Conservation law

A conservation law equates local time change of a density with the divergence of its flux.

For a scalar conservation law $u_t+q(u{)}_x=0$, an entropy condition restricts its weak solutions to physically admissible ones. For every convex entropy function $\eta$, define an entropy flux by $Q^{\prime }={\eta }^{\prime }{q}^{\prime }$. The displayed inequality is imposed in the distributional sense. Positive diffusion explains this choice: for $u_t+q(u{)}_x=ϵ[D(u)u_x{]}_x$ with $D>0$, the entropy balance is $\eta (u{)}_t+Q(u{)}_x=ϵ[{\eta }^{\prime }(u)D(u)u_x{]}_x-ϵ{\eta }^{\prime \prime }(u)D(u)u_x^2$. The last term is nonpositive, and an appropriate vanishing-diffusion limit yields the entropy inequality. For a strictly concave flux and a single shock, this selects states with incoming characteristics, rather than a discontinuity whose characteristics spread outward.

A flux records transport of a quantity through a surface per unit time. A flux density $\mathbf{j}$ gives total transport rate ${\int }_S\mathbf{j}\cdot \mathbf{n}dS$, where $\mathbf{n}$ is the unit normal vector. A conservation law relates its divergence to the local change in the quantity's density.

Fick's laws describe transport down a concentration gradient and the resulting change of concentration with time. Combining Fick's first law with a conservation law gives the diffusion equation.

A diffusive flux transports a density $u$ down its gradient according to Fick's first law. For scalar diffusion coefficient $D>0$, $\mathbf{j}=-D\nabla u$ and the conservation law $u_t+\mathrm{div}\mathbf{j}=0$ gives a diffusion equation. If $D$ depends on $u$, this becomes a nonlinear diffusion equation.

A density and flux satisfy a differential balance whose spatial integral is conserved when the boundary flux vanishes. Rapid decay provides this boundary condition for dispersive equations on the real line.

Volume conservation states that the volume of incompressible material can change only through boundary fluxes or specified sources and sinks. In a one-dimensional shallow layer of depth $h$ and depth-averaged speed $u$, it has the form $h_t+(hu{)}_x=s$.

Mass flux is the mass crossing a unit area per unit time. In a one-dimensional flow it is $\rho u$. Integrating its normal component over a surface gives the mass flow rate.

The mass loss rate is the positive rate at which a body loses mass. For a steady spherical stellar wind, the continuity equation gives $\dot{M}=4\pi r^2\rho u_r$, constant across spheres enclosing the source. It is positive for outward radial flow; an accretion convention often uses a separate positive inward rate.

Momentum flux includes advected momentum and stress. In a one-dimensional inviscid flow it is $p+\rho u^2$.

A steady uniform incompressible jet has mass flow rate $\rho Au$, momentum flux $\rho A{u}^2$ and kinetic-energy flux $\rho A{u}^3/2$. These use the area and velocity at the same cross section. A nonuniform profile instead requires integrating powers of local speed; using mean speed in a cubic flux without a profile correction is generally wrong.

Energy flux is the energy crossing a unit area per unit time. For a one-dimensional inviscid gas with internal-energy density $W$, it is

The continuity equation expresses local conservation as time change plus flux divergence equal to zero.

Concentration measures the amount of a constituent per unit amount of a mixture. An advected concentration changes when sources, sinks, diffusion, or entrainment alter the constituent relative to the carrier material.

The mass of a specified constituent divided by the total volume of a sample.

An equilibrium concentration is a constant concentration satisfying the fixed point equations of a specified dynamical model. Balance between production and loss can maintain such a value without implying thermodynamic chemical equilibrium.

Protein concentration specifies protein amount per sample volume, commonly as mass concentration or molar concentration. A molar value must state the counted entity, such as chains or oligomeric particles, and quantification must account for contaminants and optical properties.

Salinity measures dissolved salt content. A transport model must specify its units; salt mass fraction is one convenient convention. A dilute solution's ice liquidus is often approximated by $T_L=T_m-mC$. Salt-free ice has zero salinity, and the moving interface's salt balance distinguishes salt rejection during freezing from dilution during melting.

The conservation law for salt in a reservoir: the rate of change of its salt content equals inflowing minus outflowing salt transport and any explicit salt sources. Freshwater addition changes salinity without directly adding salt. For a well-mixed fixed-volume reservoir, salt content is proportional to its volume times salinity.

The salt forcing used to represent freshwater exchange while retaining a fixed reference volume. Removing freshwater at volume flux $E$ is represented by a salt input $E{S}_0$, and adding freshwater by the opposite salt forcing, with reference salinity $S_0$. Replacing the actual salinity in a compensating-flow salt budget by $S_0$ is an approximation when salinity contrasts are small.

Molar concentration is amount of substance of a named chemical species per final chemical solution volume: $c=n/V$. State whether the count describes dissolved species or a conserved analytical concentration.