## A Soft-Magnetic Slender Body in a Highly Viscous Fluid Public Deposited • March 20, 2019
Creator
• Martindale, James D.
• Affiliation: College of Arts and Sciences, Department of Mathematics
Abstract
• Theoretical, numerical, and experimental studies for a rotating soft-magnetic body in low Reynolds number flow which imitates the motion of nodal cilia are considered. This includes a discussion of the torque balance for the coupled magnetic-fluid interaction for a rod rotating in a viscous fluid, and a discussion of experimental results which have suggested alterations to the current slender body theory used to find the flow around a rotating rod. Firstly, the current state of the theory for a slender rod attached to a no-slip plane sweeping out a cone will be discussed in terms of the singularity strength distribution which allows us to calculate the forces and torques on the slender body. This theory has been developed in the works of Terry Jo Leiterman and Longhua Zhao for straight and bent rod geometries respectively. Analogous techniques of the classical fluid slender body theory are then applied for a soft-magnetic rod in free space in order to generate the appropriate strength distribution for the singularities placed along the center line of the rod. A discussion of the magnetic slender body theory is presented for a rod in free space and for a rod held fixed about a point in a uniform background field. Once the appropriate strength distribution is found through asymptotic matching, the magnetic torque on the rod may be calculated. A steady-state problem is considered where the position of the rod relative to the magnetic field may be found by balancing the fluid and magnetic torques. For a straight rod sweeping an upright cone in a uniform background magnetic field, this problem reduces to finding the solution to a polynomial whose arguments are trigonometric functions of the angles involved in describing the position of the rod and magnetic field. From this polynomial, we may construct explicit intervals in which the solution is unique. We also examine various limiting cases that should be seen in the physical experiment as a first-order check to the validity of the magnetic model. The magnetic problem is then extended to bent rod geometries, again using similar techniques derived in the fluid slender body theory. This bent rod geometry creates rather unwieldy expressions for the fluid and magnetic torque, but nonetheless, the torque balance may be solved numerically. This theory is not currently put to use in our experiment since the exact properties of our driving magnet have not been considered. Next, a discussion of the current state of our experiments for a rod rotating in a viscous fluid is presented. It is of great importance to understand the results of our Lagrangian particle tracking which were discussed for both straight and bent pin geometries by Leiterman and Zhao. Progress has been made in the procedure for tracking the rod which ensures that better measurements for the angles which describe the rod position are passed the the theory for comparison. Key differences between the theory and experimental results are presented. In order to corroborate these Lagrangian results and suggest alterations to the current fluid theory, we use a full three-dimensional particle image velocimetry (PIV) to capture an Eulerian view of the fluid flow structure in a horizontal plane slightly above the tip of the rotating rod. A discussion of the experimental setup and parameters is discussed, as well as the metrics we will use to compare the two experimental methods. Results of these experiments are then compared with the theory in various regards. Finally, alterations to the current theory including a consideration of the free surface and lubrication effects are discussed. A comparison is made between particle trajectories for a sphere in a uniform flow over a plane using lubrication theory and singularity theory to establish whether this effect will be non-negligible in our experiment.
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