Q$1$a: A spherical, rigid, oil droplet with a diameter of $0\mathrm{.}25mm$ and density of $820kg{m}^{-3}$ is settled in air $($assume that air is still$).$ The density and viscosity of air are $1\mathrm{.}2kg{m}^{-3}$ and $1\mathrm{.}81{10}^{-5}$Pas, respectively. The relationship between the Reynolds number and drag force per unit area, $R^{\text{'}},$ is shown in Figure $1.$

In Figure $1,$ the variables on the vertical axis are defined with physical properties, the Reynolds number and the terminal velocity as:

$\frac{R^{\text{'}}}{U_0^2}R{e}^2=\frac{2}{3}\frac{d_p^3({}_p-)g}{{}^2}$

$\frac{R^{\text{'}}}{U_0^2}R{e}^{-1}=\frac{2g}{3{}^2{U}_0^3}({}_p-).$

Here,

$R^{\text{'}}$: drag force per unit area,

$$: fluid density,

$U_0$: terminal velocity $of$ particle,

$d_p$: particle diameter,

${}_p$: particle density,

$g$: gravitational acceleration,

$$: fluid viscosity,

$Re$: Reynolds number.

$g$ : gravitational acceleration,

$$ : fluid viscosity,

Re : Reynolds number.

Answer the following questions:

$($i$)$ Calculate the terminal velocity of the oil droplet in $mm{s}^{-1}.$

$($ii$)$ Determine the drag force acting on the oil droplet at equilibrium.

Figure $1$: Reynolds number versus drag force per unit area.