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

1. Jul 15, 2012

### 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 [Broken]

here's the final result...

Last edited by a moderator: May 6, 2017
2. Jul 15, 2012

### 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.δr = δr

δr.δz = rcosθ

etc

Last edited: Jul 15, 2012
3. Jul 15, 2012

### 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 cant arrive at the final integral.

in spherical coordinates...

τ =
rr τ 0 ]
θr 0 0 ]
[0 0 0 ]

pδ =
[p 0 0]
[0 p 0]
[0 0 p]

so pδ+τ=
[p+τrr τ 0]
θr p 0]
[0 0 p]

are those steps correct?

4. Jul 15, 2012

### racnna

ooh nevermind...i think i got it now

5. Jul 15, 2012

### Staff: Mentor

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 = Trr δrδr + T (δrδθ + δθδr) + Tθθ δθδθ + 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 = δr $\cdot$ T = Trr δr + T δθ

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 = Trr cosθ - T sinθ

I hope this helps. Study the appendix in BSL.

Last edited by a moderator: May 6, 2017
6. Jul 16, 2012

### racnna

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

7. Jul 16, 2012

### Staff: Mentor

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

8. Jul 16, 2012

### racnna

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

9. Jul 16, 2012

### Staff: Mentor

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

Chet