Abstract
One important source of dynamic transitions and pattern formation is transition and stability problems in fluid dynamics, including in particular Rayleigh–Bénard convection, the Couette–Taylor problem, the Couette–Poiseuille–Taylor problem, and the parallel shear flow problem. The study of these basic problems leads to a better understanding of turbulent behavior in fluid flows, which in turn often leads to new insights and methods in the solution of other problems in science and engineering.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Busse, F. (1978). Non-linear properties of thermal convection. Reports on Progress in Physics 41, 1929–1967.
Caffarelli, L., R. Kohn, and L. Nirenberg (1982a). On the regularity of the solutions of navier-stokes equations. Comm. Pure Appl. Math. 35, 771–831.
Chandrasekhar, S. (1981). Hydrodynamic and Hydromagnetic Stability. Dover Publications, Inc.
Cross, M. and P. Hohenberg (1993). Pattern formation outside of equilibrium. Reviews of Modern Physics 65(3), 851–1112.
Drazin, P. and W. Reid (1981). Hydrodynamic Stability. Cambridge University Press.
Foiaş, C. and R. Temam (1979). Some analytic and geometric properties of the solutions of the evolution Navier-Stokes equations. J. Math. Pures Appl. (9) 58(3), 339–368.
Friedlander, S., M. Vishik, and V. Yudovich (2000). Unstable eigenvalues associated with inviscid fluid flows. J. Math. Fluid Mech. 2(4), 365–380.
Getling, A. V. (1997). Rayleigh-Benard convection: Structures and dynamics. Advanced Series in Nonlinear Dynamics. World Scientific.
Ghil, M. and S. Childress (1987). Topics in Geophysical Fluid Dynamics: Atmospheric Dynamics, Dynamo Theory, and Climate Dynamics. Springer-Verlag, New York.
Ghil, M., T. Ma, and S. Wang (2001). Structural bifurcation of 2-D incompressible flows. Indiana Univ. Math. J. 50(Special Issue), 159–180. Dedicated to Professors Ciprian Foias and Roger Temam (Bloomington, IN, 2000).
Ghil, M., T. Ma, and S. Wang (2005). Structural bifurcation of 2-D nondivergent flows with Dirichlet boundary conditions: applications to boundary-layer separation. SIAM J. Appl. Math. 65(5), 1576–1596 (electronic).
Goldstein, S. (1937). The stability of viscous fluid flow between rotating cylinders. Proc. Camb. Phil. Soc. 33, 41–61.
Golubitsky, M., I. Stewart, and D. G. Schaeffer (1988). Singularities and groups in bifurcation theory. Vol. II, Volume 69 of Applied Mathematical Sciences. New York: Springer-Verlag.
Hopf, E. (1951). Über die Anfangswertaufgabe für die hydrodynamischen Grundgleichungen. Math. Nachr. 4, 213–231.
Hsia, C.-H., T. Ma, and S. Wang (2007). Stratified rotating Boussinesq equations in geophysical fluid dynamics: dynamic bifurcation and periodic solutions. J. Math. Phys. 48(6), 065602, 20.
Hsia, C.-H., T. Ma, and S. Wang (2010). Rotating Boussinesq equations: dynamic stability and transitions. Discrete Contin. Dyn. Syst. 28(1), 99–130.
Iooss, G. and M. Adelmeyer (1998). Topics in bifurcation theory and applications (Second ed.), Volume 3 of Advanced Series in Nonlinear Dynamics. River Edge, NJ: World Scientific Publishing Co. Inc.
Kato, T. (1995). Perturbation theory for linear operators. Classics in Mathematics. Berlin: Springer-Verlag. Reprint of the 1980 edition.
Kirchgässner, K. (1975). Bifurcation in nonlinear hydrodynamic stability. SIAM Rev. 17(4), 652–683.
Kiselev, A. A. and O. A. Ladyženskaya (1957). On the existence and uniqueness of the solution of the nonstationary problem for a viscous, incompressible fluid. Izv. Akad. Nauk SSSR. Ser. Mat. 21, 655–680.
Koschmieder, E. L. (1993). Bénard cells and Taylor vortices. Cambridge Monographs on Mechanics. Cambridge University Press.
Ladyženskaya, O. A. (1958). On the nonstationary Navier-Stokes equations. Vestnik Leningrad. Univ. 13(19), 9–18.
Ladyzhenskaya, O. A. (1982). The finite-dimensionality of bounded invariant sets for the Navier-Stokes system and other dissipative systems. Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 115, 137–155, 308. Boundary value problems of mathematical physics and related questions in the theory of functions, 14.
Lappa, M. (2009). Thermal convection: Patterns, evolution and stability. Wiley.
Leray, J. (1933). Étude de diverses équations intégrales non linéaires et de quelques problèmes que pose l’dydrodynamique. J. Math. Pures Appl 12, 1–82.
Lions, J.-L., R. Temam, and S. Wang (1992b). On the equations of the large-scale ocean. Nonlinearity 5(5), 1007–1053.
Ma, T. and S. Wang (2000). Structural evolution of the Taylor vortices. M2AN Math. Model. Numer. Anal. 34(2), 419–437. Special issue for R. Temam’s 60th birthday.
Ma, T. and S. Wang (2001). Structure of 2D incompressible flows with the Dirichlet boundary conditions. Discrete Contin. Dyn. Syst. Ser. B 1(1), 29–41.
Ma, T. and S. Wang (2004a). Boundary layer separation and structural bifurcation for 2-D incompressible fluid flows. Discrete Contin. Dyn. Syst. 10(1–2), 459–472.
Ma, T. and S. Wang (2004b). Dynamic bifurcation and stability in the Rayleigh-Bénard convection. Commun. Math. Sci. 2(2), 159–183.
