Numerical essays on the development of Turing patterns under the effect of incompressible convective fields: An approach from the cavity problem
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Mar 15, 2011
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Abstract
This work presents a number of numerical examples of reaction-difussion equations in Turing space, modified by convective fields in incompressible flows, using a Schnakenberg reaction mechanism. Examples were made in 2D using quad elements, which have an imposed advective field derived from the cavity problem solution. The developed model consists of an uncoupled system of equations including the reaction-advection-diffusion equations and the Navier-Stokes equations for incompressible flow. This system is solved simultaneously using the finite element method. Results illustrate that complex patterns are formed, mixing dots and stripes which reach a stable state. Changes in pattern concentration in both space and time are also shown due to the effect of the advective field. Numerical examples confirm that pattern formation is independent of initial conditions and mesh.
Keywords
Reaction-advection-diffusion, Turing instabilities, cavity problemaReacción-advección-difusión, inestabilidades de Turing, problema de la cavidad
References
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SAGUÉS, F. et al. Travelling-stripe forcing of Turing patterns. Physica D: Nonlinear Phenomena, 2004, vol. 199, núms. 1-2, pp. 235-242.
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WHITE, D. The planforms and onset of convection with a temperature dependent viscosity. Journal of Fluid Mechanics, 1988, vol. 191, pp. 247-286.
YI, F.; WEI, J. y SHI, J. Bifurcation and spatiotemporal patterns in a homogeneous diffusive predator-prey system. Journal of Differential Equations, 2009, vol. 246, pp. 1944-1977.
ZHANG, L. y LIU, S. Stability and pattern formation in a coupled arbitrary order of autocatalysis system. Applied Mathematical Modelling, 2009, vol. 33, pp. 884-896.
ZIENKIEWICZ, O. y TAYLOR, R. The finite element method. The Basis, vol. I. Butterworth-Heinemann, Oxford, 2000.
ZIENKIEWICZ, O. y TAYLOR, R. Finite Element Method, vol. 3. Butterworth-Heinemann College, 2000. pp. 5-150.
ALLGOWER, E. y GEORG, K. Numerical path following. Handbook of Numerical Analysis, 1997, vol. 5, pp. 3-207.
ARDES, M.; BUSSE, F. y WICHT, J. Thermal convection in rotating spherical shells. Physics of the Earth and Planetary Interiors, 1997, vol. 99, pp. 55-67.
BALKAREI, Y. et al. Regenerative oscillations, spatial-temporal single pulses and static inhomogeneous structures in optically bistable semiconductors. Optics Communications, 1988, vol. 66, núms. 1-2, pp. 161-166.
BAURMANNA, M.; GROSS, T. y FEUDEL, U. Instabilities in spatially extended predator–prey systems: Spatio-temporal patterns in the neighborhood of Turing–Hopf bifurcations. Journal of Theoretical Biology, 2007, vol. 245, pp. 220-229.
CHUNG, T. Computational fluid dynamics. Cambridge: Cambridge University Press, 2002.
CRAUSTE, F.; LHASSAN, M. y KACHA, A. A delay reaction-diffusion model of the dynamics of botulinum in fish. Mathematical Biosciences, 2008, vol. 216, pp. 17-29.
FERREIRA, S.; MARTINS, M. y VILELA, M. Reaction-diffusion model for the growth of avascular tumor. Physical Review, 2002, vol. 65, núm. 2.
FRANCA, L. y FREY, S. Stabilized finite element methods: II. The incompressible Navier-Stokes equations. Computer Methods in Applied Mechanics and Engineering, 1992, vol. 99, pp 209-233.
FREDERIK, H. et al. Pigmentation pattern formation in butterflies: experiments and models. C. R. Biologies, 2003, vol. 326, pp. 717-727.
GARCÍA-AZNAR, J. et al. Computational simulation of fracture healing: Influence of interfragmentary movement on the callus growth. Journal of Biomechanics, 2007, vol. 40, núm. 7, pp. 1467-1476.
GARZÓN, D. Simulación de procesos de reacción-difusión: aplicación a la morfogénesis del tejido óseo. Tesis de PhD. Zaragoza: Universidad de Zaragoza, 2007.
HIRAYAMA, O. y TAKAKI, R. Thermal convection of a fluid with temperature-dependent viscosity. Fluid Dynamics Research, 1988, vol. 12, núm. 1, pp. 35-47.
HOFFMAN, J. Numerical methods for engineers and scientists. New York: Mc Graw-Hill, 1992.
HUGHES, T. The finite element method (linear static and dynamic finite element analysis). s. l.: Dover, 2000.
