Water is flowing at the rate of 0.05 m³ /s in a smooth pipe of diameter 500 mm and length of 900 m. What will be the nature of the flow? Take the kinematic viscosity of water as 0.02 stokes.

Concept:

Reynold's number:

• It is defined as the ratio of inertia force to the viscous force.
• It is a dimensionless quantity, which is used to determine the type of flow as laminar or turbulent.
• It is given as:

\(Re=\frac{\rho{V}D}{\mu}=\frac{VD}{\nu}\)

Where, V = average velocity of flowing fluid, D = characteristic dimension

\(\rho\) = density of fluid, \(\mu\) = dynamic viscosity of fluid, \(\nu\) = kinematic viscosity of fluid.

• For flow through the pipe,

\(Re<2000\Rightarrow\) Laminar flow

\(2000<{Re}<4000\Rightarrow\) Transition flow

\(Re>4000\Rightarrow\) Turbulent flow

Laminar flow: It is a type of flow in which fluid flow in a well-defined path or streamline and the streamlines are straight and parallel. The fluid flow smoothly in the form of layer over the adjacent layer.

Turbulent flow: It is a type of flow in which fluid flow in a zig-zag way, which results in formation of eddies die to which high loss of energy takes place.

Calculation:

Given:

Q = 0.05 m³/s, D = 500 mm, L = 900 m, \(\nu\) = 0.02 stokes

Then, \(Q=AV=\frac{\pi}{4}D^2V\)

\(V=\frac{4Q}{\pi{D^2}}=\frac{4\times0.05}{\pi\times(0.5^2)}=0.2546\) m/s

Now, \(Re=\frac{VD}{\nu}=\frac{0.2546\times0.5}{0.02\times10^{-4}}=63650\)

\(Re>4000\Rightarrow\) Turbulent flow.

Thus, option (1) is the correct answer.