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Consider steady spherical accretion of a gas obeying a polytropic equation of state in isentropic flow, with fixed background potential $\Phi =-A{r}^{-1/2}$, where $A=16\pi G{\Psi }_0$, with reservoir mass density ${\rho }_0$ and sound speed $c_0$ at infinity. At a regular sonic point, radial momentum balance requires $c_c^2=A/(4\sqrt{r_c})$. The Bernoulli function isSetting $u=c_s=c_c$ at the sonic point givesA finite positive critical sound speed therefore requires $1<\gamma <9/7$. Mass conservation gives $\dot{M}=4\pi r_c^2{\rho }_c{c}_c$, while the constant-entropy gas law gives ${\rho }_c/{\rho }_0=(c_c^2/c_0^2{)}^{1/(\gamma -1)}$. Since $r_c\propto A^2{c}_0^{-4}$ at fixed $\gamma$, these equations yield $\dot{M}\propto {\rho }_0{G}^4{\Psi }_0^4{c}_0^{-7}$. The steep sound-speed dependence comes from both the selected capture scale and the critical flow speed. The background potential is prescribed; gas self-gravity and finite outer boundaries require a different model.

For a background gravitating density ${\rho }_{\mathrm{dm}}={\Psi }_0{r}^{-5/2}$, the potential is $\Phi =-16\pi G{\Psi }_0{r}^{-1/2}$. A steady polytropic inflow has a regular sonic point only for $1<\gamma <9/7$, and its accretion rate scales as $\dot{M}\propto {\rho }_0{G}^4{\Psi }_0^4{c}_0^{-7}$.

A locally steady glacier nose moving at speed $v>0$ has $q=vh$ by mass conservation. At distance $s$ behind its edge, the linear basal drag law then gives $s=(\rho g/\mu v)(h^3/9+h^2/(2\beta ))$. The thin nose has $h\propto s^{1/2}$ from basal sliding; a thicker outer nose has $h\propto s^{1/3}$ from internal deformation. A thick bulk therefore still has a sliding-controlled inner tip. The divergent idealized edge slope limits the immediate validity of the shallow-ice approximation.

For two-layer sedimentation with a compression shock, the clearing, internal and deposit fronts can all meet at one event. The concentration is normalized by the packed value: initially it is $a$ above $z_m$ and $2a$ below, with $0<a<1/2$, in a column of height $H$.

Equating the clearing and deposit trajectories givesRequiring the internal front to pass through the same point givesThis also satisfies particle-volume mass conservation, $h_c=a(H+z_m)$. At $t_c$ the entire particle population has entered the stationary packed deposit and the remaining interface separates that deposit from clear fluid. The special value of $z_m$ is the boundary between the two possible orders of pairwise sedimentation shock interactions.

Immiscible fluids share pore space, with each phase's Darcy flux determined by its pressure, viscosity and relative permeability. Fluid saturation specifies how much pore space each phase occupies. A capillary pressure couples their pressures, while their individual mass conservation laws determine displacement and saturation evolution.

A conservative two-reservoir model separates fast atmospheric exchange from slow ocean adjustment. Let atmospheric carbon-dioxide inventory be $Ip$, where $p$ is atmospheric partial pressure and $I$ is mass per pressure; let ocean dissolved carbon inventory be $M_{o}C$ in moles, with ocean water mass $M_o$ and dissolved inorganic carbon concentration $C$ in moles per mass. Write $p_s=C/K$ for a linear air-sea equilibrium approximation, and take exchange $h(p-C/K)$ positive into the ocean.

Let emissions $v$ and weathering transfer $W$ be carbon-dioxide-equivalent mass rates, $M_{\mathrm{C}{\mathrm{O}}_2}$ the corresponding molar mass, and $bC$ an ocean carbon-removal rate in moles per time. The balances areMultiplying the second by $M_{\mathrm{C}{\mathrm{O}}_2}$ and adding the first gives total-inventory change $v-M_{\mathrm{C}{\mathrm{O}}_2}bC$: exchange and weathering transfer cancel, as required by mass conservation in this chosen model.

