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 DF  Nonlinear Fluid Dynamics
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 Department of Applied Physics
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 Barcelona School of Civil Engineering (ETSECCPB)
 juan.j.sanchezupc.edu
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A comparison of highorder time integrators for highly supercritical thermal convection in rotating spherical shells
Garcia Gonzalez, Fernando; Net Marce, Marta; Sanchez Umbria, Juan Jose
Lecture notes in computational science and engineering
Date of publication: 20140110
Journal article
Read the abstract View Share Reference managersThe efficiency of implicit and semiimplicit time integration codes based on backward differentiation and extrapolation formulas for the solution of the threedimensional Boussinesq thermal convection equations in rotating spherical shells was studied in Garcia et al. (J Comput Phys 229:7997¿8010, 2010) at weakly supercritical Rayleigh numbers R, moderate (103) and low (104) Ekman numbers, E, and Prandtl number s = 1. The results presented here extend the previous study and focus on the effect of s and R by analyzing the efficiency of the methods for obtaining solutions at TeX, s = 0. 1 and low and high supercritical R. In the first case (quasiperiodic solutions) the decrease of one order of magnitude does not change the results significantly. In the second case (spatiotemporal chaotic solutions) the differences in the behavior of the semiimplicit codes due to the different treatment of the Coriolis term disappear because the integration is dominated by the nonlinear terms. As in Garcia et al. (J Comput Phys 229:7997¿8010, 2010), high order methods, either with or without time step and order control, increase the efficiency of the time integrators and allow to obtain more accurate solutions. 
Exponential versus IMEX highorder time integrators for thermal convection in rotating spherical shells
Garcia Gonzalez, Fernando; Bonaventura, Luca; Net Marce, Marta; Sanchez Umbria, Juan Jose
Journal of computational physics
Date of publication: 20140124
Journal article
Read the abstract View Share Reference managersWe assess the accuracy and efficiency of several exponential time integration methods coupled to a spectral discretization of the threedimensional Boussinesq thermal convection equations in rotating spherical shells. Exponential methods are compared to implicit¿explicit (IMEX) multistep methods already studied previously in [1]. The results of a wide range of numerical simulations highlight the superior accuracy of exponential methods for a given time step, especially when employed with large time steps and at low Ekman number. However, presently available implementations of exponential methods appear to be in general computationally more expensive than those of IMEX methods and further research is needed to reduce their computational cost per time step. A physically justified extrapolation argument suggests that some exponential methods could be the most efficient option for integrating flows near Earth's outer core conditions. 
Numerical bifurcation methods and their application to fluid dynamics: analysis beyond simulation
Dijkstra, Hendrik; Wubs, Fred W.; Cliffe, Andrew K.; Doedel, Eusebius J.; Dragomirescu, Ioana Florica; Eckhardt, Bruno; Gelfgat, Alexander Yu; Hazel, Andrew L.; Lucarini, Valerio; Salinger, Andrew G.; Phipps, Erik T.; Sanchez Umbria, Juan Jose; Schuttelaars, Henk M.; Tuckerman, Laurette S.; Thiele, Uwe
Communications in computational physics
Date of publication: 201401
Journal article
Read the abstract View Share Reference managersWe provide an overview of current techniques and typical applications of numerical bifurcation analysis in fluid dynamical problems. Many of these problems are characterized by highdimensional dynamical systems which undergo transitions as parameters are changed. The computation of the critical conditions associated with these transitions, popularly referred to as 'tipping points', is important for understanding the transition mechanisms. We describe the two basic classes of methods of numerical bifurcation analysis, which differ in the explicit or implicit use of the Jacobian matrix of the dynamical system. The numerical challenges involved in both methods arementioned and possible solutions to current bottlenecks are given. To demonstrate that numerical bifurcation techniques are not restricted to relatively lowdimensional dynamical systems, we provide several examples of the application of the modern techniques to a diverse set of fluid mechanical problems. ©c 2014 GlobalScience Press. 
A parallel algorithm for the computation of invariant tori in largescale dissipative systems
Sanchez Umbria, Juan Jose; Net Marce, Marta
Physica. D, Nonlinear phenomena
Date of publication: 201306
Journal article
Read the abstract View Share Reference managersA parallelizable algorithm to compute invariant tori of highdimensional dissipative systems, obtained upon discretization of PDEs is presented. The size of the set of equations to be solved is only a small multiple of the dimension of the original system. The sequential and parallel implementations are compared with a previous method (Sánchez et al. (2010)) [11], showing that important savings in wallclock time can be achieved. In order to test it, a thermal convection problem of a binary mixture of fluids has been used. The new method can also be applied to problems with very low rotation numbers, for which the previous is not suitable. This is tested in two examples of twodimensional maps.
