5.2 Bernoulli's theorem

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

It states that the sum of pressure energy per unit volume, potential energy per unit volume and kinetic energy per unit volume is same at every point in the tube when an incompressible and non-viscous liquid flows in stream line through a tube of non-uniform cross-section.

Mathematically, Bernoulli's equation is given by, $$P + \rho gh + \frac{1}{2}\rho {v^2} = {\text{constant}}$$

Work done by pressure at point $A$, $$\begin{equation} \begin{aligned} {W_{{P_1}}} = \left( {{P_1}{A_1}} \right)\left( {{v_1}\Delta t} \right) \\ {W_{{P_1}}} = {P_1}{A_1}{v_1}\Delta t \\\end{aligned} \end{equation} $$

Work done by pressure at point $B$, $$\begin{equation} \begin{aligned} {W_{{P_2}}} = - \left( {{P_2}{A_2}} \right)\left( {{v_2}\Delta t} \right) \\ {W_{{P_2}}} = - {P_2}{A_2}{v_2}\Delta t \\\end{aligned} \end{equation} $$

So, total work done by pressure is given by, $$\begin{equation} \begin{aligned} W = {W_{{P_1}}} + {W_{{P_2}}} \\ W = {P_1}{A_1}{v_1}\Delta t - {P_2}{A_2}{v_2}\Delta t\quad ...(i) \\\end{aligned} \end{equation} $$

Potential energy at point $A$, $$\begin{equation} \begin{aligned} {U_A} = mg{h_1} \\ {U_A} = \left( {\rho {A_1}{v_1}\Delta t} \right)g{h_1} \\\end{aligned} \end{equation} $$

Potential energy at point $B$, $$\begin{equation} \begin{aligned} {U_B} = mg{h_2} \\ {U_B} = \left( {\rho {A_2}{v_2}\Delta t} \right)g{h_2} \\\end{aligned} \end{equation} $$

So, total work done by gravity is given by, $$\begin{equation} \begin{aligned} {W_g} = - \Delta U \\ {W_g} = - \left( {{U_B} - {U_A}} \right) \\ {W_g} = {U_A} - {U_B} \\ {W_g} = \rho {A_1}{v_1}\Delta tg{h_1} - \rho {A_2}{v_2}\Delta tg{h_2}\quad ...(ii) \\\end{aligned} \end{equation} $$

Kinetic energy at point $A$, $$\begin{equation} \begin{aligned} {K_A} = \frac{1}{2}m{v^2} \\ {K_A} = \frac{1}{2}\rho {A_1}{v_1}\Delta tv_1^2 \\\end{aligned} \end{equation} $$

Kinetic energy at point $B$, $${K_B} = \frac{1}{2}\rho {A_2}{v_2}\Delta tv_2^2$$

Change in kinetic energy, $$\begin{equation} \begin{aligned} \Delta K = {K_B} - {K_A} \\ \Delta K = \frac{1}{2}\rho \left( {{A_2}{v_2}\Delta tv_2^2 - {A_1}{v_1}\Delta tv_1^2} \right)\quad ...(iii) \\\end{aligned} \end{equation} $$

From work energy theorem we get, $$\begin{equation} \begin{aligned} \sum W = \Delta K \\ {W_P} + {W_g} = \Delta K \\ \left( {\rho {A_1}{v_1}\Delta t - \rho {A_2}{v_2}\Delta t} \right) + \left( {\rho {A_1}{v_1}\Delta tg{h_1} - \rho {A_2}{v_2}\Delta tg{h_2}} \right) = \frac{1}{2}\rho \left( {{A_2}{v_2}\Delta tv_2^2 - {A_1}{v_1}\Delta tv_1^2} \right) \\\end{aligned} \end{equation} $$

As we know, ${A_1}{v_1} = {A_2}{v_2}$ So, $$\begin{equation} \begin{aligned} {P_1} - {P_2} + \rho g\left( {{h_1} - {h_2}} \right) = \frac{1}{2}\rho \left( {v_2^2 - v_1^2} \right) \\ {P_1} + \rho g{h_1} + \frac{1}{2}\rho v_1^2 = {P_2} + \rho g{h_2} + \frac{1}{2}\rho v_2^2 \\\end{aligned} \end{equation} $$ or $$P + \rho gh + \frac{1}{2}\rho {v^2} = {\text{constant}}$$