Continuity equation in the context of Intensive and extensive properties

In physics, a conservation law states that a particular measurable property of an isolated physical system does not change as the system evolves over time. Exact conservation laws include conservation of mass-energy, conservation of linear momentum, conservation of angular momentum, and conservation of electric charge. There are also many approximate conservation laws, which apply to such quantities as mass, parity, lepton number, baryon number, strangeness, hypercharge, etc. These quantities are conserved in certain classes of physics processes, but not in all.

A local conservation law is usually expressed mathematically as a continuity equation, a partial differential equation which gives a relation between the amount of the quantity and the "transport" of that quantity. It states that the amount of the conserved quantity at a point or within a volume can only change by the amount of the quantity which flows in or out of the volume.

In physics, charge conservation is the principle, of experimental nature, that the total electric charge in an isolated system never changes. The net quantity of electric charge, the amount of positive charge minus the amount of negative charge in the universe, is always conserved. Charge conservation, considered as a physical conservation law, implies that the change in the amount of electric charge in any volume of space is exactly equal to the amount of charge flowing into the volume minus the amount of charge flowing out of the volume. In essence, charge conservation is an accounting relationship between the amount of charge in a region and the flow of charge into and out of that region, given by a continuity equation between charge density ${\displaystyle \rho (\mathbf {x} )}$ and current density ${\displaystyle \mathbf {J} (\mathbf {x} )}$.

This does not mean that individual positive and negative charges cannot be created or destroyed. Electric charge is carried by subatomic particles such as electrons and protons. Charged particles can be created and destroyed in elementary particle reactions. In particle physics, charge conservation means that in reactions that create charged particles, equal numbers of positive and negative particles are always created, keeping the net amount of charge unchanged. Similarly, when particles are destroyed, equal numbers of positive and negative charges are destroyed. This property is supported without exception by all empirical observations so far.

In electromagnetism, charge density is the amount of electric charge per unit length, surface area, or volume. Volume charge density (symbolized by the Greek letter ρ) is the quantity of charge per unit volume, measured in the SI system in coulombs per cubic meter (C⋅m), at any point in a volume. Surface charge density (σ) is the quantity of charge per unit area, measured in coulombs per square meter (C⋅m), at any point on a surface charge distribution on a two dimensional surface. Linear charge density (λ) is the quantity of charge per unit length, measured in coulombs per meter (C⋅m), at any point on a line charge distribution. Charge density can be either positive or negative, since electric charge can be either positive or negative.

Like mass density, charge density can vary with position. In classical electromagnetic theory charge density is idealized as a continuous scalar function of position ${\displaystyle {\boldsymbol {x}}}$, like a fluid, and ${\displaystyle \rho ({\boldsymbol {x}})}$, ${\displaystyle \sigma ({\boldsymbol {x}})}$, and ${\displaystyle \lambda ({\boldsymbol {x}})}$ are usually regarded as continuous charge distributions, even though all real charge distributions are made up of discrete charged particles. Due to the conservation of electric charge, the charge density in any volume can only change if an electric current of charge flows into or out of the volume. This is expressed by a continuity equation which links the rate of change of charge density ${\displaystyle \rho ({\boldsymbol {x}})}$ and the current density ${\displaystyle {\boldsymbol {J}}({\boldsymbol {x}})}$.

In fluid mechanics, flows in closed conduits are usually encountered in places such as drains and sewers where the liquid flows continuously in the closed channel and the channel is filled only up to a certain depth. Typical examples of such flows are flow in circular and Δ shaped channels.

Closed conduit flow differs from open channel flow only in the fact that in closed channel flow there is a closing top width while open channels have one side exposed to its immediate surroundings. Closed channel flows are generally governed by the principles of channel flow as the liquid flowing possesses free surface inside the conduit. However, the convergence of the boundary to the top imparts some special characteristics to the flow like closed channel flows have a finite depth at which maximum discharge occurs. For computational purposes, flow is taken as uniform flow. Manning's Equation, Continuity Equation (Q=AV) and channel's cross-section geometrical relations are used for the mathematical calculation of such closed channel flows.