If atmospheric relaxation $t_p=I/h$ is short compared with the ocean response, set the first derivative approximately to zero. ThenThe reduced ocean timescale is $t_C=M_o/b$. A sustained emissions increase first raises atmospheric pressure at nearly fixed $C$, then both adjust as the ocean inventory changes. This separation requires $t_p\ll t_C$ and slow forcing relative to $t_p$; the full coupled equations must be retained when those conditions fail. Weathering is represented here as an atmosphere-to-ocean transfer, not automatically as permanent burial.

Velocity divergence measures the local fractional rate of change of a moving material element's volume:Positive values indicate local expansion and negative values local contraction. For a uniform dilation $\mathbf{v}=a\mathbf{x}$ in three dimensions, it equals $3a$; a rigid rotation has zero velocity divergence.

Mass conservation gives $D\rho /Dt=-\rho \nabla \cdot \mathbf{v}$, linking local expansion to decreasing mass density. An incompressible flow has zero velocity divergence, but can still stretch one material direction while compressing another, or shear a parcel without changing its volume. A narrowing steady pipe illustrates that distinction: fluid accelerates axially while transverse contraction balances that stretching so that the total velocity divergence can remain zero.

Viscous-sheet drawing stretches a thin ribbon between rollers. Mass conservation, axial force balance, lateral surface tension, and edge kinematics jointly determine its half-thickness, half-width, and axial speed.

The volumetric flow rate measures volume crossing an oriented surface per unit time:For a planar cross-section of area $A$, define the mean normal velocity $\bar{U}=A^{-1}{\int }_A\mathbf{v}\cdot \mathbf{n}dA$; then $Q=A\bar{U}$ exactly. The units are ${\mathrm{m}}^3{\mathrm{s}}^{-1}$. A nonuniform velocity profile changes the mean, rather than invalidating this definition.

In steady incompressible flow without leaks or branches, mass conservation makes $Q$ the same at every pipe section. In a compressible stream the conserved quantity is mass flow rate, and $Q$ can change as mass density changes. Collecting discharged liquid for a measured interval gives an elementary flow-rate measurement, provided the collection captures the full flow and the volume is interpreted at the appropriate conditions.

Subtract local specific angular momentum times mass conservation from conservation of angular momentum including wind losses S,T. With outward radial mass flux $\mathcal{F}$ and outward viscous torque in an accretion disk $\mathcal{G}=-2\pi \bar{\nu }\Sigma r^3{\Omega }^{\prime }$, this gives $\mathcal{F}{h}^{\prime }+{\mathcal{G}}^{\prime }=2\pi r(Sh-T)$. For Keplerian rotation and $T=qSh$, substitution in mass conservation yields the displayed equation. A wind with q=1 removes local angular momentum but supplies no additional drift term; q>1 can drive accretion even with zero viscosity.

For a Keplerian disk, let $S$ be the total loss through both faces per unit projected area and $\dot{M}>0$ the inward accretion rate. Annular mass conservation gives ${\dot{M}}^{\prime }=2\pi RS$. A circular orbit has specific orbital energy $-GM/(2R)$, so a marginally escaping wind requires additional energy $GM/(2R)$ per unit lost mass. Equating this sink to a fraction $f$ of the two-face viscous dissipation $(9/4)\nu \Sigma {\Omega }^2$ gives the displayed relation. If the wind carries the local specific angular momentum $h=\sqrt{GMR}$, conservation of angular momentum gives $(\dot{M}h-\mathcal{G}{)}^{\prime }=h{\dot{M}}^{\prime }$, with viscous torque in an accretion disk $\mathcal{G}=3\pi \nu \Sigma h$. Eliminating the torque yields $R^2{\dot{M}}^{\prime \prime }+\frac{3}{2}R{\dot{M}}^{\prime }=\frac{3}{2}f\dot{M}$.

For the Zeldovich approximation $\mathbf{x}=\mathbf{q}-b{\nabla }_q\psi$, let ${\lambda }_i$ be the eigenvalues of the potential Hessian, with this sign convention. Before shell crossing, mass conservation gives the displayed density. The first positive eigenvalue reaches $b{\lambda }_i=1$ and collapses one direction, creating a sheetlike cosmological caustic. Extrapolating straight trajectories after crossing lets particles pass through without self-consistent gravitational trapping, so the predicted multistream sheet spreads excessively. Collisionless gravitational evolution keeps matter more tightly bound; the linear displacement extrapolation cannot determine its late thickness reliably.