A declared smooth open flow: the uniform scaling $\rho ={\rho }_0{e}^{-3\alpha t}$, $u=\alpha x$ satisfies the control-volume identities on fixed and material regions, and elbow port data illustrate the inlet/outlet decomposition

(i) Declared uniform scaling. Fix constants ${\rho }_0>0$, $\alpha \mathrm{\ne }0$ and $p_0$, with $[{\rho }_0]=\mu {\mathrm{\ell }}^{-3}$, $[\alpha ]={\tau }^{-1}$ and $[p_0]=\mu {\mathrm{\ell }}^{-1}{\tau }^{-2}$. On an open interval $I\ni 0$ use uniform scaling, meaning expansion for $\alpha >0$ and contraction for $\alpha <0$, and set $\rho (t)={\rho }_0{e}^{-3\alpha t},u(t,x)=\alpha x,\sigma (t,x)=-p(t,x)\mathrm{I}\mathrm{d},p(t,x)=p_0-\frac{\rho (t){\alpha }^2}{2}\mid x{\mid }^2,b\equiv 0,$ and fix radii $R_0>0$ and $R>0$ with units $\mathrm{\ell }$. Let the material region be a ball $B_{R_B}$ with radius $R_B>\max (R,R_0)$. Define the material motion $\chi (t,X)=e^{\alpha t}X$ on a neighbourhood of $\overline{B_{R_B}}$. More precisely, choose an open ball $O=B_{R_O}$ with $R_O>R_B$ and define $\chi :I\times O\to {\mathbb{R}}^3$ by this formula. Its restriction to $B_{R_0}$ has current radius $R(t)=R_0{e}^{\alpha t}$. Define the fixed-ball control motion by ${\psi }_R(t,X)=X$ for $X\in O$, with control region $B_R$, and the material-ball control motion by ${\psi }_{\mathrm{m}\mathrm{a}\mathrm{t}}(t,X)=e^{\alpha t}X$ for $X\in O$, with control region $B_{R_0}$. Each map is a $C^2$ motion on the open reference domain $O$ and has positive spatial Jacobian. Take the fields on a common open neighbourhood of the closed sweep of $\chi$ and the closed sweeps of these controls. The fields satisfy the adopted balances of The finite-regularity open-flow model: moving-control-region mass and momentum balances, their material and fixed special cases, and the inlet/outlet decomposition reproducing the open-body and rocket laws (the density is spatially uniform, the velocity is $\alpha x$, and the pressure is quadratic).

(ii) Declared elbow control-volume data. Take a material region $B$ with motion $\chi$ and associated spatial velocity field $u$ as in Continuum motions, spatial velocity, material acceleration and swept regions, and a material subregion $B^{\mathrm{\prime }}\subseteq B$ with $C^1$ boundary. On an open reference neighbourhood $O\supset \overline{B^{\mathrm{\prime }}}$, define the fixed control motion $\psi :I\times O\to {\mathbb{R}}^3$ by $\psi (t,X)=X$; its control region is $C_t=C:=B^{\mathrm{\prime }}$ and its boundary velocity is $w=0$. Adopt the material balances of The finite-regularity open-flow model: moving-control-region mass and momentum balances, their material and fixed special cases, and the inlet/outlet decomposition reproducing the open-body and rocket laws for this model and the local mass and momentum equations on an open field neighbourhood of the closed control region, as allowed by that theorem's local-equation alternative.

Assume $\rho =1000\mathrm{k}\mathrm{g}{\mathrm{m}}^{-3}$ is constant and the fields are time-independent on that neighbourhood. The bounded $C^1$ boundary of $C$ is partitioned into finitely many $C^1$ boundary pieces, up to surface-measure-zero edges; designate an inlet patch ${\mathrm{\Sigma }}_{\mathrm{i}\mathrm{n}}$, an outlet patch ${\mathrm{\Sigma }}_{\mathrm{o}\mathrm{u}\mathrm{t}}$, and let the remaining pieces form an impermeable remainder ${\mathrm{\Sigma }}_0$. The outward unit normals on the ports are $-\widehat{x}$ and $+\widehat{y}$. On the ports take $u=q_{\mathrm{i}\mathrm{n}}\widehat{x}$ and $u=q_{\mathrm{o}\mathrm{u}\mathrm{t}}\widehat{y}$, respectively, where each $q$ is a nonnegative, nonzero $C^1$ profile supported compactly inside its port, with units $\mathrm{\ell }{\tau }^{-1}$; take $u\cdot \nu =0$ on the remainder. These traces vanish near the port edges and are compatible with a $C^1$ field and continuous boundary normal. For either port define $Q:={\int }_{\mathrm{\Sigma }}q\mathrm{d}S>0,v_{\mathrm{e}\mathrm{f}\mathrm{f}}:=\frac{{\int }_{\mathrm{\Sigma }}{q}^2\mathrm{d}S}{Q},A_{\mathrm{e}\mathrm{f}\mathrm{f}}:=\frac{Q^2}{{\int }_{\mathrm{\Sigma }}{q}^2\mathrm{d}S},$ so $Q=A_{\mathrm{e}\mathrm{f}\mathrm{f}}{v}_{\mathrm{e}\mathrm{f}\mathrm{f}}$, the mass rate is $\rho Q$, and the flux-averaged velocity is $v_{\mathrm{e}\mathrm{f}\mathrm{f}}$ in the port direction. Declare $A_{\mathrm{e}\mathrm{f}\mathrm{f},\mathrm{i}\mathrm{n}}={10}^{-4}{\mathrm{m}}^2$, $A_{\mathrm{e}\mathrm{f}\mathrm{f},\mathrm{o}\mathrm{u}\mathrm{t}}=5\times {10}^{-5}{\mathrm{m}}^2$ and $v_{\mathrm{e}\mathrm{f}\mathrm{f},\mathrm{i}\mathrm{n}}=4\mathrm{m}{\mathrm{s}}^{-1}$; the outlet effective speed is unknown. The effective areas are defined by these integrals, not the geometric patch areas. Assume the time-independent velocity field is $C^1$ on the common neighbourhood. Set the body-force field to $b=0$ and assume the stress field is $C^1$ on the same neighbourhood. These are conditional model premises for a balance calculation; no existence claim for a boundary-value solution or decomposition of the boundary stress is made. The calculation determines only the net external resultant required by the steady balance.

• What the exact solution does and does not solve. The uniform scaling satisfies the local balances and therefore the control-volume identities; it does not satisfy any particular boundary condition on a wall, because the velocity field is defined on all of space. The material ball is a material region of the declared motion, consistent with Continuum motions, spatial velocity, material acceleration and swept regions.

• Boundary forces in the elbow. The number $3.578\mathrm{N}$ is the magnitude of the net external resultant on the fluid required by the adopted steady balance. Equal inlet and outlet pressures and inviscid-wall language would not by themselves determine the wall pressure traction; no pressure-free flux force or device force is isolated here. The example illustrates the inlet/outlet decomposition and is not a design formula.