Critical Velocity Formula

The velocity at which gravity and air resistance are equalized on the object is called the critical velocity of a free-fall object. It is the velocity at which the flow of a fluid transition from streamlined to turbulent. The critical velocity of a liquid depends on various factors such as Reynolds number, viscosity coefficient, the radius of the tube and density of the fluid flowing through the tube. It is denoted by the symbol V_c. Its unit of measurement is m/s and the dimensional formula is given by [M⁰L¹T^-1].

Formula

V_c = R_eη / ρr

where,

V_c is the critical velocity,

R_e is the ratio of inertial force to viscous force, that is, Reynolds number,

η is the coefficient of viscosity,

ρ is the density of the fluid,

r is the radius of tube.

Sample Problems

Problem 1. Calculate the critical velocity of a fluid flowing through a tube of radius 2 m. The density and coefficient of viscosity of the fluid are 1.5 kg/m³ and 2 kg/ms respectively. The value of Reynolds number is 1500.

Solution:

We have,

R_e = 1500

η = 2

ρ = 1.5

r = 2

Using the formula we get,

V_c = R_eη / ρr

= (1500) (2)/ (1.5) (2)

= 3000/3

= 1000 m/s

Problem 2. Calculate the critical velocity of a fluid flowing through a tube of radius 3 m. The density and coefficient of viscosity of the fluid are 2 kg/m³ and 1.5 kg/ms respectively. The value of Reynolds number is 2000.

Solution:

We have,

R_e = 2000

η = 1.5

ρ = 2

r = 3

Using the formula we get,

V_c = R_eη / ρr

= (2000) (1.5)/ (2) (3)

= 3000/6

= 500 m/s

Problem 3. Calculate the Reynolds number of a fluid flowing through a tube of radius 1 m. The density and coefficient of viscosity of the fluid are 3 kg/m³ and 4 kg/ms respectively. The value of critical velocity is 300 m/s.

Solution:

We have,

V_c = 300

η = 4

ρ = 3

r = 1

Using the formula we get,

V_c = R_eη / ρr

=> 300 = R_e (4) / (3) (1)

=> R_e = 900/4

=> R_e = 225

Problem 4. Calculate the Reynolds number of a fluid flowing through a tube of radius 3 m. The density and coefficient of viscosity of the fluid are 5 kg/m³ and 2 kg/ms respectively. The value of critical velocity is 400 m/s.

Solution:

We have,

V_c = 400

η = 2

ρ = 5

r = 3

Using the formula we get,

V_c = R_eη / ρr

=> 400 = R_e (2) / (5) (3)

=> R_e = 6000/2

=> R_e = 3000

Problem 5. Calculate the viscosity coefficient of a fluid flowing through a tube of radius 4 m. The density and Reynolds number of the fluid are 5 kg/m³ and 2800 respectively. The value of critical velocity is 200 m/s.

Solution:

We have,

V_c = 200

R_e = 2800

ρ = 5

r = 4

Using the formula we get,

V_c = R_eη / ρr

=> 200 = (2800) η / (5) (4)

=> η = 4000/2800

=> η = 1.42 kg/ms

Problem 6. Calculate the viscosity coefficient of a fluid flowing through a tube of radius 2 m. The density and Reynolds number of the fluid are 6 kg/m³ and 1000 respectively. The value of critical velocity is 350 m/s.

Solution:

We have,

V_c = 350

R_e = 1000

ρ = 6

r = 2

Using the formula we get,

V_c = R_eη / ρr

=> 350 = (1000) η / (6) (2)

=> η = 4200/1000

=> η = 4.2 kg/ms

Problem 7. Calculate the density of a fluid flowing through a tube of radius 6 m. The coefficient of viscosity and Reynolds number of the fluid are 5 kg/ms and 2500 respectively. The value of critical velocity is 420 m/s.

Solution:

We have,

V_c = 420

R_e = 2500

η = 5

r = 6

Using the formula we get,

V_c = R_eη / ρr

=> 420 = (2500) (5) / 6ρ

=> ρ = 12500/2520

=> ρ = 4.96 kg/m³