Question

A tube of length l and radius R carries a steady flow of fluid whose density is ρ and viscosity η.The velocity v of flow is given by

V = V₀ ( 1 -r²/R²),

where r is the distance of flowing fluid from the axis. Find

(a) Volume of fluid, flowing across the section of the tube, in unit time. 

(b) Kinetic energy of the fluid within the volume of the tube 

(c) The frictional force exerted on the tube by the fluid, and 

(d) The difference of pressures at the ends of the tube.

Answer 1

(a) Let us consider a cylindrical section at a distance of r and having thickness dr. 

The volume of fluid flowing from this section per second

So, the volume of fluid flowing across the section of the tube in unit time

(b) The kinetic energy of the fluid within the volume element of thickness dr 

So, the K.E. of fluid within the tube

Integrating, we get

(c) The viscous drag exerts a force on the tube

(d) The pressure difference ΔP is given by

ΔP = P₂ – P₁ = P

Where P₁ = 0 & P₂ = P

As we know that

P = Force (F)/ area of section of tube