Consider the flow of a real fluid past a sphere. let D be the diameter of the sphere, U is the velocity of flowing fluid of mass density ρ and viscosity μ .

(i)    If R_e ≤ 0.2 :

When the velocity of flow is very small less than 0.2 then the viscous forces are much more predominant than inertial force. C.G. stokes analyzed theoretically the flow around a sphere under very low velocities, such that Rc < 0.2. According to stokes solution, total drag is,

F_D = 3 π μ D U

Stoke further divided the total drag is given by equation, two-third is contributed by skin frication and one third by pressure difference.

Skin friction drag FDf = 2/3F_D = 2 π μ D U

Pressure drag F_D = 1/3F_D = π μ D U

Also total drag is given by, (refer equation)

F_D = C_D x ½ ρ U² x A

Equating the Equations

3 π μ D U = C_D x ½ ρ U² x π/4 D²

C_D = 24μ / ρ U D

C_D = 24 / R_e

The Equation is called as stokes law,

(ii)    For Rc between 0.2 and 5:

The stokes law is improved by oseen, increasing inertia force.

C_D = 24/R_e [1 + 3/ R_e]

(iii)    For 5 ≤ R_e ≤ 1000 : The C_D for R_e between 5 to 1000 is equal to 0.4
(iv)    For 1000 ≤ R_e ≤ 100000 : The value of C_D in this range is approximately equal to 0.5
(v)    For R_e > 10⁵ : The value of C_D is equal to 0.2 for R_e > 10⁵.

Terminal Velocity of the Body:

•    When the body falls from rest in the atmosphere, its velocity increases due to gravitational acceleration.
•    As the velocity increases, the drag force also increases,
•    When the drag force becomes to the weight of the body
•    The net external force acting on the body becomes zero and the body will move with constant velocity. This constant velocity is called as terminal velocity.

For terminal velocity,

W = F_D + F_B

Where  W = Weight of sphere
            F_D = Drag force
            F_B = Buoyant force

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