Water having kinematic viscosity of 0.01 stoke flows at a velocity of 2 m/s in a pipe of 15 cm diameter. For dynamic similarity, the velocity of oil of kinematic viscosity of 0.03 stoke in a pipe of same diameter will bea)0.33 m/sb)0.66 m/sc)2 m/sd)6 m/sCorrect answer is option 'D'. Can you explain this answer?

Question

Answer

To understand the given problem, we need to apply the concept of dynamic similarity. Dynamic similarity refers to the similarity in the flow characteristics between two different fluids or situations. In this case, we are comparing the flow of water and oil in pipes of the same diameter.

Given data:
- Kinematic viscosity of water (ν1) = 0.01 stoke
- Velocity of water (V1) = 2 m/s
- Diameter of the pipe (D) = 15 cm

We need to find the velocity of oil (V2) in a pipe of the same diameter, but with a different kinematic viscosity (ν2 = 0.03 stoke).

The concept of dynamic similarity is based on the Reynolds number (Re), which is a dimensionless quantity that determines the flow regime. It is given by the formula:

Re = (ρ * V * D) / μ

where:
- ρ is the density of the fluid
- V is the velocity of the fluid
- D is the diameter of the pipe
- μ is the dynamic viscosity of the fluid

In this case, we can assume that the density of water and oil are approximately the same, so we can neglect the density term. Now, let's calculate the Reynolds number for water:

Re1 = (ρ * V1 * D) / μ1

Similarly, let's calculate the Reynolds number for oil:

Re2 = (ρ * V2 * D) / μ2

As per the concept of dynamic similarity, the Reynolds number should be the same for both fluids in order to have similar flow characteristics. Therefore, we can set up the following equation:

Re1 = Re2

Substituting the values, we get:

(ρ * V1 * D) / μ1 = (ρ * V2 * D) / μ2

Canceling out the common terms and rearranging the equation, we get:

V2 = (V1 * μ2) / μ1

Now, substituting the given values:

V2 = (2 * 0.03) / 0.01 = 6 m/s

Hence, the velocity of oil in a pipe of the same diameter will be 6 m/s (option D).

Answer

Concept
For Dynamic similarity the ratio of corresponding forces acting at the corresponding points should be the same in the model and prototype, the ratio of these forces are called dimensionless numbers.
Therefore for the dynamic similarity between the model and prototype the dimensionless should also be the same. The models are designed on the basis of dominating forces.
Reynolds Model law is applicable for pipe flows and submarines airplanes, fully submerged bodies etc.
For dynamic similarity the Reynolds number must be the same.
Reynolds Number Re = VD/v
where V is the velocity, D is the pipe diameter, ν is the kinematic viscosity
Calculation:
Given:
ν₁= 0.01 stoke, V₁= 2 m/s, D₁ = D₂ = 15 cm, ν₂ = 0.03 stoke,
Now For dynamic similarity, Reynold’s number would be the same.

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