Ma, T. and S. Wang (2004c). Interior structural bifurcation and separation of 2D incompressible flows. J. Math. Phys. 45(5), 1762–1776.
Ma, T. and S. Wang (2005b). Bifurcation theory and applications, Volume 53 of World Scientific Series on Nonlinear Science. Series A: Monographs and Treatises. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ.
Ma, T. and S. Wang (2005d). Geometric theory of incompressible flows with applications to fluid dynamics, Volume 119 of Mathematical Surveys and Monographs. Providence, RI: American Mathematical Society.
Ma, T. and S. Wang (2006). Stability and bifurcation of the Taylor problem. Arch. Ration. Mech. Anal. 181(1), 149–176.
Ma, T. and S. Wang (2007a). Rayleigh-Bénard convection: dynamics and structure in the physical space. Commun. Math. Sci. 5(3), 553–574.
Ma, T. and S. Wang (2008d). Exchange of stabilities and dynamic transitions. Georgian Mathematics Journal 15:3, 581–590.
Ma, T. and S. Wang (2009a). Boundary-layer and interior separations in the Taylor-Couette-Poiseuille flow. J. Math. Phys. 50(3), 033101, 29.
Ma, T. and S. Wang (2010a). Dynamic transition and pattern formation in Taylor problem. Chin. Ann. Math. Ser. B 31(6), 953–974.
Ma, T. and S. Wang (2010b). Dynamic transition theory for thermohaline circulation. Phys. D 239(3–4), 167–189.
Ma, T. and S. Wang (2010c). Tropical atmospheric circulations: dynamic stability and transitions. Discrete Contin. Dyn. Syst. 26(4), 1399–1417.
Ma, T. and S. Wang (2011b). El Niño southern oscillation as sporadic oscillations between metastable states. Advances in Atmospheric Sciences 28:3, 612–622.
Mezić, I. (2001). Break-up of invariant surfaces in action-angle-angle maps and flows. Phys. D 154(1–2), 51–67.
Newton, P. K. (2001). The N-vortex problem, Volume 145 of Applied Mathematical Sciences. New York: Springer-Verlag. Analytical techniques.
Palm, E. (1975). Nonlinear thermal convection. Annual Review of Fluid Mechanics 7(1), 39–61.
Pedlosky, J. (1987). Geophysical Fluid Dynamics (second ed.). New-York: Springer-Verlag.
Rabinowitz, P. H. (1968). Existence and nonuniqueness of rectangular solutions of the Bénard problem. Arch. Rational Mech. Anal. 29, 32–57.
Raguin, L. G. and J. G. Georgiadis (2004). Kinematics of the stationary helical vortex mode in taylor-couette-poiseuille flow. J. Fluid Mech. 516, 125–154.
Rayleigh, L. (1916). On convection currents in a horizontal layer of fluid, when the higher temperature is on the under side. Phil. Mag. 32(6), 529–46.
Rogerson, A. M., P. D. Miller, L. J. Pratt, and C. K. R. T. Jones (1999). Lagrangian motion and fluid exchange in a barotropic meandering jet. J. Phys. Oceanogr. 29(10), 2635–2655.
Salby, M. L. (1996). Fundamentals of Atmospheric Physics. Academic Press.
Samelson, R. M. and S. Wiggins (2006). Lagrangian transport in geophysical jets and waves, Volume 31 of Interdisciplinary Applied Mathematics. New York: Springer. The dynamical systems approach.
Surana, A., O. Grunberg, and G. Haller (2006). Exact theory of three-dimensional flow separation. I. Steady separation. J. Fluid Mech. 564, 57–103.
Swinney, H. L. (1988). Instabilities and chaos in rotating fluids. In Nonlinear evolution and chaotic phenomena (Noto, 1987), Volume 176 of NATO Adv. Sci. Inst. Ser. B Phys., pp. 319–326. New York: Plenum.
Taylor, G. I. (1923). Stability of a viscous liquid contained between two rotating cylinders. Philos. Trans. Royl London Ser. A 223, 289–243.
Temam, R. (1997). Infinite-dimensional dynamical systems in mechanics and physics (Second ed.), Volume 68 of Applied Mathematical Sciences. New York: Springer-Verlag.
Temam, R. (2001). Navier-Stokes equations. AMS Chelsea Publishing, Providence, RI. Theory and numerical analysis, Reprint of the 1984 edition.
Velte, W. (1966). Stabilität and verzweigung stationärer lösungen der davier-stokeschen gleichungen. Arch. Rat. Mech. Anal. 22, 1–14.
Vishik, M. I. (1992). Asymptotic behaviour of solutions of evolutionary equations. Lezioni Lincee. [Lincei Lectures]. Cambridge: Cambridge University Press.
Wiggins, S. (1990). Introduction to applied nonlinear dynamical systems and chaos, Volume 2 of Texts in Applied Mathematics. New York: Springer-Verlag.
Yudovich, V. I. (1966). Secondary flows and fluid instability between rotating cylinders. Appl. Math. Mech. 30, 822–833.
Yudovich, V. I. (1967a). Free convection and bifurcation. J. Appl. Math. Mech. 31, 103–114.
Yudovich, V. I. (1967b). Stability of convection flows. J. Appl. Math. Mech. 31, 272–281.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2014 Springer Science+Business Media, LLC
About this chapter
Cite this chapter
Ma, T., Wang, S. (2014). Fluid Dynamics. In: Phase Transition Dynamics. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-8963-4_4
Download citation
DOI: https://doi.org/10.1007/978-1-4614-8963-4_4
Published:
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4614-8962-7
Online ISBN: 978-1-4614-8963-4
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)