 We can have a nice 15-minute break for the next break, actually. So, dipolar non-active and active is frozen. Shareful orientational pinning, cross-stream migration, and particle focusing will be a type of the next talk, but by Maheraza Shabagnia from our institute, IPM in Tehran. Okay, Maheraza, please proceed. Maheraza, we can hear you. I think you're muted somehow, presenting. We see the slides. But no voice, you're muted, I think. Hello, and good afternoon, everyone. My internet is a bit slow, so I might lose the connection. In the middle, I would try my best. I want to talk today about two classes of magnetic esferoids. The first category is representative of magnetotactic bacteria, active particles that passively orient themselves towards the Earth's magnetic field, and then migrate towards the oxy-anoxic transition zone. The other class of magnetic esferoids that I want to talk about are the passive magnetic esferoids. These are separated based on their shape, as you can see on the right, by focusing the prolate esferoids on the channel center line in a recent experiment. Now, in order to explain these two problems, we use a single model based on continuum probabilistic description of a Smolhovsky equation. The difference between the two cases are, of course, first, the swimmers have a constant swimming velocity, but the passive esferoids don't. And instead, the passive esferoids are driven, are translated by a lift velocity induced by the hydrodynamic interaction with the channel walls. The orientational dynamics are basically the same in a channel flow, be it a quad-flow on the top for the case of active particles or a Poiseuille flow for the case of passive particles. There are two contributions to the rotational velocity. One regarding the Geoffrey orbits and the other one regarding the magnetic torque that eventually pins the orientation of particles in a single direction. Now, going to the case of active particles, as we can see, for prolate swimmers, when we look at the fraction of swimmers in the lower half of the channel, as we can see, by increasing the strength of the field from zero, we see a linear response and a linear rate of migration towards the lower half until we reach the full migration case. However, for the up-late swimmers, we do have a linear response. But after a certain point, there is a reverse migration, meaning that by increasing the strength of the field, more and more swimmers will end up in the upper half instead of the lower half. And then at a certain threshold, all the swimmers will migrate to the lower half, and we will have a full migration. Now, this sudden migration to the lower half will become handy, as we will explain in the next slides. In order to explain this, we will consider the case of field-modified Geoffrey orbits. As we can see, the classic Geoffrey orbits in the zero magnetic field correspond to this purple curve. As we increase the magnetic field, after a certain threshold, a single stable fixed point for prolate particles develops, meaning that the orientation of the particles is stable in this direction. However, for up-late swimmers, when we increase the field, after a certain threshold, there are two stable fixed points. One representing the swimmers that are moving upward and upstream. The other one representing the swimmers that are moving downward and upstream. And when we increase the field again, after a certain other threshold, there is only one fixed point, one stable fixed point representing the case of swimmers moving slightly upstream and towards the lower wall. Now, this difference can be shown better by using these two graphs, showing the stable and unstable fixed points. As we can see, again, for prolate, as far as there is only one single fixed point as we increase the field. And this pinning direction is near the upstream direction, but slightly towards the lower wall. For the case of up-lates, the area between the two thresholds, reverse migration regime, represents the case of two swimmers, two classes of swimmers, one moving upward and upstream, and the other one moving downward and, again, upstream. Those moving downward will end up near the lower wall, unsurprisingly, and those moving upward will end up near the upper wall. Now, the consequence of that is that we can plot these phase diagrams, and based on those, we can suggest two mechanism of separations. Remember that sudden jump in the fraction of swimmers in the lower half, near the lower wall? Well, that can be used for separation of up-lates swimmers based on aspect ratio in the full migration regime. And also, because of the stable upstream direction of swimming of both prolate and up-late swimmers in the pinning regime, we can rely on the net upstream flux in order to separate the swimmers based on their different aspect ratios and also on their swimming velocity. Now, these analyses could be generalized to the case of passive particles, as we mentioned before. Again, to show the model with a difference that in place of activity, in place of swimming speed, we enter the lift velocity that is generated by the hydrodynamic interaction between the particles and the channel walls. The rotational velocity is basically the same as the case of active particles, and therefore needs no further explanation. Now, as you can see, for a strong enough magnetic fields representing this ratio, magnetic torque to dimensionless fellow Peclet number, in the case of a transverse magnetic field, we see that prolate