The Equation of Continuity

The Equation of Continuity

In order to describe the motion of a fluid, in principle one might apply Newton’s laws to a particle (a small volume element of fluid) and follow its progress in time. This is a difficult approach. Instead, we consider the properties of the fluid, such as velocity and pressure, at fixed points in space.

In order to simplify the discussion we make several assumptions:

(i) The fluid is non viscous

There is no dissipation of energy due to internal friction between adjacent layer in the fluid.

(ii)The flow is steady

The velocity and pressure at each point are constant in time

(iii)The flow is irrotational:

A tiny paddle wheel placed in the liquid will not rotate.

In rotational flow, for example, in eddies, the fluid has net angular momentum about a given point.

Since fluid does not leave the tube of flow, this mass will later pass through a cylinder of length Δl2 and area A2. The mass in this cylinder is Δm2 = ρ2A2Δl2. The lengths Δl1 and Δl2 are related to the speeds at the respective locations: Δl1 = v1Δt and Δl2 = v2Δt. Since no mass is lost or gained.

Δm ₁ = Δm ₂ , and

ρ1A ₁ v ₁ = ρ ₂ A ₂ v ₂ (12.28)

This is called the equation of continuity. It is a statement of the conservation of mass.

If the fluid is incompressible, its density remains unchanged. This is a good approximation for liquid, but not for gases. If ρ1 = ρ2, the equation (12.28) becomes,

A ₁ v ₁ = A ₁ v ₂ (12.29)

The product Av is the volume rate of flow (m3/s). Figure(12.29) shows a pipe whose cross section narrows. From equation (12.29) we conclude that the speed of a fluid is greatest where the cross-sectional area is the least. Notice that the streamlines are close together where the speed is higher.