Critical Wave Number Research Articles

INFLUENCE OF MAGNETIC FIELD ON THE STABILITY OF DOUBLE DIFFUSIVE NANOFLUID CONVECTION IN A VERTICAL HOMOGENEOUS POROUS CHANNEL

The stability analysis of double-diffusive convection of nanofluid flow in a vertical porous channel is investigated in the presence of the transverse magnetic effect. A uniform magnetic field perpendicular to the flow direction is applied. Nanofluid is modeled using a model that combines Brownian motion and thermophoresis. The boundaries of the channel are assumed to be nonpermeable, isothermal, and isosolutal. A normal mode analysis is used to obtain the eigenvalue problem for the perturbed state. The Chebyshev spectral collocation technique is used to solve the resulting eigenvalue problem. The critical Rayleigh number and corresponding critical wavenumber are presented graphically for the effect of various parameters at least stable mode on the planes (Ha, Rac) and (Ha, αc), respectively. It is observed that transverse magnetic parameters tend to have a major effect on convective instability.

The Coriolis Effect on Thermal Convection in a Rotating Sparsely Packed Porous Layer in Presence of Cross-Diffusion

The effect of rotation and cross-diffusion on convection in a horizontal sparsely packed porous layer in a thermally conducting fluid is studied using linear stability theory. The normal mode method is employed to formulate the eigenvalue problem for the given model. One-term Galerkin weighted residual method solves the eigenvalue problem for free-free boundaries. The eigenvalue problem is solved for rigid-free and rigid-rigid boundaries using the BVP4c routine in MATLAB R2020b. The critical values of the Rayleigh number and corresponding wave number for different prescribed values of other physical parameters are analyzed. It is observed that the Taylor number and Solutal Rayleigh number significantly influence the stability characteristics of the system. In contrast, the Soret parameter, Darcy number, Dufour parameter, and Lewis number destabilize the system. The critical values of wave number for different prescribed values of other physical parameters are also analyzed. It is found that critical wave number does not depend on the Soret parameter, Lewis number, Dufour parameter, and solutal Rayleigh number; hence critical wave number has no impact on the size of convection cells. Further critical wave number acts as an increasing function of Taylor number, so the size of convection cells decreases, and the size of convection cells increases because of Darcy number.

Numerical investigation of atmospherelike flows in a spherical geometry.

Buoyancy-driven convection flows play a crucial role in global heat and momentum transport in atmosphere. Simplified planetary and stellar atmospheres can be described by a spherical gap geometry with special boundary conditions for the temperature. In the spherical gap, the dielectrophoretic effect is used to synthesize the radial gravity field. Lateral thermal boundary conditions are used to model solar radiation at the equator and at the poles. The temperature reaches a maximum value at the equator and becomes colder near the poles. In the case of a rotating gap, the influence of the Coriolis and centrifugal forces are taken into account. Different regimes of the two-dimensional steady basic flow are discussed in dependence on the Taylor number and Rayleigh number and for the radii ratio η=R_{in}/R_{out}, where R_{in},R_{out} are the radii of the inner and outer surfaces, respectively. Linear instability theory is used to study when the basic flow becomes unstable. The critical Rayleigh number at which the steady axisymmetric basic flow becomes time-dependently axisymmetric or three dimensional is found to be a function of the Taylor number. Furthermore, the critical azimuthal wave number m_{c}, which is responsible for the structure of the supercritical three-dimensional flow, and the critical frequency of the perturbation ω_{c} were found. The spatial location of the perturbation helps to understand the origin of the instability.

Secondary instabilities in Taylor–Couette flow of shear-thinning fluids

The stability of the Taylor vortex flow in Newtonian and shear-thinning fluids is investigated in the case of a wide gap Taylor–Couette system. The considered radius ratio is$\eta = R_1/R_2=0.4$. The aspect ratio (length over the gap width) of experimental configuration is 32. Flow visualization and measurements of two-dimensional flow fields with particle image velocimetry are performed in a glycerol aqueous solution (Newtonian fluid) and in xanthan gum aqueous solutions (shear-thinning fluids). The experiments are accompanied by axisymmetric numerical simulations of Taylor–Couette flow in the same gap of a Newtonian and a purely viscous shear-thinning fluid described by the Carreau model. The experimentally observed critical Reynolds and wavenumbers at the onset of Taylor vortices are in very good agreement with that obtained from a linear theory assuming a purely viscous shear-thinning fluid and infinitely long cylinders. They are not affected by the viscoelasticity of the used fluids. For the Newtonian fluid, the Taylor vortex flow (TVF) regime is found to bifurcate into a wavy vortex flow with a high frequency and low amplitude of axial oscillations of the vortices at${Re} = 5.28 \, {Re}_c$. At${Re} = 6.9 \, {Re}_c$, the frequency of oscillations decreases and the amplitude increases abruptly. For the shear-thinning fluids the secondary instability conserves axisymmetry. The latter is characterized by an instability of the array of vortices leading to a continuous sequence of creation and merging of vortex pairs. Axisymmetric numerical simulations reproduce qualitatively very well the experimentally observed flow behaviour.