A parallelizable algorithm to compute invariant tori of highdimensional dissipative systems, obtained upon discretization of PDEs is presented. The size of the set of equations to be solved is only a small multiple of the dimension of the original system. The sequential and parallel implementations are compared with a previous method (Sánchez et al. (2010)) [11], showing that important savings in wallclock time can be achieved. In order to test it, a thermal convection problem of a binary mixture of fluids has been used. The new method can also be applied to problems with very low rotation numbers, for which the previous is not suitable. This is tested in two examples of twodimensional maps. 
Computation of azimuthal waves and their stability in thermal convection in rotating spherical shells with application to the study of a doubleHopf bifurcation
Sanchez Umbria, Juan Jose; Garcia Gonzalez, Fernando; Net Marce, Marta
Physical review E: statistical, nonlinear, and soft matter physics
Date of publication: 20130322
Journal article
Read the abstract Access to the full text Share Reference managersA methodology to compute azimuthal waves, appearing in thermal convection of a pure fluid contained in a rotating spherical shell, and to study their stability is presented. It is based on continuation, NewtonKrylov, and Arnoldi methods. An application to the study of a doubleHopf bifurcation of the basic state is shown for Ekman and Prandtl numbers E=104 and =0.1, respectively, radius ratios n[0.32,0.35], Rayleigh numbers R[1.8105,6105], and nonslip and perfectly conducting boundary conditions. The knowledge of the bifurcation diagrams, including the unstable solutions, allows one to understand the coexistence of stable thermal Rossby waves of different azimuthal wave numbers at some parameter regions, and the origin of some new intermittent solutions found, as trajectories close to heteroclinic chains. Moreover, the structure of the eigenfunctions at the secondary bifurcations explains the existence of the amplitude and shape modulated waves.
A methodology to compute azimuthal waves, appearing in thermal convection of a pure fluid contained in a rotating spherical shell, and to study their stability is presented. It is based on continuation, NewtonKrylov, and Arnoldi methods. An application to the study of a doubleHopf bifurcation of the basic state is shown for Ekman and Prandtl numbers E=10−4 and σ=0.1, respectively, radius ratios η∈[0.32,0.35], Rayleigh numbers R∈[1.8×105,6×105], and nonslip and perfectly conducting boundary conditions. The knowledge of the bifurcation diagrams, including the unstable solutions, allows one to understand the coexistence of stable thermal Rossby waves of different azimuthal wave numbers at some parameter regions, and the origin of some new intermittent solutions found, as trajectories close to heteroclinic chains. Moreover, the structure of the eigenfunctions at the secondary bifurcations explains the existence of the amplitude and shape modulated waves. 
Numerical study of the onset of thermosolutal convection in rotating spherical shells
Net Marce, Marta; Garcia Gonzalez, Fernando; Sanchez Umbria, Juan Jose
Physics of fluids
Date of publication: 20120601
Journal article
Read the abstract Access to the full text Share Reference managersThe influence of an externally enforced compositional gradient on the onset of convection of a mixture of two components in a rotating fluid spherical shell is studied for Ekman numbers E = 10−3 and E = 10−6, Prandtl numbers σ = 0.1, 0.001, Lewis numbers τ = 0.01, 0.1, 0.8, and radius ratio η = 0.35. The Boussinesq approximation of the governing equations is derived by taking the denser component of the mixture for the equation of the concentration. Differential and internal heating, an external compositional gradient, and the Soret and Dufour effects are included in the model. By neglecting these two last effects, and by considering only differential heating, it is found that the critical thermal Rayleigh number Rec depends strongly on the direction of the compositional gradient. The results are compared with those obtained previously for pure fluids of the same σ. The influence of the mixture becomes significant when the compositional Rayleigh number Rc is at least of the same order of magnitude as the known Rec computed without mixture. For positive and sufficiently large compositional gradients, Rec decreases and changes sign, indicating that the compositional convection becomes the main source of instability. Then the critical wave number mc decreases, and the drifting waves slow down drastically giving rise to an almost stationary pattern of convection. Negative gradients delay the onset of convection and determine a substantial increase of mc and ωc for Rc sufficiently high. Potential laws are obtained numerically from the dependence of Rec and of the critical frequency ωc on Rc, for the moderate and small Ekman numbers explored. 
