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<<
To find the radius of the oil drop that will ensure the drag force is dominated by the linear term in the speed (Regime I), we need to solve the inequality r <<.
In this case, "<<" means "much less than," indicating that the radius of the oil drop needs to be significantly smaller than some value.
To determine the value of r that satisfies this condition, we need to calculate the critical radius where the drag force transitions from being dominated by the linear term to the quadratic term.
The drag force on the oil drop can be expressed using Stoke's law as:
F_drag = 6πηrv
Where F_drag is the drag force, η is the viscosity of air, r is the radius of the oil drop, and v is its velocity.
In Regime I, the drag force is dominated by the linear term, which can be expressed as:
F_linear = C1rv
Comparing the two equations, we can set a condition for r such that the linear term dominates:
C1rv >> 6πηrv
Dividing both sides of the inequality by 6πηrv, we get:
C1 >> 6πη
Now, we can substitute the given values in the equation:
3.90 × 10^(-4) (kg/m)/s >> 6π × η
We also know the given value for the density of air, C2 = 0.73 kg/m^3.
At 20°C, the dynamic viscosity of air (η) can be approximated as:
η = C2 * sqrt(T)
Where T is the temperature in Kelvin.
Converting 20°C to Kelvin:
T = 20 + 273.15 = 293.15 K
Substituting the values, we get:
η ≈ 0.73 * sqrt(293.15) ≈ 0.73 * 17.13 ≈ 12.51 kg/(m·s)
Simplifying our condition further:
3.90 × 10^(-4) (kg/m)/s >> 6π × 12.51 kg/(m·s)
Now, we can solve for r:
3.90 × 10^(-4) (kg/m)/s ≈ 6π × 12.51 kg/(m·s) × r
Simplifying further:
r ≈ (3.90 × 10^(-4)) / (6π × 12.51)
Evaluating the expression gives you the value of r that satisfies the condition and ensures the drag force is dominated by the linear term (Regime I).