Answer
$V=\sqrt {\frac{2a^2\Delta p}{\rho(a^2-A^2)}}$
Work Step by Step
Applying the Bernoulli’s equation between the points B and A, we obtain
$p_1+\frac{1}{2}ρV^2+ρgy=p_2+\frac{1}{2}ρv^2+ρgy$
or, $\frac{1}{2}ρV^2=p_2-p_1+\frac{1}{2}ρv^2$
or, $\frac{1}{2}ρV^2=\Delta p+\frac{1}{2}ρv^2$
Applying the equation continuity between the points B and A, we obtain
$v=\frac{AV}{a}$
Thus
$\frac{1}{2}ρV^2=\Delta p+\frac{1}{2}ρ(\frac{AV}{a})^2$
or, $V^2(\frac{a^2-A^2}{a^2})=\frac{2\Delta p}{\rho}$
or, $\boxed{V=\sqrt {\frac{2a^2\Delta p}{\rho(a^2-A^2)}}}$
