Euler equations for an inviscid fluid

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

For constant mass density ${\rho }_0$, a frame with constant angular velocity $\mathbf{\Omega }$ absorbs centrifugal acceleration into reduced pressure $\Pi =p/{\rho }_0-|\mathbf{\Omega }\times \mathbf{r}{|}^2/2$. Indeed $\mathbf{\Omega }\times (\mathbf{\Omega }\times \mathbf{r})=-\nabla |\mathbf{\Omega }\times \mathbf{r}{|}^2/2$. Transforming the material acceleration in the Euler equations gives the displayed equation. The Coriolis acceleration remains velocity dependent and cannot be absorbed into a scalar potential. A conservative physical body force can be included by adding its potential to $\Pi$.

A homogeneous fluid in a prescribed gravitational field obeys $D\mathbf{u}/Dt=-\nabla (p/\rho )+\mathbf{g}$ and $\nabla \cdot \mathbf{u}=0$. Multiplying both constant mass density and pressure by the same positive factor preserves the velocity solution, provided initial and boundary data remain compatible. Thus velocity-only observations do not determine density in this model. Independently prescribed pressure gradients or the fluid's own gravity can break this invariance.

The linearized Euler equations retain only first-order perturbations about a specified inviscid fluid state. Together with the linearized continuity equation and a pressure-density constitutive relation, they describe small-amplitude acoustic waves. About a uniform resting state, ${\rho }_0{\partial }_t{\mathbf{u}}^{\prime }=-\nabla p^{\prime }$, ${\partial }_t{\rho }^{\prime }+{\rho }_0\nabla \cdot {\mathbf{u}}^{\prime }=0$, and an isentropic pressure perturbation has $p^{\prime }=c_0^2{\rho }^{\prime }$. Gradients and advection of a moving background must be retained when its variation is not negligible.

For an unmagnetized ideal gas without gravity, the pressure equation gives internal-energy balance ${\partial }_{t}e+{\partial }_x(u_{x}e)=-p{\partial }_x{u}_x$, where $e=p/(\gamma -1)$. The momentum equation gives kinetic-energy balance with source $-u_x{\partial }_{x}p$. Their sum is conservative with the displayed total-energy flux. Tangential velocities contribute to kinetic energy even when all derivatives are in the normal direction. Smooth polytropic flow advects its entropy parameter; a physical shock need not preserve that parameter across the discontinuity.

For a fixed-composition inviscid fluid, specific enthalpy satisfies $dw=Tds+dp/\rho$. Combining this identity with the vorticity cross-product identity gives the displayed equation, with $\mathbf{\omega }=\nabla \times \mathbf{u}$ and $\epsilon =u^2/2+\Phi +w$. It exposes entropy gradients as the thermodynamic source of a mismatch between Bernoulli function gradients and the vorticity term. In a steady adiabatic process, its projection along a streamline makes $\epsilon$ constant there.

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 flow with 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.

Precisely, suppose a fluid of constant positive mass density obeys the Euler equations for an inviscid fluid, its velocity field and pressure are steady, and the body force per unit mass is $-\nabla \Phi$. ThenEvery term has units of energy per unit mass. For a uniform gravitational field, Newtonian gravitational potential is $\Phi =gz$, up to an additive constant. The pressure term represents work by surrounding fluid, so acceleration can exchange pressure work with kinetic energy. Dotting the steady Euler equations with $\mathbf{u}$ gives the displayed directional derivative: this is why constancy follows a streamline. ^[ref]

Continue the horizontal water constriction example from incompressible flow. Assume negligible viscous losses, steady flow and approximately uniform flow at both measurement sections, so the streamline speeds there equal the mean speeds $0.20$ and $0.80\mathrm{m}{\mathrm{s}}^{-1}$. Equal height cancels the Newtonian gravitational potential terms. The ideal pressure drop isThis is the Venturi effect in its ideal limit: the gain in kinetic energy density is supplied by pressure work. Cambridge's treatment uses this relation to connect a measured pressure difference to volumetric flow rate. ^[ref]

