Hydrodynamic stability

A linear stability analysis was carried out for a dilute suspension of rigid spherical particles in cylindrical Couette flow. The perturbation equations for both the continuous fluid phase and the discontinuous particle phase were decomposed into normal modes resulting in an eigenvalue problem that was solved numerically. At a given radius ratio, the theoretical critical Taylor number at which Taylor vortices first appear decreases as the particle concentration increases. Increasing the ratio of particle density to fluid density above one decreases the stability. However, using an effective Taylor number based on the suspension density and viscosity largely accounts for this effect. The axial wave number is the same for a suspension as it is for a pure fluid. Experiments using neutrally buoyant particles in a Taylor-Couette apparatus show that the flow is more stable as the particle concentration increases. The reason that the theory does not fully capture the physics of the flow should be addressed in future research.

The hydrodynamic stability of a pure liquid undergoing steady rapid evaporation at reduced pressure is examined using linear stability analysis. Results show that the rapidly evaporating liquid is unstable to local variations in evaporation rate, local surface depressions being produced by the force exerted on the surface by the rapidly departing vapour and sustained liquid flows being driven by the resultant shear exerted on the liquid surface by the vapour. The coupling of this 'differential vapour recoil ' mechanism to the Marangoni effect is investigated and the importance of inertial heat transfer, fluid inertia and viscous dissipation at the interface to system stability is resolved.

There is a need to develop alternate energy sources in the coming century because fossil fuels will become depleted and their use may lead to global climate change. Inertial fusion can become such an energy source, but significant progress must be made before its promise is realized. The high-density approach to inertial fusion suggested by Nuckolls et al. leads reaction chambers compatible with civilian power production. Methods to achieve the good control of hydrodynamic stability and implosion symmetry required to achieve these high fuel densities will be discussed. Fast Ignition, a technique that achieves fusion ignition by igniting fusion fuel after it is assembled, will be described along with its gain curves. Fusion costs of energy for conventional hotspot ignition will be compared with those of Fast Ignition and their capital costs compared with advanced fission plants. Finally, techniques that may improve possible Fast Ignition gains by an order of magnitude and reduce driver scales by an order of magnitude below conventional ignition requirements are described.

A linear analysis of Kelvin-Helmholtz instability of cylindrical interface is carried out using viscous potential flow theory. In the inviscid potential flow theory, the viscous term in Navier-Stokes equation vanishes as viscosity is zero. In viscous potential flow, the viscous term in Navier-Stokes equation vanishes as vorticity is zero but viscosity is not zero. Viscosity enters through normal stress balance in viscous potential flow theory and tangential stresses are not considered. Both asymmetric and axisymmetric disturbances are considered. A dispersion relation has been obtained and stability criterion is given in the terms of the critical value of relative velocity. A comparison between inviscid potential flow and viscous potential flow has been made. It has been observed that Reynolds number and inner fluid fraction both have destabilizing effect on the stability of the system.

Fluid flows that are smooth at low speeds become unstable and then turbulent at higher speeds. This phenomenon has traditionally been investigated by linearizing the equations of flow and looking for unstable eigenvalues of the linearized problem, but the results agree poorly in many cases with experiments. Nevertheless, it has become clear in recent years that linear effects play a central role in hydrodynamic instability. A reconciliation of these findings with the traditional analysis can be obtained by considering the 'pseudospectra' of the linearized problem, which reveals that small perturbations to the smooth flow in the form of streamwise vortices may be amplified by factors on the order of 10(exp 5) by a linear mechanism, even though all the eigenmodes are stable. The same principles apply also to other problems in the mathematical sciences that involve non-orthogonal eigenfunctions.

A problem of stability of steady convective flows in rectangular cavities is revisited and studied by a second-order finite volume method. The study is motivated by further applications of the finite volume-based stability solver to more complicated applied problems, which needs an estimate of convergence of critical parameters. It is shown that for low-order methods the quantitatively correct stability results for the problems considered can be obtained only on grids having more than 100 nodes in the shortest direction, and that the results of calculations using uniform grids can be significantly improved by the Richardson's extrapolation. It is shown also that grid stretching can significantly improve the convergence, however sometimes can lead to its slowdown. It is argued that due to the sparseness of the Jacobian matrix and its large dimension it can be effective to combine Arnoldi iteration with direct sparse solvers instead of traditional Krylov-subspace-based iteration ...

