Buoyancy in Differential Forms

[stedwards]

The usual form for tension as a result of the symmetric Cauchy stress tensor is, $$\mathbf{t} = P \mathbf{\hat{n}}$$ or better $$t_i = {P_i}^j n_j$$
Buoyancy would be $$T = \int_{\partial V}{P_i}^j n_j da$$ integrated over a closed surface. I've assumed that the stress tensor $P$, is, in general, non-isotropic. (The tensor can have non-zero, off diagonal components.)

This is an incomplete expression so much as the area orientation is left to be decided. Other weaknesses: The area element is scalar, and the unit normal vector must be unitless if the stress tensor is to have units of force per unit area.

It seem that we should be able to find an antisymmetric tensor, $F_{ij}$, better suited, such that $$T=\int_{\partial V} F_{ij} dx^i dx^j= \int_{\partial V} F$$

Any thoughts, or direction?

 

[Chestermiller]

Why do you call T the buoyancy? T is the force acting on the body, and it is not just buoyancy. And why don't you have a subscript i on T? And the subscript on n in your equation for T should be j.

Also, I don't see any logical reason why T should be expressible in terms of an antisymmetric tensor.

Chet

 

[stedwards]

Why do you call T the buoyancy? T is the force acting on the body, and it is not just buoyancy. And why don't you have a subscript i on T? And the subscript on n in your equation for T should be j.

Also, I don't see any logical reason why T should be expressible in terms of an antisymmetric tensor.

Chet

I corrected the subscript. Why should T have a subscript? T is the total force.

 

[Chestermiller]

I corrected the subscript. Why should T have a subscript? T is the total force.

The integral on the right hand side of your equation correctly gives the component of the total force in the i coordinate direction. To get the magnitude of the total force, you need to contract it with itself and take the square root.

Incidentally, if the system is in hydrostatic equilibrium, the stress tensor is isotropic. Otherwise, your integral will include drag forces on the object (in addition to buoyancy).

Chet

 

[stedwards]

The integral on the right hand side of your equation correctly gives the component of the total force in the i coordinate direction.

Chet

Right, thanks, the free index got lost in the mix. Now it seems I can replace the the integrand with the tensor $Q$, with components $Q_{[ij]k}$, but unfortunately it is only antisymmetric in two indices. But it's much cleaner, and the math better matches the physics (sans further errors).

 

[fzero]

The notation is a bit nonstandard to me, but these notes, particularly sect. 15 explain that the Cauchy stress tensor can be described by a vector-valued 2-form. I think the description on pg 13 of these slides is a bit illuminating, too, as the factor of $n_b da$ is explictly related to the Hodge dual of the covector $dx^b$ related to $n_b$. Perhaps connecting the two approaches will be helpful to you.

 

[stedwards]

The notation is a bit nonstandard to me, but these notes, particularly sect. 15 explain that the Cauchy stress tensor can be described by a vector-valued 2-form. I think the description on pg 13 of these slides is a bit illuminating, too, as the factor of $n_b da$ is explictly related to the Hodge dual of the covector $dx^b$ related to $n_b$. Perhaps connecting the two approaches will be helpful to you.

I haven't had a chance to look at the notes, but have looked at the slides. I'm not sure where a "vector velocity field" comes into this, though. I would model work as $W=\int_{a}^b F_i dx^i$, where the author might be thinking of rate of work.

In any case, where he writes $dx^a \otimes (\ast x^b)$, I have used $(dx^a)\wedge(\ast dx^a)$, to get the integral of tension over a closed surface, $$T_i dx^i = \lgroup { \int_{\partial V} {P_i}^j \epsilon_{jkl} dx^k \otimes dx^l \rgroup } \otimes dx^i$$ by extrapolating from orthonormal coordinates, where the area element, $\ast dx^a$ is both oriented and normal to $\hat{n}$ = dx^a##. They combine as epsilon. epsilon is the Levi-Civita tensor.

Does this seem correct to you?

 

[fzero]

The author is definitely using rate of work. Your expression seems reasonable to me. The author assumes $t$-dependence in $\sigma$ that makes it a bit hard to derive your expression directly from the rate of work expressions on pg. 14.

 

[stedwards]

The author is definitely using rate of work. Your expression seems reasonable to me. The author assumes $t$-dependence in $\sigma$ that makes it a bit hard to derive your expression directly from the rate of work expressions on pg. 14.

Yes. That makes some sense, now. (I still need a closer look.)

The first notes are very difficult to follow, but a closer look reveals that the author only considers isotropic pressure (equation at top of page 22) I know I used "buoyancy" in the title, but I was looking forward to some sort of application of stoke's theorem to static fluids as well as anisotropic pressure, in general.

By the way, I used notation from here, Sean Carroll's Lecture Notes on General Relativity, to indicate antisymmetric, and symmetric indices. See statement following equation 1.6.9 {correction: 1.69}). So the notation, $Q_{[ij]k}$, should make more sense. I had though Carroll's was more standard usage than it now seems.