Two-dimensional mathematical framework for evaporation dynamics of respiratory droplets

We solve a problem posed recently by Gravner and Griffeath [4]: to find a finite seed A0 of 1s for a simple {0, l}-valued cellular automaton growth model on Z2 such that the occupied crystal An after n updates spreads with a two-dimensional asymptotic shape and a provably irrational density. Our solution exhibits an initial A0 of 2,392 cells for Conway’s Game Of Life from which An cover nT with asymptotic density (3 - √5/90, where T is the triangle with vertices (0,0), (-1/4,-1/4), and (1/6,0). In “Cellular Automaton Growth on Z2: Theorems, Examples, and Problems” [4], Gravner and Griffeath recently presented a mathematical framework for the study of Cellular Automata (CA) crystal growth and asymptotic shape, focusing on two-dimensional dynamics. For simplicity, at any discrete time n, each lattice site is assumed to be either empty (0) or occupied (1). Occupied sites after n updates grows linearly in each dimension, attaining an asymptotic density p within a limit shape L: . . . n-1 A → p • 1L • (1) . . . This phenomenology is developed rigorously in Gravner and Griffeath [4] for Threshold Growth, a class of monotone solidification automata (in which case p = 1), and for various nonmonotone CA which evolve recursively. The coarse-grain crystal geometry of models which do not fill the lattice completely is captured by their asymptotic density, as precisely formulated in Gravner and Griffeath [4]. It may happen that a varying “hydrodynamic” profile p(x) emerges over the normalized support L of the crystal. The most common scenario, however, would appear to be eq. (1), with some constant density p throughout L. All the asymptotic densities identified by Gravner and Griffeath are rational, corresponding to growth which is either exactly periodic in space and time, or nearly so. For instance, it is shown that Exactly 1 Solidification, in which an empty cell permanently joins the crystal if exactly one of its eight nearest (Moore) neighbors is occupied, fills the plane with density 4/9 starting from a singleton.

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Visit Probability in Space–Time Prisms Based on Binomial Random Walk

ISPRS International Journal of Geo-Information ◽

10.3390/ijgi9090555 ◽

2020 ◽

Vol 9 (9) ◽

pp. 555

Author(s):

Deepak Elias ◽

Bart Kuijpers

Keyword(s):

Random Walk ◽

Moving Objects ◽

Formal Model ◽

Dimensional Space ◽

Space Time ◽

Mathematical Framework ◽

Two Dimensional ◽

Time Points ◽

Dimensional Space Time ◽

Measured Sample

Space–time prisms are used to model the uncertainty of space–time locations of moving objects between (for instance, GPS-measured) sample points. However, not all space–time points in a prism are equally likely and we propose a simple, formal model for the so-called “visit probability” of space–time points within prisms. The proposed mathematical framework is based on a binomial random walk within one- and two-dimensional space–time prisms. Without making any assumptions on the random walks (we do not impose any distribution nor introduce any bias towards the second anchor point), we arrive at the conclusion that binomial random walk-based visit probability in space–time prisms corresponds to a hypergeometric distribution.

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A General Mathematical Framework for Modeling Two-Dimensional Wildland Fire Spread

International Journal of Wildland Fire ◽

10.1071/wf9950063 ◽

1995 ◽

Vol 5 (2) ◽

pp. 63 ◽

Cited By ~ 62

Author(s):

GD Richards

Keyword(s):

Differential Equations ◽

Wildland Fire ◽

Fire Spread ◽

Meteorological Conditions ◽

Mathematical Framework ◽

Two Dimensional ◽

Affecting Factors ◽

Modelling Framework ◽

Rate Of Spread ◽

The Relationship

This work considers the modelling of two dimensional fire spread for heterogeneous fuel and meteorological conditions. Differential equations are used as the modelling form, and a set of partial differential equations that describes fire growth in terms of the rate of spread at each point on the perimeter is derived. These equations require the specification of the rate of spread as a function of the variables affecting it, and form a general modelling framework into which such a function can be placed. To model the relationship between the rate of spread and its affecting factors an analysis of point source ignition fires for homogeneous fuel and meteorological conditions is made. Based on this analysis a spread rate model for heterogeneous conditions is proposed. The resulting differential equations require a sophisticated computer solution, however there are a number of nontrivial fire situations for which solutions can relatively easily be obtained, and example solutions are presented.

