Marginal Stability Curves Research Articles

Algebraic growth of 2D optimal perturbation of a plane Poiseuille flow in a Brinkman porous medium

The impact of various factors on the stability of a plane porous Poiseuille flow is investigated in detail. Both modal and non-modal linear stability analyses are used to study the effects of the porosity of the porous media and the ratio of effective viscosity to the fluid viscosity. The present analysis includes solving the linearized Navier-Stokes equation in the form of the Orr-Sommerfeld (O-S) type equation by applying the 2D perturbation to the basic mean flow. The Chebyshev collocation method is used to solve the Orr-Sommerfeld equation numerically. Through modal analysis, the accurate values of the critical triplets (αc,Rc,cc), the eigen-spectrum, the growth rate curves, and the marginal stability curves are studied. Then, by using non-modal analysis, the transient energy growth G(t) of two-dimensional optimal perturbations, the ε-pseudospectrum of the non-normal O-S operator (L), and the regions of stability, instability, and potential instability of the fluid flow system are investigated in detail. The collective results of both modal and non-modal analysis show that the porous parameter and the ratio of effective viscosity to the fluid viscosity have stabilizing effects on the fluid system due to the postponement of the onset of stability.

Linear Stability of a Combined Convective Flow in an Annulus

Linear stability analysis of a combined convective flow in an annulus is performed in the paper. The base flow is generated by two factors: (a) different constant wall temperatures and (b) heat release as a result of a chemical reaction that takes place in the fluid. The nonlinear boundary value problem for the distribution of the base flow temperature is analyzed using bifurcation analysis. The linear stability problem is solved numerically using a collocation method. Two separate cases are considered: Case 1 (non-zero different constant wall temperatures) and Case 2 (zero wall temperatures). Numerical calculations show that the development of instability is different for Cases 1 and 2. Multiple minima on the marginal stability curves are found for Case 1 as the Prandtl number increases. Concurrence between local minima leads to the selection of the global minimum in such a way that a finite jump in the value of the wave number is observed for some values of the Prandtl number. All marginal stability curves for Case 2 have one minimum in the range of the Prandtl numbers considered. The corresponding critical values of the Grashof number decrease monotonically as the Prandtl number grows.

COUPLED EFFECT OF SOLID FRACTION AND INTERNAL ENERGY SOURCE ON CONVECTIVE HEAT TRANSPORT IN THREE-LAYERED AIR-POROUS-AIR DOMAIN: COMPARISON WITH DIFFERENT TWO-LAYERED DOMAINS

This paper studies convective heat transport driven by a uniform internal energy source coupled with the solid fraction in the partial air-saturated porous domains bounded by the top and bottom impermeable thermally conductive surfaces. The consideration of the coupled effect distinguishes our work from the papers of previous authors. We provide a comprehensive numerical study of internal convection and compare results of the linear stability analysis for three distinct domains. The first domain is air-porous-air (APA), where the heat-generating porous matrix is between the upper and lower air layers. In the second air-porous (AP) domain, the air layer overlays the porous medium. In the third porous-air (PA) domain, the air layer underlays the porous medium. The bimodal marginal stability curves and regime map with a demarcation line between the local and large-scale convective flows are obtained only for the APA and AP domains due to the division of each domain into the upper unstably stratified and lower stably stratified parts. The local convection cannot originate in the PA domain because the air layer belongs to the lower stably stratified part. A remarkable destabilizing effect of additional air layers has been revealed. For example, at the fixed solid fraction of 0.1, we achieve a 40-fold reduction of the critical Darcy-Rayleigh number in the APA domain by increasing the depth ratio from 0 to 0.5, and by 14 times in the AP domain and only by 3.5 times in the PA domain. The destabilizing effect enhances with increases in the solid fraction.

Hydrodynamic instability of flow through a rotating channel filled with isotropic porous media

Various geophysical and engineering applications have underlying physics, comprising system rotation's effects on the dynamics and transport phenomena in porous media flows. Comprehensive knowledge of the instability in a rotating fluid-saturated porous layer is beneficial for controlling the transport phenomena and the mixing process. The present study focuses on the temporal evolution of small disturbances in a pressure-induced fluid flow through a spanwise rotating channel filled with an isotropic porous material. A Darcy–Brinkman model, including the Coriolis force term in the momentum equation, is employed to describe the developed flow. A normal mode analysis is performed, and the coupled Orr–Sommerfeld–Squire eigenvalue problem is formulated to capture the linear instability of the perturbed flow. The Chebyshev collocation technique is used to solve the fourth-order eigenvalue problem to capture the transient behavior of the finite-amplitude disturbances. The temporal growth rate and marginal stability curves related to the Coriolis force-based instabilities are investigated. The rotating porous media flow is unstable at a much lower Reynolds number than the non-rotating configuration. The analysis confirms co-existing unstable modes and mode coalescence for a specific range of parameters, which can enhance the mixing and transport inside the porous layer. The neutral stability curves show the appearance of two different unstable zones corresponding to the long and moderate waves. Moreover, the higher permeability and porosity of the porous medium have a destabilizing influence.