Wrinkling Analysis of Double Nanofilms Embedded in Compliant Substrates Based on Nonlocal Elasticity Theory

In this paper, an analytical approach for wrinkling analysis of double nanofilms embedded in compliant substrates is presented. The governing differential equations for wrinkling of double nanofilms are derived based on Eringen’s nonlocal elasticity theory and the classical plate theory (CLPT). Substrates are regarded as elastic bodies of finite thickness and the normal pressure of substrates on the films is assumed to be linear with the lateral deformation of the films. Solutions for critical wrinkling load and wave number are computed in terms of the nonlocal parameter, the elastic properties and thickness of double nanofilms, the elastic properties and thickness of the substrates, and wrinkle modes. The parametric study shows that the dimensionless critical wave number and the dimensionless critical load decrease gradually with the increase of the nonlocal parameter. Four typical wrinkling states are studied: in-phase wrinkling, out-of-phase wrinkling, single-film wrinkling and asymmetric wrinkling. In-phase wrinkling has the highest chance to occur among four wrinkle modes. In addition, the results show that the dimensionless critical load and wave number are much smaller when the films become much stiffer and thinner than the substrates.

Stability of a Buoyant Oldroyd-B Flow Saturating a Vertical Porous Layer with Open Boundaries

The performance of several engineering applications are strictly connected to the rheology of the working fluids and the Oldroyd-B model is widely employed to describe a linear viscoelastic behaviour. In the present paper, a buoyant Oldroyd-B flow in a vertical porous layer with permeable and isothermal boundaries is investigated. Seepage flow is modelled through an extended version of Darcy’s law which accounts for the Oldroyd-B rheology. The basic stationary flow is parallel to the vertical axis and describes a single-cell pattern where the cell has an infinite height. A linear stability analysis of such a basic flow is carried out to determine the onset conditions for a multicellular pattern. This analysis is performed numerically by employing the shooting method. The neutral stability curves and the values of the critical Rayleigh number are evaluated for different retardation time and relaxation time characteristics of the fluid. The study highlights the extent to which the viscoelasticity has a destabilising effect on the buoyant flow. For the limiting case of a Newtonian fluid, the known results available in the literature are recovered, namely a critical value of the Darcy–Rayleigh number equal to 197.081 and a corresponding critical wavenumber of 1.05950.

Buoyancy-driven instabilities of partially miscible fluids in inclined porous media

This study presents a buoyancy-driven stability analysis in a three-dimensional inclined porous medium with a capillary transition zone that is formed between a non-wetting and an underlying wetting phase. In this two-phase, two-component, partially miscible system, a solute from a non-wetting phase diffuses into a porous layer saturated with a wetting-phase fluid, creating a dense diffusive boundary layer beneath an established capillary transition zone. Transient concentration and gravity-driven velocity fields are derived for the wetting phase while the saturation field remains fixed. Linear stability analysis with the quasi-steady-state approximation is employed to determine the onset of solutal convective instability for buoyancy-dominant, in-transition and capillary-dominant systems. The analysis of the problem leads to a differential eigenvalue problem composed of a system of three complex-valued equations that are numerically solved to determine the critical times, critical wavenumbers and neutral stability curves as a function of inclination angle for different Bond numbers. The layer inclination is shown to play an essential role in the stability of the problem, where the gravity-driven flow removes solute concentrations in the diffusive boundary layer. The results indicate that the horizontal porous layer exhibits the fastest onset of instability, and longitudinal rolls are always more unstable than oblique and transverse rolls. The inclination angle has a more substantial impact on stabilizing the diffusive boundary layer in the buoyancy-dominant than in the capillary-dominant systems. Furthermore, for both buoyancy-dominant and capillary-dominant systems, the critical times and wavenumbers vary exponentially with inclination angle ≤ 60° and follow the Stirling model.

