Fluid Equation of continuity

Equation of continuity

Consider a non-viscous liquid in streamline flow through a tube AB of varying cross section as shown in Fig.  Let a1 and a2 be the area of cross section, v₁ and v₂ be the velocity of flow of the liquid at A and B respectively.

Volume of liquid entering per second at A = a₁v₁.

If ρ is the density of the liquid, then mass of liquid entering per second at A = a₁v₁ρ.

Similarly, mass of liquid leaving per second at B = a₂v₂ρ

If there is no loss of liquid in the tube and the flow is steady, then mass of liquid entering per second at A = mass of liquid leaving per second at B

(i.e) a₁v₁ρ = a₂v₂ρ or a₁v₁ = a₂v₂ i.e. av = constant

This is called as the equation of continuity. From this equation

v α _a¹ .

i.e. the larger the area of cross section the smaller will be the velocity of flow of liquid and vice-versa.

Total energy of a liquid

A liquid in motion possesses pressure energy, kinetic energy and potential energy.

(i) Pressure energy

It is the energy possessed by a liquid by virtue of its pressure.

(ii) Kinetic energy

It is the energy possessed by a liquid by virtue of its motion.

If m is the mass of the liquid moving with a velocity v, the kinetic energy of the liquid = ½ mv²

(iii) Potential energy

 It is the energy possessed by a liquid by virtue of its height above the ground level.

 If m is the mass of the liquid at a height h from the ground level, the potential energy of the liquid = mgh