Thermal convection in rotating spherical shells
Garcia Gonzalez, Fernando
Defense's date: 20121130
Department of Applied Physics, Universitat Politècnica de Catalunya
Theses
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Cálculo numérico de variedades invariantes en EDPS disipativas. Aplicaciones a la convección térmica
Net Marce, Marta; Garcia Gonzalez, Fernando; Sanchez Umbria, Juan Jose
Participation in a competitive project
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Computation of invariant tori by NewtonKrylov methods in largescale dissipative systems
Sanchez Umbria, Juan Jose; Net Marce, Marta; Simó, Carles
Physica. D, Nonlinear phenomena
Date of publication: 201002
Journal article
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On the multiple shooting continuation of periodic orbits by NewtonKrylov methods
Sanchez Umbria, Juan Jose; Net Marce, Marta
International journal of bifurcation and chaos
Date of publication: 2010
Journal article
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On the multiple shooting continuation of periodic orbits by NewtonKrylov methods
Sanchez Umbria, Juan Jose; Net Marce, Marta
International journal of bifurcation and chaos
Date of publication: 201001
Journal article
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A comparison of highorder time integrators for thermal convection in rotating spherical shells
Garcia Gonzalez, Fernando; Net Marce, Marta; GarciaArchilla, Bosco; Sanchez Umbria, Juan Jose
Journal of computational physics
Date of publication: 20101001
Journal article
Read the abstract Access to the full text Share Reference managersA numerical study of several time integration methods for solving the threedimensional Boussinesq thermal convection equations in rotating spherical shells is presented. Implicit and semiimplicit time integration techniques based on backward differentiation and extrapolation formulae are considered. The use of Krylov techniques allows the implicit treatment of the Coriolis term with low storage requirements. The codes are validated with a known benchmark, and their efficiency is studied. The results show that the use of high order methods, especially those with time step and order control, increase the efficiency of the time integration, and allows to obtain more accurate solutions.
Postprint (author’s final draft) 
Estudio comparativo de la eficiencia de diversos integradores para la simulación numérica de la convección térmica en geometría esférica y rotación
Garcia Gonzalez, Fernando; Net Marce, Marta; GarciaArchilla, Bosco; Sanchez Umbria, Juan Jose
Congreso de Ecuaciones Diferenciales y Aplicaciones / Congreso de Matemática Aplicada
Presentation's date: 20090924
Presentation of work at congresses
Read the abstract Access to the full text Share Reference managersThe efficiency of implicit and semiimplicit time integration codes based on bac kward differentiation and extrapolation formulae for the solution of the threedimensio nal Boussinesq thermal convection equations in rotating spherical shells is studied. The use of Krylov techniques allows the implicit treatment of the Coriolis term with low storage requirements. The results show that high order methods, either with or without time step and order control, increase the efficiency of the time integrators, and allow to obtain more accurate solutions. 
On the onset of lowPrandtlnumber convection in rotating spherical shells: nonslip boundary conditions
Net Marce, Marta; Garcia, Ferran; Sanchez Umbria, Juan Jose
Journal of fluid mechanics
Date of publication: 200804
Journal article
Read the abstract Access to the full text Share Reference managersAccurate numerical computations of the onset of thermal convection in wide rotating spherical shells are presented. LowPrandtlnumber (σ) fluids, and nonslip boundary conditions are considered. It is shown that at small Ekman numbers (E), and very low σ values, the wellknown equatorially trapped patterns of convection are superseded by multicellular outerequatoriallyattached modes. As a result, the convection spreads to higher latitudes affecting the body of the fluid, and increasing the internal viscous dissipation. Then, from E<10−5, the critical Rayleigh number (Rc) fulfils a powerlaw dependence Rc ∼E −4/3, as happens for moderate and high Prandtl numbers. However, the critical precession frequency (ωc) and the critical azimuthal wavenumber (mc) increase discontinuously, jumping when there is a change of the radial and latitudinal structure of the preferred eigenfunction. In addition, the transition between spiralling columnar (SC), and outerequatoriallyattached (OEA) modes in the (σ, E)space is studied. The evolution of the instability mechanisms with the parameters prevents multicellular modes being selected from σ 0.023. As a result, and in agreement with other authors, the spiralling columnar patterns of convection are already preferred at the Prandtl number of the liquid metals. It is also found that, out of the rapidly rotating limit, the prograde antisymmetric (with respect to the equator) modes of small mc can be preferred at the onset of the primary instability. 
