Linear Stability Problem Research Articles

A comparative study on instability of steady flows in helical pipes

A computational study of three-dimensional instability of steady flows in a helical pipe of arbitrary curvature and torsion is carried out for the first time. The problem is formulated in Germano coordinates in two equivalent but different forms of the momentum equations so that results obtained using both formulations cross verify each other. An additional formulation in the cylindrical coordinates is applied for a limiting case of the toroidal pipe. The calculations are performed by the finite volume and finite difference methods. Grid independence of the results is established for both steady flows, the eigenvalues associated with the linear stability problem, and the critical parameters. The calculated steady flows agree well with experimental measurements and previous numerical results. The computed critical Reynolds numbers corresponding to the onset of oscillatory instability agree well with the most recent experimental results, but disagree with the earlier ones. Novel results related to the parametric stability study are reported.

Effect of extension-twisting and extension-bending coupling on a compressed plate with a cut-out

This paper investigates the behaviour of thin-walled composite plates with a cut-out under compression. The effects of lamina ply orientation on the critical state and the behaviour of the plates in the post-critical range were examined. The scope of the research involved numerical modelling of linear and non-linear stability problems by the finite element method, using the simulation software ABAQUS®. The created numerical models were then verified experimentally. The study was performed on a plate with an unsymmetric stacking sequence of plies. To force buckling into a higher mode, an unsymmetric laminate with extension-twisting and extension-bending coupling was used. This approach led to obtaining the natural, lowest mode of buckling (flexural-torsional), one that ensures stable performance of the structure in the post-critical range (no transition to flexural mode).

Stability analysis of deep-water waves on a linear shear current using unsteady conformal mapping

This paper describes linear stability analysis of the two-dimensional steady motion of periodic deep-water waves with symmetric non-overhanging profiles propagating on a linear shear current, namely a vertically sheared current with constant vorticity. In order to investigate numerically with high accuracy the stability of large-amplitude waves, we adopt a formulation using conformal mapping, in which the time-varying water surface is always mapped onto the real axis of a complex plane. This formulation allows us to apply numerical methods developed for large-amplitude irrotational waves without a shear current directly to the present problem, and reduces the linear stability problem to a generalized eigenvalue problem. Numerical solutions describe both super- and sub-harmonic instabilities of the periodic waves for a wide range of wave amplitudes and clarify how the behaviours of dominant eigenvalues change with the strength of the shear current. In particular, it is shown that, even in the presence of a linear shear current, the steady periodic waves lose stability due to superharmonic disturbances at the wave amplitude where the wave energy attains an extremum, similarly to the case of irrotational waves without a shear current. It is also found that re-stabilization with an increase in wave amplitude characterizes subharmonic instability for weak shear currents, but the re-stabilization disappears for strong shear currents.

The effect of porosity and permeability variations on convective stability of a two-layer system under longitudinal vibration in zero gravity

The effect of the porosity and permeability variations on the onset of average convection in a single-component fluid within an non-uniformly heated horizontal layer partially filled with a porous zone under zero gravity conditions is studied. The system of the fluid and porous layers oscillates as a whole with high frequency and small amplitude in the longitudinal direction. Permeability is uniform within the porous zone and related to porosity by the Carman–Kozeny formula. The method of constructing a fundamental system of solutions as well as the orthogonalization of vectors for partial solutions are applied to simulate a linear stability problem with respect to the fluid quasi-equilibrium state in the layers numerically. The instability threshold relative to the short-wave and long-wave convective rolls is found. An abrupt change in the type of instability from the short-wave to long-wave ones occurs with the growth of the porosity from 0.3 to 0.8. A similar variation also presents in the case of thermal gravitational convection in layered fluid systems with porous zones under Earth conditions. Convection in weightlessness distinguishes from that in the gravitational field by a non-monotonic behavior of the instability threshold with increasing porosity at different fixed frequencies of vibration. The onset of convection delays at low frequencies and, on the contrary, speeds up at high enough frequencies, if the porosity belongs to the interval from 0.3 to 0.8. In addition, one studies the effect of two types of boundary conditions for tangential velocities near the interface between the layers on the instability threshold in the system with a low permeable porous zone.

