Abstract
A plane non-parallel vortex flow in a square fluid domain is examined. The energy dissipation of the flow is dominated by viscosity and linear friction effect of a Hartmann layer. This is a traditional Navier–Stokes flow when the linear friction effect is not involved, whereas it is a magnetohydrodynamic flow when the energy dissipation is fundamentally dominated by the friction. It is proved that linear critical values of a spectral problem are nonlinear thresholds leading to the onset of secondary steady-state flows, the nonlinear phenomenon observed in laboratory experiments.




Similar content being viewed by others
References
Bondarenko, N.F., Gak, M.Z., Dolzhanskii, F.V.: Laboratory and theoretical models of a plane periodic flow. Izv. Akad. Nauk SSSR Fiz. Atmos. Okeana 15, 1017–1026 (1979)
Cerretelli, C., Williamson, C.H.K.: The physical mechanism for vortex merging. J. Fluid Mech. 475, 41–77 (2003)
Chen, Z.M.: Bifurcating steady-state solutions of the dissipative quasi-geostrophic equation in Lagrangian formulation. Nonlinearity 29, 3132–3147 (2016)
Chen, Z.M.: Instability of two-dimensional square eddy flows. Phys. Fluids 31, 044107 (2019)
Chen, Z.M., Price, W.G.: Supercritical regimes of liquid-metal fluid motions in electromagnetic fields: wall-bounded flows. Proc. R. Soc. Lond. A 458, 2735–2757 (2002)
Chen, Z.M., Price, W.G.: Secondary fluid flows driven electromagnetically in a two-dimensional extended duct. Proc. R. Soc. Lond. A 461, 1659–1683 (2005)
Crandall, M.G., Rabinowitz, P.H.: Bifurcation from simple eigenvalues. J. Funct. Anal. 8, 321–340 (1971)
Iudovich, V.I.: Example of the generation of a secondary stationary or periodic flow when there is loss of stability of the laminar flow of a viscous incompressible fluid. J. Appl. Math. Mech. 29, 527–544 (1965)
Kraichnan, R.H.: Inertial ranges in two-dimensional turbulence. Phys. Fluids 10, 1417 (1967)
Meunier, P., Le Dizès, S., Leweke, T.: Physics of vortex merging. Comptes Rendus Physique 6, 431–450 (2005)
Orszag, S.A., Patera, A.T.: Secondary instability of wall bounded shear flows. J. Fluid Mech. 128, 347–385 (1983)
Rabinowitz, P.H.: Existence and nonuniqueness of rectangular solutions of the Bénard problem. Arch. Rational Mech. Anal. 29, 32–57 (1968)
Sommeria, J.: Experimental study of the two-dimensional inverse energy cascade in a square box. J. Fluid Mech. 170, 139–168 (1986)
Sommeria, J., Verron, J.: An investigation of nonlinear interactions in a two-dimensional recirculating flow. Phys. Fluids 27, 1918 (1984)
Thess, A.: Instabilities in two-dimensional spatially periodic flows. Part II: Square eddy lattice. Phys. Fluids A 4, 1396 (1992)
Verron, J., Sommeria, J.: Numerical simulation of a two-dimensional turbulence experiment in magnetohydrodynamics. Phys. Fluids 30, 732 (1987)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Tim Healey.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Chen, ZM. Steady-State Bifurcation of a Non-parallel Flow Involving Energy Dissipation over a Hartmann Boundary Layer. J Nonlinear Sci 31, 91 (2021). https://doi.org/10.1007/s00332-021-09752-y
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00332-021-09752-y
Keywords
- Non-parallel flows
- Navier–Stokes equations
- Bifurcation
- Vortex flows
- Linear friction effect of Hartmann layer