Nested first-passages of tracer particles in flows of blood and control suspensions: Symmetry and lorentzian transformations
Theory of molecular Taylor-Aris dispersion (TAD) is an accepted framework describing tracer dispersion in suspension flows and determining effective diffusion coefficients. Our group reported a pseudo-Lagrangian method to study dispersion in suspension flows at FEDSM-2000. Tracer motions were studied in a steadily moving inertial reference frame (SMIRF) aligned with the flow direction; increments of change of axial position of individual tracers were collected to demonstrate how the tracer moved as they, individually, interacted with similar collections of other bodies brought to and from the region. First, individual tracers with no apparent axial velocity component (NAAVC) were identified; they exhibited fixed positions in video recordings of images collected in the SMIRF. Then, time increments were measured for tracers to pass at least 5, but usually 10 pre-selected, nested distances in the up- or downstream direction laid out with respect to the zero-site in the SMIRF. Such data were richer than measurements of tracer spread over time because stations along each path were serial first-passages (FP) with probabilistic meaning. Dispersion of various types of suspension and two transformation rules for combining velocity components are discussed herein. Traditional low-speed continuum theory and particle dynamics use Galilean transforms. Yet, to recognize the limited speed in laws for channel flows, Lorentzian transformations may be appropriate. In a four-space, deterministic paths would begin at NAAVC sites and continue in time-like conical regions of four-space. Distances in this space are measured using Minkowski's metric; at the NAAVC site and on the boundary of the space-time cone, this metric has the format of the Fürth, Ornstein, and Taylor (FOT) equation when only terms to order 2t are used. Data shown at FEDSM-2000 can be reinterpreted as "prospective paths" in time-like regions that were consolidated in normalized cumulative probability distributions to provide retrospective descriptions. The ad hoc sign alteration of the FOT equation to fit the data of FEDSM-2000 is now taken as a part of measuring lengths using a Minkowski metric, which signifies a hyperbolic geometry, for which an inherent scaling constant is a negative curvature. The space also has an intrinsic distance of ℓ = δτ, obtained from fitting parameters (δτ) for the FOT equation. Integrals of the area under the FOT curve have units of volume, which are considered as describing an average volume of dispersion on S3,the 3-sphere. Path motion through this volume was kinematic dispersion, δτ2t, which was the form for effective diffusivity in continuum theory used in FEDSM-2000. Weiner and Wilmer describe transformations in four-spaces in terms of commutating rotations on orthogonal planes, a concept readily linked to symmetries in the hyperbolic space typical of Lorentzian transformations; they also describe a second order ODE like the FOT equation.
American Society of Mechanical Engineers, Fluids Engineering Division (Publication) FEDSM
Eckstein, E., Bhal, V., Lavine, J., Ma, B., Leggas, M., & Goldstein, J. (2017). Nested first-passages of tracer particles in flows of blood and control suspensions: Symmetry and lorentzian transformations. American Society of Mechanical Engineers, Fluids Engineering Division (Publication) FEDSM, 1C-2017 https://doi.org/10.1115/FEDSM2017-69549