The gradient of potential vorticity, quaternions and an orthonormal frame for fluid particles

The gradient of potential vorticity (PV) is an important quantity because of the way PV (denoted as $q$) tends to accumulate locally in the oceans and atmospheres. Recent analysis by the authors has shown that the vector quantity \bdB = \bnabla q\times \bnabla\theta for the three-dimensional incompressible rotating Euler equations evolves according to the same stretching equation as for \bom the vorticity and \bB, the magnetic field in magnetohydrodynamics (MHD). The \bdB-vector therefore acts like the vorticity \bom in Euler's equations and the \bB-field in MHD. For example, it allows various analogies, such as stretching dynamics, helicity, superhelicity and cross helicity. In addition, using quaternionic analysis, the dynamics of the \bdB-vector naturally allow the construction of an orthonormal frame attached to fluid particles\,; this is designated as a quaternion frame. The alignment dynamics of this frame are particularly relevant to the three-axis rotations that particles undergo as they traverse regions of a flow when the PV gradient \bnabla q is large.Comment: Dedicated to Raymond Hide on the occasion of his 80th birthda

How efficiently can one untangle a double-twist? Waving is believing!

It has long been known to mathematicians and physicists that while a full rotation in three-dimensional Euclidean space causes tangling, two rotations can be untangled. Formally, an untangling is a based nullhomotopy of the double-twist loop in the special orthogonal group of rotations. We study a particularly simple, geometrically defined untangling procedure, leading to new conclusions regarding the minimum possible complexity of untanglings. We animate and analyze how our untangling operates on frames in 3-space, and teach readers in a video how to wave the nullhomotopy with their hands.Comment: To appear in The Mathematical Intelligencer. For supplemental videos, see http://www.math.iupui.edu/~dramras/double-tip.html , or https://www.youtube.com/playlist?list=PLAfnEXvHU52ldJaOye-8kZV_C1CjxGx2C . For a supplemental virtual reality experience, see http://meglab.wikidot.com/visualizatio

Validated reduced order models for simulating trajectories of bio-inspired artificial micro-swimmers

Autonomous micro-swimming robots can be utilized to perform specialized procedures such as in vitro or in vivo medical tasks as well as chemical surveillance or micro manipulation. Maneuverability of the robot is one of the requirements that ensure successful completion of its task. In micro fluidic environments, dynamic trajectories of active micro-swimming robots must be predicted reliably and the response of control inputs must be well-understood. In this work, a reduced-order model, which is based on the resistive force theory, is used to predict the transient, coupled rigid body dynamics and hydrodynamic behavior of bio-inspired artificial micro-swimmers. Conceptual design of the micro-swimmer is biologically inspired: it is composed of a body that carries a payload, control and actuation mechanisms, and a long flagellum either such as an inextensible whip like tail-actuator that deforms and propagates sinusoidal planar waves similar to spermatozoa, or of a rotating rigid helix similar to many bacteria, such as E. Coli. In the reduced-order model of the microswimmer, fluid’s resistance to the motion of the body and the tail are computed from resistive force theory, which breaks up the resistance coefficients to local normal and tangential components. Using rotational transformations between a fixed world frame, body frame and the local Frenet-Serret coordinates on the helical tail we obtain the full 6 degrees-of-freedom relationship between the resistive forces and torques and the linear and rotational motions of the swimmer. In the model, only the tail’s frequency (angular velocity for helical tail) is used as a control input in the dynamic equations of the micro-swimming robot. The reduced-order model is validated by means of direct observations of natural micro swimmers presented earlier in the literature and against; results show very good agreement. Three-dimensional, transient CFD simulations of a single degree of freedom swimmer is used to predict resistive force coefficients of a micro-swimmer with a spherical body and flexible tail actuator that uses traveling plane wave deformations for propulsion. Modified coefficients show a very good agreement between the predicted and actual time-dependent swimming speeds, as well as forces and torques along all axes

