Preferred Mode Of Instability Research Articles

Linear stability of longitudinal convective rolls in a non-Darcy porous layer filled with nanofluid due to viscous dissipation effect: A realistic approach

The onset of longitudinal convective rolls of nanofluid flow inside a horizontal porous material using Brinkman theory due to the viscous dissipation effect has been addressed numerically. An infinitely long horizontal porous region has been considered which is confined between two rigid boundaries. The boundaries of the channel are considered adiabatic (lower) and isothermal (upper). Brownian motion and thermophoresis properties are taken into consideration for nanofluid. In the linear stability analysis, oblique roll disturbances with arbitrary orientation are taken into account, where the preferred mode of instability i.e., longitudinal rolls (when P=0 or ϑ=π/2) are shown. Finally, the resulting eigenvalue problem is solved in MATLAB using the bvp4c technique. The impact of improving the modified diffusivity ratio (NA<10), Lewis number (Le<,=,>1), concentration Rayleigh number (Rn) and modified particle density increment (NB≈10−2−10−5) is to accelerate the instability boundaries, while the switching parameter (ξ) has a dual character on it. Further, the pattern of the streamlines are slightly compressed in the channel’s center for large values of ξ, whereas no markable changes are noticed for higher values of NA. The isotherms lines are closer near the adiabatic boundary when ξ takes small values.

Modified stability analysis of double‐diffusive convection in viscoelastic fluid layer saturating porous media

AbstractThe present analysis mathematically investigates the thermohaline convection problem in viscoelastic fluid layer saturating porous media by utilizing the modified Boussinesq approximation. By performing linear stability analysis, the Darcy–Rayleigh numbers for stationary and oscillatory modes of convection are derived. The effects of different parameters describing the problem are studied numerically. In nonlinear stability analysis, the heat and mass transfer rates in the form of Nusselt and Sherwood numbers, respectively, are obtained for oscillatory convection using the derived Ginzburg–Landau equation. From the results, it is observed that overstability is the preferred mode of instability in linear stability. It is found that in linear double‐diffusive convection problems, the stress relaxation imparts a destabilizing effect whereas the strain retardation time, the coefficient of specific heat variation due to temperature, and the concentration gradient have a stabilizing effect on the system's stability. The numerical values of heat and mass transfer rates varied with the coefficient of specific heat showing that the heat transport decreases while the mass transport increases. Also, the stress relaxation time, the concentration gradient, and the gravity modulation's amplitude increase while the strain retardation time decreases the heat and mass transfer rates. The wavelength of oscillations remains unaltered with the variation of specific heat variation due to temperature. The modulation frequency does not affect the heat/mass transfer rate; though, the wavelength of oscillations decreases with increasing frequency.

Effect Of MFD Viscosity On Ferroconvection In A Fluid Saturated Porous Medium With Variable Gravity

&lt;p&gt;Convective instability of a horizontal ferromagnetic fluid saturated porous layer with magnetic field dependent (MFD) viscosity subjected to gravity field varying with distance in the layer is investigated. The fluid motion is described by the Brinkman model. The method of small perturbation is applied and the resulting eigenvalue problem is solved using the higher order Galerkin technique. The stationary instability is shown to be the preferred mode of instability and the resulting eigenvalue problem is solved by taking into account the realistic rigidrigid- isothermal boundary conditions. The study reveals that the effect of MFD viscosity is to delay the onset of ferroconvection and the stabilizing effect of MFD viscosity is reduced when the magnetic Rayleigh number is sufficiently large. In the presence of variable gravity, the effect of magnetic and non-magnetic parameters on ferroconvective instability is also discussed.&lt;/p&gt;

Surface deformations in dynamic thermocapillary convection under partial slip.

