Question

Water flows through a horizontal tubes as shown in figure (13-E9). If the difference of heights of water column in the vertical tubes is $2cm$, and the areas of cross section at $A$ and $B$ are $4c{m}^2$ and $2c{m}^2$ respectively, find the rate of flow of water across any section.

Answer 1

Let the velocity of flow at the two sections be $v_1$ and $v_2$ respectively. Then by equation of continuity we have: $A_1{v}_1=A_2{v}_2$

$\Rightarrow 4v_1=2v_2\to \left(1\right)$

Also applying Bernoulli's equation at the two sections we have:

$\frac{1}{2}\rho v_1^2+P_1=\frac{1}{2}\rho v_2^2+P_2$

also $P_1-P_2=\rho g(h_2-h_1)=2\rho g$

$\Rightarrow \frac{1}{2}\rho (v_2^2-v_1^2)=2\rho g$

$v_2^2-v_1^2=4g\to \left(2\right)$

From equation $1$ and $2$:

$(2v_1{)}^2-v_1^2=4g$

$\Rightarrow 3v_1^2=4g$ or $v_1=\sqrt{\frac{4g}{3}}$ and $v_2=2\sqrt{\frac{4g}{3}}$.