Hydrodynamics of granular gases of inelastic and rough hard disks or spheres. II. Stability analysis.

Conditions for the stability under linear perturbations around the homogeneous cooling state are studied for dilute granular gases of inelastic and rough hard disks or spheres with constant coefficients of normal (α) and tangential (β) restitution. After a formally exact linear stability analysis of the Navier-Stokes-Fourier hydrodynamic equations in terms of the translational (d_{t}) and rotational (d_{r}) degrees of freedom, the transport coefficients derived in the companion paper [A. Megías and A. Santos, "Hydrodynamics of granular gases of inelastic and rough hard disks or spheres. I. Transport coefficients" Phys. Rev. E 104, 034901 (2021)10.1103/PhysRevE.104.034901] are employed. Known results for hard spheres [Garzó, Santos, and Kremer, Phys. Rev. E 97, 052901 (2018)10.1103/PhysRevE.97.052901] are recovered by setting d_{t}=d_{r}=3, while novel results for hard disks (d_{t}=2,d_{r}=1) are obtained. In the latter case, a high-inelasticity peculiar region in the (α,β) parameter space is found, inside which the critical wave number associated with the longitudinal modes diverges. Comparison with event-driven molecular dynamics simulations for dilute systems of hard disks at α=0.2 shows that this theoretical region of absolute instability may be an artifact of the extrapolation to high inelasticity of the approximations made in the derivation of the transport coefficients, although it signals a shrinking of the conditions for stability. In the case of moderate inelasticity (α=0.7), however, a good agreement between the theoretical predictions and the simulation results is found.

Thermovibrational instability in a nanofluid porous medium

Thermovibrational convection in a water based metallic (Cu-water and Ag-water) or metallic oxide (Al2O3-water and TiO2-water) nanofluid saturated porous medium is addressed incorporating the Brinkman boundary correction. The nanofluid behaves more like a fluid rather than a conventional solid–fluid mixture and is studied through Khanafer–Vafai–Lightstone (KVL) model. A linear stability analysis is carried out through the Floquet theory to find the instability thresholds for convection onset in terms of critical Rayleigh number and critical wavenumber. The results corresponding to synchronous as well as subharmonic modes are obtained for the Darcy and Brinkman regimes. A rise in particle concentration is found to cause stabilizing response in general. However, it leaves an opposite effect in the Brinkman regime when the medium is filled with metallic nanofluid and is heated from above. Under this situation, the use of oblate-spheroidal particles in achieving optimal convective heat transfer enhancement is highlighted.

Horizontal pressure gradient and Soret effects on the onset of thermosolutal porous convection

AbstractThe intricacies of a constant horizontal pressure gradient on the onset of Soret‐driven thermosolutal porous convection have been investigated. The resulting generalized eigenvalue problem is solved numerically using the Galerkin method and also the condition for the onset is obtained in a closed‐form using a single‐term Galerkin method with trigonometric trial function. The results obtained from both methods are found to be in good agreement. The effect of increasing horizontal pressure gradient, Lewis number, Soret parameter, and the Vadasz number is to hasten, while the increase in the solute Darcy–Rayleigh number is to delay the onset of oscillatory convection. The presence of the horizontal pressure gradient is found to decrease the threshold value of solute Darcy–Rayleigh number beyond which the instability sets in as oscillatory. Moreover, the horizontal pressure gradient imparts a conflicting behavior on the critical wave number and critical frequency of oscillations. The numerical results attained under the limiting cases are shown to be in excellent agreement with the published ones.

Creation of streaks using heating patterns

Streaks and rolls are of interest in mixing intensification. It is shown that they can be created in a controlled manner in fully developed shear layers using spatially distributed heating with their spatial distribution dictated by the heating pattern. The method works for any Reynolds number and any heating intensity. The energy costs of streak formation were determined for laminar flows both in terms of additional pressure losses required to drive the same flow rate in the heated and isothermal channels and in terms of the reduction of the flow rate if the pressure gradient remained unaltered. Streak-increased heat transfer across the shear layer has been determined. The creation of streaks using the Rayleigh–Bénard instability was studied for completeness—a heating intensity exceeding the critical Rayleigh number was required, with the spatial structure of the streaks dictated by the critical wavenumber.