Numerical Methods in Calculating Eigenvalues: Case Studies in Stability of Euler Columns

A comparative analysis of well renowned “Shooting Method” with another numerical method “Complementary Functions Method” (CFM) is presented for calculating eigenvalue (λ). Contrary to the shooting method hit and trial approach, CFM exploits the properties of linear ordinary differential equation (LODE). In the case of linear eigenvalue Boundary value Problem (BVP), CFM generates an algebraic equation system with one unknown “λ” and, alone root finding method is sufficient to give required eigenvalue. However, the Shooting Method create a system of algebraic equations containing two unknowns “λ” and “missing initial conditions”, that demands an additional numerical technique along with root finding method. These radical differences between two approaches, sets the basis for this comparative investigation. As a case study in Linear Elastic Stability, different cases of Euler columns are investigated by finding eigenvalues for each case numerically, under both methods. Comparison is performed on the basis of results accuracy and cost effectiveness for both numerical techniques while solving linear stability problems.

Existence and stability of discrete solitons in nonlinear Schrödinger lattices with hard potentials

In this paper we study the discrete solitons in nonlinear Schrödinger lattices with hard potentials. We first derive some sufficient conditions for the existence of discrete solitons in the anti-continuum limit. We then establish some spectral stability and instability results for discrete solitons in the anti-continuum limit using the regular perturbation theory. We also develop asymptotic approximations of the eigenvalues of the linearized stability problem.

Comparative Study on Sixth Order Boundary Value Problems with Application to Linear Hydrodynamic Stability Problem and Benard Layer Eigenvalue Problem

Numerical estimation for higher order eigenvalue problems are promising and has accomplished significant importance, mainly due to existence of higher order derivatives and boundary conditions relating to higher order derivatives of the unknown functions. In this article, we perform a numerical study of linear hydrodynamic stability of a fluid motion caused by an erratic gravity field. We employ two methods, collocation and spectral collocation based on Bernstein and Legendre polynomials to solve the linear hydrodynamic stability problems and Benard type convection problems. In order to handle boundary conditions, our techniques state all the unknown coefficients of boundary conditions derivatives in terms of known co-efficient. The schemes have been carried out to several test problems to establish the efficiency of the two methods.

On linear stability of shear flows of an ideal stratified fluid: research methods and new results

The results, that obtained by the spectral method with use of integral relations for the problem of linear stability of steady-state shear plane-parallel flows of an inviscid stratified incompressible fluid in the gravity field with respect to plane perturbations in the Boussinesq approximation and without it, are specified, complemented and developed by the most powerful analytical method of the modern mathematical theory of hydrodynamic stability – the second (or direct) Lyapunov method. In both case, the new analytical method made it possible to prove that given steady-state flows of stratified fluid are absolutely unstable in theoretical sense with respect to small plane perturbations and to obtain the sufficient conditions for practical linear instability of considered flows. The illustrative analytical examples of given steady-state flows and small plane perturbations as normal waves imposed on them are constructed. Using the asymptotic method, it is proved that constructed perturbations grow in time irrespective of the fact whether the Miles-Howard and the Miles-type theorems are valid or not.

Boundary linear stabilization of the modified generalized Korteweg\u2013de Vries\u2013Burgers equation

The linear stabilization problem of the modified generalized Korteweg–de Vries–Burgers equation (MGKdVB) is considered when the spatial variable lies in [0,1]. First, the existence and uniqueness of global solutions are proved. Next, the exponential stability of the equation is established in L^{2} (0,1). Then, a linear adaptive boundary control is put forward. Finally, numerical simulations for both non-adaptive and adaptive problems are provided to illustrate the analytical outcomes.

Thermoelastic stability of closed cylindrical shell in supersonic gas flow

In this paper the problem of linear stability of a closed cylindrical shell under the action of both non-uniform temperature field and supersonic gas flow is considered. The stability conditions for the unperturbed state of the aerothermoelastic system are obtained. It is shown that, by the combined action of the temperature field and the ambient supersonic flow, the process of linear stability can be controlled and the temperature field affects significantly the critical flutter speed.

