1. Bernoulli’s equation for an ideal fluid flowing steadily is

$P + \rho g h + \frac{1}{2}\rho v^2 = \text{constant}$

where:

• $P$ = pressure energy per unit volume
• $\rho g h$ = potential energy per unit volume
• $\frac{1}{2}\rho v^2$ = kinetic energy per unit volume

2. Check each option:

• Option A: $P+\frac{1}{2}\rho g h+\frac{1}{2}\rho v^2=\text{constant}$ This is incorrect because the gravitational term should be $\rho g h$, not $\frac{1}{2}\rho g h$.

• Option B: $P+mgh+\frac{1}{2}mv^2=\text{constant}$ This is incorrect because Bernoulli’s equation is written per unit volume, so density $\rho$ appears, not mass $m$.

• Option C: $P+\rho g h+\rho v^2=\text{constant}$ This is incorrect because the kinetic term should be $\frac{1}{2}\rho v^2$, not $\rho v^2$.

• Option D: $P+\rho g h+\frac{1}{2}\rho v^2=\text{constant}$ This matches the correct Bernoulli’s equation.

3. Final answer:

Therefore, the correct option is D.