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

Explanation

Consider a pipe AB of varying cross sectional area `a_(1)` and `a_(2)` such that `a_(1) gt a_(2)`. A non-viscous and incompressible liquid flows steadily through the pipe, with velocities `v_(1)` and `v_(2)` in area `a_(1)` and `a_(2)` respectively, as shown in figure.

let `m_(1)` be the mass of fluid flowing through section A in time `Delta t, m_(1) = (a_(1)v_(1)Delta t) rho`
Let `m_()` be the mass of fluid flowing through section B in time `Delta t, m_(2) = (a_(2)v_(2)Delta t) rho`
For an imcompressible liquid, mass is conserved `m_(1) = m_(2)`
`a_(1)v_(1) Delta t rho = a_(2)v_(2) Delta t rho`
`a_(1)v_(1) = a_(2)v_(2) rArr 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, greater will be velocity of the fluid.