On a convergent nonlinear perturbation theory without small denominators or secular terms

1976 ◽  
Vol 17 (1) ◽  
pp. 121-140 ◽  
Author(s):  
Charles R. Eminhizer ◽  
Robert H. G. Helleman ◽  
Elliott W. Montroll
Keyword(s):  
Perturbation Theory ◽  
Nonlinear Perturbation ◽  
Small Denominators ◽  
Secular Terms

On a nonlinear perturbation theory without secular terms

Physica ◽  
1974 ◽  
Vol 74 (1) ◽  
pp. 22-74 ◽  
Author(s):  
R.H.G. Helleman ◽  
E.W. Montroll
Keyword(s):  
Perturbation Theory ◽  
Nonlinear Perturbation ◽  
Secular Terms

Nonlinear Perturbation Theory of the Incompressible Richtmyer-Meshkov Instability

1996 ◽  
Vol 76 (17) ◽  
pp. 3112-3115 ◽  
Author(s):  
Alexander L. Velikovich ◽  
Guy Dimonte
Keyword(s):  
Perturbation Theory ◽  
Nonlinear Perturbation

Thermosolutal bifurcation phenomena in porous enclosures subject to vertical temperature and concentration gradients

1999 ◽  
Vol 395 ◽  
pp. 61-87 ◽  
Author(s):  
M. MAMOU ◽  
P. VASSEUR
Keyword(s):  
Perturbation Theory ◽  
Travelling Waves ◽  
Neumann Boundary ◽  
Lewis Number ◽  
Parallel Flow ◽  
Nonlinear Perturbation ◽  
Concentration Gradients ◽  
Bifurcation Phenomena ◽  
Flow Approximation ◽  
Vertical Walls

The Darcy model with the Boussinesq approximations is used to study double-diffusive instability in a horizontal rectangular porous enclosure subject to two sources of buoyancy. The two vertical walls of the cavity are impermeable and adiabatic while Dirichlet or Neumann boundary conditions on temperature and solute are imposed on the horizontal walls. The onset and development of convection are first investigated using the linear and nonlinear perturbation theories. Depending on the governing parameters of the problem, four different regimes are found to exist, namely the stable diffusive, the subcritical convective, the oscillatory and the augmenting direct regimes. The governing parameters are the thermal Rayleigh number, RT, buoyancy ratio, N, Lewis number, Le, normalized porosity of the porous medium, ε, aspect ratio of the enclosure, A, and the thermal and solutal boundary condition type, κ, applied on the horizontal walls. On the basis of the nonlinear perturbation theory and the parallel flow approximation (for slender or shallow enclosures), analytical solutions are derived to predict the flow behaviour. A finite element numerical method is introduced to solve the full governing equations. The results indicate that steady convection can arise at Rayleigh numbers below the supercritical value, indicating the development of subcritical flows. At the vicinity of the threshold of supercritical convection the nonlinear perturbation theory and the parallel flow approximation results are found to agree well with the numerical solution. In the overstable regime, the existence of multiple solutions, for a given set of the governing parameters, is demonstrated. Also, numerical results indicate the possible occurrence of travelling waves in an infinite horizontal enclosure.

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