Weakly nonlinear thermohaline rotating convection in a sparsely packed porous medium

This paper discusses the analytical study of linear and weakly nonlinear stability of thermohaline convection in a sparsely packed porous due to rotation. In the linear stability analysis, we used the normal mode technique to find the critical Rayleigh number and it is a function of q,Ta,Λ,R2, and L. Marginal stability curves were traced corresponding to thermal Rayleigh number and studied bifurcation points. In the weakly nonlinear analysis, a nonlinear two-dimensional Landau Ginzburg equation was derived at the onset of stationary convection, studied the secondary instabilities and heat transport due to convection. One dimensional coupled Landau Ginzburg equations are derived at the onset of oscillatory convection and analyzed the stability regions of steady-state, standing waves, and travelling waves.

Влияние размера воздушной области на порог тепловой конвекции в системе «воздух–тепловыделяющая пористая среда–воздух»

The problem concerning convective stability of the motionless air state in a layered air–heat-generating porous medium–air system under the gravitational field is simulated numerically in the paper. We take into consideration the heat source strength dependent on solid volume fraction in the porous sublayer. Marginal stability curves are presented at different values of the relative depth of the air sublayer. It has been shown that the onset value of internal convection decreases with increasing this depth. Besides, the convective roller flow is of a large-scale character at low relative depths, but it can be localized within the upper air sublayer at high enough depths. A comparison of the threshold internal Rayleigh-Darcy numbers at equal ratios of the air to porous depths has shown that the enhancement of convection (or lowering its threshold) is more effective in the three-layered air–heat-generating porous medium–air system as compared to that of the two-layered air–heat-generating porous medium system.

Instability of the plane parallel flow through a saturated porous medium in the presence of a longitudinal magnetic field using the Chebyshev collocation method

The onset of instability against small disturbances of the pressure driven laminar fluid flow through a saturated porous medium in the presence of horizontal magnetic field between the two parallel impermeable plates is studied numerically. Assuming the channel walls have zero conductivity. The Chebyshev collocation method is used to solve the 6th order generalized Orr–Sommerfeld equation. The critical values of the Reynolds number Rec, the wave number αc and the wave speed cc are computed for various sets of the values of the porous parameter σp, magnetic Reynolds number Rm and Alfven number A. The marginal stability curves are drawn for various values of the physical parameters present in the problem. The onset of instability is also discussed in detail using the curves of growth rate of the disturbances and the critical values for various parameters present in the problem. It is observed that the role of σp, Rm and A are to delay the onset of instability of the system. The degree of stabilization is moderate up to certain value of A but it is so rapid when the Alfven number exceeds that value is observed. It is also seen that, the role of the combined influence of porosity of the porous medium and the strength of the magnetic field together suppress the growth of the disturbances very significantly. In this study the results computed are also compared with the existing literature as a particular case for ordinary Newtonian fluids.

A linear stability analysis of two-layer moist convection with a saturation interface

The linear convective instability of a mixture of dry air, water vapour and liquid water, with a stable unsaturated layer residing on an unstable saturated layer, is studied. It may serve as a prototype model for understanding the instability that causes mixing at the top of stratocumulus cloud or fog. Such a cloud-clear air interface is modelled as an infinitely thin saturation interface where radiative and evaporative cooling take place. The interface position is determined by the Clausius–Clapeyron equation, and can undulate with the evolution of moisture and temperature. In the small-amplitude regime two physical mechanisms are revealed. First, the interface undulation leads to the undulation of the cooling source, which destabilizes the system by superposing a vertical dipole heating anomaly on the convective cell. Second, the evolution of the moisture field induces non-uniform evaporation at the interface, which stabilizes the system by introducing a stronger evaporative cooling in the ascending region andvice versain the descending region. These two mechanisms are competing, and their relative contribution to the instability is quantified by theoretically estimating their relative contribution to buoyancy flux tendency. When there is only evaporative cooling, the two mechanisms break even, and the marginal stability curve remains the same as the classic two-layer Rayleigh–Bénard convection with a fixed cooling source.

