Unstable Steady-state Solutions Research Articles

Local exact controllability of the two-dimensional Navier-Stokes equations

Let be a bounded domain with boundary consisting of two disjoint closed curves and such that is connected and . The Navier-Stokes system , is considered in with boundary and initial conditions and (here , , and is the outward normal to ). Let be a solution of this system such that satisfies the indicated boundary conditions on and , where . Then the existence of a control on with the following properties is proved: the solution of the Navier-Stokes system such that , and , coincides with for , that is, . In particular, if and do not depend on and is an unstable steady-state solution, then it follows from the above result that one can suppress the occurrence of turbulence by some control on . An analogous result is established in the case when and is a distributed control concentrated in an arbitrary subdomain .

Dynamical Approach Study of Spurious Steady-State Numerical Solutions of Nonlinear Differential Equations, Part III. The Effects of Nonlinear Source Terms in Reaction-Convection Equations

Abstract The objective of this paper is to investigate how well numerical steady-state solutions can mimic true steady-states of a model nonlinear reaction-convection. In order to assess the possible'sourcesoferrors,slowconvergence and nonconvergence of steady-state numerical solutions when using the time-dependent approach, six different numerical aspects are addressed. First, spurious stable and unstable steady-state numerical solutions (numerically irrelevant solutions) can be independently introduced by spatial and temporal discretizations satisfying the same boundary condition and initial data. Second, the various numerical treatments of the reaction term can affect the stability of the spurious as well as the exact steady-state solutions. Third, the time discretization can destabilize the spurious stable steady-state numerical solutions that are introduced by the spatial discretizations or vice versa. Fourth, for a given finite difference method, the strength of the coefficient of the convection term and stiffness of the nonlinear source term can strongly affect the permissibility of spurious steady-state numerical solutions. Fifth, the possiblecause of convergence to a spurious steady-state and a suggestion to avoid spurious steady-states are discussed. Finally, the numerical phenomenon of incorrect propagation speeds of discontinuities may be linked to the existence of some spurious stable steady-state numerical solutions.

On the stability of the shear flow of a viscoelastic fluid with slip along the fixed wall

In this paper we solve the time-dependent shear flow of an Oldroyd-B fluid with slip along the fixed wall. We use a non-linear slip model relating the shear stress to the velocity at the wall and exhibiting a maximum and a minimum. We assume that the material parameters in the slip equation are such that multiple steady-state solutions do not exist. The stability of the steady-state solutions is investigated by means of a one-dimensional linear stability analysis and by numerical calculations. The instability regimes are always within or coincide with the negative-slope regime of the slip equation. As expected, the numerical results show that the instability regimes are much broader than those predicted by the linear stability analysis. Under our assumptions for the slip equation, the Newtonian solutions are stable everywhere. The interval of instability grows as one moves from the Newtonian to the upper-convected Maxwell model. Perturbing an unstable steady-state solution leads to periodic solutions. The amplitude and the period of the oscillations increase with elasticity.

Inhibitory Substrate Utilization by Steady‐State Biofilms

Numerical simulations were performed that examined the utilization of inhibitory substrates by steady-state biofilms. Based on kinetic and energetic constraints, the growth of a monolayer of cells into a steady-state biofilm requires that substrate concentrations be between \IS\N\dm\di\dn and \IS\N\dm\da\dx. The continued exposure of a monolayer or dispersed cells to substrate concentrations less than \IS\N\dm\di\dn or greater than \IS\N\dm\da\dx will not lead to the development of a self-perpetuating biological treatment process. However, once developed, a steady-state biofilm can tolerate bulk substrate concentrations greater than \IS\N\dm\da\dx, because the inner layers of the biofilm are exposed to substrate concentrations less than \IS\N\dm\da\dx due to mass transport resistance and substrate utilization in the outer layers of the biofilm. In such situations, maximal rates of substrate utilization and microbial growth occur in the middle or at the inner surface of the steady-state biofilm. The simulations also indicated that there are two sets of steady-state biofilms: one associated with \IS\N\dm\di\dn (stable steady-state solutions) and the other associated with \IS\N\dm\da\dx (unstable steady-state solutions).

Stability of annular flow

A simple transient model using an unsteady state continuity equation and a quasi steady momentum equation is used to analyse the stability of co-current and counter-current annular flows. It is shown that unstable steady state solutions may take place in both cases. Application of the theory is demonstrated for the problem of flooding and flow reversal.

Isothermal concentration oscillations on catalytic surfaces

It is shown that surface heterogeneity can induce sustained kinetic oscillations, when a simple, isothermal reaction occurs on a catalytic surface. Moreover, the proposed model demonstrates that these oscillations may occur for realistic values of surface heterogeneity. Simple criteria have been developed, for predicting the kinetic parameters for which a unique, unstable steady-state solution exists.