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Kinematics for an Actuated Flexible n-Manifold

Journal of Mechanisms and Robotics ◽

10.1115/1.4031301 ◽

2015 ◽

Vol 8 (2) ◽

Cited By ~ 10

Author(s):

Oded Medina ◽

Amir Shapiro ◽

Nir Shvalb

Keyword(s):

Smooth Function ◽

Inverse Kinematics ◽

Degrees Of Freedom ◽

Three Dimensional ◽

Mathematical Framework ◽

Two Dimensional ◽

Rectangular Grid ◽

Flexible Robots ◽

Forward And Inverse Kinematics ◽

Grid Surface

Recent years show an increasing interest in flexible robots due to their adaptability merits. This paper introduces a novel set of hyper-redundant flexible robots which we call actuated flexible manifold (AFM). The AFM is a two-dimensional hyper-redundant grid surface embedded in ℝ2 or ℝ3. Theoretically, such a mechanism can be manipulated into any continuous smooth function. We introduce the mathematical framework for the kinematics of an AFM. We prove that for a nonsingular configuration, the number of degrees of freedom (DOF) of an AFM is simply the number of its grid segments. We also show that for a planar rectangular grid, every nonsingular configuration that is also energetically stable is isolated. We show how to calculate the forward and inverse kinematics for such a mechanism. Our analysis is also applicable for three-dimensional hyper-redundant structures as well. Finally, we demonstrate our solution on some actuated flexible grid-shaped surfaces.

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Analytical Framework of Forced Convective Two-dimensional MHD Flow of Jeffrey Nano Liquid With Impact of Thermal Radiation and Activation Energy

10.21203/rs.3.rs-672606/v1 ◽

2021 ◽

Author(s):

Haroon Ur Rasheed ◽

Saeed Islam ◽

Zeeshan Khan ◽

Waris Khan

Keyword(s):

Activation Energy ◽

Thermal Radiation ◽

Radiation Effects ◽

Mhd Flow ◽

Stability And Convergence ◽

Field Energy ◽

Mathematical Framework ◽

Two Dimensional ◽

Buongiorno’S Model ◽

Jeffrey Nanofluid

Abstract The present communication owns to address the mathematical framework of two-dimensional electrically conductive and thermally radiative Jeffrey nanofluid flow by a curved surface. The interaction of a periodic magnetic field with the suspended nanoparticles and mixed convection are critically important due to its application in a broad spectrum. Buongiorno’s model, incorporates the effect of thermophoretic force and Brownian movement, describes the nature of Jeffrey nanofluid. The influence of activation energy, viscous dissipation, and thermal radiation effects are reserved. The dimensionless system of differential equations has been diminished from the modeled equations via transformation framework which is solved analytically versus homotopic algorithm. The stability and convergence analysis has been carried out to optimize system parameters and accuracy of the system. The effect of physical constraints on flow field, energy, and concentration of nanoparticles are portrayed via plotted graphs and debated.

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Interscale transfer in two-dimensional compact vortices

Journal of Fluid Mechanics ◽

10.1017/s0022112099007302 ◽

2000 ◽

Vol 406 ◽

pp. 109-129 ◽

Cited By ~ 4

Author(s):

GIANNI PEDRIZZETTI ◽

J. C. VASSILICOS

Keyword(s):

Large Time ◽

Vortex Sheet ◽

Passive Scalar ◽

Velocity Fields ◽

Mathematical Framework ◽

Two Dimensional ◽

Leading Order ◽

New Approach ◽

Scale Interactions ◽

Spiral Vortex

The property of transfer between different scales of motion in evolving two-dimensional compact vortices is studied here, and a general mathematical framework is developed to describe the transfer between scales inside compact structures. This new approach is applied to the case of an axisymmetric advection which represents the leading-order (large time) approximation for Lundgren's family of two-dimensional vortices. It is also generalized to passive scalar advection by non-axisymmetric velocity fields. It is shown that scale interactions generated by an axisymmetric advection are essentially local and dominated by distant triadic interactions: in the case of an evolving spiral vortex sheet this result is confirmed even when non-axisymmetric corrections are included. A physical interpretation of the results is given, which can be summarized by saying that locality of scale interactions is caused by the uniformity of shear at a given scale and is therefore increasingly natural at small lengthscales. Local interactions are shown to arise in axisymmetric advection but to be uncommon in non-axisymmetric advection.