Linear Stability of a Steady Convective Flow between Permeable Cylinders

Linear stability analysis of a steady convective flow in a tall vertical annulus caused by nonlinear heat sources is conducted in the paper. Heat sources are generated as a result of a chemical reaction. The effect of radial cross-flow through permeable porous walls of the annulus is analyzed. The problem is relevant to biomass thermal conversion. The base flow solution is obtained by solving nonlinear boundary value problem. Linear stability analysis is performed, using collocation method. The calculations show that radial inward or outward flow has a stabilizing effect on the flow, while the increase in the Frank–Kamenetskii parameter (proportional to the intensity of the chemical reaction) destabilizes the flow. The increase in the Reynolds number based on the radial velocity leads to the appearance of the second minimum on the marginal stability curves. The rate of increase in the critical Grashof number with respect to the Reynolds number is different for inward and outward radial flows.

Localized patterns in a generalized Swift–Hohenberg equation with a quartic marginal stability curve

Abstract In some pattern-forming systems, for some parameter values, patterns form with two wavelengths, while for other parameter values, there is only one wavelength. The transition between these can be organized by a codimension-three point at which the marginal stability curve has a quartic minimum. We develop a model equation to explore this situation, based on the Swift–Hohenberg equation; the model contains, amongst other things, snaking branches of patterns of one wavelength localized in a background of patterns of another wavelength. In the small-amplitude limit, the amplitude equation for the model is a generalized Ginzburg–Landau equation with fourth-order spatial derivatives, which can take the form of a complex Swift–Hohenberg equation with real coefficients. Localized solutions in this amplitude equation help interpret the localized patterns in the model. This work extends recent efforts to investigate snaking behaviour in pattern-forming systems where two different stable non-trivial patterns exist at the same parameter values.

Modal stability and Squire’s theorem for an inhomogeneous viscoelastic suspension

We study the linear stability of the Poiseuille flow of a viscoelastic upper convected Maxwell fluid in which the rheological parameters depend on the concentration of particles suspended in the fluid (dense suspension with negligible diffusion). After determining the basic flow and basic concentration profile we consider a temporal three dimensional perturbation in the form of a stream-wise and span-wise wave. We derive the linearized perturbed equation and prove the validity of Squire’s theorem, extending the result of Tlapa and Bernstein (1970) in which the theorem was proved for constant rheological parameters. We discuss the relation between the Weissenberg and the Reynolds numbers. We finally study the 2D eigenvalue problem for the case of constant coefficients and for non-constant coefficients with low Weissenberg number. We solve the problem numerically by means of a spectral collocation method and we plot the marginal stability curves discussing how stability depends on the fluid rheology.

Hydroelastic response of a floating plate on the falling film: A stability analysis

The role of a floating elastic plate on the hydrodynamic instability of a gravity-driven flow down an inclined plane is examined in the linear and nonlinear regimes. Linear instability of the system with respect to infinitesimal disturbances is captured using normal-mode analysis. The critical conditions for instability are obtained analytically utilizing the small film aspect ratio. The bifurcation of the nonlinear evolution equation is analyzed using weakly nonlinear stability analysis. The time evolution of the surface elevation is analyzed using the nonlinear analysis. The Orr–Sommerfeld boundary value problem corresponding to the perturbed flow is derived, and it is solved numerically using the spectral collocation method. The behavior of the marginal stability curves and temporal growth of the unstable waves are portrayed for a range of dimensionless flow parameters. Moreover, the pressure acting on the surface are calculated and analyzed for various structural parameters, applicable to different flow configurations. The study reveals that the structural parameters such as rigidity and mass per unit length play a crucial role in suppressing and facilitating the unstable surface waves of the flow. However, the compressive force acting on the plate results in the flexural destabilization. Thus, the plate parameters are more efficient in damping the shorter wave disturbances. Numerical observations imply that the floating elastic plate assists in stabilizing the free-falling flow and dampens the high amplitude waves.