A prescribed, horizontal temperature gradient is imposed upon a horizontal liquid layer bounded from above by a deformable, liquid-gas interface and bounded from below by a partial-slip, rigid surface. A steady shear flow driven by thermocapillary motion emerges. This dynamic liquid layer is susceptible to the onset of oblique three-dimensional hydrothermal waves, purely two-dimensional hydrothermal waves, longitudinal traveling waves, and longitudinal rolls depending on the capillary number. A low capillary number analysis finds that surface deformations are destabilizing for all modes of instability. There is a preference for two-dimensional hydrothermal waves when there are surface deformations. Though longitudinal traveling waves are never selected as the preferred mode of instability, these waves offer a convenient way to understand the behavior of oblique hydrothermal waves, which are near-longitudinal. This is especially the case for low capillary numbers, but oblique hydrothermal waves instead tend to align themselves with the direction of flow as the capillary number increases. Surface deformations affect longitudinal waves most significantly out of all the modes of instability, especially for low Prandtl numbers. The typical length scales shorten and the critical Marangoni numbers increase with the capillary number for all types of modes. Notably, the system selects long waves near a critical Prandtl number when the interface is nondeformable and when the layer is subject to partial slip, but this is no longer the case when the upper surface is deformable.

Linear Stability of Horizontal Throughflow in a Brinkman Porous Medium with Viscous Dissipation and Soret Effect

The onset of double-diffusive convective instability of a horizontal throughflow induced by viscous dissipation in a fluid-saturated porous layer of high permeability is investigated. The porous layer is infinitely long along the horizontal direction and is bounded by two rigid surfaces maintained at constant, but different solute concentrations. The lower surface is thermally insulated, whereas the upper surface is considered to be isothermal. The Darcy–Brinkman model is adopted for deriving the equations governing flow in the medium, and the Soret effect is considered to persist in the flow. The instability in the base flow is considered to be induced by the non-negligible viscous heating. Disturbances in the base flow are assumed in the form of oblique rolls, where the longitudinal and the transverse rolls are at two extreme inclinations. The disturbance functions are assumed to be of O(1). It is considered that $$Ge\ll 1$$ and $$|Pe|\gg 1$$ , where Ge is the Gebhart number and Pe is the Peclet number. The eigenvalue problem with coupled ordinary differential equations governing the disturbances in the flow is solved numerically using bvp4c in MATLAB. Results obtained depict that the flow is most stable in the Brinkman regime and the longitudinal rolls are the preferred mode of instability. The solute concentration gradient and the Soret parameter have both stabilizing and destabilizing effect on the flow in the medium, when the values of both are either positive or negative. However, they have either monotonically stabilizing or monotonically destabilizing effect on the flow, when the values of both have opposite signs.

Local thermal non-equilibrium effects on thermal convection in a rotating anisotropic porous layer

Effects of local thermal non-equilibrium (LTNE) on thermal convection in a rotating fluid-saturated anisotropic porous layer are investigated. The analysis has been carried out by constructing a simplified model consisting of six coupled nonlinear ordinary differential equations. The study reveals the equivalence of linear and nonlinear stability boundaries indicating the linearized instability theory captures completely the physics of the onset of convection. Results show that the presence of rotation is to introduce oscillatory convection once the Taylor number exceeds a threshold value. The preferred mode of instability is found to be influenced significantly by the mechanical anisotropy parameter as well and it is demonstrated that it has both stabilizing and destabilizing effects on the steady onset in the presence of rotation. Besides, asymptotic analyses for small and large values of the interphase heat transfer coefficient are presented. Heat transport is calculated in terms of Nusselt number. Also, the coupled nonlinear ordinary differential equations are solved numerically using Runge–Kutta method and the transient behavior of Nusselt number is demonstrated for various values of physical parameters.

