In which direction a spherical particle moves under torque?

[Eric]

Consider a small rigid spherical particle of radius a, fully immersed in a viscous incompressible Newtonian fluid of shear viscosity η above a hard-wall with stick (no-slip) boundary conditions, located at the plane z = 0. A constant positive (external) torque Tx is applied on the particle.

My question is:
Assuming low Reynolds number hydrodynamics, the particle will move along the y direction due to hydrodynamic interactions. But, does the velocity along y have a positive or negative sign?

Any help would be highly appreciated.

Thank you
Eric

 

[dreens]

You need to tell us how the torque is oriented relative to the no-slip plane. Maybe you're trying to do that by calling the torque "Tx". Do you mean the torque is parallel to the x direction, and pointing in the positive x direction? This would make the bottom of the marble want to move toward +y and the top of the marble want to move toward -y, since positive torque is counter-clockwise.

 

[Eric]

Thanks for your reply.
Here the torque is oriented in the positive direction (contour-clockwise) leading to particle motion in the y direction due to hydrodynamic interactions.
Also the wall is stationary (fixed). My calculations lead to motion in y+ while intuitively one would rather expect hat the particle will move in y- (analogy with a particle rolling on a wall) I am totally confused..
It would be great if you could please provide with some hints that may help a bit to figure out what is going on.
Thank you
E

 

[dreens]

I agree with your intuition. Motion should be in -y direction. I don't think you can solve Navier Stokes very easily by hand in this geometry... what calculations are you doing?

 

[Eric]

I agree with your intuition. Motion should be in -y direction. I don't think you can solve Navier Stokes very easily by hand in this geometry... what calculations are you doing?

Here only the Stokes equation are solved (nonlinear term is dropped out). The mobility calculations are used, see e.g. Swan,[/PLAIN] J. W. and Brady, J. F., Phys. Fluids 19, 113306 (2007)

 

[dreens]

Wow that's an interesting paper. I'm impressed that this can be solved, even for a pair of particles as in the appendix. I don't have the heart to look at it closely enough to try and figure out why the math is giving you the wrong sign. Not that it's worth more than two cents, but I'll bet 90:10 odds that the sign error lies somewhere in the math or your implementation of it, and not that the particle really rolls backward when you torque it near a wall.

If you really need a faith boost before you start hunting back through your math, maybe try this out with a xylophone mallet (sphere with attached stick so you can torque it) and a bowl of syrup haha.

 

[Eric]

Wow that's an interesting paper. I'm impressed that this can be solved, even for a pair of particles as in the appendix. I don't have the heart to look at it closely enough to try and figure out why the math is giving you the wrong sign. Not that it's worth more than two cents, but I'll bet 90:10 odds that the sign error lies somewhere in the math or your implementation of it, and not that the particle really rolls backward when you torque it near a wall.

If you really need a faith boost before you start hunting back through your math, maybe try this out with a xylophone mallet (sphere with attached stick so you can torque it) and a bowl of syrup haha.

Thanks dreens for your helpful insights. The equation stated above corresponds to Eq. (B2) of Swan's paper (the RT tensor is the transpose of TR tensor) and clearly the sign is positive, i.e. counter-intuitively (The equation seems to be a known result and it was first derived by Goldman some decades ago [Slow viscous motion of a sphere parallel to a plane wall- I Motion through a quiescent fluid, Chem. Eng. Sci. 1967]) I will think about it

Thanks again
Kindest regards,
Er