Onset of instability in Hadley–Prats flow in a weakly heterogeneous porous layer with viscous dissipation

The stability of a flow subjected to an inclined temperature gradient (Hadley-type flow) in a horizontal porous media is studied in the presence of a basic horizontal mass flow (Prats flow). Therefore, the basic flow is called the Hadley–Prats flow. A weak vertical heterogeneity in permeability and conductivity is considered. The effect of viscous dissipation is taken to be non-negligible. The Rayleigh number corresponding to the vertical thermal gradient RaC is considered as an eigenvalue. Other parameters are the Péclet number (Pe) associated with the horizontal through flow, horizontal Rayleigh number (RaH) associated with the horizontal temperature gradient, Gebhart number (Ge) associated with viscous dissipation; parameters γ1 and γ2 represent the changes in permeability and conductivity, respectively. A linear stability analysis is done and the governing equations are solved numerically to obtain the critical Rayleigh number and wave number. Longitudinal and transverse rolls are discussed. Longitudinal rolls are the preferred modes for instability in most scenarios. It is found that when throughflow is present, the heterogeneity in permeability can show a stabilizing effect for longitudinal rolls but destabilizing effect for transverse rolls and vice versa depending on the direction of the throughflow. Increase in conductivity also may stabilize or destabilize the flow depending on the mass flow and viscous heating. The horizontal thermal gradient shows interesting effects in the presence of weak heterogeneity and horizontal throughflow. Significant change in the critical Rayleigh number is observed even for small values of the horizontal Rayleigh number.

Oscillatory thermocapillary convection in deformed half zone liquid bridges of low Prandtl number fluids

A series of three-dimensional (3D) transient simulations are carried out to study the onset and features of oscillatory thermocapillary convection in deformed half zone liquid bridges of low Prandtl number fluids (Pr = 0.01) with unit aspect ratio (As). The critical conditions for different volume ratios on ground and under microgravity are determined. It turns out that the threshold of the second bifurcation is more sensitive to volume ratio than the first bifurcation. The thresholds in deformed liquid bridges are usually higher than those in cylindrical liquid bridges. The critical azimuthal wavenumber increases from 2 to 3 with volume ratio. Gravity affects the flow through three factors. Compared with the hydrostatic deformation and heating orientation, buoyancy is marginal to the flow transition. It only affects the threshold and the deviation caused by buoyancy is within 2%. The effects of gravity on the critical conditions are dependent on volume ratio. Except for the threshold, gravity may also affect the critical azimuthal wavenumber and the critical oscillation mode. 3D oscillatory flow regimes with two different oscillation modes are present. A new type of “Rocking” mode is discovered in the present study.

Liquid crystal elastomers wrinkling

When a liquid crystal elastomer layer is bonded to an elastic layer, it creates a bilayer with interesting properties that can be activated by applying traction at the boundaries or by optothermal stimulation. Here, we examine wrinkling responses in three-dimensional nonlinear systems containing a monodomain liquid crystal elastomer layer and a homogeneous isotropic incompressible hyperelastic layer, such that one layer is thin compared to the other. The wrinkling is caused by a combination of mechanical forces and external stimuli. To illustrate the general theory, which is valid for a range of bilayer systems and deformations, we assume that the nematic director is uniformly aligned parallel to the interface between the two layers, and that biaxial forces act either parallel or perpendicular to the director. We then perform a linear stability analysis and determine the critical wave number and stretch ratio for the onset of wrinkling. In addition, we demonstrate that a plate model for the thin layer is also applicable when this is much stiffer than the substrate.

Stability of an oscillatory Taylor–Couette flow in an upper convected Maxwell fluid

The stability of pulsed bi-dimensional flow between two co-oscillating cylinders in a linear Maxwell fluid was studied by Riahi et al. [J. Soc. Rheol. 42, 321–327 (2014)]. In the present paper, we revisit this flow configuration with emphasis on the effect of the non-linear terms in the constitutive equation of the model, measured by the Weissenberg number, on the dynamics of the system. Under these assumptions and using the upper convected Maxwell derivative, we examine this model to large amplitude oscillatory shear giving rise to the appearance, in comparison to the linear Maxwell model, of the azimuthal normal stress in the basic state. Using the spectral method and the Floquet theory for the spatiotemporal resolution of the obtained eigenvalue problem, numerical results exhibit numerous classes of Taylor vortex flows depending on the order of magnitude of the fluid elasticity. The resulting stability diagram consists of several branches intersecting at specific frequencies where two different Taylor vortex flows simultaneously branch off from the basic state. This feature is accompanied by the occurrence of several co-dimension two bifurcation points besides jumps/drops in the corresponding critical wave number. In addition, it turns out that the elasticity produces strong destabilizing and stabilizing effects in the limit of high and low frequency regimes, respectively, attributed solely to the non-linearities considered by the rheological model.

