Approximation of attractors for nongradient systems

1995 ◽  
Vol 125 (4) ◽  
pp. 739-758
Author(s):  
I. N. Kostin
Keyword(s):  
Large Class ◽  
Metric Space ◽  
Numerical Approximation ◽  
Stokes Equations ◽  
Hausdorff Metric ◽  
Navier Stokes ◽  
Lorenz Equations ◽  
Navier Stokes Equations ◽  
Suggested Procedure ◽  
Nongradient Systems

The problem of approximation of attractors for semidynamical systems (SDSs) in a metric space is studied. Let some (exact) SDS possessing an attractor M be inaccurately defined, i.e. let another (approximate) SDS, which is close in some sense to the exact one, be given. The problem is to construct a set , which is close to M in the Hausdorff metric.The suggested procedure for constructing is finite, which makes it possible to use it in computations. The results obtained are suitable for numerical approximation of attractors for a rather large class of semidynamical systems, including ones generated by the Lorenz equations and the Navier–Stokes equations.

Mathematical study of a single leukocyte in microchannel flow

2018 ◽  
Vol 13 (5) ◽  
pp. 43 ◽  
Author(s):  
S. Boujena ◽  
O. Kafi ◽  
A. Sequeira
Keyword(s):  
Mathematical Model ◽  
Numerical Approximation ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Microchannel Flow ◽  
Navier Stokes Equations ◽  
Coupled Problem ◽  
Inflammatory Drugs ◽  
The Mathematical Model ◽  
Rate Type

The recruitment of leukocytes and subsequent rolling, activation, adhesion and transmigration are essential stages of an inflammatory response. Chronic inflammation may entail atherosclerosis, one of the most devastating cardiovascular diseases. Understanding this mechanism is of crucial importance in immunology and in the development of anti-inflammatory drugs. Micropipette aspiration experiments show that leukocytes behave as viscoelastic drops during suction. The flow of non-Newtonian viscoelastic fluids can be described by differential, integral and rate-type constitutive equations. In this study, the rate-type Oldroyd-B model is used to capture the viscoelasticity of the leukocyte which is considered as a drop. Our main goal is to analyze a mathematical model describing the deformation and flow of an individual leukocyte in a microchannel flow. In this model we consider a coupled problem between a simplified Oldroyd-B system and a transport equation which describes the density considered as non constant in the Navier–Stokes equations. First we present the mathematical model and we prove the existence of solution, then we describe its numerical approximation using the level set method. Through the numerical simulations we analyze the hemodynamic effects of three inlet velocity values. We note that the hydrodynamic forces pushing the cell become higher with increasing inlet velocities.

scholarly journals Error estimates for a numerical approximation to the compressible barotropic Navier–Stokes equations

2015 ◽  
Vol 36 (2) ◽  
pp. 543-592 ◽  
Author(s):  
Thierry Gallouët ◽  
Raphaèle Herbin ◽  
David Maltese ◽  
Antonin Novotny
Keyword(s):  
Error Estimates ◽  
Numerical Approximation ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Navier Stokes Equations

scholarly journals Unstable periodic orbits in the Lorenz attractor

2011 ◽  
Vol 369 (1944) ◽  
pp. 2345-2353 ◽  
Author(s):  
Bruce M. Boghosian ◽  
Aaron Brown ◽  
Jonas Lätt ◽  
Hui Tang ◽  
Luis M. Fazendeiro ◽  
...  
Keyword(s):  
Periodic Orbits ◽  
Stokes Equations ◽  
Traditional Approach ◽  
Navier Stokes ◽  
Lorenz Attractor ◽  
Time Averaging ◽  
Unstable Periodic Orbits ◽  
Lorenz Equations ◽  
Navier Stokes Equations ◽  
Generic Orbit

We apply a new method for the determination of periodic orbits of general dynamical systems to the Lorenz equations. The accuracy of the expectation values obtained using this approach is shown to be much larger and have better convergence properties than the more traditional approach of time averaging over a generic orbit. Finally, we discuss the relevance of the present work to the computation of unstable periodic orbits of the driven Navier–Stokes equations, which can be simulated using the lattice Boltzmann method.

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