Continuity equation is based on the principle of conservation of mass. It states, “The mass accumulated is equals the rate of change of mass between the two sections”.

Consider a stream tube as shown in, Fluid flow enters in the tube at section 1-1 and fluid flow exists from the tube at section 2-2.Two sections at ‘ds’ distance apart in a stream tube.

Rate of increase of mass within the body = Mass flow rate enters-Mass flow rate exists

= ρAV - ( ρAV + ∂ / ∂_s (ρAV) d_s )

= - ∂ / ∂_s (ρAV) d_s

Now rate of increase of mass between the two section ∂ / ∂_s mass = ∂ / ∂t ρ.  volume

= ∂ / ∂t ρAd_s

By law of conservation of mass, Equating equations

∂ / ∂t ρAd_s  = - ∂ / ∂_s (ρAV) d_s

∂ / ∂t ρAd_s + ∂ / ∂_s (ρAV) d_s  = 0

∂ / ∂t ρA + ∂ / ∂_s (ρAV)   =  0

This is the general continuity equation for one-dimensional flow based on stream tube concept.

For steady flow        ∂ / ∂t ρA = 0

For incompressible flow ρ is constant,

∂ / ∂_s (AV) = 0

Integrating w.r.t. ds,   AV  = constant
                                      A₁V₁ = A₂V₂

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