# Do you recognize this equation? (Fluid mechanics)

Hmm, lost the OP somehow.

Here it is:

GonzoGal’s doing an independent study in modeling fluid flow, and her Professor gave her the following equation:

v=(3a/4)(vbar/|x|+x(x.vbar)/|x|^3+(a^2/3)(vbar/|x|^3-3x(x.vbar)/|x|^5))+a^3*cross(x,omegabar)/|x|^3

Where vbar is apparently the velocity of a particle, omegabar is the angular velocity of the particle, and x is the distance from the particle. a is the radius of the particle, cross() is the cross product, and v is the velocity of the fluid.

I’ll try my best to make it look nice:

``````

v= 3a(vbar+x(x.vbar)+a[sup]2[/sup]*(vbar-3x(x.vbar))))+a[sup]3[/sup]*(x X omegabar)
4  |x|     |x|[sup]3[/sup]  3   |x|[sup]3[/sup]    |x|[sup]5[/sup]             |x|[sup]3[/sup]

``````

We’ve been scouring the textbooks and the net for this equation, but can’t find it anywhere. We don’t know where he got it from, and we’ve got suspicions that it’s not the right formula, or he wrote it wrong. (when vbar and omegabar are 0 or when x is large, v goes to 0, but we expect it to go to some v[sub]infinity[/sub] which doesn’t even appear in the equation.)

I know this is obscure, but maybe there’s a fluid mechanics expert in the house… if so:
a) where does this equation come from, and does it have a name?
b) does it have any typos or errors in it?
c) Is there a way to express it in terms of the force on the particle?

That looks sort of like a vortex element-based equation for velocity, but I’m not sure. See if this context sounds at all familiar…

Among the ways of simulating fluid flows is to keep track of regions of vorticity. Equations can be derived showing the ways in which these vortex elements influence one another’s motion, as well as what the velocity at any point in the domain is at any time.

Usually, the relation for v(x,t) has integrals (or summations) in it - you’re basically integrating the effect of each of the vortex elements (which are characterized by a size a and vorticity strength omega.

All those dot products with x, normalized by the distance or the distance cubed look very familiar…

I don’t have the proper text handy, but some of that looks vaguely familiar. Any other context clues you might be able to provide would probably help, too.

Yes! That’s sounding very close…

Here’s some more context:

The model is of an arbitrarily shaped object in a creeping flow at steady state. She’s modeling the object shape as a collection of particles. (voxels).

What the professor has told her is to take that equation,
and the stokes drag equation for force on a particle:

F[sub]e[/sub] = 6pi mu a [v[sub]e[/sub] - SIGMA v(xe-xm)]

Where F[sub]e[/sub] is the force exerted by the flow on particle e, a is the radius of particle e, v[sub]e[/sub] is the velocity of particle e, and that SIGMA is the sum over all the particles m != e of the flow velocity due to that particle.

There’s a similar function for Tau[sub]e[/sub], the torque on particle e.
What she’s supposed to do is take the F[sub]e[/sub] equations and put them in a matrix to solve for the F’s and Taus:

``````

[Big Matrix of "knowns"][ F1             [another matrix of knowns]
F2
...       =
Tau1
Tau2...]

``````

But this means we have to put the Fe’s in a system of linear equations. It’s supposed to be steady-state, so there’s no t, so we don’t see how to relate Vbar and Omegabat to Force and Torque to get linear equations.

Looks an awful lot like a sediment transport formula to me. Darned if I can find one on the 'net.

This might be what makes me think that, though.

I finally found this: this equation

Could this stress be your “force on the particle” you are looking for?