Continuity Equation:

A continuity equation is a differential equation that describes the conservative transport of some kind of quantity. Since mass, energy, momentum, and other natural quantities are conserved, a vast variety of physics may be described with continuity equations. Continuity equation is an equation obeyed by any conserved, indestructible quantity such as mass, electric charge, thermal energy, electrical energy, or quantum-mechanical probability, which is essentially a statement that the rate of increase of the quantity in any region equals the total current flowing into the region. It is also known as equation of continuity. Continuity equation is defined as the mean velocities at all cross sections having equal areas are then equal, and if the areas are not equal, the velocities are inversely proportional to the areas of the respective cross sections. The equation of continuity is a statement of mass conversion. The Law of Conservation of Mass states that mass can be neither created nor destroyed. The equation of continuity states that for an incompressible fluid flowing in a tube of varying cross-section, the mass flow rate is the same everywhere in the tube. The mass flow rate is simply the rate at which mass flows past a given point, so it's the total mass flowing past divided by the time interval. The equation of continuity can be reduced to:

ρ₁A₁ν₁ = ρ₂A₂ν₂,( ρ = density, A = cross-sectonal area, ν = velocity )

This equation is called the continuity equation for steady one-dimensional flow. For a steady flow through a control volume with many inlets and outlets, the net mass flow must be zero, where inflows are negative and outflows are positive. The continuity equation can also be stated in any steady state process as the rate at which mass enters a system is equal to the rate at which mass leaves the system. In electromagnetic theory, the continuity equation can either be regarded as an empirical law expressing charge conservation which can be derived as a consequence of Maxwell's equations. It states that the divergence of the current density is equal to the negative rate of change of the charge density.