Electrohydrodynamic instability of a rotating couple stress dielectric fluid layer

The simultaneous effect of rotation and a vertical AC electric field on the onset of electrohydrodynamic instability in a horizontal couple stress dielectric fluid layer caused by the dielectrophoretic force due to the variation in the dielectric constant with temperature is investigated. The boundaries of the fluid layer are considered to be either free–free or rigid–rigid or lower rigid–upper free which are either isothermal or insulated to temperature perturbations. The resulting eigenvalue problem is solved exactly for free-free isothermal boundaries. It is observed that oscillatory convection is not a preferred mode of instability as far as the dielectric fluids are concerned and the necessary conditions for its occurrence are found to be independent of applied vertical AC electric field. For the other combinations of velocity and temperature boundary conditions, the problem is solved numerically using the Galerkin method. In contrast to the non-rotating case, it is shown that a rotating fluid layer becomes destabilizing in the presence of couple stress for all the boundary conditions considered. The effect of increasing AC electric Rayleigh number is to increase the transfer of heat more effectively and hastens the onset of convection, while rotation inhibits the onset of electrohydrodynamic instability. Although the rigid-rigid boundaries enhance the stability when compared to free-free boundaries up to certain values of Taylor number, the situation gets reversed at high Taylor number domain which in turn depends on the value of couple stress parameter and boundary conditions. Isothermal boundaries display more stabilizing effect than insulated ones.

The dynamics of two interacting compositional plumes in the presence of a magnetic field

The dynamics of two compositionally buoyant columns of fluid rising in an infinite less buoyant fluid is studied in the presence of a uniform magnetic field, B 0. The fluid is thermally stably stratified and has a viscosity, ν, a thermal diffusivity, κ and magnetic diffusivity, η. The stability of the mean state to infinitesimal disturbances is governed by the seven dimensionless parameters: the Reynolds number, R (=UL/ν, where U, L are characteristic velocity and length respectively) which measures the strength of the compositional buoyancy; the dimensionless measures x 0, x 1, d of the thickness of the two plumes and the distance between them, respectively; the ratio Γ of the strengths of the two plumes (as measured by their basic concentration of light material); the Chandrasekhar number, Qc (=, in which μ is the magnetic permeability, ρ0 the fluid density and B 0 a characteristic unit of magnetic field), is a measure of the magnitude of the magnetic field and the normalized horizontal projection of the magnetic field, where θ measures the inclination of the magnetic field to the vertical. The stability is examined for small values of R. The preferred mode of instability is studied in the parameter space (x 0, x 1, d, ). It is shown that the influence of the magnetic field does not change the order of the magnitude of the growth rate from O(R 0) of the two non-magnetic interacting plumes and it does not introduce any new modes to the stability problem. However, the presence of the magnetic field introduces novel features to the stability problem. For any fixed set x 0, x 1, d, Γ, Qc, the growth rate can either increase with or initially decrease reaching a minimum before it increases again. As Qc increases, with fixed, the growth rate can assume one of four different behaviours: (i) it maintains the same value of the non-magnetic case with the disturbance propagating along field lines; (ii) it decreases steadily with Qc; (iii) it maintains the same value as in the absence of the field until a value is reached when it starts to increase to a maximum before it decreases to zero for large values of Qc and (iv) it increases from its value for Qc = 0 reaching a maximum before it decreases steadily to zero at some value of Qc dependent on the other parameters. The helicity and α-effect have also been studied to find that the unstable motions can produce mean helicity and α-effect.

Thermoconvective instabilities in an inclined porous channel heated from below

The thermoconvective instability in an inclined rectangular channel filled with a fluid saturated porous medium and heated from below with a uniform flux is investigated. A stationary parallel buoyant flow with a linear temperature change in the transverse direction is considered. The linear stability to transverse and longitudinal roll disturbances of this basic state is examined. The thermoconvective instability onset of transverse rolls occurs when the Darcy–Rayleigh number exceeds a critical value, that increases with the inclination angle. The critical Darcy–Rayleigh number is discontinuous at the inclination angle 23.4749° above the horizontal. It is shown that, when the inclination angle exceeds 31.3618°, either the transverse rolls are stable or a second discontinuous transition to a higher branch of instability occurs. The longitudinal rolls may be unstable for every inclination except for the vertical. A stability diagram is sketched displaying the geometrical conditions (inclination angle, aspect ratio) such that either the transverse rolls or the longitudinal rolls is the preferred mode of instability.