Weakly nonlinear mode interactions in spherical Rayleigh–Bénard convection

In an annular spherical domain with separation $d$, the onset of convective motion occurs at a critical Rayleigh number $Ra=Ra_{c}$. Solving the axisymmetric linear stability problem shows that degenerate points $(d=d_{c},Ra_{c})$ exist where two modes simultaneously become unstable. Considering the weakly nonlinear evolution of these two modes, it is found that spatial resonances play a crucial role in determining the preferred convection pattern for neighbouring modes $(\ell ,\ell \pm 1)$ and non-neighbouring even modes $(\ell ,\ell \pm 2)$. Deriving coupled amplitude equations relevant to all degeneracies, we outline the possible solutions and the influence of changes in $d,Ra$ and Prandtl number $Pr$. Using direct numerical simulation (DNS) to verify all results, time periodic solutions are also outlined for small $Pr$. The $2:1$ periodic signature observed to be general for oscillations in a spherical annulus is explained using the structure of the equations. The relevance of all solutions presented is determined by computing their stability with respect to non-axisymmetric perturbations at large $Pr$.

The N-flagella problem: elastohydrodynamic motility transition of multi-flagellated bacteria.

Peritrichous bacteria such as Escherichia coli swim in viscous fluids by forming a helical bundle of flagellar filaments. The filaments are spatially distributed around the cell body to which they are connected via a flexible hook. To understand how the swimming direction of the cell is determined, we theoretically investigate the elastohydrodynamic motility problem of a multi-flagellated bacterium. Specifically, we consider a spherical cell body with a number N of flagella which are initially symmetrically arranged in a plane in order to provide an equilibrium state. We solve the linear stability problem analytically and find that at most six modes can be unstable and that these correspond to the degrees of freedom for the rigid-body motion of the cell body. Although there exists a rotation-dominated mode that generates negligible locomotion, we show that for the typical morphological parameters of bacteria the most unstable mode results in linear swimming in one direction accompanied by rotation around the same axis, as observed experimentally.

Linear Stability Analysis of Time-Dependent Electrodeposition in Charged Porous Media

We study the linear stability analysis of time-dependent electrodeposition in a charged random porous medium, whose pore surface charges can generally be of any sign, that is flanked by a pair of planar metal electrodes. Discretization of the linear stability problem results in a generalized eigenvalue problem for the dispersion relation that is solved numerically, which agrees well with the analytical approximation obtained from a boundary layer analysis valid at high wavenumbers. Under galvanostatic conditions in which an overlimiting current is applied, in the classical case of zero pore surface charges, the voltage and electric field at the cathode diverge when the bulk electrolyte concentration there vanishes at Sand's time. The same phenomenon happens for positive surface charges but at a time earlier than Sand's time. In contrast, negative surface charges allow the electrochemical system to sustain an overlimiting current via surface conduction past Sand's time, keeping the voltage and electric field bounded. Therefore, at Sand's time, negative surface charges greatly reduce the electrode surface instabilities while zero and positive surface charges magnify them. We compare theoretical predictions for overall electrode surface stabilization from the linear stability analysis with published experimental data for copper electrodeposition in cellulose nitrate membranes and demonstrate good agreement between theory and experiment. We also use the linear stability analysis as a tool to analyze how the crystal grain size changes with duty cycle during pulse electroplating.

A long-wave estimation for the damping coefficient at a flat water–water vapour interface with a phase transition

Gravitational–capillary waves, arising in a system of thick layers of two aggregate states of a substance, are studied using the example of water and its vapour. Their initially flat interface is in hydrostatic and thermodynamic equilibrium at atmospheric pressure and the corresponding saturation temperature$100\,^{\circ }\text{C}$, provided that a balance is maintained for heat fluxes transported through immovable phases in the process of heat conduction. From the problem of linear stability for small perturbations, estimation relations for their damping coefficient and eigenfrequency shift are obtained, including the factors of viscosities of the media and phase transition. The first two contributions, proportional to the kinematic viscosity of the liquid and the square root of the kinematic viscosity of its vapour, respectively, are consistent, accurate to additionally accounted for capillary forces, with the result of Dore’s analysis, which is a development of the weakly viscous Lamb theory for ‘deep water’. Numerical calculations have shown that, with increasing wavelength, the accuracy of the proposed approximation increases, especially for the wave damping coefficient, and the contribution of the viscosity of the light phase increases in percentage terms. As for the phase transition effect, which was previously overestimated using the quasi-equilibrium approximation, it remains insignificant at the level of heat fluxes acceptable in the model of thick layers of homogeneous media. The factor of thermocapillarity is found to be very weak; it becomes qualitatively noticeable only on the background of disappearing heating. Here, together with the factor of dependence of the saturation temperature on the phase pressure, it determines a non-zero correction to the wave damping coefficient.

