9,045 research outputs found
Nonlinear stability of flock solutions in second-order swarming models
In this paper we consider interacting particle systems which are frequently
used to model collective behavior in animal swarms and other applications. We
study the stability of orientationally aligned formations called flock
solutions, one of the typical patterns emerging from such dynamics. We provide
an analysis showing that the nonlinear stability of flocks in second-order
models entirely depends on the linear stability of the first-order aggregation
equation. Flocks are shown to be nonlinearly stable as a family of states under
reasonable assumptions on the interaction potential. Furthermore, we
numerically verify that commonly used potentials satisfy these hypotheses and
investigate the nonlinear stability of flocks by an extensive case-study of
uniform perturbations.Comment: 22 pages, 1 figure, 1 tabl
Blood Flow Through an Artery in the Presence of a Stenosis
We consider blood flow through an artery in the form of a cylindrical pipe in the presence of a stenosis. Here blood is treated as an incompressible, viscous and non-Newtonian Bingham plastic fluid. We derive the equation of continuity and the momentum equation which are obtained using mass conservation law and momentum conservation law, respectively. Assuming that the flow is due to the pressure drop and wall shear stress, we derive the expressions for the velocity component in the axial direction and the volumetric flow rate in an artery. Computational results for the axial velocity and flow rate are obtained using MATLAB and presented in tabular and graphical forms to analyze the effects of the slip velocity, pressure difference, yield stress, and stenosis height. Results obtained through our computations indicate that dependent variables (axial velocity and flow rate) increase with the increase in pressure difference and decrease with the increase in yield stress. As stenosis height increases, the dependent variables display a decrease. Both the dependent variables are minimum when the stenosis height is maximum. Increasing the slip velocity enhances the axial velocity and the flow rate
Convergence of a linearly transformed particle method for aggregation equations
We study a linearly transformed particle method for the aggregation equation
with smooth or singular interaction forces. For the smooth interaction forces,
we provide convergence estimates in and norms depending on the
regularity of the initial data. Moreover, we give convergence estimates in
bounded Lipschitz distance for measure valued solutions. For singular
interaction forces, we establish the convergence of the error between the
approximated and exact flows up to the existence time of the solutions in norm
Explicit flock solutions for Quasi-Morse potentials
We consider interacting particle systems and their mean-field limits, which
are frequently used to model collective aggregation and are known to
demonstrate a rich variety of pattern formations. The interaction is based on a
pairwise potential combining short-range repulsion and long-range attraction.
We study particular solutions, that are referred to as flocks in the
second-order models, for the specific choice of the Quasi-Morse interaction
potential. Our main result is a rigorous analysis of continuous, compactly
supported flock profiles for the biologically relevant parameter regime.
Existence and uniqueness is proven for three space dimension, whilst existence
is shown for the two-dimensional case. Furthermore, we numerically investigate
additional Morse-like interactions to complete the understanding of this class
of potentials.Comment: 26 page
A new interaction potential for swarming models
We consider a self-propelled particle system which has been used to describe
certain types of collective motion of animals, such as fish schools and bird
flocks. Interactions between particles are specified by means of a pairwise
potential, repulsive at short ranges and attractive at longer ranges. The
exponentially decaying Morse potential is a typical choice, and is known to
reproduce certain types of collective motion observed in nature, particularly
aligned flocks and rotating mills. We introduce a class of interaction
potentials, that we call Quasi-Morse, for which flock and rotating mills states
are also observed numerically, however in that case the corresponding
macroscopic equations allow for explicit solutions in terms of special
functions, with coefficients that can be obtained numerically without solving
the particle evolution. We compare thus obtained solutions with long-time
dynamics of the particle systems and find a close agreement for several types
of flock and mill solutions.Comment: 23 pages, 8 figure
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