Instability and Viscous Dissipation in the Horizontal Brinkman Flow through a Porous Medium

The convective instability activated by the sole effect of viscous dissipation in a fluid saturated porous layer is studied. The basic parallel flow in a highly permeable porous medium is analysed by considering the viscous heating contribution in the local energy balance, by assuming a thermally insulated lower boundary and an isothermal upper boundary. The Brinkman model of momentum transfer is adopted. Arbitrarily oriented oblique roll disturbances are considered in the linear stability analysis. Among them, the longitudinal rolls, having axis parallel to the basic flow direction, are shown to be the preferred mode of instability. Some considerations on the reliability of the Brinkman model, when the viscous dissipation contribution is not negligible and when the flow conditions are close to the limiting case of a clear fluid, are finally expressed.

Coriolis effect on thermal convection in a couple-stress fluid-saturated rotating rigid porous layer

Both linear and weakly nonlinear stability analyses are performed to study thermal convection in a rotating couple-stress fluid-saturated rigid porous layer. In the case of linear stability analysis, conditions for the occurrence of possible bifurcations are obtained. It is shown that Hopf bifurcation is possible due to Coriolis force, and it occurs at a lower value of the Rayleigh number at which the simple bifurcation occurs. In contrast to the nonrotating case, it is found that the couple-stress parameter plays a dual role in deciding the stability characteristics of the system, depending on the strength of rotation. Nonlinear stability analysis is carried out by constructing a set of coupled nonlinear ordinary differential equations using truncated representation of Fourier series. Sub-critical finite amplitude steady motions occur depending on the choice of physical parameters but at higher rotation rates oscillatory convection is found to be the preferred mode of instability. Besides, the stability of steady bifurcating equilibrium solution is discussed using modified perturbation theory. Heat transfer is calculated in terms of Nusselt number. Also, the transient behavior of the Nusselt number is investigated by solving the nonlinear differential equations numerically using the Runge–Kutta–Gill method. It is noted that increase in the value of Taylor number and the couple-stress parameter is to dampen the oscillations of Nusselt number and thereby to decrease the heat transfer.

The dynamics of two interacting compositional plumes

The dynamics of two compositionally buoyant columns of fluid rising in an infinite less buoyant fluid is studied. The fluid within and outside the columns is thermally stably stratified and has a kinematic viscosity, ν, and a thermal diffusivity, κ. The mean vertical flow and the associated temperature profiles depending on the horizontal coordinate normal to the surfaces of the columns are obtained and the associated buoyancy, heat and material fluxes are discussed. The stability of the mean state to infinitesimal disturbances is governed by the five dimensionless parameters: the number R (=UL/ν, where U and L are characteristic velocity and length, respectively) which measures the strength of the compositional buoyancy, the dimensionless measures of the thickness of the two plumes and the distance between them, respectively, and Γ representing the ratio of the strengths of the two plumes. The stability is examined for small values of R. The preferred mode of instability is studied in the space (, Γ). When a single plume is present (and x 1, d, Γ = 0), it is already known that the perturbations are neutral at O(R 0) and linearly unstable at O(R 1) for all values of x 0 and Prandtl number, σ(=ν/κ). It is shown here that at , the presence of a second interacting plume has the significant effect that the system of two plumes becomes unstable for a certain range of the parameters while it remains neutrally stable for the others. In contrast with the instability of the single plume, the instability of the two interacting plumes does not depend, at the level, on the Prandtl number of the fluid. However, the neutrally stable two plumes might suffer instability at and that instability may depend on the Prandtl number. The instability is found to be driven by the gradients of all of the concentration of light material, the flow and temperature of the basic state. The presence of the second plume destroys the definite parity of the perturbations' solution and this can lead to non-zero total helicity.

