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Differentiating the Stokeslet gives its rate-of-strain tensor and vorticity, displayed above for $r=|\mathbf{x}|>0$. The pressure is $p=\mathbf{F}\cdot \mathbf{x}/(4\pi r^3)$. Integrating the Newtonian fluid stress tensor over a sphere enclosing the origin gives traction resultant $-\mathbf{F}$, fixing the point-force normalization.

Under critical balance in a rapidly rotating fluid, define ${\mathbf{u}}_{\perp }^{(0)}=\hat{\mathbf{z}}\times {\nabla }_{\perp }\Phi$ and $D_{\perp }={\partial }_t+\{\Phi ,\cdot \}$. Leading horizontal incompressibility gives zero divergence, but at the next order ${\nabla }_{\perp }\cdot {\mathbf{u}}_{\perp }^{(1)}=-{\partial }_{z}w$. Its Coriolis acceleration contribution survives in the vertical vorticity equation, producing the displayed coupled system. The vertical momentum equation uses the leading pressure $p_0=2\Omega \Phi$. A flow that stays purely horizontal is independent of $z$ in this reduced system; the assertion is not an exact theorem for arbitrary fast solutions of the full rotating Euler equations.

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.

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.

Let a sphere of radius $a$ descend along the $z$ axis through a stationary unbounded Newtonian fluid. A second sphere with centre $(\ell ,0,0)$ is held against translation but has zero applied torque. In the prescribed isolated-sphere incident flow, Faxén's rotational law and the vorticity of translating-sphere Stokes flow giveThe speed $U$ cancels on integrating along the complete passage. Since ${\int }_{-\infty }^{\infty }dZ/({\ell }^2+Z^2{)}^{3/2}=2/{\ell }^2$, the result is the displayed angle about $-{\mathbf{e}}_y$. The target radius is absent from the rotational Faxén response to this harmonic incident vorticity. This is a one-way incident-flow calculation; nonoverlap alone does not justify ignoring mutual hydrodynamic interactions between finite spheres. A fully coupled encounter requires the method of reflections for Stokes flow or an exact two-body solution.

With $u={\psi }_y$ and $v=-{\psi }_x$, this steady incompressible solution of the Euler equations has velocity $(-\sin x\sin y,-\cos x\cos y)$ and vorticity $2\psi$. Its pressure is $C+\rho (\cos 2x-\cos 2y)/4$. The Bernoulli function is $C_0-{\psi }^2$, so it is constant along streamlines but generally varies between them.

For light along ${\mathbf{e}}_x$, vorticity along ${\mathbf{e}}_y$ and $\epsilon =\omega /(2D_r)$, spherical coordinates $\mathbf{p}=(\cos \theta ,\sin \theta \cos \varphi ,\sin \theta \sin \varphi )$ give a first orientation-density correction proportional to $\sin \varphi g(\theta )$. The linearized alignment-diffusion operator preserves azimuthal harmonics and the rotation of the axisymmetric equilibrium forces only this sine harmonic. Angular integration makes the first correction to the mean purely vertical. Uniqueness follows from the weighted identity $\mathcal{A}(f_{0}h)={\nabla }_p\cdot (f_0{\nabla }_{p}h)$: integration by parts shows the only homogeneous smooth solution is a multiple of $f_0$, removed by normalization.

The Squire equation governs wall-normal vorticity $\eta ={\partial }_{z}u-{\partial }_{x}w$ in the same normal mode convention as the Orr-Sommerfeld equation. It follows by taking ${\partial }_z$ of the streamwise momentum equation minus ${\partial }_x$ of the spanwise momentum equation. At a rigid no-slip wall, $\hat{\eta }=0$.

A Squire mode has zero wall-normal velocity and nonzero wall-normal vorticity, satisfying the homogeneous Squire equation. For real base velocity, positive finite Reynolds number, and zero boundary terms,for a nontrivial mode in a finite channel with no-slip walls. This follows by multiplying the homogeneous equation by $\overline{\eta }$, integrating by parts and taking the real part. Such modes are damped even when the coupled velocity-vorticity system can exhibit transient growth from non-normal modes.

For a two-mode tensorial squirmer flow of radius $a$ and a passive equal sphere held at $X=Re$, with $a/R\ll 1$, let $s=e\cdot Be$. Faxén's first law gives holding force $F_h=-9\pi \mu a^{3}se/R^2+O(\mu a^4|A|/R^3+\mu a^5\Vert B\Vert /R^4)$. Its reflected Stokeslet changes swimming velocity by $-9a^{3}se/(4R^3)$. Since this leading force is radial, its vorticity at the swimmer vanishes. The next holding force, generated by the potential dipole, yields rotation $a^{4}A\times X/(4R^6)$. Terms from the passive sphere's finite radius, reflected stresslet and rotlet are needed at higher order. These formulas assume fixed slip and no externally applied swimmer force or torque.

