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For initially uniform bubbles with upward speed $V_s(1-\varphi )$, no new gas production and free escape at the top, gas conservation gives flux $V_s\varphi (1-\varphi )$. A clearing shock rises from the impermeable bottom with speed $V_s(1-{\varphi }_0)$ by the Rankine-Hugoniot condition. The displayed mean void fraction averages the remaining uniform region over the whole liquid depth $H$. This assumes the dilute regime and no retained foam layer.

For a quasihydrostatic bubbly mixture with gas void fraction $\alpha$, volume averaging gives the displayed mass density. A body displacing mixture volume $V$ has approximate buoyant force ${\rho }_{\mathrm{mix}}gV$. A floating body of mass $m$ therefore needs $V=m/{\rho }_{\mathrm{mix}}$; gas release can increase its draft until the required displacement exceeds its available watertight volume. This mechanism explains a possible loss of flotation without making sinking inevitable in every gas plume. Upward liquid flow, turbulence and nonuniform bubble distributions require additional hydrodynamic forces beyond this simple mixture model.

Take $z=0$ at the base and assume gas leaves freely at the liquid surface, without being retained as foam. There is no new bubble production after filling. Neglecting the liquid velocity, gas-volume conservation is a scalar conservation lawAn impermeable base has no incoming bubbles. Thus a clear region develops below the uniform bubbly region. The clearing discontinuity satisfies the Rankine-Hugoniot condition, with states $0$ and ${\varphi }_0$:For the dilute regime $0<{\varphi }_0<1/2$, the characteristic speeds obey $j^{\prime }(0)>s>j^{\prime }({\varphi }_0)$, so this is the admissible compressive shock. Before it reaches the surface,The remaining bubbles move upwards at $s$ and escape at the top. Averaging the void fraction over the entire cylindrical volume givesThis bubble clearing in a cylindrical vessel describes bubbles in the liquid, not a hypothetical retained gas layer above it. A closed surface or long-lived foam would require a different upper boundary condition.

A large gas volume can reduce the effective mass density of the water surrounding the hull. In the simple quasihydrostatic model for the buoyancy of an aerated liquid, gas void fraction $\alpha$ givesBy Archimedes' principle, a boat displacing volume $V_{\mathrm{disp}}$ then has buoyant force ${\rho }_{\mathrm{mix}}g{V}_{\mathrm{disp}}$. To support the same boat mass $m$, it needs a larger displaced volume $m/{\rho }_{\mathrm{mix}}$, so its draft increases. If that exceeds the available watertight displacement, it can lose flotation or become swamped:Uneven aeration and strong plume motion can also disturb stability. This explains a possible hazard, not a claim that any bubble plume must sink a boat: upward water flow, turbulence and the distribution of bubbles can exert substantial additional forces.

Choose $x$ vertically upward. Let $G(x,t)$ be the gas-occupied part of a pipe cross-section, of area $\alpha A$. Define the phase mean mass density, mass-weighted velocity, second moment of velocity, and phase mean pressure byThe void fraction $\alpha$ is a volume fraction, not a mass fraction. The gas momentum in a slice of length $dx$ is $A\alpha {\bar{\rho }}_g\bar{v}dx$; its advected momentum flux is $A\alpha {\bar{\rho }}_g\overline{v^2}$. The two end pressures give the net force $-A{\partial }_x(\alpha {\bar{p}}_g)dx$. Include every other vertical force on the gas, including interfacial forces, in $M_{g}dx$. Newton's second law for this fixed slice, followed by division by $Adx$, therefore givesThis is a cross-section averaged phase momentum balance. In particular, mass weighting makes the momentum and its flux exact averages; one must not replace $\overline{v^2}$ by ${\bar{v}}^2$ without an additional closure. With liquid mean velocity $\bar{u}$, constant liquid mass density ${\rho }_{\ell }$, and liquid second moment $\overline{u^2}$, the same calculation givesWith the common pressure $p={\bar{p}}_g={\bar{p}}_{\ell }$, the specified pressure contributions to $M_g,M_{\ell }$ cancel the phase-fraction derivatives from the product rule. The remaining pressure forces per unit mixture volume are $-\alpha p_x$ and $-(1-\alpha )p_x$.

Assume that the bubbles at a given height have the stated radius $a$. Their number per unit mixture volume is $N=3\alpha /(4\pi a^3)$. The drag force on a rising bubble opposes its relative velocity, so multiplication of the single-bubble law by $N$ gives the signed magnitudeGravity acts downward; the drag force on the liquid is the opposite of that on the gas. With no wall force,A bubble retains its mass in the absence of transfer between phases. Consequently $a^3{\bar{\rho }}_g=a_0^3{\rho }_0$ andFor clarity suppress bars on $p,{\rho }_g,v$ in the steady calculation. On a macroscopic variation length $\ell$, gas inertia per unit volume scales like $\alpha {\rho }_g{v}^2/\ell$, whereas the quadratic drag force scales like $\alpha {\rho }_{\ell }{v}^2/a$. Their ratio is $O[({\rho }_g/{\rho }_{\ell })(a/\ell )]$. Thus ${\rho }_g=O({\rho }_0)\ll {\rho }_{\ell }$, together with the scale separation $a\ll \ell$ inherent in this averaging model, justifies the leading-order neglect of gas inertia. Small reference density alone does not justify it in an arbitrarily short or strongly compressed region. Neglect gas weight at the same order, with the liquid fraction not vanishing. For a stationary liquid, summing the two momentum equations givesThe leading gas balance is $0=-\alpha p_x-D_v$. Substitution of the bubble-radius law yieldsThe gas continuity equation, with constant pipe area and no mass exchange, is ${\partial }_t(\alpha {\rho }_g)+{\partial }_x(\alpha {\rho }_{g}v)=0$. Hence the steady gas mass flux $J=\alpha {\rho }_{g}v$ is constant. For $0<\alpha <1$ the pressure equation selects $v>0$, andEvaluate the latter constant at a reference point ${\rho }_g={\rho }_0$, $\alpha ={\alpha }_0$ and divide. The terminal-drift density law for compressible bubbles isThe exponent $3/5$ comes from combining mass conservation with the density dependence of bubble radius; no gas equation of state was needed.

The void fraction is the fraction of a mixture's local volume occupied by gas. For a horizontally uniform bubbly liquid it is the transported concentration in a one-dimensional gas-volume continuity equation. It is distinct from gas mass fraction because gas and liquid have different mass densities.