Reynold’s Number:

Experimentally, it is found that the critical velocity \( v_c \) of a liquid depends upon its coefficient of viscosity \( \eta \), density of the liquid \( \rho \), and the diameter \( r \) of the cylindrical tube into which the liquid flows.

Therefore, \( v_c=R{\eta}^a{\rho}^b{r}^c \)

where, \( R \) is the dimension less constant called Reynold’s number, a,b,c are dimensionless number to which \( \eta \), \( \rho \) and \( r \) are raised respectively.

Writing the dimensions of the above quantities, we get

\( [LT^{-1}]={[ML^{-1}]T^{-1}]}^a\cdot{[ML^{-3}]}^b\cdot{[L]}^c \)

Equating the power of [L], [M], [T] respectively, we get

\( 1=-a-3b+c \\ 0=a+b \\ -1=-a \)

Therefore, \( a=1 \), \( b=-1 \) and \( c=1+a+3b=-1 \).

Therefore, \( v_c=R\eta{\rho}^{-1}r^{-1} \)

or, \( \displaystyle{R=v_c\frac{\rho{r}}{\eta}} \)

So the experiment shows that there is a combination of \( \rho \), \( \eta \), \( v_c \) and \( r \) by the relation \( R=v_c\frac{\rho{r}}{\eta} \) which is a dimensionless number, whose value determines the nature of motion, – streamline or turbulent.

\( R \) is Reynold’s number.

Significance of Reynold’s number:

The Reynold’s number determines the nature of the motion of the liquid and when this number attains a certain value the flow of liquid changes from streamline flow to turbulent flow. The value of \( R \) is approximately 1000 for the narrow tubes. If the value of \( R \) lies between 0 and 2000, then the motion is streamline motion. If \( R>3000 \), then the motion is turbulent. If the value of \( R \) lies between 2000 and 3000 then we can conclusively predict about its nature, i.e, transition region, in which the motion may change at any time from streamline to turbulent motion.

The critical velocity of a liquid \( v_c= \frac{R\eta}{\rho{r}} \) is directly proportional to its viscosiy, inversely proportional to its density and inversely proportional to the radius of the tube. So narrow tube, low density and high viscosity help in producing orderly or streamline motion and wise tube, high density and low viscosity tend to produce turbulent motion.