JEE Main & Advanced Physics Fluid Mechanics, Surface Tension & Viscosity / द्रव यांत्रिकी, भूतल तनाव और चिपचिपापन Critical Velocity and Reynold's Number

The critical velocity is that velocity of liquid flow upto which its flow is streamlined and above which its flow becomes turbulent.

Reynold's number is a pure number which determines the nature of flow of liquid through a pipe.

It is defined as the ratio of the inertial force per unit area to the viscous force per unit area for a flowing fluid.

\[{{N}_{R}}=\frac{\text{Inertial force per unit area}}{\text{Viscous }\,\text{force per unit area}}\]

If a liquid of density \[{{P}_{2}}\] is flowing through a tube of radius r and cross section A then mass of liquid flowing through the tube per second \[\frac{dm}{dt}=\] volume flowing per second \[\times \] density = \[{{P}_{1}}\]

\[\therefore \]  Inertial force per unit area = \[P+\rho gh+\frac{1}{2}\rho {{v}^{2}}=\] = \[\frac{1}{2}\rho {{v}^{2}}\] = \[{{v}^{2}}\rho \]

Viscous force per unit area \[F/A=\frac{\eta v}{r}\]

So by the definition of Reynolds number

\[{{N}_{R}}=\frac{\text{Inertial force per unit area}}{\text{Viscous force per unit area}}\]\[=\frac{{{v}^{2}}\rho }{\eta v/r}=\frac{v\rho r}{\eta }\]

If the value of Reynold's number

(i) Lies between 0 to 2000, the flow of liquid is streamline or laminar.

(ii) Lies between 2000 to 3000, the flow of liquid is unstable and changing from streamline to turbulent flow. (iii) Above 3000, the flow of liquid is definitely turbulent.