourier sine series has $a^2={\lambda }^2+n^2{\pi }^2$, where $\lambda$ is horizontal wavenumber. The coupled vertical velocity and temperature amplitudes satisfy the displayed quadratic for their growth rate $\sigma$. Its roots are real for positive Rayleigh number, while both roots have negative real parts for negative Rayleigh number. Setting $\sigma =0$ gives the free-slip convection neutral curve.

In nonrotating Rayleigh-Bénard convection with fixed-temperature, stress-free boundary conditions, the stress-free convection growth-rate polynomial has real roots on the destabilizing side. Consequently the first instability is stationary, rather than oscillatory. Minimizing the free-slip convection neutral curve selects ${\lambda }_c=\pi /\sqrt{2}$ and ${\mathrm{Ra}}_c=27{\pi }^4/4$.

Using layer depth $d$ and thermal diffusivity $\kappa$ gives time scale $d^2/\kappa$, velocity scale $\kappa /d$, and perturbation pressure scale ${\rho }_0{\kappa }^2/d^2$. With temperature scale $\Delta T$, the Linearized Boussinesq equations have viscous coefficient Prandtl number $\mathrm{Pr}=\nu /\kappa$ and buoyancy coefficient $\mathrm{Ra}\mathrm{Pr}$, where Rayleigh number $\mathrm{Ra}=g{\alpha }_T\Delta T{d}^3/(\nu \kappa )$.

The conductive state is the no-flow equilibrium in which heat conduction alone maintains a linear temperature profile across a layer with fixed boundary temperatures. Its hydrostatic pressure follows by integrating $P_b^{\prime }=-g{\rho }_b$, with mass density obtained from the equation of state. Under the Boussinesq approximation, deviations from this state are the variables used in the linear Rayleigh-Bénard convection problem.

A thermally driven fluid layer rotating about the vertical axis experiences a Coriolis force that couples vertical velocity to vertical vorticity. Its onset depends on the Rayleigh number, Prandtl number and Taylor number, as well as thermal and mechanical boundary conditions. The force does no mechanical work, but changes the linear mode balance and nonlinear convection roll competition.

Near an oscillatory onset of rotating Rayleigh-Bénard convection, opposite travelling roll waves have coupled Hopf bifurcation amplitudes. Symmetry gives the displayed cubic amplitude equations, with generally complex self-coupling and cross-coupling coefficients. Write $a_s=\mathrm{Re}{g}_s$ and $a_c=\mathrm{Re}{g}_c$. For $r>0,a_s>0$, a single travelling wave has intensity $r/a_s$ and is stable against its counterpropagating competitor when $a_c>a_s$. Equal standing-wave intensities are $r/(a_s+a_c)$; with $r>0$ and positive denominator they are stable to intensity differences when $a_s>a_c$. Phase symmetries give orbital, rather than strict phase-decay, stability. Spatial modulation, mean flows and oblique convection rolls require additional stability tests.

A saturated convection roll is not stable merely because its single-amplitude cubic coefficient is positive. Perturbations include phase sidebands, transverse bending, differently oriented convection rolls, oscillatory modes and large-scale Eulerian mean flow. Their growth rates follow from the corresponding enlarged amplitude equations or the full linearization about the convection roll. Küppers–Lortz instability and small-angle instability of rotating convection rolls are distinct orientation-dependent mechanisms.

For finite Prandtl number and stress-free boundaries, nearly parallel convection rolls can interact with a weakly damped large-scale Eulerian mean flow. The ordinary regular two-roll amplitude expansion becomes nonuniform as the angle tends to zero, so the mean-flow mode must be retained. This mechanism is not automatically the finite-angle Küppers–Lortz instability, and thresholds depend on the stated boundary and parameter regime.

For an existing convection roll of intensity $r/g_0$, the amplitude of a new orientation at relative angle $\vartheta$ grows at rate $r[1-g(\vartheta )/g_0]$. Rotation permits $g(\vartheta )\ne g(-\vartheta )$. If one oblique orientation has $g(\vartheta )<g_0$, the convection roll is unstable immediately above a supercritical stationary threshold. This is the Küppers–Lortz mechanism. Successive orientation replacements can produce time-dependent convection roll switching; a universal attractor does not follow from the two-mode calculation. The perturbing orientation requires three-dimensional disturbances even if the initial convection roll is two-dimensional.

