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Drag on a Sphere

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 Re ≤ 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,

FD = 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 = 2/3FD = 2 π μ D U

Pressure drag FD = 1/3FD = π μ D U

Also total drag is given by, (refer equation)

FD = CD x ½ ρ U2 x A

Equating the Equations

3 π μ D U = CD x ½ ρ U2 x π/4 D2

CD = 24μ / ρ U D

CD = 24 / Re

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.

CD = 24/Re [1 + 3/ Re]

(iii)    For 5 ≤ Re ≤ 1000 : The CD for Re between 5 to 1000 is equal to 0.4
(iv)    For 1000 ≤ Re ≤ 100000 : The value of CD in this range is approximately equal to 0.5
(v)    For Re > 105 : The value of CD is equal to 0.2 for Re > 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 = FD + FB

Where  W = Weight of sphere
            FD = Drag force
            FB = Buoyant force

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