HUGHES, T.; FRANCA, L. y BALESTRA, M. A new finite element formulation for computational and fluid dynamics: V. Circumventing the Babuska-Brezzi condition: a stable Petrov-Galerkin formulation of the Stokes problema accommodating equal-order interpolations. Computer Methods in Applied Mechanics and Engineering, 1986, vol. 59, pp. 85-99.
JAVIERRE, E. et al. Numerical modeling of a mechano-chemical theory for wound contraction analysis. International Journal of Solids and Structures, 2009, vol. 46, núm. 20, pp. 3597-3606.
KRINSKY, V. I. Self-organisation: Auto-waves and structures far from equilibrium. Philadelphia: Springer, 1984.
LIR, J. y LIN, T. Visualization of roll patterns in Rayleigh–Bénard convection of air in rectangular shallow cavity. International Journal of Heat and Mass Transfer, 2001, vol. 44, pp. 2889-2902.
MADZVAMUSE, A. A numerical approach to the study of spatial pattern formation in the ligaments of arcoid bivalves. Bulletin of Mathematical Biology, 2002, vol. 64, pp. 501-530.
MADZVAMUSE, A. A numerical approach to the study of spatial pattern formation. D. Phil. Thesis. Oxford: Oxford University, 2000.
MADZVAMUSE, A. Turing instability conditions for growing domains with divergence free mesh velocity. Nonlinear Analysis: Theory, Methods & Applications, 2009, vol. 71, núm. 12, pp. 2250-2257.
MADZVAMUSE, A.; WATHEN, A. y MAINI, P. A moving grid finite element method applied to a model biological pattern generator. Journal of Computational Physics, 2003, vol. 190, pp. 478-500.
MEI, Z. Numerical bifurcation analysis for reaction-diffusion equations. Berlin: Springer-Verlag, 2000.
NOZAKURA, T. e IKEUCHI, S. Formation of dissipative structures in galaxies. Astrophysical Journal, 1984, vol. 279, pp. 40-52.
PAINTER, K; MAINI, P. y OTHMER, H. A chemotactic model for the advance and retreat of the primitive streak in avian development. Bulletin of Mathematical Biology, 2000, vol. 62, pp. 501-525.
PAINTER, K.; OTHMER, H. y MAINI, P. Stripe formation in juvenile Pomacanthus via chemotactic response to a reaction-diffusion mechanism. Proceedings of National Academy Sciences USA, 1999, vol. 96, núm. 10, pp. 5549-5554.
ROSSI, F. et al. Dynamics of pattern formation in biomimetic systems. Journal of Theoretical Biology, 2008, vol. 255, pp. 404-412.
ROTHSCHILD, B. y AULT, J. Population-dynamic instability as a cause of patch structure. Ecological Modelling, 1996, vol. 93, pp. 237-239.
RÜDIGER, S. et al. Theory of pattern forming systems under traveling-wave forcing. Physics Reports, 2007, vol. 447, núms. 3-6, pp. 73-111.
SAGUÉS, F. et al. Travelling-stripe forcing of Turing patterns. Physica D: Nonlinear Phenomena, 2004, vol. 199, núms. 1-2, pp. 235-242.
TU, J.; YEOH, G. H. y LIU, C. Computational fluid dynamics: a practical approach. Oxford: Elsevier, 2008.
WHITE, D. The planforms and onset of convection with a temperature dependent viscosity. Journal of Fluid Mechanics, 1988, vol. 191, pp. 247-286.
YI, F.; WEI, J. y SHI, J. Bifurcation and spatiotemporal patterns in a homogeneous diffusive predator-prey system. Journal of Differential Equations, 2009, vol. 246, pp. 1944-1977.
ZHANG, L. y LIU, S. Stability and pattern formation in a coupled arbitrary order of autocatalysis system. Applied Mathematical Modelling, 2009, vol. 33, pp. 884-896.
ZIENKIEWICZ, O. y TAYLOR, R. The finite element method. The Basis, vol. I. Butterworth-Heinemann, Oxford, 2000.
ZIENKIEWICZ, O. y TAYLOR, R. Finite Element Method, vol. 3. Butterworth-Heinemann College, 2000. pp. 5-150.
How to Cite
Garzón-Alvarado, D. A., Galeano-Urueña, C. H., & Mantilla-González, J. M. (2011). Numerical essays on the development of Turing patterns under the effect of incompressible convective fields: An approach from the cavity problem. Ingenieria Y Universidad, 14(2), 239. https://doi.org/10.11144/Javeriana.iyu14-2.ensf
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