Hydrodynamic instability of free river bars

In this paper, we explore the hydrodynamic instability of free river bars driven by a weakly varying turbulent flow in a straight alluvial channel with erodible bed and non-erodible banks. We employ linear stability analysis in the framework of depth-averaged formulations for the hydrodynamics and the sediment transport. A significant fraction of the sediment flux is considered to be in suspension. The analysis is performed for the alternate pattern of river bars at the leading order followed by the next order, covering the effects of flow regime. We find that the unstable region bounded by a marginal stability curve depends significantly on the shear Reynolds number, which demarcates different flow regimes, and the Shields number and the relative roughness (particle size to flow depth ratio). The results at the next order stabilize the bars with longer wavenumbers. The variations of threshold aspect ratio with Shields number and relative roughness are studied for different flow regimes. In addition, for a given Shields number and relative roughness, the diagram of threshold aspect ratio vs shear Reynolds number is explained. Unlike the conventional theories of bar instability, the analysis reveals limiting values of Shields number and relative roughness beyond which the theoretical results at the next order produce infeasible regions of instability. The limiting values of Shields number and relative roughness appear to reduce, as the shear Reynolds number increases.

String baryon in four-dimensional N=2 supersymmetric QCD from the 2D-4D correspondence

We study non-Abelian vortex strings in four-dimensional (4D) $\mathcal{N} = 2$ supersymmetric QCD with U$(N=2)$ gauge group and $N_f=4$ flavors of quark hypermultiplets. It has been recently shown that these vortices behave as critical superstrings. The spectrum of closed string states in the associated string theory was found and interpreted as a spectrum of hadrons in 4D $\mathcal{N} = 2$ supersymmetric QCD. In particular, the lowest string state appears to be a massless BPS "baryon." Here we show the occurrence of this stringy baryon using a purely field-theoretic method. To this end we study the conformal world-sheet theory on the non-Abelian string -- the so called weighted $\mathcal{N} = (2,2)$ supersymmetric $\mathbb{CP}$ model. Its target space is given by the six-dimensional non-compact Calabi-Yau space $Y_6$, the conifold. We use mirror description of the model to study the BPS kink spectrum and its transformations on curves (walls) of marginal stability. Then we use the 2D-4D correspondence to show that the deformation of the complex structure of the conifold is associated with the emergence of a non-perturbative Higgs branch in 4D theory which opens up at strong coupling. The modulus parameter on this Higgs branch is the vacuum expectation value of the massless BPS "baryon" previously found in string theory.

Thermally unstable throughflow of a power–law fluid in a vertical porous cylinder with arbitrary cross–section

The present paper investigates how the cross–sectional shape of a vertical porous cylinder affects the onset of thermoconvective instability of the Rayleigh–Bénard type. The fluid saturating the porous medium is assumed to be a non–Newtonian power–law fluid. A linear stability analysis of the vertical thorughflow is carried out. Three special shapes of the cylinder cross–section are analysed: square, circular and elliptical. The effect of changing the power–law index is investigated. The stability of a steady base state with vertical throughflow is analysed. The resulting stability problem is a differential eigenvalue problem that is solved numerically through the shooting method. The dimensionless numbers here considered are the non–Newtonian version of the Darcy–Rayleigh number, Ra, the Péclet number, Pe, and the power–law index, n. Results are presented in the form of marginal stability curves with Ra plotted as a function of the cylinder aspect ratio, by assuming different values of Pe and n. The critical values of Ra are also computed. Results show that the critical Rayleigh number Rac for instability depends only on Pe and n, and is independent of the shape of the cylinder cross–section. The geometry of the sidewall just contributes the selection of the allowed wavenumbers.

Effect of a floating elastic membrane for stabilizing the film flow down a porous inclined plane

AbstractThe effect of a floating elastic membrane on a gravity‐driven flow down an inclined porous plane is examined. The present work extends the earlier work of Anjalaiah et al. [Phys. Fluid, Vol. 25, 022101 (2013)] by including the thin membrane and excluding the insoluble surfactant at the top surface of the flow. The membrane tension emergence from the elasticity, is used to monitoring the surface tension on the top surface of the fluid layer. The fluid flow below the elastic membrane is governed by the Navier–Stokes equations and flow through the porous medium is governed by the Darcy–Brickmann equations. Using the normal mode analysis, two Orr–Sommerfeld systems are derived for each layers and solved by using the spectral collocation method to capture the linear instability. Further, the temporal growth rate and marginal stability curve are analyzed for different set of structural parameters. In general, the instability of free surface flow is more due to the presence of the unstable Yih mode. Both, the membrane tension and the uniform mass distribution rate along the length of the membrane assure to stabilize the surface waves by lowering the temporal growth rate of the disturbances. Consequently, the floating elastic membrane and the porosity of the inclined substrate pretend to suppress the unstable Yih mode and stabilize the flow anatomy. Hence, the floating elastic membrane can be disposed for the lethargic control of the free‐surface flows' instability.

