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The steady Bernoulli equation isAt the regular sonic point, part a givesDefineThe isentropic flow relation $c_s^2\propto {\rho }^{\gamma -1}$ and mass conservation between the stellar surface and the sonic point giveEquating the surface and sonic values of the Bernoulli function therefore yields the required relationAn outflow reaching infinity with positive terminal kinetic energy requires $E>0$, henceAt $\gamma =5/3$ the terminal state is marginal with $E=0$; larger $\gamma$ cannot support the stipulated wind to infinity.

The steady Euler equations for an inviscid fluid areTaking the scalar product with $u$ and usinggivesThusThe Bernoulli function $H$ is constant along each streamline: fluid particles in steady flow exchange pressure, kinetic, and potential energy without changing their total mechanical energy density.

Let $y(t)$ be the downward displacement of the free surface from its initial level, and let $U$ be the speed in the tube. Conservation of volume givesThe free surface and the outlet are both at atmospheric pressure, and their vertical separation is $H+h_0-y$. Applying the Bernoulli equation between them, while retaining the small free-surface speed, givesThereforeThe surface reaches the upper tube end when $y=h_0$. Hence the draining time of a uniform tank through a siphon is

The radial equation can be writtenThus the Bernoulli function is constant on each streamline:The boundary conditions $u\to 0$ and $\rho \to {\rho }_0$ as $r\to \infty$ fix $\mathcal{B}=c_s^2\ln {\rho }_0$.

At the Bondi sonic point, $u_s=c_s$ and $GM/r_s=2c_s^2$. Evaluating the Bernoulli function there givesand henceThe conserved mass accretion rate is thereforeThus

With the dimensionless variables $x=r/r_s$ and $y=u/c_s$, mass conservation and part (c) giveSubstitution into the Bernoulli function, together with $GM/r=2c_s^2/x$, eliminates every dimensional parameter and gives the transcendental equationEquivalently,orThe transonic branch passes through $(x,y)=(1,1)$.

Although the Lorentz force can exchange energy between poloidal and toroidal motion, it does no net work in the frame rotating with a magnetic surface. Dotting steady momentum with the poloidal field and using the mass-loading, isorotation, and angular-momentum invariants combines the magnetic work with the toroidal kinetic term to giveThereforeis constant on each magnetic surface. Hereis the gravitational potential plus the centrifugal potential in the frame corotating at the field-line angular velocity. This is the modified Bernoulli function of the wind.

A steady spherical polytropic flow escaping a point-mass potential conserves the outward mass loss rate and the Bernoulli function $C=u^2/2+c_s^2/(\gamma -1)-GM/r$. For positive terminal speed, a finite-radius smooth sonic point requires $C>0$ and $1<\gamma <5/3$. Its sound speed obeys $c_{s,c}^2=2C(\gamma -1)/(5-3\gamma )$, and evaluating the conserved flux there selects the transonic mass loss rate. Other parameter limits need separate treatment.

Relativistic spherical accretion is steady radial gas inflow in Schwarzschild spacetime, with conserved rest-mass flux and relativistic Bernoulli function. A regular solution passes a finite sonic point outside the event horizon. For an isentropic gas with $\gamma =5/3$ and cold asymptotic sound speed $c_{\infty }\ll c$, the critical radius is approximately $3GM/(4c{c}_{\infty })$ and the leading mass accretion rate is $\pi G^2{M}^2{\rho }_{\infty }/c_{\infty }^3$. Relativistic corrections resolve the zero-radius critical point of the Newtonian $\gamma =5/3$ model without eliminating accretion.

A smooth radial compressible flow with a fixed adiabatic exponent conserves $r^2\rho u$ and $p/{\rho }^{\gamma }$. In a prescribed Newtonian gravitational potential, its Bernoulli function is $u^2/2+c^2/(\gamma -1)+\Phi$, with $c$ the adiabatic sound speed. Eliminating the density derivative yields $(u-c^2/u)u^{\prime }=2c^2/r-{\Phi }^{\prime }$.

A smooth transonic branch in a power-law gravitational potential has $u_{\ast }^2=c_{\ast }^2=A\beta /(2r_{\ast }^{\beta })$ at its sonic point. Substitution into the Bernoulli function gives $\mathcal{B}=c_{\ast }^2[1/(\gamma -1)+1/2-2/\beta ]$. The local sonic-point slope discriminant distinguishes a genuine crossing from a degenerate everywhere-sonic scaling solution.