Computation of invariant manifolds in largescale dissipative systems
Sanchez Umbria, Juan Jose
Workshop Computational Dynamics
Presentation's date: 20080616
Presentation of work at congresses
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Antisymmetric polar modes of thermal convection in rotating spherical fluid shells at high Taylor numbers
Garcia Gonzalez, Fernando; Net Marce, Marta; Sanchez Umbria, Juan Jose
Physical review letters
Date of publication: 20081103
Journal article
Read the abstract Access to the full text Share Reference managersThe onset of thermal convection in a rotating spherical shell of intermediate radius ratio ¼ 0:4 is studied numerically for Taylor numbers Ta 1011 and the Prandtl number of the liquid sodium ( ¼ 0:01). For the first time, it is shown that at very high Taylor numbers the first unstable mode can be antisymmetric with respect to the equator and confined inside a cylinder tangent to the inner sphere at the equator (polar mode). The exponent of the power law determined from the asymptotic dependence of the critical Rayleigh number for very high Ta is 0.57, lower than 2=3, given theoretically for the spiraling columnar modes, and than 0.63, found numerically for the outer equatorially attached modes 
Computation of invariant manifolds in largescale dissipative systems
Sanchez Umbria, Juan Jose
World Congress on Computational Mechanis and European Congres on Computational Methods in Applied Sciences and Engineering
Presentation's date: 20080630
Presentation of work at congresses
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Convección térmica de fluidos puros y binarios en una capa de fluido esférica sometida a rotación elevada
Sanchez Umbria, Juan Jose; Net Marce, Marta
Participation in a competitive project
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NewtonKrylov continuation of periodic orbits in largescale dissipative systems
Sanchez Umbria, Juan Jose
SIAM Conference on Applications of Dynamical Systems
Presentation's date: 20070528
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Numerical techniques for the onset of thermal convection in spherical geometry
Sanchez Umbria, Juan Jose
SIAM Conference on Applications of Dynamical Systems
Presentation's date: 20070528
Presentation of work at congresses
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Amplitude equations close to a triple(+1) bifurcation point of D~4symmetric periodic orbits in O(2)equivariant systems
Sanchez Umbria, Juan Jose; Net Marce, Marta; Vega, J M
Discrete and continuous dynamical systems. Series B
Date of publication: 200612
Journal article
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Krylov methods and determinants for detecting bifurcations in one parameter dependent partial differential equations
GarciaArchilla, B; Sanchez Umbria, Juan Jose; SIMO, C
Bit numerical mathematics
Date of publication: 200612
Journal article
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Computation of the normal form near a triple +1 bifurcation of a periodic orbit in a large scale O(2)equivariant system.
Sanchez Umbria, Juan Jose
Carles Simó Fest (60th birthday). Conference on Dynamical Systems
Presentation's date: 20060529
Presentation of work at congresses
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Calculo de una forma normal alrededor de una orbita periodica multicritica en un problema de Mecanica de fluidos
Sanchez Umbria, Juan Jose
Ddays2006 (Reunion de la red DANCE)
Presentation's date: 20061018
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Computation of a normal form near a multicritical periodic orbit in a thermal convection problem.
Sanchez Umbria, Juan Jose
Applied Dynamical Systems Workshop. Advanced numerical methods for mathematical modelling.
Presentation's date: 20060622
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Thermal convection in rotating spherical shells
Sanchez Umbria, Juan Jose
Jornades de doctorat del programa Física Aplicada i Simulació en Ciències
Presentation's date: 20060623
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Metodos de Krylov, determinantes y bifurcaciones en EDPs.
Sanchez Umbria, Juan Jose
Ddays2006 (Reunion de la red DANCE)
Presentation's date: 20061018
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Symmetric periodic orbits and global dynamics of quasiperiodic flows in a O(2) equivariant system: Twodimensional thermal convection
Net Marce, Marta; Sanchez Umbria, Juan Jose
International journal of bifurcation and chaos
Date of publication: 200512
Journal article
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Continuation of periodic orbits in largescale dissipative systems
Sanchez Umbria, Juan Jose; Net Marce, Marta; GarcíaArchilla, B; Simó, C
Date of publication: 200512
Book chapter
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AJUTS DE SUPORT ALS GRUPS DE RECERCA DE CATALUNYA
Sanchez Umbria, Juan Jose; JORBA, ANGEL
Participation in a competitive project
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Ajuts de suport als grups de recerca de Catalunya: Grup de Sistemes Dinàmics
Jorba Monte, Angel; Baldoma Barraca, Inmaculada Concepcion; Sanchez Umbria, Juan Jose; Net Marce, Marta
Participation in a competitive project
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Numerical determination of the coefficients of the amplitude equations near a triple +1 bifurcation of a periodic orbit in a largescale O(2) system.