spheroids are focused by the lift velocity, which is shown by the blue arrows, in the middle of the channel and on the channel center line. Around this curve, which is the pinning curve, representing the stable fixed points in the orientational space, i.e. the pinning orientations. Now, this will cause the prolate particles of actually different aspect ratios to be focused around the channel center line with different levels of the spread, of course, in the transverse magnetic field. But the spherical particles, because generally they don't experience any lift velocity inside the channel, end up near the walls due to thermal effects, which are only present in our model and not the previous theoretical models. And this, as we can see, closely resembles the case of experimental studies. That being that, as I said in the beginning, prolate particles are focused in the middle of the channel and spheres end up near the lower wall. And the reason for the difference between the two cases, as you can see here, that the experimental observation only covers the lower half is that the particles are fed into the channel only from the inlet near the bottom wall. Now, in order to separate particles of different aspect ratios, this setup isn't efficient because all the prolate particles in this field direction, transverse field direction, end up near the middle of the channel. So we change the field direction. And in less acute field directions, we can separate the oblique particles in the lower half of the channel, as you can see here in the concentration plots. These peaks are very, very well separated. So the particles can be separated oblique particles. But prolate particles have intertwined peaks. So we need to change the field direction again to more acute field directions in order to separate the prolate particles. But this time, although the prolate particles are separated very nicely, as can be seen from these concentration graphs, the oblique particles are overlapping. Therefore, they cannot be separated. And the trick is to choose the right field direction in order to separate the particles with the maximum efficiency. Thank you very much. Well, thank you very much, especially for the sake of time. Basically, we are now ahead of time. We have time for questions and a nice long break. OK, may I ask a question? Yes, Reza, please. Say, just I'm wondering that if you don't have magnetic field as a control, how the particles are separated from each other in the channel? Say the question. Sorry, I did not realize the question. I'm just asking, in the absence of magnetic field, control the direction of the particles. Yes. If you put the particles inside the channel, how they are separated just because of the shape? Without a magnetic field. This is the question. Without a magnetic field. Because of a magnetic field. But the control signal, that's in the absence of the magnet. Because it could be nicer that we compare the results of the magnetic field with the case of the without magnetic field. Yes. If we're speaking about the passive spheroids, in the absence of magnetic fields, the lift velocity that is generated by dynamic interaction between the spheroids and the walls only causes the particles to move back and forth between the channel walls, but basically end up at the same position that they had at the inlet. And between the inlet and the outlet, they don't change their position very much. In order to break this symmetry and force the particles to be limited to a single orientation and end up in a single point across the channel, IEB focus, we need the magnetic field orientation, the magnetic field, and also a strong magnetic field in the pinning regime. I hope I answered the question. OK, thanks. Thanks. Yeah, it's basically just to add a comment. It's basically the problem of Taylor dispersion. If you don't have a magnetic field, they will be really mixed. And this is a known thing for coaxing in a graphically channel. Yes, that's very well-luckened. Thank you. Any other questions? So can I ask a question? Huh? Please. So how was the activity modeled in the Smolchowski equations here? Are you Mr. Sorry? The active model? Yeah, for the active models. Yes. In the Smolchowski equation for the active model, we use a simple translation of velocity, meaning that because of the symmetry, the convection by the flow is removed from the model, and only the swimming velocity with certain direction enters into the Smolchowski equation. And the rotational velocities are basically the same as of the passive case, one part due to Geoffrey orbits in the shear flow, and the other one due to the magnetic torque. So in this case, it's just like a constant velocity. So there is no change in the direction due to the active forces? No, no. When I say constant velocity, I mean the magnitude of velocity is fixed, but the orientation changes in the two-dimensional plane of the flow. OK, so it's like a rotational equilibrium model. Yeah, so it's active Brownian particle, but except that the particles have an elongated shape like a spheroidal shape. But otherwise, if they were points like spheres, they were exactly ABPs, active Brownian particle model. Yes, if it weren't for these anisotropic diffusion constants, it would be the same as the ABP model, as Professor Nagy just said. Thank you very much. Thank you. Thank you for the questions. Still have time, I think, for one more question, I guess. Well, actually, we are over time. No, we are set for the break.