Pressure drag acting on a sphere

[Zar139]

Homework Statement

I am trying to show that the pressure drag acting on a sphere is 2πμaU by integrating around the surface of the sphere where U is the speed of the fluid the sphere is in, a is its radius, and μ is the viscosity.

Homework Equations

The pressure at a given position r can be written as:
P-P0 = -1.5μUacos(Θ)/r²

The Attempt at a Solution

I have tried integrating the above equation for Θ from 0 to 2π over the surface of the sphere but my answer is 0 which is clearly not the correct answer. I feel like this should be a fairly easy integration but I do not know how else to go about it. Any suggestions would be greatly appreciated.

Thank you!

 

[Chestermiller]

Homework Statement

I am trying to show that the pressure drag acting on a sphere is 2πμaU by integrating around the surface of the sphere where U is the speed of the fluid the sphere is in, a is its radius, and μ is the viscosity.

Homework Equations

The pressure at a given position r can be written as:
P-P0 = -1.5μUacos(Θ)/r²

The Attempt at a Solution

I have tried integrating the above equation for Θ from 0 to 2π over the surface of the sphere but my answer is 0 which is clearly not the correct answer. I feel like this should be a fairly easy integration but I do not know how else to go about it. Any suggestions would be greatly appreciated.

Thank you!

The pressure acts normal to the surface of the sphere at all locations. So, you have to include this directionality in your determination of the drag force.

 

[Zar139]

Does this mean that I should multiply the area element dS by the unit normal vector?

 

[Chestermiller]

Does this mean that I should multiply the area element dS by the unit normal vector?

You definitely have to integrate the forces vectoriallly. How you do this depends on how you want to approach it.