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A sphere heads east from Los Angeles... ("centre of hydrodynamic 'mass'" problem)
This is actually from research, but it's one of those "this should be an easy problem and I'm wasting too much time making it too difficult" problems, so I thought it fit better under math/calculus--at its heart it is a math-y problem rather than a physics-y one.
The problem is this: I'm doing numeric simulations of small hydrodynamic swimmers composed of pairs of spheres (dumbbells); for notation, each sphere has position x_1 and x_2, and radius a_1 and a_2. Each dumbbell has a time-varying force applied in opposite directions to each sphere.
So as you can imagine, you basically have something that expands and contracts in a periodic fashion. We model the effect of force on a sphere by Stokes drag, ie
v=\frac{1}{6\pi\eta a}F
(where \eta is viscosity and F is the force on the sphere) and hydrodynamic interactions between spheres with the Oseen tensor
\frac{1}{8\pi\eta r}(\delta_{\alpha\beta} + \hat{r_\alpha}\hat{r_\beta})
(actually we use the Rotne-Prager tensor, but an Oseen approximation may be good enough). In one dimensions, it's fairly simple (see sec 2).
So finally, here's the problem: What is the "position" of a dumbbell? I want some function of the position of the two spheres (x_1 and x_2) that does not vary over the duration of a cycle. For symmetric dumbbells, the forces are equal and we can say
x_{cm} = \frac{x_1+x_2}{2}
...but this naive version fails with asymmetric spheres and the quantity varies over time. A naive update based on the Stokes force is
x_{cm} = \frac{ a_1 x_1 + a_2 x_2 }{a_1+a_2}
The amplitude of this varies over time as well, according to simulation, although to much less degree. In our simulations, we've taken to using the naive version and just measuring at the beginning of each period, which gives acceptable results that agree with analytics, and no one else seems interested in this question, but now it is bugging me.
What is some equation for x_{cm} that would not vary over time? If you stretched a line from x_1 to x_2, what point on that line would not move?
We have a coupled system of ODEs. In one dimension, the stokes force and Oseen interactions reduce to
\frac{d}{dt}x_1 = \frac{F}{6\pi\eta a_1} - \frac{F}{4\pi\eta(x_2-x_1)}
\frac{d}{dt}x_2 = \frac{F}{4\pi\eta(x_2-x_1)} - \frac{F}{6\pi \eta a_2}
Find some point x_{cm}(x_1, x_2) which is invariant with respect to time, representing the "position" of the dumbbell.
This seems like the sort of thing which would be quite easy, but for some reason it's vexing me. I'm having difficulty framing the question; I feel like it might be easy once I frame it properly.
One way to think about it is "If the force was always contractile (presuming that x_2 > x_1), at what point x_{cm} would the spheres come into contact?" -- I think that's a valid idea, because that point should be the same no matter where in the sinusoidal cycle the dumbbell was. Unfortunately I haven't made much progress here. I've also thought about, as mentioned above, imagining a line stretched between the spheres and finding a point on it which doesn't move, but that hasn't led me far either.
Another way might be to imagine some function x_{cm}(x_1, x_2) such that x_{cm}(x_1(t), x_2(t)) = x_{cm}(x_1(t+dt), x_2(t+dt) ), but since I'm trying to guess the form of x_{cm}, that hasn't helped much.
I thought comparing the derivatives, might be helpful, in that
\frac{ \frac{d}{dt}x_1 }{ \frac{d}{dt}x_2 } = - \frac{ a_2( 2( x_2 - x_1 ) - 3 a_1 ) }{ a_1( 2( x_2 - x_1 ) - 3 a_2 ) }
That's a nice-looking useful ratio, but I can't seem to do much useful with it.
I'm also more interested in the general approach, since the off-diagonal terms will change slightly if we use the RPY tensor instead of Oseen... and the equation also must extend to 3d coordinates.
This is really rather embarassing since moving to computers from my analytic undergraduate days... I can simulate hundreds of these guys in parallel using CUDA and applied stochastic noise, but I can't even frame the above correctly. What horribly obvious thing am I missing?
