Vectors/Tensors-spherical coordinates. z component of force of fluid on a sphere

[racnna]

i am a chemical engineer but this is fluid mechanics stuff so i figured you physics geniuses would know this stuff

so to find the z component of force exerted by fluid on the surface of the sphere they find the normal force acting on a surface element of the sphere, integrated over the entire surface of the sphere. then dot it with unit vector in z direction. i am STUMPED as to how they found all the dot product in this integral. This is spherical coordinates

http://img716.imageshack.us/img716/6281/forcer.png

here's the final result...
http://img685.imageshack.us/img685/5597/answern.png

please let me know if you need any additional information

 

[tiny-tim]

hi racnna!

(δ and τ are tensors (matrices))

(i don't really approve of the "dot" between the tensor and the vector )

δ.δ_r = δ_r

δ_r.δ_z = rcosθ

etc

 

[racnna]

hey tim you mind walking me through it with a little more detail. i have been performing these dot products for hours hours and still can't arrive at the final integral.

in spherical coordinates...

τ =
[τ_rr τ_rθ 0 ]
[τ_θr 0 0 ]
[0 0 0 ]

pδ =
[p 0 0]
[0 p 0]
[0 0 p]

so pδ+τ=
[p+τ_rr τ_rθ 0]
[τ_θr p 0]
[0 0 p]

are those steps correct?

 

[racnna]

ooh nevermind...i think i got it now

 

[Chestermiller]

i am a chemical engineer but this is fluid mechanics stuff so i figured you physics geniuses would know this stuff

so to find the z component of force exerted by fluid on the surface of the sphere they find the normal force acting on a surface element of the sphere, integrated over the entire surface of the sphere. then dot it with unit vector in z direction. i am STUMPED as to how they found all the dot product in this integral. This is spherical coordinates

http://img716.imageshack.us/img716/6281/forcer.png

here's the final result...
http://img685.imageshack.us/img685/5597/answern.png

please let me know if you need any additional information

This is a response from a fellow chemical engineer. You seem to be using Bird, Stewart, and Lightfoot as your textbook. BSL has an appendix that teaches how to work with second order tensors, like the stress tensor. Here is a brief summary, customized to your problem:

For this problem, the stress tensor is given by

pδ + τ = T = T_rr δ_rδ_r + T_rθ (δ_rδ_θ + δ_θδ_r) + T_θθ δ_θδ_θ + T_$\phi\phi$ δ_$\phi$δ_$\phi$

According to Cauchy's Stress relationship, the stress vector σ (i.e., vectorial force per unit area) acting on the surface of the sphere is equal to the dot product of the stress tensor with a unit normal to the surface (in this case, the unit vector in the radial direction):

σ = δ_r $\cdot$ T = T_rr δ_r + T_rθ δ_θ

The component of the force per unit area in the z direction is obtained by dotting the stress vector on the surface σ with a unit vector in the z direction:

σ $\cdot$ _δz = T_rr cosθ - T_rθ sinθ

I hope this helps. Study the appendix in BSL.

 

[racnna]

hey there fellow ChE ...thanks!...just curious are you in grad school?

 

[Chestermiller]

hey there fellow ChE ...thanks!...just curious are you in grad school?

Not exactly. I'm a 70 year old retiree who used the original edition of BSL, believe it or not, in 1962. The more recent editions of BSL reference a paper I did with my thesis advisor Joe Goddard (still active at UCSD) in the late 1960's.

Chet

 

[racnna]

oh wow...thats amazing. How would you compare the original version with the current version?

 

[Chestermiller]

oh wow...thats amazing. How would you compare the original version with the current version?

I'll answer you in a private message so that we don't compel everyone else to be occupied in our chit chat.

Chet