## 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 R

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

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

Pressure drag F

Also total drag is given by, (refer equation)

F

Equating the Equations

3 π μ D U = C

C

C

The Equation is called as stokes law,

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

C

• 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

Where W = Weight of sphere

F

F

For more help in

(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 UStoke 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 UPressure drag F

_{D}= 1/3F_{D}= π μ D UAlso total drag is given by, (refer equation)

F

_{D}= C_{D}x ½ ρ U^{2}x AEquating the Equations

3 π μ D U = C

_{D}x ½ ρ U^{2}x π/4 D^{2}C

_{D}= 24μ / ρ U DC

_{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**The C_{e}≤ 1000 :_{D}for R_{e}between 5 to 1000 is equal to 0.4**(iv) For 1000 ≤ R**The value of C_{e}≤ 100000 :_{D }in this range is approximately equal to 0.5**(v) For R**The value of C_{e}> 10^{5}:_{D}is equal to 0.2 for R_{e }> 10^{5}.## 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 forceF

_{B }= Buoyant forceFor more help in

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