Pressure, kinetic and potential energy of liquids

As it flows steadily, a liquid can possess three distinct kinds of energy simultaneously: kinetic energy, potential energy, and pressure energy. (i) KINETIC ENERGY: for a mass m of liquid moving with velocity v, $\text{KE}=\tfrac12 mv^2$; the kinetic energy PER UNIT MASS is $\dfrac{\text{KE}}{m}=\tfrac12 v^2$, and (multiplying by density $\rho=m/V$) the kinetic energy PER UNIT VOLUME is $\tfrac12\rho v^2$. (ii) POTENTIAL ENERGY: for a mass m of liquid at height h above some reference level, $\text{PE}=mgh$; the potential energy PER UNIT MASS is $gh$, and the potential energy PER UNIT VOLUME is $\rho gh$. (iii) PRESSURE ENERGY: this is the energy a fluid acquires purely because pressure is applied to push it along. Since $\text{Force}=\text{Pressure}\times\text{Area}$, the work done in pushing a volume V of fluid (cross-sectional area A) through a distance d is $F\times d=(PA)\times d=P(A\times d)=PV$ -- so the PRESSURE ENERGY of a fluid volume V at pressure P is $E_P=PV$. The pressure energy PER UNIT MASS is $\dfrac{E_P}{m}=\dfrac{PV}{m}=\dfrac{P}{\rho}$ (since $m=\rho V$), and the pressure energy PER UNIT VOLUME is simply $\dfrac{E_P}{V}=P$. These three energies -- kine …