Stream function

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.

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 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.

Using $\mathbf{u}=(-{\psi }_y,{\psi }_x)$, the displayed stream function gives ${\mathbf{u}}_0=(\sin x\cos y,-\cos x\sin y)$. Its streamlines are level sets of ${\psi }_0$. At $(n\pi ,m\pi )$, the Jacobian matrix is $\mathrm{diag}((-1{)}^{n+m},-(-1{)}^{n+m})$, so the points are saddle equilibria. At $(\pi /2+n\pi ,\pi /2+m\pi )$ the eigenvalues are $\pm i$, giving center equilibria surrounded by closed streamlines. The grid lines $x=n\pi$ and $y=m\pi$ form heteroclinic orbits between adjacent saddles; circulation alternates between cells.

Perturb the sinusoidal cellular flow by $ϵ\cos x\cos y\sin (\omega t)$ in its stream function. On $y=0$, parameterizing the unperturbed orbit with $q_0(0)=(X_0,0)$ gives $x(s)=2\arctan e^{s+a}$, $a=\log \tan (X_0/2)$. The heteroclinic Melnikov function for a periodic planar flow has integrand ${\mathrm{sech}}^2(t-t_0+a)\sin (\omega t)$. Its odd part vanishes and the Fourier transform of ${\mathrm{sech}}^2$ gives the displayed result. For each fixed nonzero frequency its simple zeros produce transverse intersections at sufficiently small perturbation, hence exchange of fluid lobes across the former cell boundary. The leading splitting is exponentially small at high frequency; that estimate is not uniform in frequency and perturbation size.

The incompressible planar field $(\alpha x-\beta y,\beta x-\alpha y)$ has the displayed stream function under $(u,v)=({\psi }_y,-{\psi }_x)$ and vorticity $2\beta$. In rotated coordinates $s=(x+y)/\sqrt{2}$, $t=(x-y)/\sqrt{2}$, $\psi =((\alpha -\beta )s^2-(\alpha +\beta )t^2)/2$. For positive parameters its nonstationary streamlines are closed ellipses if $\alpha <\beta$, hyperbolas and straight separatrices if $\alpha >\beta$, and parallel straight lines with a stationary diagonal if $\alpha =\beta$.

For a steady axisymmetric compressible fluid flow, mass conservation permits this representation. The mass flow through an annulus between two stream surfaces is $2\pi \Delta \psi$. In a steady adiabatic wind without magnetic forces, the specific entropy, specific angular momentum and Bernoulli function are locally functions of $\psi$ on connected regular stream surfaces.

For axisymmetric incompressible flow without swirl, $u_r=-{\psi }_z/r$ and $u_z={\psi }_r/r$. These identities enforce incompressibility, and $2\pi \psi$ measures axial volume flux from the symmetry axis when its additive constant is chosen there.

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.

Suppose the incident axisymmetric flow is an incompressible flow and an irrotational flow, with Stokes streamfunction $\Psi$ regular inside $r<a$. The displayed transformation adds an impermeable sphere at $r=a$ without new finite exterior singularities. Writing $s=a^2/r$ and $G=(r/a)\Psi (s,\theta )$, direct differentiation showsHence the Stokes streamfunction equation for irrotational flow is preserved. At $r=a$, $\tilde{\Psi }=0$, making the radial velocity zero. Exterior points sample only the regular interior under inversion. If $\Psi$ vanishes at the origin and is $O(r)$ there with smooth regular derivatives, the induced velocity decays at infinity. This is an inversion construction for an axisymmetric streamfunction, distinct from a scalar Kelvin transform.

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.

The incompressible linear flow $(u,v)=(ϵx-\gamma y,\gamma x-ϵy)$ has streamfunction $\psi =ϵxy-\gamma (x^2+y^2)/2$. Its velocity matrix squares to $({ϵ}^2-{\gamma }^2)I$. Thus strain-dominated flow has saddle trajectories, rotation-dominated flow has closed elliptical trajectories, and equal nonzero strain and rotation gives simple shear flow along parallel lines with a stationary line. Pure rotation is counterclockwise for positive $\gamma$.