Stability of an MHD shear flow with a piecewise linear velocity profile

In this paper we present the results of the stability analysis of a simple shear flow of an incompressible fluid with a piecewise linear velocity profile in the presence of a magnetic field. In the flow, a finite transitional magnetic-free layer with a linear velocity profile is sandwiched by two semi-infinite regions. One of these regions is magnetic-free and the flow velocity in the region is constant. The other region is magnetic and the fluid in it is quiescent. The magnetic field is constant and parallel to the flow in the transitional layer. The fluid density is constant both in the magnetic as well as the magnetic-free regions, while it has a jump-type discontinuity at the boundary between the transitional layer and the magnetic region. The effect of gravity is included in the model, and it is assumed that the lighter fluid is overlaying the heavier one, thus no Rayleigh-Taylor instability is present. The dispersion equation governing the normal-mode stability of the flow is derived and its properties are analysed. We study stability of two cases: (i) magnetic-free flow in the presence of gravity, and (ii) magnetic flow without gravity. In the first case, the flow stability is controlled by the Rayleigh number, R. In the second case, the control parameter is the inverse squared Alfvénic Mach number, H . Stability of a particular monochromatic perturbation also depends on its dimensionless wavenumber α. We combine the analytical and numerical approaches to obtain the neutral stability curves in the (α,R)-plane in the case of the magnetic-free flow, and in the (α,H)-plane in the case of the magnetic flow. The dependence of the instability increment on R in the first case, and on H in the second case is treated. We apply the results of the analysis to the stability of a strongly subsonic portion of the heliopause. Our main conclusion is as follows: The inclusion of a transitional layer near the heliopause into the model increases by an order of magnitude the strength of the interstellar magnetic field required to stabilize this portion of the heliopause in comparison with the corresponding stabilizing strength of the magnetic field required when modelling the heliopause as a tangential discontinuity

Traveling Wave Solutions in a Generalized Theory for Macroscopic Capillarity

One-dimensional traveling wave solutions for imbibition processes into a homogeneous porous medium are found within a recent generalized theory of macroscopic capillarity. The generalized theory is based on the hydrodynamic differences between percolating and nonpercolating fluid parts. The traveling wave solutions are obtained using a dynamical systems approach. An exhaustive study of all smooth traveling wave solutions for primary and secondary imbibition processes is reported here. It is made possible by introducing two novel methods of reduced graphical representation. In the first method the integration constant of the dynamical system is related graphically to the boundary data and the wave velocity. In the second representation the wave velocity is plotted as a function of the boundary data. Each of these two graphical representations provides an exhaustive overview over all one-dimensional and smooth solutions of traveling wave type, that can arise in primary and secondary imbibition. Analogous representations are possible for other systems, solution classes, and processes.</p

Analytic Formalism of 3-D Unstable Linear Wave Packets.

The analytic formalism for studying absolute and convective instabilities and spatially amplifying waves in 3-D flows of general nature is developed. The criterion by which a point (k, l, omega) in the wave number-frequency space contributes to the instability is fomulated in terms of the dispersion relation function D (k, l, omega) of the problem. Consequently, the causality condition for spatially amplifying waves is obtained. A procedure for finding a direction of maximal spatial amplification is presented. Application of the technique to computing the N-factor in the e exponent N method is discussed

On the convection in a porous medium with inclined temperature gradient and vertical throughflow. Part II. absolute and convective instabilities, and spatially amplifying waves.

In this second part of our analysis of the destabilization of transverse modes in an extended horizontal layer of a saturated porous medium with inclined temperature gradient and vertical throughflow, we apply the mathematical formalism of absolute and convective instabilities to studying the nature of the transition to instability of such modes by assuming on physical grounds that the transition is triggered by growing localized wavepackets. It is revealed that in most of the parameter cases treated in the first part of the analysis (Brevdo and Ruderman 2009), at the transition point the evolving instability is convective. Only in the cases of zero horizontal thermal gradient, and in the cases of zero vertical throughflow and the horizontal Rayleigh number R h < 49, the instability is absolute implying that, as the vertical Rayleigh number, R v, increases passing through its critical value, R vc, the destabilization tends to affect the base state throughout and eventually destroys it at every point in space. For the parameter values considered, for which the destabilization has the nature of convective instability, we found that, as R v, increases beyond the critical value, while the horizontal Rayleigh number, R h, and the Péclet number, Q v, are kept fixed, the flow experiences a transition from convective to absolute instability. The values of the vertical Rayleigh number, R v, at the transition from convective to absolute instability are computed. For convectively unstable, but absolutely stable cases, the spatially amplifying responses to localized oscillatory perturbations, i.e., signaling, are treated and it is found that the amplification is always in the direction of the applied horizontal thermal gradient

Transition from convective to absolute instability in a porous layer with either horizontal or vertical solutal and inclined thermal gradients, and horizontal throughflow

International audienc

Local and global instabilities of spatially developing flows: cautionary examples

In the analysis of the linear stability of basic states in fluid mechanics that are slowly varying in space, the quasi-homogeneous hypothesis is often invoked, where the stability exponents are defined locally and treated as slowly varying functions of a spatial coordinate. The set of local stability exponents is then used to predict the global perturbation dynamics and an implicit hypothesis is that the local analysis provides at least a conservative estimate of the global stability properties of the flow. In this paper cautionary examples are presented that demonstrate a contradiction between the results of the local and global analyses. For example, a local analysis may predict stability everywhere even when the exact PDE with non-constant coefficients is ill-posed, demonstrating that global stability exponents are not, in general, bounded by the maximal local stability exponents. A key observation in this paper is the importance of distinguishing between the discrete spectrum and the continuous spectrum when comparing global and local stability exponents. This distinction is particularly significant for spatially periodic flows where, for the global flow, only the continuous spectrum is present and, hence, instability arises always in the absence of discrete spectra. New exact definitions for global absolute and convective instabilities are also given for a class of spatially periodic basic states and applied to an example based on the complex Ginzburg–Landau equation. The consequences of this example, and of the argument involved for basic states that are slowly varying in space but non-periodic, and for some problems in fluid mechanics are also presented.</p

On the convection in a porous medium with inclined temperature gradient and vertical throughflow. Part I. Normal modes

In this analysis, we apply the methods of the theory of linear absolute and convective instabilities to studying the destabilization of transverse modes in a model of convection in an extended horizontal layer of a saturated porous medium with inclined temperature gradient and vertical throughflow. In this first part of the analysis, normal modes are treated and neutral curves are obtained for a variety of values of the horizontal Rayleigh number, R h, and the Péclet number, Q v. The computations are performed by using a high-precision pseudo-spectral Chebyshev-collocation method. Our results compare well with the results found in the literature for the critical values of the vertical Rayleigh number. It is shown that the horizontal temperature gradient effect, inducing a Hadley circulation, is stabilizing for any fixed value of the throughflow velocity. The throughflow effect is stabilizing, for each of the values of R h = 0, 10, 20, 30. For higher values of R h = 40, 50, 60 considered, the influence of increasing throughflow on the stability is mixed. For a vanishing horizontal temperature gradient the critical normal mode is non-oscillatory, for all the values of throughflow. In all the cases of a non-zero horizontal temperature gradient and a non-zero throughflow considered, the critical normal mode is oscillatory, and the oscillatory frequency is an increasing function of both R h and Q v