The self-gravitating instability of a fluid cylinder pervaded by magnetic field and endowed with surface tension has been discussed. The dispersion relation is derived and some reported works are recovered as limiting cases from it. The capillary force is destabilizing only in the small axisymmetric domain and stabilizing otherwise. The magnetic field has a strong stabilizing effect in all modes of perturbation for all wavelengths. The self-gravitating force is destabilizing in the axisymmetric perturbation. However the magnetic field effect modified a lot the destabilizing character of the model and could overcome the capillary and self-gravitating instability of the model for all short and long wavelengths.

Special cases of linear eighth-order boundary-value problems have been solved using polynomial splines. However, divergent results were obtained at points adjacent to boundary points. This paper presents an accurate and general approach to solve this class of problems, utilizing the generalized di erential quadrature rule (GDQR) proposed recently by the authors. Explicit weighting coe cients are formulated to implement the GDQR for eighth-order di erential equations. A mathematically important by-product of this paper is that a new kind of Hermite interpolation functions is derived explicitly for the ÿrst time. Linear and non-linear illustrations are given to show the practical usefulness of the approach developed. Using Frechet derivatives, non-linear eighth-order problems are also solved for the ÿrst time. Numerical results obtained using even only seven sampling points are of excellent accuracy and convergence in an entire domain. The present GDQR has shown clear advantages over the existing methods and demonstrated itself as a general, stable, and accurate numerical method to solve high-order di erential equations. ordinary di erential equation. Recently, Siddiqi and Twizell [6] used polynomial splines of degree six to solve some special cases of linear eighth-order boundary-value problems. However, the obtained results were divergent at points adjacent to the boundary. Twizell and his coworkers have also solved some other higher-order problems and encountered the same problems. The divergent results are due to the use of lower order test functions in the spline methods. In order to obtain accurate results in an entire domain, it is natural to apply di erential quadrature methods (DQM), which utilize high-order test functions in a global domain. However, the DQM usually dealt with di erential equations of order no more than two . A -point technique should be used for higher-order di erential equations . Two review papers [1,2] gave a detailed description of the development and application of the DQM.

The authors examine the stability of an outwardly propagating spherical flame accounting for both hydrodynamic and thermodiffusive effects. For Lewis numbers less than a critical value Le* <1, disturbances of the flame front grow during the initial phase of propagation, i.e. when the radius is comparable to the flame thickness. However, for Le > Le*, the flame, which is stable to thermodiffusive effects, becomes unstable only after a critical size is reached. This instability is hydrodynamic in nature and is caused by the thermal expansion of the gas. In this study the authors provide an expression for the determination of the critical size, or a critical Peclet number, which depends on the thermal expansion coefficient and on the Lewis number. The explicit dependence on all the relevant physiochemical parameters enables comparison of results with experimental data.

To illustrate the proposed method, shows calculated and experimental liquid thermal conductivity data for the system methanol water at 0" and 60°C. exemplifies a limitation of Equation (1) in that it fails to predict the minimum thermal conductivities for the azeotropic liquid mixture, CC&/tert-butanol.

The chemical reactivity of NO and NO2 is so rapid that their fluxes and concentrations can be considerably modified from that expected for conserved variables in the atmospheric surface layer, even as low as a meter above the surface. Fitzjarrald and Lenschow (1983) have calculated flux and mean concentration profiles for NO, NO 2 and 03 in the surface layer using numerical techniques. However, their solutions do not approach the photostationary state at large heights. Here we solve a simpler set of equations analytically (i.e. we assume a constant 03 concentration and neutral hydrodynamic stability), and are able to show how the flux profiles behave at large heights assuming that the concentrations approach their photostationary values. We find, for example, that at large heights the ratio of the flux of NO to that of NO 2 is equal to the ratio of their concentrations. These results are relevant to estimating surface fluxes of NO and NO 2, and are most applicable to nonurban environments where NO and NO2 concentrations are usually much less than 03 concentration.