Elastic effects on Rayleigh-Bénard convection in liquids with temperature-dependent viscosity

A linear stability analysis of convection in viscoelastic liquids with temperature-dependent viscosity is studied using normal modes and Galerkin method. Stationary convection is shown to be the preferred mode of instability when the ratio of strain retardation parameter to stress relaxation parameter is greater than unity. When the ratio is less than unity then the possibility of oscillatory convection is shown to arise. Oscillatory convection is studied numerically for Rivlin-Ericksen, Maxwell and Jeffreys liquids by considering free-free, rigid-rigid and rigid-free isothermal/adiabatic boundaries. The effect of variable viscosity parameter is shown to destabilize the system. The problem reveals the stabilizing nature of strain retardation parameter and destabilizing nature of stress relaxation parameter, on the onset of convection. The Maxwell liquids are found to be more unstable than the one subscribing to Jeffreys description whereas the Rivlin-Ericksen liquid is comparatively more stable. Free-free adiabatic boundary combination is found to give rise to a most unstable system, whereas the rigid isothermal rigid adiabatic combination gives rise to a most stable system. The problem has applications in non-isothermal systems having viscoelastic liquids as working media.

Compositional convection in the presence of a magnetic field. II. Cartesian plume

The analysis of part I, dealing with the morphological instability of a single interface in a fluid of infinite extent, is extended to the case of a Cartesian plume of compositionally buoyant fluid, of thickness 2 x 0 , enclosed between two vertical interfaces. The problem depends on six dimensionless parameters: the Prandtl number, σ ; the magnetic Prandtl number, σ m ; the Chandrasekhar number, Q c ; the Reynolds number, Re ; the ratio, B v , of vertical to horizontal components of the ambient magnetic field and the dimensionless plume thickness. Attention is focused on the preferred mode of instability, which occurs in the limit Re ≪1 for all values of the parameters. This mode can be either sinuous or varicose with the wavenumber vector either vertical or oblique , comprising four types. The regions of preference of these four modes are represented in regime diagrams in the ( x 0 , σ ) plane for different values of σ m , Q c , B v . These regions are strongly dependent on the field inclination and field strength and, to a lesser extent, on magnetic diffusion. The overall maximum growth rate for any prescribed set of the parameters σ m , Q c , B v , occurs when 1.3&lt; x 0 &lt;1.7, and is sinuous for small σ and varicose for large σ . The magnetic field can enhance instability for a certain range of thickness of the plume. The enhancement of instability is due to the interaction of the field with viscous diffusion resulting in a reverse role for viscosity. The dependence of the helicity and α -effect on the parameters is also discussed.

On the non-parallel instability of sediment-carrying channels of slowly varying width

The flow of a river in a channel of slowly varying width is investigated using an asymptotic approach. The work was motivated by a recent experimental investigation of this problem. The river transports sediment as bedload and is susceptible to an instability which causes variations in the depth of the river. The asymptotic theory is, in the first instance, used to describe steady-state flows in channels of varying width and it is found to give excellent agreement with experimental observations on this flow. The theory shows conclusively that a river of slowly increasing width will form central bars. Secondly, the approach is used to investigate the instability of the flow. The major result obtained is that in a symmetric channel which diverges from a width where the flow is unstable to one where instability is possible, then the preferred mode of instability is likely to be a central bar rather than an alternating bar as is the case in straight channels.

Instability of the Marangoni convection in a liquid bridge with liquid encapsulation under microgravity condition

Linear stability theory is used to analyze the stability of the basic state solution of Marangoni convection in a liquid bridge with liquid encapsulation. By processing the linear disturbance equations numerically, stability analysis can be evolved to a complex general eigenvalue problem. Inverse iteration algorithm combined with LZ algorithm is employed to solve the complex generalized eigenvalue problem. The results show that the stability of the system can be enhanced greatly by choosing reasonable matching parameters of the two fluid layers. The preferred mode of instability is axial wave number α=2,3 and 4, which means that the system is more sensitive to the disturbance with larger wave lengths.

