Branches Of Steady Solutions Research Articles

Determination of Critical Ambient Temperatures for Thermal Ignition

This paper investigates the single heat-balance equation (with temperature, rescaled to make it dimensionless without using the ambient temperature). Here reactant consumption has been neglected, and this means that the upper branches of steady solutions are of less physical interest

Singular Secondary Bifurcation

A bifurcation problem is analyzed for a Brussellator boundary value problem, which is a typical reaction-diffusion system. The bifurcation parameter $\lambda $ is proportional to the length of the system. We employ previously developed perturbation methods to analyze the secondary bifurcation of steady solutions that arise from the splitting of multiple primary bifurcation points. The resulting bifurcation equations are nonlinear algebraic equations to determine the amplitudes of the solutions. The coefficients in these equations depend upon the system parameters in the Brussellator problem, such as the two prescribed constant reactants A and B. For critical values of these parameters the solutions of the algebraic system are singular, and hence the perturbation method is invalid for parameter values at and near these critical values. A new perturbation method is employed yielding new branches of steady solutions and possible tertiary bifurcations to time-periodic solutions. Thus the analysis of singularities in the bifurcation equations reveals new mechanisms for the occurrence of steady and time-periodic solutions. Some implications of the results for chemical morphogenesis are discussed.

Oscillatory and steady convection in a magnetic field

Two-dimensional convection in a Boussinesq ffuid in the presence of an imposed magnetic field is described in terms of a simplified model, which is exact to second order in the amplitude of the motion and appears to be qualitatively correct for larger amplitudes. If the ratio of the magnetic diffusivity to the thermal diffusivity is SUEciently small and the imposed magnetic field is sufficiently large, convection sets in when r = r(O) as overstable oscillations, which grow in amplitude as the normalized Rayleigh number r is increased. There is also a branch of steady solutions that bifurcates from the static equilibrium at r = r' > do) and stable steady solutions exist for r > rmin. For certain choices of parameters subcritical steady convection, with rmin < de), is found and the oscillatory branch ends on the unstable portion of the steady branch, where the period of the oscillations becomes infinite. In some circumstances there may be a bifurcation from symmetrical to asymmetrical oscillations, followed by a sequence of bifurcations at each of which the period doubles. Other choices of parameters allow only supercritical convection with r increasing monotonically on the steady branch; if convection first appears as overstable oscillations the steady branch is then unstable for rce) < r < rmin and there is a Hopf bifurcation at r = rmin. This complicated pattern of behaviour is consistent with the results of numerical experiments on the full two-dimensional problem.

Oscillations in double-diffusive convection

We have studied the transition between oscillatory and steady convection in a simplified model of two-dimensional thermosolutal convection. This model is exact to second order in the amplitude of the motion and is qualitatively accurate for larger amplitudes. If the ratio of the solutal diffusivity to the thermal diffusivity is sufficiently small and the solutal Rayleigh number, RS, sufficiently large, convection sets in as overstable oscillations, and these oscillations grow in amplitude as the thermal Rayleigh number, RT, is increased. In addition to this oscillatory branch, there is a branch of steady solutions that bifurcates from the static equilibrium towards lower values of RT; this subcritical branch is initially unstable but acquires stability as it turns round towards increasing values of RT. For moderate values of RS the oscillatory branch ends on the unstable (subcritical) portion of the steady branch, where the period of the oscillations becomes infinite. For larger values of RS a birfurcation from symmetrical to asymmetrical oscillations is followed by a succession of bifurcations, at each of which the period doubles, until the motion becomes aperiodic at some finite value of RT. The chaotic solutions persist as RT is further increased but eventually they lose stability and there is a transition to the stable steady branch. These results are consistent with the behaviour of solutions of the full two-dimensional problem and suggest that period-doubling, followed by the appearance of a strange attractor, is a characteristic feature of double-diffusive convection.

Nonlinear periodic convection in double-diffusive systems

We study two examples of two-dimensional nonlinear double-diffusive convection (thermohaline convection, and convection in an imposed vertical magnetic field) in the limit where the onset of marginal overstability just precedes the exchange of stabilities. In this limit nonlinear solutions can be found analytically. The branch of oscillatory solutions always terminates on the steady solution branch. If the steady solution branch is subcritical this occurs when the period of the oscillation becomes infinite, while if it is supercritical, it occurs via a Hopf bifurcation. A detailed discussion of the stability of the oscillations is given. The results are in broad agreement with the largeramplitude results obtained previously by numerical techniques.

Flow Instability through a Curved Duct with Square Cross Section

Flow instability through a curved duct with square cross section is numerically studied by using the spectral method over a wide range of the Dean number 0≤Dn≤5000 for the curvature δ= 0.1. A temperature difference is applied between the vertical sidewalls for the Grashof number Gr=100, where the outer wall is heated and the inner wall is cooled. After a comprehensive survey over the parametric ranges, two branches of asymmetric steady solutions are obtained by the Newton-Raphson iteration method. Linear stability of the steady solutions is then investigated. It is found that only the first branch is linearly stable in a couple of interval of Dn while the other branch is linearly unstable. Steady values of the Nusselt numbers, Nu, are also calculated for two differentially heated vertical sidewalls. When there is no stable steady solution, time evolution of Nu is obtained and it is found in the unstable region the flow undergoes through various flow instabilities, if Dn is increased. Key words: Curved square duct; Steady solutions; Time evolution; Dean number; Nusselt number GANIT J. Bangladesh Math. Soc. (ISSN 1606-3694) 29 (2009) 71-85 DOI: http://dx.doi.org/10.3329/ganit.v29i0.8517