Слабо-нелинейный анализ устойчивости термоакустического адвективного течения

The influence of a traveling sound wave on the stability of an advective flow with respect to spiral perturbations is studied. A weak nonlinear stability analysis and direct numerical simulation were performed. From the condition of solvability of a third-order problem, a differential equation is obtained, which describes the evolution of amplitude functions. At the same time, to calculate the constants in this equation, it is necessary to determine the eigenfunctions of the linear problem of stability and the solution of some inhomogeneous linear problems. The shooting method was used using the orthogonalization procedure. It is shown that supercritical regimes appear soft and stable near the threshold with respect to modulation. Direct numerical simulation of the problem was carried out using the finite difference method. A study of convective flow with a significant distance from the stability threshold was carried out. It has been established that for a certain value of parameters in the layer, oscillations of a complex structure occur.

Linear Stability Analysis of Transient Electrodeposition in Charged Porous Media: Suppression of Dendritic Growth by Surface Conduction

We study the linear stability of transient electrodeposition in a charged random porous medium, whose pore surface charges can be of any sign, flanked by a pair of planar metal electrodes. Discretization of the linear stability problem results in a generalized eigenvalue problem for the dispersion relation that is solved numerically, which agrees well with the analytical approximation obtained from a boundary layer analysis valid at high wavenumbers. Under galvanostatic conditions in which an overlimiting current is applied, in the classical case of zero surface charges, the electric field at the cathode diverges at Sand’s time due to electrolyte depletion. The same phenomenon happens for positive charges but earlier than Sand’s time. However, negative charges allow the system to sustain an overlimiting current via surface conduction past Sand’s time, keeping the electric field bounded. Therefore, at Sand’s time, negative charges greatly reduce surface instabilities and suppress dendritic growth, while zero and positive charges magnify them. We compare theoretical predictions for overall surface stabilization with published experimental data for copper electrodeposition in cellulose nitrate membranes and demonstrate good agreement between theory and experiment. We also apply the stability analysis to how crystal grain size varies with duty cycle during pulse electroplating.

Устойчивость плоскопараллельных и двумерных течений жидкости над пористой средой

The paper presents a comparative study of linear stability problems for the plane-parallel and two-dimensional stationary flow in double layer system of fluid and porous medium. An incompressible fluid flows over homogeneous saturated porous layer. The fluid top is a free rigid boundary. Porous medium is described by the Brinkman model. Two-dimensional stationary solutions are built numerically by iterative procedure of establishment method. Sets of the neutral curves are obtained by the shooting method; perturbations are assumed to be flat linear waves travelling down the flow. The neutral curves are compared for plane-parallel and two-dimensional stationary flow for different porosity and thickness of porous layer. The critical Reynolds numbers are almost twice larger than the stability threshold of plane parallel problem while the critical wave numbers close these two cases. If the porous layer is thick, the two-dimensional problem rapidly destabilized. Its stability threshold becomes lower that for plane-parallel flow. The long-wave instability realizes. Critical wave numbers are small and tends to zero at high porosities. The neutral curve become bimodal at low porosity.

