5.1 Equation of continuity

1.1 Ideal liquid
1.2 Density of a liquid $\left( \rho \right)$
1.3 Relative density of a liquid $(RD)$
1.4 Density of a mixture of two or more liquid
1.5 Density variation with temperature
1.6 Density variation with pressure

2.1 Atmospheric pressure
2.2 Pressure variation with depth
2.3 Measurement of pressure
2.4 Pressure difference in accelerating fluids

5.1 Equation of continuity
5.2 Bernoulli's theorem
5.3 Application of Bernoulli's theorem

7.1 Terminal velocity $\left( {{v_T}} \right)$

8.1 Surface tension
8.2 Variation in surface tension
8.3 Surface energy $(E)$
8.4 Excess pressure

9.1 Capillarity
9.2 Rise of a liquid in a tube of insufficient lengthz

The product of the area of cross-section and the velocity of flow remains constant at every point in the tube when an incompressible and non-viscous liquid flows in stream line motion through a tube of non-uniform cross-section.

Mathematically, $${A_1}{v_1} = {A_2}{v_2} = {A_3}{v_3} = {A_4}{v_4}$$ So, $$Av = {\text{constant}}$$ Also, $${\text{Area }}(A) \propto \frac{1}{{{\text{velocity }}(v)}}$$

The equation of continuity represents the conservation of mass in case of moving fluids.

Let us consider a tube of varying cross-section as shown in the figure.

Let $v_1$ and $v_2$ be the velocities at point $P$ and $Q$ respectively.

Consider an incompressible and non-viscous liquid flows through the tube such that mass of the liquid flowing though point $P$ in a time interval of $\Delta t$ is equal to the mass of liquid flowing through point $Q$ in a time interval of $\Delta t$

Mathematically, $$\begin{equation} \begin{aligned} {m_P}\Delta t = {m_Q}\Delta t \\ \left( {{A_1}{v_1}} \right)\rho \Delta t = \left( {{A_2}{v_2}} \right)\rho \Delta t \\\end{aligned} \end{equation} $$ So, $${A_1}{v_1} = {A_2}{v_2}$$ or $$\frac{{{A_1}}}{{{A_2}}} = \frac{{{v_2}}}{{{v_1}}}$$

As, ${A_1} > {A_2}$ then ${v_2} > {v_1}$

So, the velocity of flow of the liquid is greater in the smaller cross-section.

Therefore, in a steady flow through tube of varying cross-section, the velocity of flow increases when area of cross-section of tube decreases and vice versa.