The effect of a uniform vertical magnetic field on the onset of oscillatory marangoni convection in a horizontal layer of conducting fluid

In this paper we use classical linear stability theory to undertake a detailed investigation of the effect of a uniform vertical magnetic field on the onset of oscillatory thermocapillary-driven Marangoni convection in a horizontal layer of quiescent, electrically conducting fluid heated from below. For simplicity we restrict our attention to the simplest case of a fluid layer with a non-deformable free surface and perfectly electrically conducting boundaries in which the onset of convection is always steady in the absence of the magnetic field. The present numerical calculations show that the presence of the magnetic field can cause the preferred mode of instability to be oscillatory rather than steady convection. Nevertheless, in all the cases investigated the effect of the magnetic field is always to stabilise the layer relative to the case of no magnetic field.

On the stability of vertical double-diffusive interfaces. Part 2. Two parallel interfaces

This is the second part of a three-part study of the stability of vertically oriented double-diffusive interfaces having an imposed vertical stable temperature gradient. In this study, flow is forced within a fluid of infinite extent by a prescribed excess of compositionally buoyant material between two parallel interfaces. Compositional diffusivity is ignored while thermal diffusivity and viscosity are finite. The stability of the interfaces is analysed first in the limit that they are close together (compared with the salt-finger lengthscale), then for general spacing. Attention is focused on whether the preferred mode of instability is varicose or sinuous and whether its wavevector is vertical or oblique.The interfaces are found to be unstable for some wavenumber for all values of the Prandtl number and interface spacing. The preferred mode of instability for closely spaced interfaces is varicose and vertical for Prandtl number less than about 9, sinuous oblique for Prandtl number between 9 and 15 and sinuous vertical for larger Prandtl number. For general spacing each of the four possible modes of instability is preferred for some range of Prandtl number and interface separation, with no clear pattern of preference, except that the sinuous oblique mode is preferred for widely separated interfaces. The growth rate of the preferred mode is largest for interfaces having separations of from 1 to 3 salt-finger lengths.

Gravity wave breaking in two and three dimensions: 1. Model description and comparison of two‐dimensional evolutions

A nonlinear, compressible, spectral collocation code is employed to examine gravity wave breaking in two and three spatial dimensions. Two‐dimensional results exhibit a structure consistent with previous efforts and suggest wave instability occurs via convective rolls aligned normal to the gravity wave motion (uniform in the spanwise direction). Three‐dimensional results demonstrate, in contrast, that the preferred mode of instability is a series of counterrotating vortices oriented along the gravity wave motion, elongated in the streamwise direction, and confined to the region of convective instability within the wave field. Comparison of the two‐dimensional results (averaged spanwise) for both two‐ and three‐dimensional simulations reveals that vortex generation contributes to much more rapid wave field evolution and decay, with rapid restoration of near‐adiabatic lapse rates and stronger constraints on wave energy and momentum fluxes. These results also demonstrate that two‐dimensional models are unable to describe properly the physics or the consequences of the wave breaking process, at least for the flow parameters examined in this study. The evolution and structure of the three‐dimensional instability, its influences on the gravity wave field, and the subsequent transition to quasi‐isotropic small‐scale motions are the subjects of companion papers by Fritts et al. (this issue) and Isler et al. (this issue).

Three-dimensional axisymmetric flow in perfectly plastic metals

A method is presented for solving the three-dimensional axisymmet-ric field equations for a perfectly plastic material which obeys the von-Mises yieldcriterion and the Levy-Mises flow law. The method is used for the particular case inwhich a small axisymmetric perturbed flow is superposed on a uniform flow withoutflow reversal taking place. The method then leads to solving a fourth order differen-tial equation for the velocity potential. The special case of a thick cylindrical shellunder compressive flow is examined. The solution so obtained, being derived fromthe three dimensional theory, includes a correct treatment of transverse shear distor-tion. A preferred mode of instability is identified having a wave-length in reasonableagreement with that obtained experimentally by other workers.