The Orr-Sommerfeld operator's eigenvalues determine the stability of exponentially growing disturbances in parallel and quasi-parallel flows. This work assesses the sensitivity of these eigenvalues to modifications of the base flow, which need not be infinitesimally small. Such base flow variations may represent differences between the laboratory flow and its ideal, theoretical counterpart. The worst case, i.e. the change in base flow with the most destabilizing effect on the eigenvalues, is found using variational techniques for the plane Couette flow. Relatively small changes in the base flow are shown to be destabilizing, although the ideal flow is unconditionally stable according to linear theory. These observations inspire a velocity-based definition of pseudospectra in the hydrodynamic stability context.

A linear stability analysis was carried out for a dilute suspension of rigid spherical particles in cylindrical Couette flow. The perturbation equations for both the continuous fluid phase and the discontinuous particle phase were decomposed into normal modes resulting in an eigenvalue problem that was solved numerically. At a given radius ratio, the theoretical critical Taylor number at which Taylor vortices first appear decreases as the particle concentration increases. Increasing the ratio of particle density to fluid density above one decreases the stability. However, using an effective Taylor number based on the suspension density and viscosity largely accounts for this effect. The axial wave number is the same for a suspension as it is for a pure fluid. Experiments using neutrally buoyant particles in a Taylor-Couette apparatus show that the flow is more stable as the particle concentration increases. The reason that the theory does not fully capture the physics of the flow should be addressed in future research.

Nafion-coated bismuth film electrodes (NCBFEs) and Nafion-coated mercury film electrodes (NCMFEs) were used to electrochemically preconcentrate metal analytes for subsequent analysis by inductively coupled plasma-mass spectrometry (ICP-MS). Either type of electrodes is part of a thin-layer electrochemical flow cell that is positioned upstream of a microconcentric nebulizer for the ICP-MS. Performances of these electrodes were compared in terms of the analytical "figures of merit" (e.g., dynamic ranges, reproducibility, hydrodynamic stability, and elimination of matrix effects detrimental to ICP-MS). The coupled technique (ASV-ICP-MS) is found to possess a wide dynamic range (at least 4 to 5 orders of magnitude) and to be reproducible. Both electrodes are much more stable than the thin mercury film electrode (TMFE) traditionally used for ASV-ICP-MS, with the lifetime of the NCBFE exceeding 8 h. Adopting these electrodes for ASV-ICP-MS overcomes the problems associated with a TMFE, the erosion of which decreases the sample throughput, affects the analysis precision, and contaminates conventional glass nebulizers and spray chambers of the spectrometer. The medium exchange procedure inherent in ASV is successfully implemented with a two-valve flow injection system for the accumulation of trace Cd 2ϩ into the electrode from a certified seawater sample, followed by stripping Cd into a solution that is compatible to the ICP-MS operation. (J Am Soc Mass Spectrom 2006, 17, 945-952)

We investigate the relationship between the local and global bending motion of fluid conveying pipes on an elastic foundation. The local approach refers to an infinite pipe without taking into account its finite ends while in the global approach we consider a pipe of finite length with a given set of boundary conditions. Several kinds of propagating disturbances are identified

We derive a rigorous bound of the solution of the resolvent equation for plane Couette flow in three space dimensions. We combine analytical techniques with numerical computations. Compared to earlier results, our analytical techniques cover a larger part of the parameter domain consisting of wave numbers in two space directions and the Reynolds number. Numerical computations are needed only in a compact subset of the parameter domain. . 76E05, 35Q35, 47N20, 35P05.

The centrifugal instability mechanism in boundary layers flows over concave surfaces is responsible for the development of streamwise counter-rotating vortices, known as Görtler vortices. These Vortices create two regions in the spanwise direction, the upwash and downwash regions. The downwash region is responsible to compress the boundary layer towards the wall, increasing drag and heat transfer rates. The upwash region does the opposite. In the nonlinear development of the Görtler vortices the upwash region becomes narrow, and the average drag and heat transfer rate is higher than that for a Blasius boundary layer. In the present research, using a Spatial Direct Numerical Simulation, it is analyzed the influence of the Görtler Vortices spanwise wavelength in heat transfer rates. Different wavelengths are analyzed and compared with experiments. 1 The results show that steady Görtler flow can reach heat transfer rates higher than the turbulent values, even without introducing secondary instabilities.