Linearizing inviscid barotropic flow about uniform rest gives ${\partial }_t(\nabla \times u)=0$, not automatically $\nabla \times u=0$. Irrotational linear acoustics therefore requires zero initial vorticity or restriction to the longitudinal acoustic mode. A time-independent divergence-free field, with zero pressure and density perturbations, is a nonacoustic solution of the linearized equations.

For a smooth planar incompressible flow, use $u={\psi }_y$, $v=-{\psi }_x$, so its scalar vorticity is $\omega =-\Delta \psi$. If $\omega =\lambda \psi$ for a constant $\lambda$, the flow solves the steady constant-density Euler equations with a suitable pressure.

Indeed the vector identity for the convective acceleration givesTherefore the pressure can be chosen so that the Bernoulli function satisfies $p/\rho +|\mathbf{u}{|}^2/2=C-\lambda {\psi }^2/2$. Its gradient is $-\omega \nabla \psi$, exactly the steady momentum equation. It is constant along each streamline, since the stream function is constant there.

For example $\psi =\sin x\cos y$ satisfies $\omega =2\psi$. Its pressure is $p_0-\rho {\sin }^{2}x/2+\rho {\sin }^{2}y/2$. Thus a rotational flow can have an explicit pressure and a streamline-dependent Bernoulli constant.

In a steady inviscid planar incompressible flow, vorticity is constant along streamlines. A regular stream function supplies a local transverse coordinate, so locally $\Delta \psi$ depends only on $\psi$. A global single-valued function needs the same value on every component of each level set. This can fail: $\psi =x^3-x$ gives the steady Euler shear $(u,v)=(0,1-3x^2)$ with constant pressure, but $\psi (0)=\psi (1)=0$ while $\Delta \psi$ is respectively $0$ and $6$.

For an axisymmetric flow without azimuthal velocity, the spherical Stokes streamfunction gives $u_r={\Psi }_{\theta }/(r^2\sin \theta )$ and $u_{\theta }=-{\Psi }_r/(r\sin \theta )$. These formulas enforce incompressibility. The only possibly nonzero component of vorticity isThus the displayed equation is precisely the condition for irrotational flow, away from coordinate poles, with limits interpreted through regularity of the physical velocity field. It differs from the scalar spherical Laplace equation: $\Psi$ is a Stokes streamfunction, not a velocity potential.

For a forced sphere and a distant equal force-free, torque-free sphere, the leading incident Stokeslet strain is proportional to $(\mathbf{U}\cdot \mathbf{n})(I-3\mathbf{nn})$. Its reflected stresslet is axisymmetric about the line of centers and has zero vorticity on that line. The nominal fifth-order self-spin coefficient therefore vanishes. Finite-size incident-strain corrections and higher force-free scattering multipoles first supply a generic seventh-order angular velocity. Purely longitudinal forcing has zero spin exactly by axial symmetry. The leading translation correction is fourth order for a generic orientation, but sixth order for purely transverse forcing. These cancellations matter when using the method of reflections for Stokes flow to infer orders from far-field decay alone.

For an axial rotlet dipole parallel to a flat free surface, its reflected image gives the displayed normal vorticity at height $h$. A spherical torque-free body responds with angular velocity ${\omega }_z/2$ and an elongated body also responds to strain. At constant height, self-propulsion plus yaw gives a circular path; the signed moment sets its handedness.

This three-layer shear flow has $U=\Delta Uz/h$ and density ${\rho }_0$ between $z=\pm h/2$. Above and below, the velocity is the corresponding constant $\pm \Delta U/2$, and the mass density is ${\rho }_0\mp \Delta \rho /2$. The density drops and vorticity jumps coincide at both interfaces. Its hydrodynamic stability involves coupled gravity-vorticity interface waves, unlike a profile with uniform shear extending through all three layers.

The Townsend-Betchov cascade cartoon organizes inertial range vorticity into sheets by bi-axial strain, then uses sheet instability and roll-up to generate finer vortex tubes. Dissipative tube cores balance stretching with kinematic viscosity. Such differing mechanisms across scales challenge literal identical-step, memoryless cascade models without contradicting a mean energy flux.

For inviscid planar incompressible flow, write velocity as $(-{\psi }_y,{\psi }_x)$ and vorticity as $\zeta ={\nabla }^2\psi$. Taking curl of the Euler equations for an inviscid fluid removes pressure and gives the displayed transport equation, with $J(\psi ,\zeta )={\psi }_x{\zeta }_y-{\psi }_y{\zeta }_x$. The Poisson equation then recovers the streamfunction at each instant. A velocity tangent to the plane cannot stretch its perpendicular vorticity vector.

For equal-depth layers coupled by an interface,The Laplacian terms are relative vorticity; the streamfunction differences encode opposite layer-thickness changes. Each material derivative follows that layer's own velocity. The sum has a barotropic mode without interface displacement, while the difference contains a baroclinic mode.