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Steady spherical continuity equation gives $r^2\rho u=\text{constant}$. Combining this with Euler equations for an inviscid fluid and $c_s^2=dp/d\rho$ gives the displayed equation. Smooth transonic passage requires $u^2=c_s^2$ and ${\Phi }^{\prime }=2c_s^2/r$ simultaneously. The barotropic enthalpy function gives the integral $u^2/2+h+\Phi =\text{constant}$.

The local vector-calculus identityseparates the convective acceleration into a kinetic-energy gradient and a vorticity cross-product contribution. Substitution into the steady Euler equations for an inviscid fluid yields the vorticity form of the Bernoulli equation.

An inviscid flow neglects viscosity in the fluid flow. Its momentum balance is described by the Euler equations for an inviscid fluid.

For perturbations $({\rho }^{\prime },p^{\prime },\mathbf{u})$ about a uniform fluid of density ${\rho }_0$, the linearized continuity equation, momentum equation, and homentropic pressure perturbation relation areFor irrotational flow $\mathbf{u}=\nabla \varphi$, they imply $p^{\prime }=-{\rho }_0{\varphi }_t$ and ${\varphi }_{tt}=c_0^2{\nabla }^2\varphi$.

For a homogeneous pressureless fluid in comoving coordinates, the Euler equation giveswhile the Poisson equation givesTaking the divergence of the first relation and comparing them yieldsThis local fluid derivation describes a homogeneous infinite universe without choosing a physical centre of expansion.