Solutions for nonlinear convection in the presence of a lateral boundary

Daniels, P. G.; Ho, D.; Skeldon, Anne C.

doi:10.1016/s0167-2789(02)00789-3

Cited by 3 publications

(7 citation statements)

References 20 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Stable steady-state solutions are limited to the range 0.797 ≤ q ≤ 0.871 with the end points corresponding to the positions where dq/dφ = 0. This is similar to the behaviour in the one-dimensional case reported by Daniels et al (2003) where it was shown that dual solutions exist at internal points of the range, one of which is unstable. The same is expected in the two-dimensional case, but the method of solution adopted here precludes the determination of the unstable branch.…”

Section: Nonlinear Solutionssupporting

confidence: 88%

“…They showed that even if the lateral walls are far (many roll widths) apart, they severely restrict the band of allowed wavenumbers in the bulk of the fluid compared with that which exists for the corresponding infinite layer. These results were extended to two values of ǫ in the nonlinear regime by Kramer & Hohenberg (1984) and to a range of values of ǫ by Daniels et al (2003) who showed how the waveband is related to the phase shift of the periodic form relative to the wall. In the present paper this analysis is extended to the two-dimensional Swift-Hohenberg equation for a channel −a ≤ y ≤ a with the equivalent of no-slip boundary conditions on the sidewalls: u = ∂u ∂y = 0 at y = ±a.…”

Section: Introductionmentioning

confidence: 81%

“…Using the same procedure as that described for the one-dimensional case by Daniels et al (2003), the present results then also determine the families of steady-state solutions even or odd in x (together with their interconnecting branches) that exist in long finite channels.…”

Section: Discussionmentioning

confidence: 99%

See 2 more Smart Citations

Non-linear wavelength selection for pattern-forming systems in channels

Daniels

Skeldon

2008

IMA Journal of Applied Mathematics

Self Cite

View full text Add to dashboard Cite

A method is described for calculating nonlinear steady-state patterns in channels taking into account the effect of an end wall across the channel. The key feature is the determination of the phase shift of the nonlinear periodic form distant from the end wall as a function of wavelength. This is found by analysing the solution close to the end wall, where Floquet theory is used to describe the departure of the solution from its periodic form and to locate the Eckhaus stability boundary. A restricted band of wavelengths is identified, within which solutions for the phase shift are found by numerical computation in the fully nonlinear regime and by asymptotic analysis in the weakly nonlinear regime. Results are presented here for the two-dimensional Swift-Hohenberg equation but in principle the method can be applied to more general pattern-forming systems. Near onset, it is shown that for channel widths less than a certain critical value the restricted band includes both subcritical and supercritical wavelengths, whereas for wider channels only subcritical wavelengths are allowed.

show abstract

Section: Nonlinear Solutionssupporting

confidence: 88%

Section: Introductionmentioning

confidence: 81%

See 1 more Smart Citation

Non-linear wavelength selection for pattern-forming systems in channels

Daniels

Skeldon

2008

IMA Journal of Applied Mathematics

Self Cite

View full text Add to dashboard Cite

show abstract

“…In section 2, we introduce the Swift-Hohenberg equations as a model for the Taylor-Couette flow. We note that the modification of pattern formation due to the presence of weak forcing at lateral boundaries in Swift-Hohenberg equations has been addressed in the work of Daniels and co-workers [18,19] with application to Rayleigh-Bénard convection in mind. In contrast with their work, we include an O(1) boundary condition that forces the flow strongly.…”

Section: Introductionmentioning

confidence: 99%

Boundary effects and the onset of Taylor vortices

Rucklidge

Champneys

2004

Physica D: Nonlinear Phenomena

View full text Add to dashboard Cite

It is well established that the onset of spatially periodic vortex states in the TaylorCouette flow between rotating cylinders occurs at the value of Reynolds number predicted by local bifurcation theory. However, the symmetry breaking induced by the top and bottom plates means that the true situation should be a disconnected pitchfork. Indeed, experiments have shown that the fold on the disconnected branch can occur at more than double the Reynolds number of onset. This leads to an apparent contradiction: why should Taylor vortices set in so sharply at the Reynolds number predicted by the symmetric theory, given such large symmetry-breaking effects caused by the boundary conditions? This paper offers a generic explanation. The details are worked out using a Swift-Hohenberg pattern formation model that shares the same qualitative features as the Taylor-Couette flow. Onset occurs via a wall mode whose exponential tail penetrates further into the bulk of the domain as the driving parameter increases. In a large domain of length L, we show that the wall mode creates significant amplitude in the centre at parameter values that are O(L −2 ) away from the value of onset in the problem with ideal boundary conditions. We explain this as being due to a Hamiltonian Hopf bifurcation in space, which occurs at the same parameter value as the pitchfork bifurcation of the temporal dynamics. The disconnected anomalous branch remains O(1) away from the onset parameter since it does not arise as a bifurcation from the wall mode.

show abstract

“…This observation of the dependence of the bifurcation about SHE on L have been made by many authors. For instance, see [3,12,13]. One can refer to Chapter 9 of [8] for the dynamic bifurcation for KSE.…”

Section: Preliminariesmentioning

confidence: 99%

Dynamic Bifurcation of the Periodic Swift-Hohenberg Equation

Han¹,

Yari²

2012

Bulletin of the Korean Mathematical Society

View full text Add to dashboard Cite

Abstract. In this paper we study the dynamic bifurcation of the SwiftHohenberg equation on a periodic cell Ω = [−L, L]. It is shown that the equations bifurcates from the trivial solution to an attractor A λ when the control parameter λ crosses the critical value. In the odd periodic case, A λ is homeomorphic to S 1 and consists of eight singular points and their connecting orbits. In the periodic case, A λ is homeomorphic to S 1 , and contains a torus and two circles which consist of singular points.

show abstract

Solutions for nonlinear convection in the presence of a lateral boundary

Cited by 3 publications

References 20 publications

Non-linear wavelength selection for pattern-forming systems in channels

Non-linear wavelength selection for pattern-forming systems in channels

Boundary effects and the onset of Taylor vortices

Dynamic Bifurcation of the Periodic Swift-Hohenberg Equation

Contact Info

Product

Resources

About