A linear stability analysis was carried out for a dilute suspension of rigid spherical particles in cylindrical Couette flow. The perturbation equations for both the continuous fluid phase and the discontinuous particle phase were decomposed into normal modes resulting in an eigenvalue problem that was solved numerically. At a given radius ratio, the theoretical critical Taylor number at which Taylor vortices first appear decreases as the particle concentration increases. Increasing the ratio of particle density to fluid density above one decreases the stability. However, using an effective Taylor number based on the suspension density and viscosity largely accounts for this effect. The axial wave number is the same for a suspension as it is for a pure fluid. Experiments using neutrally buoyant particles in a Taylor-Couette apparatus show that the flow is more stable as the particle concentration increases. The reason that the theory does not fully capture the physics of the flow should be addressed in future research.

A dense, pellicular UpFront adsorbent (ϱ=1.5 g/cm3, UpFront Chromatography, Cophenhagen, Denmark) was characterized in terms of hydrodynamic properties and protein adsorption performance in expanded bed chromatography. Cibacron Blue 3GA was immobilised into the adsorbent and protein adsorption of bovine serum albumin (BSA) was selected to test the setup. The Bodenstein number and axial dispersion coefficient estimated for this dense pellicular adsorbent was 54 and 1.63×10−5 m2/s, respectively, indicating a stable expanded bed. It could be shown that the BSA protein was captured by the adsorbent in the presence of 30% (w/v) of whole-yeast cells with an estimated dynamic binding capacity (C/C 0=0.01) of approximately 6.5 mg/mL adsorbent.

A revised formal solution of the vibrating ribbon problem of hydrodynamic stability is presented. The initial formulation of Gaster (1965) is modified by application of the Briggs method and a careful treatment of the complex double Fourier transform inversions. Expressions are obtained in a natural way for the discrete spectrum as well as for the four branches of the continuous spectra. These correspond to discrete and branch-cut singularities in the complex wavenumber plane. The solutions from the continuous spectra decay both upstream and downstream of the ribbon, with the decay in the upstream direction being much more rapid than that in the downstream direction. Comments and clarification of related prior work are made.

An analysis of the hydrodynamic stability of a fluid near its near critical pointinitially at rest and in thermodynamic equilibrium -is considered in the Rayleigh-Bénard configuration, i.e. heated from below. The geometry is a two-dimensional square cavity and the top and bottom walls are maintained at constant temperatures while the sidewalls are insulated. Owing to the homogeneous thermo-acoustic heating (piston effect), the thermal field exhibits a very specific structure in the vertical direction. A very thin hot thermal boundary layer is formed at the bottom, then a homogeneously heated bulk settles in the core at a lower temperature; at the top, a cooler boundary layer forms in order to continuously match the bulk temperature with the colder temperature of the upper wall. We analyse the stability of the two boundary layers by numerically solving the Navier-Stokes equations appropriate for a van der Waals' gas slightly above its critical point. A finite-volume method is used together with an acoustic filtering procedure. The onset of the instabilities in the two different layers is discussed with respect to the results of the theoretical stability analyses available in the literature and stability diagrams are derived. By accounting for the piston effect the present results can be put within the framework of the stability analysis of Gitterman and Steinberg for a single layer subjected to a uniform, steady temperature gradient.

The linear stability analysis of a pressure-driven two-layer channel flow of two immiscible, Newtonian and incompressible fluids is considered. The walls of the channel are maintained at different constant temperatures and Nahme's law is applied to model the temperature dependence of the fluid viscosity. A modified Orr-Sommerfeld equation for the disturbance streamfunction coupled to a linearized energy equation is derived and solved using a spectral collocation method. Our results indicate that increasing the dimensionless top wall temperature has a non-monotonic effect on the linear stability characteristics. We also found that increasing the thermal conductivity and density ratios stabilise the flow for the set of parameter values considered; the viscosity ratio has a non-monotonic effect on the maximal growth rate. An energy 'budget' analysis shows that the most dangerous mode is of 'interfacial' type.

