Continuity Equation

 The continuity equation is based on conservation of mass. Let’s look at a volume ν with surface S,

which is fixed in space. The mass flow out of this volume B is equal to the decrease of mass inside the

volume C.

The mass flow through a certain area dS is ρV · dS. Since dS points outward, we’re looking at the mass

flowing outward. To find the total mass flowing outward, we just integrate over the surface S, to find

that

B =

ZZ

S

ρV · dS. (1.1)

Now let’s find C. The mass in a small volume dν is ρ dν. The total mass in the volume ν can be found

by a triple integral. But we’re not looking for the total mass, but for the rate of mass decrease. So we

simply take a time derivative of the mass. This gives

C = −

∂

∂t ZZZ

ν

ρ dν. (1.2)

Note that the minus is there, because we’re looking for the rate of mass decrease. (Not increase!) Using

B = C we can find the continuity equation

∂

∂t ZZZ

ν

ρ dν +

ZZ

S

ρV · dS = 0. (1.3)

Since the control volume is fixed, we can pull ∂

∂t within the integral. And by using Gauss’ divergence

theorem, we can rewrite this to

ZZZ

ν

∂ρ

∂t dν +

ZZZ

ν

∇ · (ρV) dν =

ZZZ

ν



∂ρ

∂t + ∇ · (ρV)



dν = 0. (1.4)

Now it may be assumed that, for every small volume dν in the volume ν, the integrand is zero:

∂ρ

∂t + ∇ · (ρV) = 0. (1.5)

Note that in the case of a steady flow ∂ρ

∂t = 0, so also ∇ ·(ρV) = 0. And if the flow is also incompressible,

then ∇ · V = 0. The value ∇ · V occurs relatively often in equations and will be discussed later.