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

Explanation

Let us consider a pipe AB of varying cross sectional area `a_(1)anda_(2)` such that `a_(1)gta_(2)`. A non-viscous and incompressible liquid flows steadily through the pipe, with velocities `v_(1)andv_(2)` in area `a_(1)anda_(2)`, respectively as shown in Figure.

Let `m_(1)` be the mass of fluid flowing through section A in time `Deltat,m_(1)=(a_(1)v_(1)Deltat)rho`
Let `m_(2)` be the mass of fluid flowing through section B in time `Deltat,m_(2)=(a_(2)v_(2)Deltat)rho`
For an incompressible liquid, mass is conserved `m_(1)=m_(2)`
`a_(1)v_(1)Deltatrho=a_(2)v_(2)Deltatrho`
`a_(1)v_(1)=a_(2)v_(2)rArrav="constant"`
Which is called the equation of continuity. It is based on conservation of mass in the flow of fluids. In general, av = constant.