Question

If it takes 5 minutes to fill a 15 L bucket from a water tap of diameter $\dfrac{2}{\sqrt{\pi }}$ 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

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, $\left[ 1L={{10}^{-3}}m \right]$
$R=\dfrac{\rho \nu d}{\eta }$ 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,
$R=\dfrac{\rho \nu d}{\eta }$
Here, $\nu$ is the fluid velocity which can be obtained by following,
$\nu =\dfrac{4Q}{\pi {{d}^{2}}}$
Q is the volume of water flowing out per second which is given by ,
$Q=\dfrac{15L}{5\min }$
$Q=\dfrac{15\times {{10}^{-3}}{{m}^{3}}}{5\times 60s}$ $\left[ 1L={{10}^{-3}}m \right]$ 1 min =60 sec
$Q=\dfrac{3\times {{10}^{-3}}{{m}^{3}}}{60s}$
$Q=5\times {{10}^{-5}}{{m}^{3}}$
Now, d is diameter = $\dfrac{2}{\sqrt{\pi }}$ cm = $\dfrac{2}{\sqrt{\pi }}$ ×10-2m
Reynolds’s number is given by,
$R=\dfrac{\rho \nu d}{\eta }$ = $\dfrac{\rho }{\eta }\dfrac{4Qd}{\pi {{d}^{2}}}$
$R=\dfrac{\rho 4Q}{\eta \pi d}$
Put all the values in above equation
$R=\dfrac{{{10}^{3}}\times 4\times 5\times {{10}^{-5}}}{{{10}^{-3}}\times 3.14\times \dfrac{2}{\sqrt{3.14}}\times {{10}^{-2}}}$ ….. Use $\left[ \pi =3.14 \right]$
= $\dfrac{4\times 5}{\sqrt{3.14}\text{ }\times \text{2}}\times \dfrac{{{10}^{-2}}}{{{10}^{-5}}}$
= $\dfrac{20}{\text{ 2}\times \text{1}\text{.773}}\times {{10}^{3}}$
$R\approx 5500$ , 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,