Water is flowing in a pipe as shown in the figure. If the diameter of the pipe at point 1 and point 2 are D and 2D respectively, then find the ratio of the velocity of the water at point 1 to point 2.

CONCEPT:

Continuity equation:

• According to the continuity equation, the mass flow rate at every point in the fluid flow remains constant.

⇒ ρ₁A₁v₁ = ρ2A2v2

• If the flow is incompressible i.e. ρ1 = ρ2,

⇒ A1v1 = A2v2

CALCULATION:

Given d₁ = D and d₂ = 2D

• Area of the pipe at point 1,

\(\Rightarrow A_1=\frac{\pi}{4}{D_1^2}\)

\(\Rightarrow A_1=\frac{\pi}{4}{D^2}\)

• Area of the pipe at point 2,

\(\Rightarrow A_2=\frac{\pi}{4}{D_2^2}\)

\(\Rightarrow A_2=\frac{\pi}{4}{(2D)^2}\)

\(\Rightarrow A_2=4\times\frac{\pi}{4}{D^2}\)

According to continuity equation,

⇒ A1v1 = A2v2

• Hence, option 4 is correct.