Question

What will be the critical velocity of water in a tube of radius 1m, if the coefficient of viscosity of water is 1793 poise and Reynolds’s number is 1000?
$\begin{array}{rl}& \text{A}\text{. }7.93cm/s\\ & \text{B}\text{. }17.93\times {10}^{-2}cm/s\\ & \text{C}\text{. }17.93\times {10}^{-4}cm/s\\ & \text{D}\text{. }17.93\times {10}^{-3}cm/s\end{array}$

Answer 1

Hint: Use the formula of critical velocity. The physical quantity, critical velocity depends on the radius of the tube. Coefficient of viscosity, Reynolds number and density of the liquid. Units given in options are in the C.G.S. system. Therefore convert a unit of all quantity in the C.G.S. system.

Formula used:
The formula of critical velocity is given by,
$V_c={\displaystyle \frac{\mu N_r}{\rho R}}$
where,
${\text{V}}_c$ represents the critical velocity
$\text{R}$ represents the radius of the tube
$\rho$ represents the density of the liquid
${\text{N}}_r$ represents Reynolds’s number,
and $\mu$ represents the coefficient of viscosity

Complete step by step answer:
Critical velocity is the speed and direction at which the flow of water or liquid through a tube changes from smooth also called as laminar to turbulent which is given by,

$V_c={\displaystyle \frac{\mu N_r}{\rho R}}$
where,
${\text{V}}_c$ represents the critical velocity
$\text{R}$ represents the radius of the tube
$\rho$ represents the density of the liquid
${\text{N}}_r$ represents Reynolds’s number,
and $\mu$ represents the coefficient of viscosity

Given that coefficient of viscosity=1793
Reynolds number=1000
Radius=1m
$\begin{array}{rl}& \mu =1793poise=179.3N.s/m^2,\\ & N_r=1000\\ & R=1\\ & \rho =1000kg{m}^{-3}\end{array}$
Put values in the above equation
$\begin{array}{rl}& V_c={\displaystyle \frac{\left(179.3\times 1000\right)}{1000\times 1}}\\ & V_c=17.9\times {10}^{-3}cm{s}^{-1}\end{array}$
Critical velocity of water in a tube of radius 1m is $17.9\times {10}^{-3}cm{s}^{-1}$.
Therefore option (D) is the correct option.

Additional Information:
Reynolds number is a pure number, so it has no unit and dimensions. It is denoted by N. if N lies between 0 to 2000 then the flow of liquid is laminar.
If N lies between 2000 to 3000, then laminar flow may change to turbulent flow.
If N is greater than 3000 then the flow of liquid is turbulent.
In the question unit of coefficient of viscosity is poise which is defined as if a force of one dyne per square centimeter exists between two layers of fluid with unit velocity gradient, its coefficient of viscosity is one poise.

Note:
Calculating critical velocity depends on multiple variables like Reynolds’s number, the density of the liquid, coefficient of friction, the radius of the tube but it is the Reynolds number that characterizes the flow of the liquid through a tube as either turbulent or laminar. The Reynolds number is a dimensionless variable, meaning it has no units attached to it. The flow of liquid inside a tube remains to streamline, if its velocity is less than a critical value, laminar flow converts into turbulence.