Drag On A Sphere
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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.
• 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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(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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