For the free-slip plane layer, onset selects the smallest Rayleigh number across all admissible stationary and oscillatory normal modes. The first vertical mode wins because increasing vertical index raises viscous cost and reduces oscillatory admissibility. Oscillatory minimization replaces the right-hand side of the stationary wave-number equation by $P^2\mathrm{Ta}{\pi }^2/(1+P{)}^2$. For the oscillatory minimum also require positive angular frequency squared. Where both selected curves are admissible, at rapid rotation the ratio of the minimized thresholds is $R_{o,c}/R_{s,c}\sim 2P^{4/3}/(1+P{)}^{1/3}$. Equality gives $8P^4-P-1=0$, with positive root about $0.67660$; below it sufficiently rapid rotation can select oscillatory primary onset.

For stress-free impermeable fixed-temperature plates, thermal-time growth rate $s$, horizontal wavenumber $k$ and vertical index $n$, put $a=k^2+n^2{\pi }^2$ and $P=\mathrm{Pr}$. Vertical velocity, temperature and vertical vorticity amplitudes satisfy $a(s/P+a)W+\sqrt{\mathrm{Ta}}n\pi Z=R{k}^2\Theta$, $(s/P+a)Z=\sqrt{\mathrm{Ta}}n\pi W$, $(s+a)\Theta =W$. Taking their determinant givesThis is a cubic and remains valid without dividing by a potentially zero factor. Setting $s=0$ or $s=i\omega$ determines the stationary or admissible oscillatory neutral curves.

For $s=i\omega$, real-imaginary separation of the rotating-convection cubic givesThe curve is admissible only if ${\omega }^2>0$, so $P<1$ and sufficient rotation are necessary. A formal minimum with nonpositive angular frequency squared is not a Hopf bifurcation. At zero angular frequency this curve meets the stationary curve for that fixed wavenumber in a double-zero limit.

Substitution of zero growth rate into the rotating-convection cubic gives the displayed Rayleigh number. For the first vertical mode, setting $t=k^2$, $m={\pi }^2$, differentiating with respect to $t$ gives $(t+m{)}^2(2t-m)=\mathrm{Ta}m$. At rapid rotation, $k_c\sim (\mathrm{Ta}{\pi }^2/2{)}^{1/6}$ and $R_c\sim 3(\mathrm{Ta}{\pi }^2/2{)}^{2/3}$. Discrete allowed wavenumbers or different plate conditions require a different minimization.

For nonrotating convection under the Boussinesq approximation with fixed-temperature impermeable stress-free boundary conditions, a marginal vertical mode $\sin (n\pi z)$ has neutral Rayleigh number $(k^2+n^2{\pi }^2{)}^3/k^2$. The minimum is at $n=1$, $k=\pi /\sqrt{2}$, and $R_c=27{\pi }^4/4$. These follow by eliminating the temperature from $(D^2-k^2{)}^{2}w=R{k}^2\theta$ and $(D^2-k^2)\theta =-w$, then minimizing over $k$.

In thermal-time units, taking the Prandtl number large removes momentum inertia at leading order. The Stokes flow momentum equation then determines velocity instantaneously from the temperature perturbation, while the nonlinear heat equation retains its time derivative. Small Prandtl number does not generally give this model. An exact contrasting shear in a free-slip layer is $u=U_0{e}^{-\mathrm{Pr}{\pi }^{2}t}\cos \pi z$, $w=\theta =0$: it decays slowly as the Prandtl number tends to zero but is excluded by the instantaneous Stokes equation.

For a nonzero horizontal Fourier mode $k$, solve $(D^2-k^2{)}^{2}w=R{k}^2\theta$, $w=w^{\prime \prime }=0$, to obtain $w={\mathcal{W}}_R\theta$. Under homogeneous temperature Dirichlet boundary conditions, the sine modes diagonalize this self-adjoint operator, with eigenvalues $R{k}^2/(k^2+n^2{\pi }^2{)}^2$. The temperature linear operator is ${\mathcal{L}}_R=D^2-k^2+{\mathcal{W}}_R$. At a marginal mode $g=\sin (n\pi z)$ it has ${\mathcal{W}}_{R}g=(k^2+n^2{\pi }^2)g$.

A damped-driven Navier-Stokes equation adds linear damping and external forcing. In two-dimensional steady vorticity form it reads

For a steady damped vorticity equation with $\gamma >0$, divergence-free transport, and bounded forcing, multiplication by $\omega |\omega {|}^{q-2}$ gives $\gamma \Vert \omega {\Vert }_q\le \Vert g{\Vert }_q$. Sending $q$ to infinity yields

A vanishing-viscosity limit studies solutions as the viscosity coefficient tends to zero. Uniform estimates and compactness can make the viscous term disappear weakly while retaining enough strong convergence to pass nonlinear transport terms to an inviscid equation.