Continuity equation

The fluid continuity equation is based on the Law of Conservation of Mass and the principle of streamlines.

Consider a tube with a small diameter at one end and a large diameter at the other - like a wind tunnel, or like a funnel, but longer. Because there can be no flow across streamlines, within a tube of expanding diameter in steady flow (invariant with time), the mass flowing through a given cross sectional area must be equal to the mass flowing through every other cross sectional area in the same amount of time. This is known as mass flow, and can be represented as the product of density, velocity, and area - or the derivative of mass with respect to time.

Hence, the continuity equation consists of the relation:

ρ₁A₁V₁ = ρ₂A₂V₂

where ρ is density, A is area, and V is velocity.

For highly incompressible fluids, such as water, density is constant and need not be accounted for in the equation.

The continuity equation is used in physics classes and by aeronautical and marine engineers.

The continuity equation is a statement of the Law of Conservation of Mass applied to fluid flow. It is used often in fluid mechanics to help solve a variety of problems, and is also part of the explanation of the phenomenon of lift.

As a conservation law, it summarizes to "what goes in, must come out". In this case, inside a finite amount of space, any mass flowing in (or out) is balanced by other mass flowing out (or in) or an increase (or decrease) in density inside that space. For example, a box completely full of foam cannot have any more foam put in it unless you either remove some foam or compress it (increasing its density).

The derivation of the full integral form of the equation is long and arduous, and thus will not be repeated here. The final equation is shown below, with time represented by t, density by ρ, the volume of interest V with boundary surface A, and velocity vector v and unit surface normal vector n

∂/∂t ∫∫∫_V ρ dV + ∫∫_A -ρv⋅n dA = 0

The interpretation of this equation is that the time rate of change of mass within the volume (the triple integral) and the mass leaving the volume (double integral, representing mass flux) must sum to zero.

In many practical applications, one or more assumptions is made to simplify the use of the continuity equation. The most common is to assume the flow is a steady-state flow, which means the vector field of velocity does not change with respect to time. This makes the first integral a constant with respect to time, so its time partial derivative is zero, leaving:

∫∫_A -ρv⋅n dA = 0

This means that there is a dynamic equilibrium of fluid mass in the volume. If it is arranged that there is some finite number of inlets and some (possibly different) finite number of outlets for the fluid, then it means that the mass of fluid that goes in the inlets must immediately also go out the outlets, because the mass inside the volume cannot change.

Simplifying further for the case where there is exactly one inlet, labeled 1; exactly one outlet, labeled 2; and both of these are flat surfaces with area A₁ and A₂, respectively, the continuity equation becomes:

ρ₁A₁v₁=ρ₂A₂v₂

This is the most common version used in elementary fluid mechanics problems. It translates to, "The mass of fluid flowing past station one per unit time is the same as the mass of fluid flowing past station two per unit time." A further simplification can be made if the flow is incompressible; this implies that density is constant so ρ₁=&rho₂ and they can be divided off the above equation with impunity.

The equation can also be written in the form of a differential equation, which can be used to work with an infinitesimal volume, in other words, a point. This is derived from the full integral form using the divergence theorem.

∂ρ/∂t + ∇⋅(ρV) = 0