We release an oil drop of radius r in air. The density of the oil is 640 kg/m3. C1 and C2 for 1 atmosphere air at 20∘ C are 3.90 × 10−4 (kg/m)/sec and 0.73 kg/m3, respectively.
How small should the oil drop be so that the drag force is dominated by the linear term in the speed (in lectures we called this Regime I). In this regime, the terminal velocity is mg/C1r. [m is the mass of the drop].
r<<
1.768e-4
r^3 << (3*(C_1)^2)/(4*pi*C_2*oil density*g)
r^3 << (3)*(2.30e-4)^2 / (4)(3.14)(0.88)(660)(10)
r << 1.3e-4
To determine the size of the oil drop that would allow the drag force to be dominated by the linear term in the speed (Regime I), we can use the equation for the terminal velocity in this regime, which is given as:
v_terminal = (m * g) / (C1 * r)
Where:
- v_terminal is the terminal velocity of the oil drop
- m is the mass of the oil drop
- g is the acceleration due to gravity
- C1 is the coefficient related to the drag force
- r is the radius of the oil drop
In order for the linear term in the speed to dominate, we want the radius of the oil drop (r) to be much smaller than the other values. Specifically, we want the radius to be significantly smaller than the characteristic length scale in the problem.
In this case, the characteristic length scale would be determined by the values of C1 and C2, which are related to the properties of the air. Since C1 is given in units of (kg/m)/sec and C2 in kg/m^3, we can see that C1 has units of velocity, while C2 has units of density. Therefore, in order for r to be much smaller than C1 and C2, it must be much smaller than their respective length scales.
To find the length scales associated with C1 and C2, we can consider their definitions:
C1 = (m_air / r_air) / tau_air
C2 = rho_air / r_air
Where:
- m_air is the mass of a molecule of air
- r_air is the typical size of an air molecule
- tau_air is the mean time between molecular collisions in air
- rho_air is the density of air
Since C1 has units of (kg/m)/sec, we can see that r_air and tau_air must be on the order of meters, while m_air is on the order of kilograms. Similarly, since C2 has units of kg/m^3, rho_air must be on the order of kilograms per cubic meter.
Therefore, for the oil drop to be in Regime I, where the linear term in the speed dominates, the radius (r) of the oil drop must be much smaller than the characteristic length scales associated with C1 and C2, which are on the order of meters.
In practical terms, this means that the radius of the oil drop should be much smaller than the typical size of air molecules and the mean free path between molecular collisions in air, which are both on the order of meters. Typically, a radius much smaller than 1 micrometer (10^-6 meters) would satisfy this condition.