Question

Derive an expression for critical velocity of a liquid prior to turbulent flow using method of dimensions

Answer 1

Dear Student,

Critical velocity is the velocity of a liquid flow upto which its flow is streamlined and after which its flow becomes turbulent. The critical velocity depends on coefficient of viscosity 'n', density of the liquid 'r', and radius of the tube (r).

η = coefficient of viscosity

ρ = density of fluid

r = radius of tube.
Dimensional Formula of coefficient of viscosity (η) = M¹L^-1T^-1
Dimensional Formula of Density of fluid (ρ) =ML^-3T⁰
Dimensional Formula of radius (r) = M⁰L¹T⁰

$$\begin{gathered} v=k\left[{\eta }^a{r}^b{\rho }^c\right] \\ \left[L{T}^{-1}\right]={\left[M{L}^{-1}{T}^{-1}\right]}^a{\left[L\right]}^b{\left[M{L}^{-3}\right]}^c \\ comparethepowers \\ -a+b-3c=1 \\ a+c=0 \\ -1=-a \\ a=1 \\ c=-1 \\ b=-1 \\ criticalvelocityv=\frac{k\eta }{r\rho }herekisreynoldsnoifwereplacerwithdiameterD \\ Regards \end{gathered}$$