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

Explanation

The mass flow rate through a pipe, it is 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 constnat with respect to time. Under this condition, the path taken by the fluid particle is a streamline.
Consider a pipe AB of varying cross sectional area `a_1 and a_2` such that `a_1 > 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.
Let `m_1` be the mass of fluid flowing through section A in time `Deltat, m_1 = (a_1v_1Deltat) rho`
Let `m_2` be the mass of fluid flowing through section B in time `Deltat, m_2 = (a_2v_2 Deltat) rho`
For an imcompressible liquid, mass is conserved `m_1 = m_2`
`a_1v_1 Deltat rho = a_2v_2 Deltat rho`
`a_1 v_1 = a_2v_2 implies av = "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 rate remains constant throughout the pipe. In other words , the smaller the cross section, greater will be the velocity of the fluid.