Reynolds Number and Flow Regimes:

The Reynolds number (Re) is a dimensionless quantity used to predict the flow regime of a fluid. It is defined as the ratio of inertial forces to viscous forces in a fluid flow and is given by the formula:

Re = (ρ * V * D) / μ

Where:
- Re is the Reynolds number
- ρ is the density of the fluid
- V is the velocity of the fluid
- D is the characteristic length or diameter of the flow
- μ is the dynamic viscosity of the fluid

In fluid mechanics, the flow regime can be categorized into two main types: laminar flow and turbulent flow.

Laminar Flow:
Laminar flow occurs when the fluid flows smoothly in parallel layers with minimal mixing between the layers. In this regime, the fluid particles move in an orderly manner, following well-defined streamlines. The velocity profiles are symmetrical and the flow is characterized by low levels of turbulence and mixing. Laminar flow is typically observed at low Reynolds numbers.

Turbulent Flow:
Turbulent flow, on the other hand, is characterized by chaotic and irregular motion of fluid particles. The flow becomes highly turbulent and turbulent eddies form, resulting in significant mixing and fluctuations in velocity and pressure. Turbulent flow is observed at high Reynolds numbers.

Transition from Laminar to Turbulent Flow:

The transition from laminar to turbulent flow occurs when the Reynolds number exceeds a critical value. At this critical Reynolds number, the laminar flow becomes unstable and small disturbances in the flow start to amplify, leading to the formation of turbulent eddies.

The critical Reynolds number for the transition from laminar to turbulent flow is influenced by various factors such as the geometry of the flow, surface roughness, and flow conditions. However, for a smooth pipe with no surface roughness, the critical Reynolds number is generally accepted to be around 2000.

Hence, the correct answer is option 'A' (1800), as it is the closest value to the typical critical Reynolds number for the transition from laminar to turbulent flow in a smooth pipe.