Question

If it takes 5 minutes to fill a 15 L bucket from a water tap of diameter ${\displaystyle \frac{2}{\sqrt{\pi }}}$ cm then the Reynolds number for the flow is ( density of water = ${10}^{3}kg/m^3$ and viscosity of water $=\text{1}{0}^{-\text{3}}$ pa.s) close to
(A) 11,000
(B) 550
(C) 1100
(D) 5500

Answer 1

Hint: Reynolds number is a dimensionless value which is applied in fluid mechanics to represent whether the fluid flow in a duct or part of a body is steady or turbulent. Reynolds number is given by ratio of inertial force to viscous force. Use the following formula, $[1L={10}^{-3}m]$
 $R={\displaystyle \frac{\rho \nu d}{\eta }}$ Here, $\rho \to$ Fluid density
 $v\to$ Fluid velocity
 $\eta \to$ Fluid viscosity
 $d\to$ Diameter or Length of fluid.

Complete step by step solution
We have given,
( $\rho$ ) density of water = ${10}^{3}kg/m^3$
( $\eta$ ) viscosity of water= 10-3 pa.s
We have to find, Reynolds’s number which is given by,
 $R={\displaystyle \frac{\rho \nu d}{\eta }}$
Here, $\nu$ is the fluid velocity which can be obtained by following,
 $\nu ={\displaystyle \frac{4Q}{\pi d^2}}$
Q is the volume of water flowing out per second which is given by ,
 $Q={\displaystyle \frac{15L}{5min}}$
 $Q={\displaystyle \frac{15\times {10}^{-3}{m}^3}{5\times 60s}}$ $[1L={10}^{-3}m]$ 1 min =60 sec
 $Q={\displaystyle \frac{3\times {10}^{-3}{m}^3}{60s}}$
 $Q=5\times {10}^{-5}{m}^3$
Now, d is diameter = ${\displaystyle \frac{2}{\sqrt{\pi }}}$ cm = ${\displaystyle \frac{2}{\sqrt{\pi }}}$ ×10-2m
Reynolds’s number is given by,
 $R={\displaystyle \frac{\rho \nu d}{\eta }}$ = ${\displaystyle \frac{\rho }{\eta }}{\displaystyle \frac{4Qd}{\pi d^2}}$
 $R={\displaystyle \frac{\rho 4Q}{\eta \pi d}}$
Put all the values in above equation
 $R={\displaystyle \frac{{10}^3\times 4\times 5\times {10}^{-5}}{{10}^{-3}\times 3.14\times {\displaystyle \frac{2}{\sqrt{3.14}}}\times {10}^{-2}}}$ ….. Use $[\pi =3.14]$
= ${\displaystyle \frac{4\times 5}{\sqrt{3.14}\times \text{2}}}\times {\displaystyle \frac{{10}^{-2}}{{10}^{-5}}}$
= ${\displaystyle \frac{20}{\text{ 2}\times \text{1}\text{.773}}}\times {10}^3$
 $R\approx 5500$ , This is the approximate value of Reynolds’s number.

Note
Reynolds number formula is used to determine the diameter, velocity and viscosity of the fluid
If Re $\text{ 2}000$ , the flow is called Laminar
If Re $>4000$ , the flow is called turbulent
If 2000 $<$ Re $<4000$ , the flow is called transition.
Here, Re is Reynolds number,