Euler equations for an inviscid fluid

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

For a planar Euler velocity obtained from vorticity in $L_{\mathrm{loc}}^{\infty }(dt;L^1\cap L^{\infty })$, the planar vorticity velocity kernel gives a locally integrable speed bound and log-Lipschitz modulus. Spatial mollification gives existence of global trajectories; the Osgood uniqueness criterion gives uniqueness and continuous dependence. Measurability in time and an initial trace are needed. This establishes the flow for a fixed Euler solution; nonlinear uniqueness of the Euler solution itself requires further comparison arguments.

The Bernoulli principle relates pressure, kinetic energy per unit volume and Newtonian gravitational potential along a streamline of steady inviscid flow. For constant mass density, the Bernoulli equation gives $p/\rho +u^2/2+\Phi$ constant along that streamline. Constancy across different streamlines requires further hypotheses, such as irrotational flow.

For plates $z=\pm h(t)$, incompressibility, impermeability and regularity on the axis determine a z-independent radial velocity $u_r=-r\dot{h}/(2h)$ and vertical velocity $u_z=z\dot{h}/h$. A possible azimuthal velocity requires additional rotational data; continuity alone does not determine it. For a smooth regular initial profile $U(r)$ independent of $z$, its azimuthal Euler equation transports it as $u_{\theta }(r,t)=\lambda U(\lambda r)$, where $\lambda =\sqrt{h/h_0}$. If the rotation is solid-body with angular velocity $\Omega (t)$, the vorticity equation gives $\Omega ={\Omega }_{0}h/h_0$. Along a particle path, $r^{2}h$ and $r^2\Omega$ are constant, illustrating conservation of angular momentum.

Radial velocity is the component of a velocity field along the line from a chosen centre to the point of observation.

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 $C(t)=0$.

An inviscid uniform tube plug driven by entrance excess pressure $KV$ obeys $\rho L\dot{u}=KV$ and $\dot{V}=-\pi a^{2}u$. From rest, the first zero of $V$ occurs after a quarter harmonic period, independent of its initial volume. The maximum speed is proportional to that initial volume. Entrance losses, a reservoir kinetic head or independently prescribed initial flow change this ideal model.

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.

With the streamfunction convention $(u,v)=({\psi }_y,-{\psi }_x)$, constant scalar vorticity gives $\mathbf{u}\times \mathbf{\omega }=-\omega \nabla \psi$. For steady inviscid constant-density flow without body force, this integrates tothroughout a connected fluid region. Reversing the streamfunction convention changes the sign of its term; the declared convention matters.

For steady flow of constant-density fluid in a uniformly rotating frame with conservative body forces, the displayed quantity is constant along each relative streamline. The centrifugal potential contributes the negative quadratic term, and the Coriolis force contributes no work because it is perpendicular to the relative velocity. The result follows by dotting the rotating Euler equations for an inviscid fluid with that velocity.

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.