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Significance of Vocal Tract Geometrical Variations and Loudness on Airflow and Droplet Dispersion in a Two-Dimensional Representation of [F]

10.1115/fedsm2021-65485 ◽

2021 ◽

Author(s):

Amir A. Mofakham ◽

Brian T. Helenbrook ◽

Tanvir Ahmed ◽

Byron D. Erath ◽

Andrea R. Ferro ◽

...

Keyword(s):

Vocal Tract ◽

Two Dimensional ◽

Discrete Phase Model ◽

Ansys Fluent ◽

Lower Lip ◽

Large Size ◽

Discrete Phase ◽

Respiratory Droplets ◽

Droplet Transmission ◽

Shed Light

Abstract The significance of respiratory droplet transmission in spreading respiratory diseases such as COVID-19 has been identified by researchers. Although one cough or sneeze generates a large number of respiratory droplets, they are usually infrequent. In comparison, speaking and singing generate fewer droplets, but occur much more often, highlighting their potential as a vector for airborne transmission. However, the flow dynamics of speech and the transmission of speech droplets have not been fully investigated. To shed light on this topic, two-dimensional geometries of a vocal tract for a labiodental fricative [f] were generated based on real-time MRI of a subject during pronouncing [f]. In these models, two different curvatures were considered for the tip tongue shape and the lower lip to highlight the effects of the articulator geometries on transmission dynamics. The commercial ANSYS-Fluent CFD software was used to solve the complex expiratory speech airflow trajectories. Simultaneously, the discrete phase model of the software was used to track submicron and large size respiratory droplets exhaled during [f] utterance. The simulations were performed for high, normal, and low lung pressures to explore the influence of loud, normal, and soft utterances, respectively, on the airflow dynamics. The presented results demonstrate the variability of the airflow and droplet propagation as a function of the vocal tract geometrical characteristics and loudness.

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Self-Organization of Vocabularies under Different Interaction Orders

Artificial Life ◽

10.1162/artl_a_00230 ◽

2017 ◽

Vol 23 (2) ◽

pp. 287-294

Author(s):

Javier Vera

Keyword(s):

Computer Simulations ◽

Discrete Time ◽

Word Meaning ◽

Self Organization ◽

Mathematical Framework ◽

Two Dimensional ◽

Time Step ◽

Agent Based ◽

Automata Networks ◽

First Approximation

Traditionally, the formation of vocabularies has been studied by agent-based models (primarily, the naming game) in which random pairs of agents negotiate word-meaning associations at each discrete time step. This article proposes a first approximation to a novel question: To what extent is the negotiation of word-meaning associations influenced by the order in which agents interact? Automata networks provide the adequate mathematical framework to explore this question. Computer simulations suggest that on two-dimensional lattices the typical features of the formation of word-meaning associations are recovered under random schemes that update small fractions of the population at the same time; by contrast, if larger subsets of the population are updated, a periodic behavior may appear.

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Results from the multi-species Benchmark Problem 3 (BM3) using two-dimensional models

Water Science & Technology ◽

10.2166/wst.2004.0833 ◽

2004 ◽

Vol 49 (11-12) ◽

pp. 169-176 ◽

Cited By ~ 15

Author(s):

D.R. Noguera ◽

C. Picioreanu

Keyword(s):

Numerical Solutions ◽

Mathematical Framework ◽

Two Dimensional ◽

Benchmark Problem ◽

Biofilm Growth ◽

One Dimensional ◽

Bulk Substrate ◽

The One ◽

Definition Of ◽

Multidimensional Models

In addition to the one-dimensional solutions of a multi-species benchmark problem (BM3) presented earlier (Rittmann et al., 2004), we offer solutions using two-dimensional (2-D) models. Both 2-D models (called here DN and CP) used numerical solutions to BM3 based on a similar mathematical framework of the one-dimensional AQUASIM-built models submitted by Wanner (model W) and Morgenroth (model M1), described in detail elsewhere (Rittmann et al., 2004). The CP model used differential equations to simulate substrate gradients and biomass growth and a particle-based approach to describe biomass division and biofilm growth. The DN model simulated substrate and biomass using a cellular automaton approach. For several conditions stipulated in BM3, the multidimensional models provided very similar results to the 1-D models in terms of bulk substrate concentrations and fluxes into the biofilm. The similarity can be attributed to the definition of BM3, which restricted the problem to a flat biofilm in contact with a completely mixed liquid phase, and therefore, without any salient characteristics to be captured in a multidimensional domain. On the other hand, the models predicted significantly different accumulations of the different types of biomass, likely reflecting differences in the way biomass spread within the biofilm is simulated.

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