In the context of the French Laser-Mégajoule fusion-research programme, a direct-drive target design is developed. It is based on the use of a CH-foam ablator filled with cryogenic deuterium-tritium. One-dimensional optimization leads to a potential gain of 60 with a 1.5 MJ laser. The hydrodynamic stability of the implosion is investigated at the ablation front and the hot-spot surface by means of modelling and two-dimensional simulations. The effect of irradiation non-uniformities on low-mode (time-dependent) implosion asymmetries is studied by two-dimensional hydrodynamics simulations with three-dimensional laser-light raytracing. The effect of beam focal shapes on hot spot low-mode asymmetries is also addressed.

The collapse of turbulence in a plane channel flow is studied, as a simple analogy of stably stratified atmospheric flow. Turbulence is parameterized by first-order closure and the surface heat flux is prescribed, together with the wind speed and temperature at the model top. To study the collapse phenomenon both numerical simulations and linear stability analysis are used. The stability analysis is nonclassical in a sense that the stability of a parameterized set of equations of a turbulent flow is analyzed instead of a particular laminar flow solution. The analytical theory predicts a collapse of turbulence when a certain critical value of the stability parameter δ/L (typically O(0.5-1)) is exceeded, with δ the depth of the channel and L the Obukhov length. The exact critical value depends on channel roughness to depth ratio z 0 /δ. The analytical predictions are validated by the numerical simulations, and good agreement is found. As such, for the flow configuration considered, the present framework provides both a tool and a physical explanation for the collapse phenomenon.

The hydrodynamic stability problem of two-fluid boundary layers is studied. Using the linear stability theory, a fourth order Ordinary Differential Equation (the Orr-Sommerfeld equation) is derived for each of the fluid layers starting from the two-dimensional, incompressible Navier-Stokes equations. The equations, boundary and interface conditions result in an Eigenvalue problem that is solved using an efficient shoot-search technique. The Eigenvalues of the problem are the Reynolds number, wave number, and complex wave velocity. Unlike the single fluid problem, which results in a single unstable solution (Tollmien-Schlichting mode), the two fluid problem has multiple unstable solutions (Tollmien-Schlichting and Interfacial modes). The parameters that influence the stability problem are the viscosity and density ratios of the two fluids, thickness of the sheared fluid layer, surface tension and gravity. For certain combinations of viscosity ratio and fluid thickness the Tollmien-Sc...

We study a non-dissipative hydrodynamical mechanism that can stabilize the spin of the accretor in an ultra-compact double white dwarf binary. This novel synchronization mechanism relies on a nonlinear wave interaction spinning down the background star. The essential physics of the synchronization mechanism is summarized as follows. As the compact binary coalesces due to gravitational wave emission, the largest star eventually fills its Roche lobe and accretion starts. The accretor then spins up due to infalling material and eventually reaches a spin frequency where a normal mode of the star is resonantly driven by the gravitational tidal field of the companion. If the resonating mode satisfies a set of specific criteria, which we elucidate in this paper, it exchanges angular momentum with the background star at a rate such that the spin of the accretor locks at this resonant frequency, even though accretion is ongoing. Some of the accreted angular momentum that would otherwise spin up the accretor is fed back into the orbit through this resonant tidal interaction. In this paper we solve analytically a simple dynamical system that captures the essential features of this mechanism. Our analytical study allows us to identify two candidate Rossby modes that may stabilize the spin of an accreting white dwarf in an ultra-compact binary. These two modes are the l = 4, m = 2 and l = 5, m = 3 CFS unstable hybrid r-modes, which stabilize the spin of the accretor at frequency 2.6 ω orb and 1.54 ω orb respectively, where ω orb is the binary's orbital frequency. Since the stabilization mechanism relies on continuously driving a mode at resonance, its lifetime is limited since eventually the mode amplitude saturates due to non-linear mode-mode coupling. Rough estimates of the lifetime of the effect lie from a few orbits to possibly thousands of years. We argue that one must include this hydrodynamical stabilization effect to understand stability and survival rate of ultra-compact binaries, which is relevant in predicting the galactic white dwarf gravitational background that LISA will observe.

