Barotropic inviscid momentum in gradient form, Bernoulli along streamlines and the irrotational constant

Let $I\subseteq \mathbb{R}$ be a nonempty open interval and $O\subseteq {\mathbb{R}}^3$ open. Let $u:I\times O\to {\mathbb{R}}^3$ and $\rho :I\times O\to (0,\mathrm{\infty })$ be $C^1$, let $p:(0,\mathrm{\infty })\to \mathbb{R}$ be $C^1$ with enthalpy $h({\rho }_0)={\int }_1^{{\rho }_0}{p}^{\mathrm{\prime }}(s)\mathrm{/}sds$, and let $V:I\times O\to \mathbb{R}$ be $C^1$ (Incompressible Newtonian and barotropic inviscid fluid constitutive models). Write $\sigma :=-p(\rho )I_3$ and suppose that, on $I\times O$,

$$\begin{array}{cccc}& {\mathrm{\partial }}_t\rho +\mathrm{div}(\rho u)=0\text{and}\rho D_{t}u=\rho (-\mathrm{\nabla }V)+g,g_i:=\sum _j{\mathrm{\partial }}_j{\sigma }_{ij}\mathrm{.}& & \text{(M)}\end{array}$$

(The local equation (M) is the conclusion of Local momentum balance: $\rho D_{t}u=\rho b+\mathrm{div}\sigma$ from the integral balance on admissible material subregions under its integral balance and continuity hypotheses; here it is the adopted local form of the barotropic inviscid model.)

(i) On $I\times O$,

$D_{t}u=-\mathrm{\nabla }(h(\rho )+V)\mathrm{.}$

(ii) Suppose in addition that the flow is steady: ${\mathrm{\partial }}_{t}u=0$, ${\mathrm{\partial }}_t\rho =0$ and ${\mathrm{\partial }}_{t}V=0$ on $I\times O$. A streamline on an open interval $J\subseteq I$ is a $C^1$ curve $\gamma :J\to O$ satisfying ${\gamma }^{\mathrm{\prime }}(t)=u(t,\gamma (t))$. For every streamline on a nonempty $J$,

$\frac{d}{dt}(h(\rho )+V+\frac{1}{2}\mathrm{\mid }u{\mathrm{\mid }}_2^2)(t,\gamma (t))=0,$

equivalently $u\cdot \mathrm{\nabla }(h(\rho )+V+\frac{1}{2}\mathrm{\mid }u{\mathrm{\mid }}_2^2)=0$ on $I\times O$: the quantity is constant along every streamline.

(iii) If the steady hypotheses of (ii) hold and in addition the spatial velocity gradient $Du(t,x)$ is symmetric at every $(t,x)$ (the irrotational condition used in this collection), then

$\mathrm{\nabla }(h(\rho )+V+\frac{1}{2}\mathrm{\mid }u{\mathrm{\mid }}_2^2)=0\text{on }I\times O,$

so the full derivative of this quantity vanishes and it is constant on every Euclidean ball contained in $I\times O$, hence on each connected component of $I\times O$.

• The steady hypotheses of (ii) and (iii) are part of the statement. A time-dependent or sheared barotropic flow need not satisfy Bernoulli along streamlines. The term “irrotational” in (iii) is defined here by the explicit condition $Du=(Du{)}^{\mathsf{T}}$; no curl convention is needed. No global constant is asserted in (ii): different streamlines may carry different constants.

• Only the stated smoothness and the local equation (M) are used. The physical reading — an inviscid barotropic fluid with conservative body force — is the model branch of Incompressible Newtonian and barotropic inviscid fluid constitutive models; the theorem itself is a conditional consequence of (M).