The $300\mathrm{Pa}$ result is conditional, rather than a prediction of the actual viscous pipe. Mass conservation fixes $Q$ and the ratio of mean speeds even with viscosity, but nonuniform velocity profiles require an integrated energy balance, and wall friction dissipates mechanical energy. A pump would supply additional work. In particular, constant mean speed in a straight pipe does not imply constant pressure: Poiseuille flow needs a continuing pressure gradient to overcome viscosity. ^[ref]

In ideal steady flow, a contraction of a pipe increases the speed and lowers the pressure. For constant mass density, equal body-force potential at the ends and approximately uniform endpoint speeds, conservation of mass and the Bernoulli equation give $\dot{M}=S_1{S}_2\sqrt{2\rho (p_1-p_2)/(S_1^2-S_2^2)}$ for flow from a wider to a narrower section. Elevation changes contribute a body-force potential difference, and nonuniform profiles or viscous losses require additional information.

// Target: probability-and-statistics.bigb

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 $C(t)=0$.

This is the unsteady potential-flow version of Bernoulli's principle, with a time-dependent potential gauge.

A spherical balloon feeding an inviscid uniform tube plug, with entrance excess pressure $2\gamma /R$, satisfies $L\dot{u}=2\gamma /(\rho R)$ and $4\pi R^2\dot{R}=-\pi a^{2}u$. Eliminating $u$ yields the displayed equation. Its first integral is $R^4{\dot{R}}^2+K{R}^2=C$. From rest at radius $R_0$, $C=K{R}_0^2$; integration gives $t=R_0^2{K}^{-1/2}(\pi /4-\theta /2+\sin (2\theta )/4)$, where $\theta =\arcsin (R/R_0)$. The ideal emptying time is $\pi R_0^2/(4\sqrt{K})$. Independently prescribed initial flow changes $C$ and therefore this time.

For a slender tube of length $L$ supplied from a large reservoir with excess pressure $P$, negligible gravity and entrance inertia, the Unsteady Bernoulli equation gives $L\dot{U}+U^2/2=P/\rho$. A stationary initial liquid column has the displayed speed and volume flux $\pi a^{2}U$ for tube radius $a$. The inertial term controls the early linear acceleration; the kinetic-energy term sets the terminal inviscid speed. This is a leading slender-tube model while the reservoir remains large and the tube full.

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.

In a steady adiabatic inviscid fluid with a conservative body force, the Bernoulli invariant is the displayed specific energy, constant along each streamline. Dotting the Euler equations with velocity and using the specific enthalpy identity gives $\mathbf{u}\cdot \nabla b=0$. In ideal magnetohydrodynamics, magnetic work need not vanish; the planar ideal-MHD Bernoulli invariant includes the corresponding electromagnetic correction.

A slowly varying, steady, inviscid channel flow has constant discharge $Q=uh$ per unit width and the displayed Bernoulli equation. The bed-height function of depth has its maximum at $h_c=(Q^2/g{)}^{1/3}$, where the Froude number is one. At a nondegenerate critical crest, a differentiable profile changes between the subcritical flow and supercritical flow branches. With subcritical upstream squared Froude number $F<1$ and upstream depth $h_1$, the shock-free supercritical downstream depth is $h_1(F+\sqrt{F^2+8F})/4$.

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.

For steady ideal flow through a horizontal contracting nozzle, Bernoulli equation gives inlet gauge pressure $p_0=\rho Q^2(A_1^{-2}-A_0^{-2})/2$. The integral momentum equation gives the fluid's force on the nozzle as $p_0{A}_0-\rho Q^2(A_1^{-1}-A_0^{-1})$ in the outlet direction. Simplification yields the displayed nonnegative magnitude. The external holding force has equal magnitude and the opposite direction.

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