Question

With what terminal velocity will an air bubble of diameter $0.8mm$ rise in a liquid of density $900kg{\text{ }}{m^{ - 3}}$ and viscosity $0.1N{\text{ s}}{{\text{m}}^{ - 2}}$ ? What will be the terminal velocity of the same bubble in water? Ignore density of air.

Answer 1

Hint: For this question, you have to know concepts related to terminal velocity which is defined as the maximum velocity attainable by an object as it falls through a fluid. We will use the equation of terminal velocity in terms of radius of the bubble, specific gravity, density of air and viscosity to calculate the problem. We also know that viscosity is a measure of a fluid's resistance to flow.
Formula used:
${V_t} = \dfrac{{2{r^2}(\delta - \sigma )g}}{{9\eta }}$
Where,
$\delta $ is the density of particle,
$\sigma $ is the density of fluid,
$\eta $ is the coefficient of viscosity,
$r$ is the radius of particle and
$g$ is the acceleration due to gravity.

Complete step-by-step solution:
According to the question it is given that the
Diameter of bubble is $0.8mm$
Density of liquid is $900kg{\text{ }}{m^{ - 3}}$ and its viscosity is $0.15N{\text{ }}s{m^{ - 2}}$
We know that terminal velocity is given by
${V_t} = \dfrac{{2{r^2}(\delta - \sigma )g}}{{9\eta }}$
$\because $ no density of air is given $\delta = 0$
Now, putting all the values in above equation:
$ {V_t} = \dfrac{{2{{(0.4 \times {{10}^{ - 3}})}^2} \times 900 \times 9.8}}{{9 \times 0.15}} \\
   = 0.0020m{\text{ }}{s^{ - 1}} \\
  \therefore {V_t} = 0.20cm{\text{ }}{s^{ - 1}} $
And now for water,
$
  \dfrac{{{{({V_T})}_{liq}}}}{{{{({V_T})}_{({H_2}0)}}}} = \dfrac{{{\sigma _1}}}{{{\sigma _2}}} \\
  \therefore {({V_T})_{({H_2}0)}} = {({V_T})_{liq}} \times \dfrac{{{\sigma _1}}}{{{\sigma _2}}} \\
   = 0.002 \times \dfrac{{900}}{{997}} \\
   \Rightarrow {({V_T})_{({H_2}0)}} = 0.18cm{\text{ }}{s^{ - 1}} $
So, the terminal velocity of air bubble in air is $0.20cm{\text{ }}{s^{ - 1}}$ and terminal velocity of air bubble in water is $0.18cm{\text{ }}{s^{ - 1}}$.

Note: We should use the formula correctly without any confusion and also the units. Terminal velocity occurs when the sum of the drag force and the buoyancy is equal to the downward force of gravity acting on the object. Since the net force on the object is zero, the object has zero acceleration and drag depends on the projected area.