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Consider steady isentropic flow toward a Newtonian gravitational potential $-GM/r$ through a tube of cross-sectional area $A(r)=C{r}^n$. Write $v>0$ for the inward speed and use a polytropic equation of state with specific-heat ratio $\gamma >1$. Mass conservation and the Euler equations for an inviscid fluid givewhere $c_s$ is the adiabatic sound speed. A regular sonic point therefore has $v_s=c_s$ and $r_s=GM/(n{c}_s^2)$. The Bernoulli equation, matched to a nearly stationary reservoir with sound speed $c_0$, givesFor $n>1/2$, a finite positive sonic point requires $1<\gamma <(2n+1)/(2n-1)$. Differentiating the flow equation at that point, with $x=r_s{v}_s^{\prime }/c_s$, givesIts discriminant is $4n[(2n+1)-(2n-1)\gamma ]$, so the same bound permits real regular slopes. The branch on which the Mach number rises inward selects the negative sign inFor a spherical tube $n=2$, this recovers the $5/3$ threshold of Bondi accretion; for a dipolar flux-tube area, $n=3$ gives $7/5$. The tube approximation must remain valid between the reservoir matching region and the accretor.

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

A vortex of circulation $\Gamma$ distance $d$ from a plane wall has an opposite-sign image across the wall. The wall velocity has magnitude $|\Gamma |d/[\pi (d^2+y^2)]$, giving a pressure reduction $\rho {\Gamma }^2{d}^2/[2{\pi }^2(d^2+y^2{)}^2]$ by the Bernoulli equation. Integrating this deficit gives the displayed force per unit width, directed from the wall toward the fluid and vortex. The uniform ambient pressure is subtracted to isolate the finite vortex-induced force.