Question

Obtain an equation of continuity for a flow of fluid on the basis of conservation of mass.

Answer 1

The mass flow rate through a pipe, it is necessary to assume that the flow of fluid is steady, the flow of the fluid is said to be steady if, at any given point, the velocity of each passing fluid particle remains constant with respect to time.

A streamlined flow of fluid through a pipe of varying cross-sectional area

Under this condition, the path taken by the fluid particle is a streamline.

Consider a pipe AB of varying cross-sectional area a₁ and a₂ such that a₁ > a₂. A non-viscous and incompressible liquid flows steadily through the pipe, with velocities v₁ and v₂ in areas a₁ and a₂, respectively.

Let m₁ be the mass of fluid flowing through section A in time ∆t, m₁ = (a₁v₁∆t)ρ

Let m₂ be the mass of fluid flowing through section B in time ∆t, m₂ = (a₂v₂∆t)ρ

For an incompressible liquid, mass is conserved m₁ = m₂

a₁v₁∆tρ = a₂v₂∆tρ

a₁v₁ = a₂v₂ ⇒ a v = constant

which is called the equation of continuity and it is a statement of conservation of mass in the flow of fluids.
In general, a v = constant, which means that the volume flux or flow rate remains constant throughout the pipe. In other words, the smaller the cross-section, the greater will be the velocity of the fluid.