Suppression of the Kapitza instability in confined falling liquid films

We revisit the linear stability of a falling liquid film flowing through an inclined narrow channel in interaction with a gas phase. We focus on a particular region of parameter space, small inclination and very strong confinement, where we have found the gas to strongly stabilize the film, up to the point of fully suppressing the long-wave interfacial instability attributed to Kapitza (Zh. Eksp. Teor. Fiz., vol. 18 (1), 1948, pp. 3–28). The stabilization occurs both when the gas is merely subject to an aerostatic pressure difference, i.e. when the pressure difference balances the weight of the gas column, and when it flows counter-currently. In the latter case, the degree of stabilization increases with the gas velocity. Our investigation is based on a numerical solution of the Orr–Sommerfeld temporal linear stability problem as well as stability experiments that clearly confirm the observed effect. We quantify the degree of stabilization by comparing the linear stability threshold with its passive-gas limit, and perform a parametric study, varying the relative confinement, the Reynolds number, the inclination angle and the Kapitza number. For example, we find a 30 % reduction of the cutoff wavenumber of the instability for a water film in contact with air, flowing through a channel inclined at$3^{\circ }$and of height 2.8 times the film thickness. We also identify the critical conditions for the full suppression of the instability in terms of the governing parameters. The stabilization is caused by the strong confinement of the gas, which produces perturbations of the adverse interfacial tangential shear stress that are shifted by half a wavelength with respect to the wavy film surface. This tends to reduce flow-rate variations within the film, thus attenuating the inertia-based driving mechanism of the Kapitza instability.

Stability of Thermovibrational Convection of a Pseudoplastic Fluid in a Plane Vertical Layer

Based on the thermovibrational convection equations, we have investigated the structure of the averaged plane-parallel convective flow in a plane vertical layer of Williamson fluid executing high-frequency linearly polarized vibrations along the layer. We show that as the vibrations are intensified, the nonlinear viscous properties of a pseudoplastic fluid cease to affect the structure and intensity of its main flow, and it becomes similar to a flow of ordinary Newtonian fluid. The linear problem of stability of an averaged plane-parallel flow of pseudoplastic Williamson fluid has been formulated and solved for the case of longitudinal high-frequency linearly polarized vibrations for small periodic perturbations along the layer. Numerical calculations have shown that, as in a Newtonian fluid, the monotonic hydrodynamic perturbations are most dangerous at low Prandtl numbers. As the Prandtl number increases, the thermal instability modes begin to exert an undesirable effect. An enhancement of pseudoplastic fluid properties leads to destabilization of the main flow for both types of perturbations. Similarly to a Newtonian fluid, an additional vibrational instability mode to which small Grashof numbers correspond appears in the presence of vibrations. The influence of this vibrational mode on the stability of the main flow is determined by the vibration frequency and the temperature gradient. An intensification of the vibrations destabilizes the flow for all of the investigated instability modes. For a given set of rheological parameters of the Williamson model, there are critical values of the modified and vibrational Grashof numbers at which the averaged flow completely loses its stability with respect to the types of perturbations under consideration. Absolute destabilization of the main flow in a pseudoplastic fluid occurs at higher values of the vibrational Grashof number than those in a Newtonian fluid.

Reduction of the linear stability problem for general zonal flows with any Rossby and Richardson numbers to a PDE

The stability to linear disturbances of a general laterally and/or vertically sheared zonal flow in thermal wind balance is considered for any Rossby number and any Richardson number. The problem for normal mode perturbations is reduced to a second‐order partial differential equation (PDE) for a single dependent variable, the pressure perturbation, in which the complex phase speed is an eigenvalue. It is shown that the boundary conditions of no normal flow are expressible in terms of the pressure and its first‐order derivatives, and that the PDE can be derived from the linearized equation for conservation of the Ertel potential vorticity. The derivations are performed first for an incompressible Boussinesq fluid in Cartesian coordinates allowing the perturbations to be either hydrostatic or non‐hydrostatic. They are shown to generalize to a perfect gas and to motions on a sphere. A PDE for general (non‐normal mode) linear perturbations can also be derived. The Ertel potential vorticity of the zonal flow is shown to arise in the component of the momentum equation parallel to the zonal flow in a generalized Coriolis acceleration term. An ordinary differential equation (ODE) is also derived for the vertical velocity perturbation to laterally uniform zonal flows which is consistent with previous results. The differences between the singularities of the equations for pressure and vertical velocity perturbations are discussed. Integral constraints on the perturbations are used to prove the standard necessary condition for symmetric instabilities and a generalization of the Charney–Stern–Pedlosky integral constraints.