M. Miklavčič

Viscous flow due to a shrinking sheet

The viscous flow induced by a shrinking sheet is studied. Existence and (non)uniqueness are proved. Exact solutions, both numerical and in closed form, are found.

A sharp condition for existence of an inertial manifold

It is shown that a perturbation argument that guarantees persistence of inertial (invariant and exponentially attracting) manifolds for linear perturbations of linear evolution equations applies also when the perturbation is nonlinear. This gives a simple but sharp condition for existence of inertial manifolds for semi-linear parabolic as well as for some nonlinear hyperbolic equations. Fourier transform of the explicitly given equation for the tracking solution together with the Plancherels theorem for Banach valued functions are used.

Impulsive stretching of a surface in a viscous fluid

The induced transient viscous flow due to a suddenly stretched surface is studied. After a similarity transform, the unsteady Navier--Stokes equation is solved by several methods, including perturbation for small times, numerical integration, and asymptotic analysis for large times. It is found that the validity of the small-time series can be greatly extended and the approach to steady state is exponential.

Oscillations and island evolution in radiating diffusion flames

We show how an island (isola) evolves out of the usual S-curve of steady states of diffusion flames when radiation losses are accounted for and how it eventually disappears when radiation increases further. At small activation temperatures there are never any islands. We show that stable oscillations evolve first out of perturbations of steady states on the S-curve at large Damköhler numbers. Only if the activation temperature is large enough do they also appear on the islands. The region of the stable oscillations grows larger as activation temperature decreases.

On the stability of one-dimensional diffusion flames established between plane, parallel, porous walls

A one-dimensional, non-premixed flame stability analysis is undertaken.Oscillatory and cellular flame instabilities are identified by a careful studyof the numerically calculated eigenvalues of the linearized system of equations. The numerical investigation details the critical locations for changes in flame behaviour, as well as the critical values of variousparameters that affect flame stability. A critical Lewis number, greaterthan unity, is identified as the value where unstable oscillations maybegin to appear (Le > Le c) and for which cellular flames can exist(Le < Le c). Some prior discussions are clarified regarding theaforementioned critical values, as well as the role of convection inproducing flame instabilities. The methodology of the stability analysis isdiscussed in detail.

A natural modal expansion for the flexible robot arm problem via a self-adjoint formulation

The equations of motion of a flexible robot arm consist of a coupled partial differential equation describing the arms transverse vibrations and an ordinary differential equation describing the hubs rigid motion. Many researchers obtained a solution using a modal expansion based on the arms equation alone, which has erroneous eigenfunctions and eigenvalues. A novel method is presented for obtaining an equivalent but self-adjoint form for the problem. This self-adjoint form leads to a natural modal expansion, where the equations decouple. This method is used to show that the effect of the hub-arm model coupling depends exclusively on the hub-inertia-to-arm-mass ratio. The need for a self-adjoint form arises in many control applications. This is because, typically, the control design is based on approximate models, and in order to guarantee robust performance, a prior estimate of the approximation error is required. When a self-adjoint form is available, obtaining approximate modes and the associated error bounds becomes an easy task. >

Single-mode saturation of a linearly unstable plasma

The nonlinear oscillations of an electron plasma described by the collisionless Vlasov equation are studied using a perturbation technique previously applied by Simon and Rosenbluth [Phys. Fluids 19, 1567 (1976)]. It is proved by a characteristic argument that the plasma is globally stable, so that Bogoliuboff’s method of ‘‘secular regularization’’ is applicable. Assuming the plasma is confined in a box, and that only the lowest mode is unstable, it is shown that the ‘‘eigenmode dominance’’ approximation of Simon and Rosenbluth fails to conserve energy, but that energy and momentum conservation can be regained by considering interaction between the discrete and continuum modes. A formula is derived for the amplitude and phase of the saturated nonlinear oscillations. In a subsidiary result, it is shown that nonlinear effects damp the steady‐state oscillations predicted by linearized theory for some stable plasmas.

Stability of mean flows over an infinite flat plate

The present investigation is mainly concerned with a stability analysis for the linearized Navier-Stokes equations for parallel and nonparallel mean flows over an infinite flat plate. The system of equations for parallel flows is presented. The system is viewed as a generalized Orr-Sommerfeld equation. Attention is given to an explicit criterion characterizing the case when the stability of all physically reasonable solutions is determined by the eigenvalues. The proof given in the investigation is applicable to both the generalized Orr-Somerfeld equations and the modified equations for nonparallel flow. The fact that the criterion is independent of the completeness or incompleteness of eigenfunctions is contrary to some expectations.

Period doubling cascade in diffusion flames

Here it is shown that chaotic oscillations can appear after a series of period doublings in radiating diffusion flames when the activation temperature is high enough. It is also shown that period doubling cascades appear typically in very small regions and that they may not be observable if one starts with small perturbations of a steady flame.

Asymptotic periodicity of the iterates of positivity preserving operators

Soit X un espace de Banach reel; soit X + un sous-ensemble ferme de X tel que: 1) si x∈X + , y∈X + , α∈[0, ∞) alors x+y∈X + et αx∈X ± ; 2) il existe M 0 ∈(0, ∞) tel que pour x∈X il existe x + ∈X + et x - ∈X + qui satisfont: x=x + −x − , ∥x + ∥≤M 0 ∥x∥≤M 0 ∥x∥ ∥x−∥ et si x=y + −y − pour y + ∈X + , y − ∈X + alors y + −x + ∈X + ; 3) si x∈X + , y∈X + alors ∥x∥≤∥x+y∥; soit B un operateur lineaire borne sur X; soit BX + CX + ; F0 est un sous-ensemble compact non vide de X et lim n→∞ dist (B n x, F 0 )=0 quand x∈X + et ∥x∥=1. Alors B n x est asymptotiquement periodique pour x∈X