Dust-Acoustic Rogue Waves in an Electron-Positron-Ion-Dust Plasma Medium

In this work, the modulational instability of dust-acoustic (DA) waves (DAWs) is theoretically studied in a four-component plasma medium with electrons, positrons, ions, and negative dust grains. The nonlinear and dispersive coefficients of the nonlinear Schrödinger equation (NLSE) are used to recognize the stable and unstable parametric regimes of the DAWs. It can be seen from the numerical analysis that the amplitude of the DA rogue waves decreases with increasing populations of positrons and ions. It is also observed that the direction of the variation of the critical wave number is independent (dependent) of the sign (magnitude) of q. The applications of the outcomes from the present investigation are briefly addressed.

Modulation theory for pattern forming systems with a spatial 1:2-resonance.

It is the purpose of this paper to justify the use of modulation equations for pattern forming systems in the case of multiple Turing instabilities with critical wave numbers having a ratio 1:2 by proving approximation results, presenting attractivity results, and discussing the existence of modulating fronts.

Post-buckling of an elastic half-space coated by double layers

We investigate the buckling and post-buckling properties of a hyperelastic half-space coated by two hyperelastic layers when the composite structure is subjected to a uniaxial compression. In the case of a half-space coated with a single layer, it is known that when the shear modulus of the layer is larger than the shear modulus of the half-space, a linear analysis predicts the existence of a critical stretch and wave number, whereas a weakly nonlinear analysis predicts the existence of a threshold value of the modulus ratio below which the buckling is super-critical and above which the buckling is sub-critical. It is shown that when another layer is added, a larger variety of behaviour can be observed. For instance, buckling can occur at a preferred wavenumber super-critically even if both layers are softer than the half-space although the top layer would need to be harder than the bottom layer. When the shear modulus of the bottom layer lies in a certain interval, the super-critical to sub-critical transition can happen a number of times as the shear modulus of the top layer is increased gradually. Thus, an extra layer imparts more flexibility in producing wrinkling patterns with desired properties, and our weakly nonlinear analysis provides a road map on the parameter regimes where this can be achieved.

MHD instability of the pressure‐driven plane laminar flow in the presence of the uniform coplanar magnetic field: Linear stability analysis

AbstractThe influence of the uniform longitudinal magnetic field on the stability against small disturbances of an electrically conducting Newtonian fluid flow between two parallel horizontal plates is investigated. The sixth‐order system of disturbance equations is solved by the Chebyshev collocation method, and the critical Reynolds number , the critical wave number , and the critical wave speed are computed for a wide range of the magnetic Reynolds number and Alfven number A. Curves of wave number against Reynolds number for neutral stability are presented for different values of the parameters. The onset of instability is also discussed in detail using the growth rate curves for various parameters of the problem. It is observed that the effect of both conductivity of the fluid and the strength of the magnetic field is to decay the onset of instability. A comprehensive study is carried out at the critical state of the fluid using the graph of , , and with respect to for various values of A. The critical values at the onset of instability are also presented for both the Galerkin method and the Chebyshev collocation method.

Elastic Wave Propagation in Strongly Nonlinear Lattices and Its Active Control

Abstract This work studies elastic wave propagation in strongly nonlinear periodic systems and its active control with specific attention to an infinite mass-in-mass lattice. Piezoelectric materials are applied to it to provide active control loads to manipulate band structures of the lattice. Governing equations of the active mass-in-mass lattice with cubic nonlinearities are established. The control loads are modeled by using linear piezoelectric springs. Due to phase differences among vibrations of different cells during wave propagation, a series of delay functions with different delays are used to represent the steady-state of a traveling wave. The incremental harmonic balance method for delay dynamic systems is employed in this case to calculate periodic solutions of the lattice. The fast Fourier transform is employed to construct the Jacobian matrix of the Newton–Raphson iteration to avoid a large number of Galerkin integrations, and thus, the efficiency is significantly improved. Amplitude-dependent dispersion curves are calculated using results of the linearized system as an initial guess for the iteration. The results are compared with existing results in the literature, which demonstrates that the present method is efficient for wave propagation analysis of strongly nonlinear structures. Effects of nonlinearities, the mass ratio, and different control actions on band structures of the mass-in-mass lattice are investigated through a comprehensive parametric study. Numerical results show that the band structures can be influenced by nonlinearities of the lattice. New stopbands and critical wave numbers can be created by the control actions.