Sanchez Umbria, Juan Jose; Net Marce, Marta
Qualitative Numerical Analysis of Highdimensional Nonlinear Systems
Presentation's date: 20050321
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NewtonKrylov continuation of periodic orbits for NavierStokes flows
Sanchez Umbria, Juan Jose; Net Marce, Marta; GarcíaArchilla, B; Simó, C
Journal of computational physics
Date of publication: 200411
Journal article
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Continuation of periodic orbits in a thermal convection problem.
Sanchez Umbria, Juan Jose
Workshop on Analysis and Continuation of Bifurcations 2004.
Presentation's date: 20040519
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Estudio numérico de los primeros estadios de la convección térmica en una capa esférica de fluido en rotación
Sanchez Umbria, Juan Jose; Net Marce, Marta
Participation in a competitive project
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From stationary to complex timedependent flows at moderate rayleigh numbers in twodimensional annular thermal convection
Net Marce, Marta; Alonso Maleta, Maria Aranzazu; Sanchez Umbria, Juan Jose
Physics of fluids
Date of publication: 200305
Journal article
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Continuation of periodic orbits in largescale dissipative systems
Sanchez Umbria, Juan Jose
EQUADIFF 2003. International Conference of Differential Equations
Presentation's date: 20030722
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Spatially resonant interactions in annular convection
Alonso Maleta, Maria Aranzazu; Sanchez Umbria, Juan Jose; Net Marce, Marta
Date of publication: 200307
Book chapter
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Cluster de cálculo paralelo de altas prestaciones
Albareda Tiana, Alfonso; Sanchez Umbria, Juan Jose; Vilaseca Alavedra, Ramon; Net Marce, Marta
Participation in a competitive project
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On stable Taylor vortices above the transition to wavy vortices
Antonijuan Rull, Josefina; Sanchez Umbria, Juan Jose
Physics of fluids
Date of publication: 200205
Journal article
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A continuation and bifurcation technique for NavierStokes flows
Sanchez Umbria, Juan Jose; Marques Truyol, Francisco; López, J
Journal of computational physics
Date of publication: 200210
Journal article
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Dinamica espaciotemporal compleja inducida por las simetrias de reflexion y rotacion en un problema de conveccion termica
Sanchez Umbria, Juan Jose
NoLineal 2002
Presentation's date: 20020605
Presentation of work at congresses
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Thermal Rossby waves in a rotating annulus. Their stability
Pino Gonzalez, David; Net Marce, Marta; Sanchez Umbria, Juan Jose; Mercader Calvo, Maria Isabel
Physical review E, statistical physics, plasmas, fluids, and related interdisciplinary topics
Date of publication: 200104
Journal article
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Oscillatory modes in an enclosed swirling flow
LOPEZ, J M; Marques Truyol, Francisco; Sanchez Umbria, Juan Jose
Journal of fluid mechanics
Date of publication: 200106
Journal article
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Low Prandtl number thermal convection between two corotating cylinders
Alonso Maleta, Maria Aranzazu; Sanchez Umbria, Juan Jose; Net Marce, Marta
12 International Couette Taylor Workshop
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Flujos oscilatorios en sistemas en rotación: convección térmica y problema de TaylorCouette
Alonso Maleta, Maria Aranzazu; Net Marce, Marta; Sanchez Umbria, Juan Jose
Participation in a competitive project
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Transition to wavy vortices for intermediate values of the radius ratio
Sanchez Umbria, Juan Jose
12 International Couette Taylor Workshop
Presentation's date: 20010906
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Multiple unsteady solutions and symmetry breaking in a confined swirling flow
LOPEZ, J M; Marques Truyol, Francisco; Sanchez Umbria, Juan Jose; Blackburn, H.M.
Bulletin of the American Physical Society
Date of publication: 200011
Journal article
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Continuation techniques applied to the TaylorCouette problem. Spiral flow and the stability of Taylor vortices
Antonijuan Rull, Josefina; Sanchez Umbria, Juan Jose
Notes on numerical fluid mechanics
Date of publication: 200006
Journal article
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Transition to Temporal Chaos in an O(2)Symmetric Convective System for Low Prandtl Numbers
Alonso Maleta, Maria Aranzazu; Sanchez Umbria, Juan Jose; Net Marce, Marta
Progress of theoretical physics supplement
Date of publication: 200002
Journal article
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