We present an experimental investigation of an instability triggered by a fast chemical reaction in a low inertia parallel flow in a small channel. Two fluids evenly injected in a straight channel react creating a strongly stratified distribution of viscosity near their interface, which destabilizes the flow. Depending on the flow rates and the aspect ratio of the flow channel, several unstable regimes are observed: mixing regime, weakly stratified regime and a stable stratified regime. In channels of 1:1 aspect ratio we find most efficient mixing (and highest flow resistance) at an intermediate pressure drops, as the flow transitions between a fully 3D regime and more stratified regime. For channels of small aspect ratio the non-monotonicity is less evident: higher pressure drops lead to increased mixing.

Hydrodynamic stability during advection combined with di usion and adsorption of a binary gas mixture in porous media is considered in this work. The general framework is that of pressure swing adsorption (PSA) on beds of zeolite beads. The dispersion=adsorption problem is studied at Darcy's scale on a 1D porous column from a numerical point of view while complete time-periodic cycles are considered. The regular solution of the problem is ÿrst sought using a ÿnite volume scheme and stability analysis is performed on this time-dependent solution. While instability evidence is shown on a direct 2D numerical simulation of the problem with pressure and concentration disturbances at the entrance of the column, the stability analysis is carried out in 1D by making use of the linear amplitude equation method. Results clearly show the stability conditions which are quite di erent depending on whether pressure or concentration disturbances are considered as initial conditions or maintained as boundary conditions at column entrance. ? : S 0 0 0 9 -2 5 0 9 ( 0 1 ) 0 0 0 1 6 -1

Chemical oxygen-iodine Laser (COIL) is one of the fast emerging high power laser source for near Infrared (?=1.315tm) laser generation. The heart of the system is the singlet oxygen generator (SOG) which is a pumping source for this laser. A Jet type SOG with a novel approach was designed and fabricated. Singlet oxygen was taken out of the SOC at an angle of 4g0 thus avoiding the carry over of droplets, which is one of the major drawbacks of horizontal system, The preliminary results have hccn reported in our earlier publication. The present paper discusses the performance of this generator for various operational conditions viz. diluent's gas nitrogen / helium, basic hydrogen peroxide composition, generator pressure and gas velocity. Further, conditions for the stable operation from generator as well as chlorine injection point of view have been identified.

This work addresses the transition from 2D steady to 2D unsteady laminar flow for a fully developed regime in a symmetric wavy channel geometry. We investigate the existence and characteristics of the spatio-temporal structure of the fully developed unsteady laminar flow for those particular geometries for which the steady flow presents a periodic variation of the main stream velocity component. We perform a 2D global linear stability analysis of the fully developed steady laminar flow, and we show that, for all the geometries studied, the transition is triggered by a Hopf bifurcation associated with the breaking of the symmetries and the invariance of the steady flow. Critical Reynolds numbers, most unstable modes and their characteristics are presented for large ranges of the geometric parameters, namely wavenumber α from 0.3 to 5 and amplitude from 0 (straight channel) to 0.5. We show that it is possible to define geometries for which the wavenumber is proportional to the most unstable mode wavenumber for the critical Reynolds number. From this modal study we address a weakly nonlinear stability analysis with a view to obtaining the Landau coefficient g, and then the sub-or supercritical nature of the first bifurcation characterising the transition. We show that a critical geometric amplitude beyond which the first bifurcation is supercritical is associated with each geometric wavenumber.

The linear stability of incompressible flows is investigated on the basis of the finite element method. The two-dimensional base flows computed numerically over a range of Reynolds numbers are perturbed with three-dimensional disturbances. The three-dimensionality in the flow associated with the secondary instability is identified precisely. First, by using linear stability theory and normal mode analysis, the partial differential equations

This paper examines the linear hydrodynamic stability of an inviscid compound jet. We perform the temporal and the spatial analyses in a unified framework in terms of transforms. The two analyses agree in the limit of large jet velocity. The dispersion equation is explicit in the growth rate, affording an analytical solution. In the temporal analysis, there are two growing modes, stretching and squeezing. Thin film asymptotic expressions provide insight into the instability mechanism. The spatial analysis shows that the compound jet is absolutely unstable for small jet velocities and admits a convec-tively growing instability for larger velocities. We study the effect of the system parameters on the temporal growth rate and that of the jet velocity on the spatial growth rate. Predictions of both the temporal and the spatial theories compare well with experiment.

