Derive the equation of continuity for the steady flow of an incompressible fluid.

(N/A) In a steady flow,if we draw streamlines for every particle,they form a tube-like structure known as a tube of flow. No fluid particle enters or leaves this tube through its sides.
The figure shows a tube of flow with points $P$,$R$,and $Q$.
Let $v_{P}$,$v_{Q}$,and $v_{R}$ be the velocities of the fluid particles at points $P$,$Q$,and $R$ respectively,and let $A_{P}$,$A_{Q}$,and $A_{R}$ be the corresponding cross-sectional areas.
Since the flow is steady,the mass of fluid passing through any cross-section in a given time interval $\Delta t$ must be constant.
The mass of fluid $m_{P}$ passing through cross-section $A_{P}$ with velocity $v_{P}$ in time $\Delta t$ is given by:
$m_{P} = A_{P} v_{P} \Delta t \rho$,where $\rho$ is the density of the incompressible fluid.
Similarly,the masses of fluid passing through $Q$ and $R$ in the same time interval $\Delta t$ are:
$m_{Q} = A_{Q} v_{Q} \Delta t \rho$ and $m_{R} = A_{R} v_{R} \Delta t \rho$.
Since the fluid is incompressible and the flow is steady,the mass entering the tube must equal the mass leaving it. Therefore:
$m_{P} = m_{R} = m_{Q}$
Substituting the expressions:
$A_{P} v_{P} \Delta t \rho = A_{R} v_{R} \Delta t \rho = A_{Q} v_{Q} \Delta t \rho$
Canceling common terms $\Delta t$ and $\rho$:
$A_{P} v_{P} = A_{R} v_{R} = A_{Q} v_{Q}$
This is the equation of continuity for steady flow,which represents the law of conservation of mass for an incompressible fluid.
In general,$A v = \text{constant}$,which implies $v \propto \frac{1}{A}$.
The product $Av$ represents the volume flux or flow rate,which remains constant throughout the tube of flow. At narrower portions,the streamlines are closely spaced,indicating higher velocity,while at wider portions,the streamlines are widely spaced,indicating lower velocity.