Skip to main content

Fluid Dynamics

  • Chapter
  • First Online:
Book cover Phase Transition Dynamics

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.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 89.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 119.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  • Busse, F. (1978). Non-linear properties of thermal convection. Reports on Progress in Physics 41, 1929–1967.

    Article  Google Scholar 

  • 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.

    Article  MathSciNet  MATH  Google Scholar 

  • Chandrasekhar, S. (1981). Hydrodynamic and Hydromagnetic Stability. Dover Publications, Inc.

    Google Scholar 

  • Cross, M. and P. Hohenberg (1993). Pattern formation outside of equilibrium. Reviews of Modern Physics 65(3), 851–1112.

    Article  Google Scholar 

  • Drazin, P. and W. Reid (1981). Hydrodynamic Stability. Cambridge University Press.

    Google Scholar 

  • 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.

    Google Scholar 

  • Friedlander, S., M. Vishik, and V. Yudovich (2000). Unstable eigenvalues associated with inviscid fluid flows. J. Math. Fluid Mech. 2(4), 365–380.

    Article  MathSciNet  MATH  Google Scholar 

  • Getling, A. V. (1997). Rayleigh-Benard convection: Structures and dynamics. Advanced Series in Nonlinear Dynamics. World Scientific.

    Google Scholar 

  • Ghil, M. and S. Childress (1987). Topics in Geophysical Fluid Dynamics: Atmospheric Dynamics, Dynamo Theory, and Climate Dynamics. Springer-Verlag, New York.

    Book  MATH  Google Scholar 

  • 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).

    Google Scholar 

  • 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).

    Google Scholar 

  • Goldstein, S. (1937). The stability of viscous fluid flow between rotating cylinders. Proc. Camb. Phil. Soc. 33, 41–61.

    Article  MATH  Google Scholar 

  • 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.

    Google Scholar 

  • Hopf, E. (1951). Über die Anfangswertaufgabe für die hydrodynamischen Grundgleichungen. Math. Nachr. 4, 213–231.

    Article  MathSciNet  MATH  Google Scholar 

  • 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.

    Google Scholar 

  • Hsia, C.-H., T. Ma, and S. Wang (2010). Rotating Boussinesq equations: dynamic stability and transitions. Discrete Contin. Dyn. Syst. 28(1), 99–130.

    Article  MathSciNet  MATH  Google Scholar 

  • 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.

    Google Scholar 

  • Kato, T. (1995). Perturbation theory for linear operators. Classics in Mathematics. Berlin: Springer-Verlag. Reprint of the 1980 edition.

    Google Scholar 

  • Kirchgässner, K. (1975). Bifurcation in nonlinear hydrodynamic stability. SIAM Rev. 17(4), 652–683.

    Article  MathSciNet  MATH  Google Scholar 

  • 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.

    MathSciNet  MATH  Google Scholar 

  • Koschmieder, E. L. (1993). Bénard cells and Taylor vortices. Cambridge Monographs on Mechanics. Cambridge University Press.

    MATH  Google Scholar 

  • Ladyženskaya, O. A. (1958). On the nonstationary Navier-Stokes equations. Vestnik Leningrad. Univ. 13(19), 9–18.

    MathSciNet  Google Scholar 

  • 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.

    Google Scholar 

  • Lappa, M. (2009). Thermal convection: Patterns, evolution and stability. Wiley.

    Google Scholar 

  • 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.

    MathSciNet  Google Scholar 

  • Lions, J.-L., R. Temam, and S. Wang (1992b). On the equations of the large-scale ocean. Nonlinearity 5(5), 1007–1053.

    Article  MathSciNet  MATH  Google Scholar 

  • 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.

    Google Scholar 

  • 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.

    Article  MathSciNet  MATH  Google Scholar 

  • 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.

    MathSciNet  MATH  Google Scholar 

  • Ma, T. and S. Wang (2004b). Dynamic bifurcation and stability in the Rayleigh-Bénard convection. Commun. Math. Sci. 2(2), 159–183.

    Article  MathSciNet  MATH  Google Scholar 

  • Ma, T. and S. Wang (2004c). Interior structural bifurcation and separation of 2D incompressible flows. J. Math. Phys. 45(5), 1762–1776.

    Article  MathSciNet  MATH  Google Scholar 

  • 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.

    Google Scholar 

  • 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.

    Google Scholar 

  • Ma, T. and S. Wang (2006). Stability and bifurcation of the Taylor problem. Arch. Ration. Mech. Anal. 181(1), 149–176.

    Article  MathSciNet  MATH  Google Scholar 

  • Ma, T. and S. Wang (2007a). Rayleigh-Bénard convection: dynamics and structure in the physical space. Commun. Math. Sci. 5(3), 553–574.