Aims. We present a qualitative analysis of key (but yet unappreciated) linear phenomena in stratified hydrodynamic Keplerian flows: (i) the occurrence of a vortex mode, as a consequence of strato-rotational balance, with its transient dynamics; (ii) the generation of spiral-density waves (also called inertia-gravity or gΩ waves) by the vortex mode through linear mode coupling in shear flows. Methods. Non-modal analysis of linearized Boussinesq equations were written in the shearing sheet approximation of accretion disk flows. Results. It is shown that the combined action of rotation and stratification introduces a new degree of freedom, vortex mode perturbation, which is in turn linearly coupled with the spiral-density waves. These two modes are jointly able to extract energy from the background flow, and they govern the disk dynamics in the small-scale range. The transient behavior of these modes is determined by the non-normality of the Keplerian shear flow. Tightly leading vortex mode perturbations undergo substantial transient growth, then, becoming trailing, inevitably generate trailing spiral-density waves by linear mode coupling. This course of events-transient growth plus coupling-is particularly pronounced for perturbation harmonics with comparable azimuthal and vertical scales, and it renders the energy dynamics similar to the 3D unbounded plane Couette flow case. Conclusions. Our investigation strongly suggests that the so-called bypass concept of turbulence, which has been recently developed by the hydrodynamic community for spectrally stable shear flows, can also be applied to Keplerian disks. This conjecture may be confirmed by appropriate numerical simulations that take the vertical stratification and consequent mode coupling into account in the high Reynolds number regime.

A numerical model describing the unsteady, two dimensional hydrodynamics, mass transfer and heat transfer in continuous flow electrophoresis is developed in order to predict the hydrodynamic stability of the carrier buffer flow. The model takes into account of the importance of the coupling between the different phenomena and is capable of predicting undesirable effects, due to natural convection, such as backflow. 0

The eigenvalue sensitivity for hydrodynamic stability operators is investigated. Classical matrix perturbation techniques as well as the concept of 8-pseudospectra are applied to show that parts of the spectrum are highly sensitive to small perturbations. Applications are drawn from incompressible plane Couette flow, trailing line vortex flow, and compressible Blasius boundarylayer flow. Parameter studies indicate a monotonically increasing effect of the Reynolds number on the sensitivity. The phenomenon of eigenvalue sensitivity is due to the nonnormality of the operators and their discrete matrix analogs and may be associated with large transient growth of the corresponding initial value problem.

The linear stability analysis of a pressure-driven two-layer channel flow of two immiscible, Newtonian and incompressible fluids is considered. The walls of the channel are maintained at different constant temperatures and Nahme's law is applied to model the temperature dependence of the fluid viscosity. A modified Orr-Sommerfeld equation for the disturbance streamfunction coupled to a linearized energy equation is derived and solved using a spectral collocation method. Our results indicate that increasing the dimensionless top wall temperature has a non-monotonic effect on the linear stability characteristics. We also found that increasing the thermal conductivity and density ratios stabilise the flow for the set of parameter values considered; the viscosity ratio has a non-monotonic effect on the maximal growth rate. An energy 'budget' analysis shows that the most dangerous mode is of 'interfacial' type.

Several problems on three-dimensional instability of axisymmetric steady flows driven by convection or rotation or both are studied by a second-order finite volume method combined with the Fourier decomposition in the periodic azimuthal direction. The study is focused on the convergence of the critical parameters with mesh refinement. The calculations are done on the uniform and stretched grids with variation of the stretching. Converged results are reported for all the problems considered and are compared with the previously published data. Some of the calculated critical parameters are reported for the first time. The convergence studies show that the three-dimensional instability of axisymmetric flows can be computed with a good accuracy only on fine enough grids having about 100 nodes in the shortest spatial direction. It is argued that a combination of fine uniform grids with the Richardson extrapolation can be a good replacement for a grid stretching. It is shown once more that the sparseness of the Jacobian matrices produced by the finite volume method allows one to enhance performance of the Newton and Arnoldi iteration procedures by combining them with a direct sparse linear solver instead of using the Krylov-subspace-based iteration methods.