Homework Statement
This is actually from research, but it's one of those "this should be an easy problem and I'm wasting too much time making it too difficult" problems, so I thought it fit better under math/calculus--at its heart it is a math-y problem rather than a physics-y one.
The problem is this: I'm doing numeric simulations of small hydrodynamic swimmers composed of pairs of spheres (dumbbells); for notation, each sphere has position x_1 and x_2, and radius a_1 and a_2. Each dumbbell has a time-varying force applied in opposite directions to each sphere.
So as you can imagine, you basically have something that expands and contracts in a periodic fashion. We model the effect of force on a sphere by Stokes drag, ie
v=\frac{1}{6\pi\eta a}F
(where \eta is viscosity and F is the force on the sphere) and hydrodynamic interactions between spheres with the Oseen tensor
\frac{1}{8\pi\eta r}(\delta_{\alpha\beta} + \hat{r_\alpha}\hat{r_\beta})
(actually we use the Rotne-Prager tensor, but an Oseen approximation may be good enough). In one dimensions, it's fairly simple (see sec 2).
So finally, here's the problem: What is the "position" of a dumbbell? I want some function of the position of the two spheres (x_1 and x_2) that does not vary over the duration of a cycle. For symmetric dumbbells, the forces are equal and we can say
x_{cm} = \frac{x_1+x_2}{2}
...but this naive version fails with asymmetric spheres and the quantity varies over time. A naive update based on the Stokes force is
x_{cm} = \frac{ a_1 x_1 + a_2 x_2 }{a_1+a_2}
The amplitude of this varies over time as well, according to simulation, although to much less degree. In our simulations, we've taken to using the naive version and just measuring at the beginning of each period, which gives acceptable results that agree with analytics, and no one else seems interested in this question, but now it is bugging me.
What is some equation for x_{cm} that would not vary over time? If you stretched a line from x_1 to x_2, what point on that line would not move?
Homework Equations
We have a coupled system of ODEs. In one dimension, the stokes force and Oseen interactions reduce to
\frac{d}{dt}x_1 = \frac{F}{6\pi\eta a_1} - \frac{F}{4\pi\eta(x_2-x_1)}
\frac{d}{dt}x_2 = \frac{F}{4\pi\eta(x_2-x_1)} - \frac{F}{6\pi \eta a_2}
Find some point x_{cm}(x_1, x_2) which is invariant with respect to time, representing the "position" of the dumbbell.
The Attempt at a Solution
This seems like the sort of thing which would be quite easy, but for some reason it's vexing me. I'm having difficulty framing the question; I feel like it might be easy once I frame it properly.
One way to think about it is "If the force was always contractile (presuming that x_2 > x_1), at what point x_{cm} would the spheres come into contact?" -- I think that's a valid idea, because that point should be the same no matter where in the sinusoidal cycle the dumbbell was. Unfortunately I haven't made much progress here. I've also thought about, as mentioned above, imagining a line stretched between the spheres and finding a point on it which doesn't move, but that hasn't led me far either.
Another way might be to imagine some function x_{cm}(x_1, x_2) such that x_{cm}(x_1(t), x_2(t)) = x_{cm}(x_1(t+dt), x_2(t+dt) ), but since I'm trying to guess the form of x_{cm}, that hasn't helped much.
I thought comparing the derivatives, might be helpful, in that
\frac{ \frac{d}{dt}x_1 }{ \frac{d}{dt}x_2 } = - \frac{ a_2( 2( x_2 - x_1 ) - 3 a_1 ) }{ a_1( 2( x_2 - x_1 ) - 3 a_2 ) }
That's a nice-looking useful ratio, but I can't seem to do much useful with it.
I'm also more interested in the general approach, since the off-diagonal terms will change slightly if we use the RPY tensor instead of Oseen... and the equation also must extend to 3d coordinates.
This is really rather embarassing since moving to computers from my analytic undergraduate days... I can simulate hundreds of these guys in parallel using CUDA and applied stochastic noise, but I can't even frame the above correctly. What horribly obvious thing am I missing?