Lagrangian particle paths and ortho-normal quaternion frames

Experimentalists now measure intense rotations of Lagrangian particles in turbulent flows by tracking their trajectories and Lagrangian-average velocity gradients at high Reynolds numbers. This paper formulates the dynamics of an orthonormal frame attached to each Lagrangian fluid particle undergoing three-axis rotations, by using quaternions in combination with Ertel's theorem for frozen-in vorticity. The method is applicable to a wide range of Lagrangian flows including the three-dimensional Euler equations and its variants such as ideal MHD. The applicability of the quaterionic frame description to Lagrangian averaged velocity gradient dynamics is also demonstrated.Comment: 9 pages, one figure, revise

Curve and surface framing for scientific visualization and domain dependent navigation

Thesis (Ph.D.) - Indiana University, Computer Science, 1996Curves and surfaces are two of the most fundamental types of objects in computer graphics. Most existing systems use only the 3D positions of the curves and surfaces, and the 3D normal directions of the surfaces, in the visualization process. In this dissertation, we attach moving coordinate frames to curves and surfaces, and explore several applications of these frames in computer graphics and scientific visualization. Curves in space are difficult to perceive and analyze, especially when they are densely clustered, as is typical in computational fluid dynamics and volume deformation applications. Coordinate frames are useful for exposing the similarities and differences between curves. They are also useful for constructing ribbons, tubes and smooth camera orientations along curves. In many 3D systems, users interactively move the camera around the objects with a mouse or other device. But all the camera control is done independently of the properties of the objects being viewed, as if the user is flying freely in space. This type of domain-independent navigation is frequently inappropriate in visualization applications and is sometimes quite difficult for the user to control. Another productive approach is to look at domain-specific constraints and thus to create a new class of navigation strategies. Based on attached frames on surfaces, we can constrain the camera gaze direction to be always parallel (or at a fixed angle) to the surface normal. Then users will get a feeling of driving on the object instead of flying through the space. The user's mental model of the environment being visualized can be greatly enhanced by the use of these constraints in the interactive interface. Many of our research ideas have been implemented in Mesh View, an interactive system for viewing and manipulating geometric objects. It contains a general purpose C++ library for nD geometry and supports a winged-edge based data structure. Dozens of examples of scientifically interesting surfaces have been constructed and included with the system

Lagrangian analysis of alignment dynamics for isentropic compressible magnetohydrodynamics

After a review of the isentropic compressible magnetohydrodynamics (ICMHD) equations, a quaternionic framework for studying the alignment dynamics of a general fluid flow is explained and applied to the ICMHD equations.Comment: 12 pages, 2 figures, submitted to a Focus Issue of New Journal of Physics on "Magnetohydrodynamics and the Dynamo Problem" J-F Pinton, A Pouquet, E Dormy and S Cowley, editor

Comparison on experimental and numerical results for helical swimmers inside channels

Swimming micro robots are becoming feasible in biomedical applications such as targeted drug delivery, opening clogged arteries and diagnosis owing to recent developments in micro and nano manufacturing technologies. It has been demonstrated at various scales that micro helices with magnetic coating or attached to a magnet can move in fluids with the application of external rotating magnetic fields. The motion of micro swimmers interacting with flow inside channels needs to be well understood especially for medical applications where the motion of micro robots inside arteries and conduits in the body become pertinent. In this work, swimming of helical micro robots with magnetic heads inside tubes is modeled with the resistive force theory (RFT) and validated with experiments conducted in glycerin filled mini glass channels placed in rotational magnetic fields. The time-averaged forward velocities of magnetically driven micro swimmers that are calculated by the RFT model agree very well with experimental results

BioVEC: A program for Biomolecule Visualization with Ellipsoidal Coarse-graining

Biomolecule Visualization with Ellipsoidal Coarse-graining (BioVEC) is a tool for visualizing molecular dynamics simulation data while allowing coarse-grained residues to be rendered as ellipsoids. BioVEC reads in configuration files, which may be output from molecular dynamics simulations that include orientation output in either quaternion or ANISOU format, and can render frames of the trajectory in several common image formats for subsequent concatenation into a movie file. The BioVEC program is written in C++, uses the OpenGL API for rendering, and is open source. It is lightweight, allows for user-defined settings for and texture, and runs on either Windows or Linux platforms