Two Modes of Vibrational Double-Diffusive Instability in a Superposed Fluid-Porous Layer Heated from Below: The Effect of Buoyancy Ratio

The convective double-diffusive instability in a two-layer system consisting of a binary fluid layer overlaying a fluid-saturated porous layer is investigated numerically. The system moves periodically in a vertical direction under the conditions of gravity field. The layers are heated from below and oscillate with high frequency and small amplitude. The instability threshold for the convection onset and the wavelength of convective rolls, which induce as soon as the fluid equilibrium state has lost its stability, are found. The buoyancy ratio and vibrational parameter effects are studied. A region of parameters for the bimodal marginal stability curves is obtained and divided it into parts for the local and large-scale convection onset values. It is revealed that wavelength of stationary convection rolls varies non-monotonically as the buoyancy ratio decreases. The longitudinal scale of rolls reduces sharply at first and then increases gradually. Vibration produces the greatest stabilizing effect on the fluid equilibrium state in a narrow range of small negative and positive buoyancy ratios, which are responsible for a considerable change in the wavelength of rolls. The buoyancy ratio effect on the oscillatory convection rolls is rather weak. The vibration effect on their marginal stability threshold and wavelength is more pronounced.

Onset of thermal convection in a superposed fluid-porous layer subjected to high-frequency longitudinal vibration in weightlessness

The onset of thermal vibrational convection in a fluid layer overlaying a fluid-saturated porous layer is studied in zero gravity. A thermal gradient is imposed transverse to the layers. The two-layer system performs high-frequency small-amplitude oscillations perpendicular to the thermal gradient. We solve a linear stability problem with respect to the fluid quasi-equilibrium state. The basic closed-looped oscillatory flow with zero average velocity characterizes the quasi-equilibrium state. The average roll convection is generated if the Rayleigh number exceeds its threshold value. The bimodal marginal stability curves are revealed at different ratios of the fluid layer thickness to that of the porous layer. They are somewhat similar to the curves first obtained by Chen and Chen for the onset of thermal convection in a superposed fluid-porous layer in a gravity field. We find that short-wave convection replaces the long-wave one with an increase in the ratio of layer thicknesses. The transition is accompanied by a jump-wise increase in the wave number of the most dangerous convective rolls. However, the critical wave number of the vibrational convection rolls can change by several orders of magnitude in contrast to that of the gravitational convection rolls. So, one can have super short-wave convection in zero gravity. A value of the instability threshold varies non-monotonically. The convection onset delays rather weakly at small thickness ratios. On the contrary, sufficiently high thickness ratios make the convection onset speed up strongly.

Linear stability analysis of mixed convection flow in a horizontal channel filled with nanofluids

AbstractThe convective instability driven by buoyancy in the Poiseuille–Rayleigh–Bénard flow through two infinite parallel horizontal plates filled with nanofluids is investigated using linear stability analysis. We considered water‐based nanofluids with different volume fractions of aluminum () and silver () nanoparticles. A spectral collocation method founded on Chebyshev polynomials is implemented and the obtained algebraic eigenvalue problem is solved. In this study, we have numerically determined the critical Rayleigh number of the onset of longitudinal and transversal rolls and the results are represented in the form of marginal stability curves. Critical wave numbers that describe the size of convective cells in the flow are also presented, analyzed, and compared with those of the Poiseuille–Rayleigh–Bénard flow without nanoparticles. The effects of the type and nanoparticle volume fractions on the onset of both longitudinal and transversal rolls are investigated.

Convection in a Horizontal Porous Layer with Vertical Pressure Gradient Saturated by a Power-Law Fluid

The onset of convection in a porous layer saturated by a power-law fluid is here investigated. The walls are considered to be isothermal, isobaric and permeable in such a way that a vertical throughflow is described. The threshold for a buoyancy-driven cellular flow is investigated by means of a linear stability analysis. This study consists in introducing disturbances with small amplitude. The disturbances are plane waves, i.e. a normal modes stability analysis of the basic stationary solution is performed. The resulting problem is an ordinary differential equation eigenvalue problem which is solved numerically by coupling the Runge–Kutta method with the shooting method. Results are presented in the form of marginal stability curves and their critical points representing the values of the control parameters such that the growth rate of the disturbances is zero. It is found that, among roll disturbances, the most unstable modes are stationary and uniform with infinite wavelength. For this reason, an asymptotic analysis for vanishing wave numbers is carried out. The results of this asymptotic analysis are obtained analytically displaying a very good agreement with the numerical solution. It is found that vertical throughflow plays a destabilising role for pseudoplastic fluids and a stabilising role for dilatant fluids.