Bernoulli Principle -
Equation Of Continuity Derivation

The equation of continuity is derived assuming the flow rate is the same everywhere. Therefore, the flow rate is the same at any two points.

dm(a) / dt = dm(b) / dt

where
dm - mass that flows past a point (kilograms)
dt - time interval (seconds)

Consider thin cross-sectional slices of fluid that flow past points a and b during some time interval. The mass of each slice can be replaced by the density and volume.

rho dV(a) / dt = rho dV(b) / dt

where
rho - density at a point (kilograms / meter3)
dV - volume that flows past a point (meters3)

The volume of the slice equals the cross-sectional area times the distance that the slice travels during the time interval.

rho A(a) dx(a) / dt = rho A(b) dx(b) / dt

where
A - cross-sectional area at a point (meter2)
dx - distance traveled (meters)

The ratio of distance over time is simply the velocity. This gives the equation of continuity.

rho A(a) v(a) / dt = rho A(b) v(b) / dt

where
v - velocity past a point (meters / second)