If ρ is the density and η is coefficient of viscosity of fluid which flows with a speed v in the pipe of diameter d, the correct formula for Reynolds number Re is:
Reynolds number (\(R_e\)) is a dimensionless quantity used in fluid mechanics to predict the flow patterns in different fluid flow situations. It is used to determine whether the flow will be laminar or turbulent. The Reynolds number is calculated using the formula:
\(R_e = \frac{\rho v d}{\eta}\)
Where:
\(\rho\) is the density of the fluid.
\(v\) is the velocity of the fluid.
\(d\) is the diameter of the pipe.
\(\eta\) is the dynamic viscosity of the fluid.
Let's evaluate each of the provided options:
Option 1: \(R_e = \frac{\eta d }{\rho v}\) - This is incorrect as the viscosity should be in the denominator along with the density, not in the numerator.
Option 2: \(R_e = \frac{\rho v}{\eta d}\) - This is incorrect because the diameter \(d\) should be in the numerator.
Option 3: \(R_e = \frac{\rho v d}{\eta}\) - This is correct as it aligns with the standard formula for Reynolds number.
Option 4: \(R_e = \frac{\eta}{\rho v d}\) - This is incorrect as all terms are misplaced in the formula.
Therefore, the correct formula for the Reynolds number is \(R_e = \frac{\rho v d}{\eta}\). This formula is useful for determining the nature of the fluid flow: if \(R_e < 2000\), the flow is generally laminar; if \(R_e > 4000\), the flow is usually turbulent.