    Article  MathSciNet  MATH  Google Scholar 

  • Ma, T. and S. Wang (2008d). Exchange of stabilities and dynamic transitions. Georgian Mathematics Journal 15:3, 581–590.

    Google Scholar 

  • Ma, T. and S. Wang (2009a). Boundary-layer and interior separations in the Taylor-Couette-Poiseuille flow. J. Math. Phys. 50(3), 033101, 29.

    Google Scholar 

  • Ma, T. and S. Wang (2010a). Dynamic transition and pattern formation in Taylor problem. Chin. Ann. Math. Ser. B 31(6), 953–974.

    Article  MathSciNet  MATH  Google Scholar 

  • Ma, T. and S. Wang (2010b). Dynamic transition theory for thermohaline circulation. Phys. D 239(3–4), 167–189.

    Article  MathSciNet  MATH  Google Scholar 

  • Ma, T. and S. Wang (2010c). Tropical atmospheric circulations: dynamic stability and transitions. Discrete Contin. Dyn. Syst. 26(4), 1399–1417.

    Article  MathSciNet  MATH  Google Scholar 

  • Ma, T. and S. Wang (2011b). El Niño southern oscillation as sporadic oscillations between metastable states. Advances in Atmospheric Sciences 28:3, 612–622.

    Google Scholar 

  • Mezić, I. (2001). Break-up of invariant surfaces in action-angle-angle maps and flows. Phys. D 154(1–2), 51–67.

    MathSciNet  MATH  Google Scholar 

  • Newton, P. K. (2001). The N-vortex problem, Volume 145 of Applied Mathematical Sciences. New York: Springer-Verlag. Analytical techniques.

    Google Scholar 

  • Palm, E. (1975). Nonlinear thermal convection. Annual Review of Fluid Mechanics 7(1), 39–61.

    Article  MathSciNet  Google Scholar 

  • Pedlosky, J. (1987). Geophysical Fluid Dynamics (second ed.). New-York: Springer-Verlag.

    Book  MATH  Google Scholar 

  • Rabinowitz, P. H. (1968). Existence and nonuniqueness of rectangular solutions of the Bénard problem. Arch. Rational Mech. Anal. 29, 32–57.

    Article  MathSciNet  MATH  Google Scholar 

  • 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.

    Article  MathSciNet  MATH  Google Scholar 

  • 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.

    Article  Google Scholar 

  • 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.

    Article  MathSciNet  Google Scholar 

  • Salby, M. L. (1996). Fundamentals of Atmospheric Physics. Academic Press.

    Google Scholar 

  • 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.

    Google Scholar 

  • Surana, A., O. Grunberg, and G. Haller (2006). Exact theory of three-dimensional flow separation. I. Steady separation. J. Fluid Mech. 564, 57–103.

    Google Scholar 

  • 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.

    Google Scholar 

  • Taylor, G. I. (1923). Stability of a viscous liquid contained between two rotating cylinders. Philos. Trans. Royl London Ser. A 223, 289–243.

    Article  MATH  Google Scholar 

  • Temam, R. (1997). Infinite-dimensional dynamical systems in mechanics and physics (Second ed.), Volume 68 of Applied Mathematical Sciences. New York: Springer-Verlag.

    Google Scholar 

  • Temam, R. (2001). Navier-Stokes equations. AMS Chelsea Publishing, Providence, RI. Theory and numerical analysis, Reprint of the 1984 edition.

    Google Scholar 

  • Velte, W. (1966). Stabilität and verzweigung stationärer lösungen der davier-stokeschen gleichungen. Arch. Rat. Mech. Anal. 22, 1–14.

    Article  MathSciNet  MATH  Google Scholar 

  • Vishik, M. I. (1992). Asymptotic behaviour of solutions of evolutionary equations. Lezioni Lincee. [Lincei Lectures]. Cambridge: Cambridge University Press.

    Google Scholar 

  • Wiggins, S. (1990). Introduction to applied nonlinear dynamical systems and chaos, Volume 2 of Texts in Applied Mathematics. New York: Springer-Verlag.

    Google Scholar 

  • Yudovich, V. I. (1966). Secondary flows and fluid instability between rotating cylinders. Appl. Math. Mech. 30, 822–833.

    Article  MathSciNet  MATH  Google Scholar 

  • Yudovich, V. I. (1967a). Free convection and bifurcation. J. Appl. Math. Mech. 31, 103–114.

    Article  Google Scholar 

  • Yudovich, V. I. (1967b). Stability of convection flows. J. Appl. Math. Mech. 31, 272–281.

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints 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

Publish with us

Policies and ethics