Critical Wave Number Kc Research Articles

Faraday instability of viscous liquid films on a heated substrate with Maxwell–Cattaneo heat flux

Faraday instability of viscous liquid films with Maxwell–Cattaneo (MC) heat flux on an infinite, heated horizontal substrate subject to vertical time-varying periodic vibration is investigated theoretically. The MC effect means that the response of the heat flux to a temperature gradient obeys a relaxation time law rather than a classical Fourier time law. Applying the classic Floquet theory to linear analysis, a recursive relation is obtained. When considering the MC effect, a new phenomenon appears at a large wave number k. The neutral stability curves branch new tongues that turn left rather than right as before, but the tongues still move up and right as the wave number increases. Furthermore, typical harmonic (H) and subharmonic (SH) alternation behavior continues to exist. Interestingly, the first tongue of a branch is H or SH, implying that there is a transition following the branches. However, near the critical wave number kc of a branch, the SH and H almost overlap. As Cattaneo number C increases, the tongue-like unstable zones of branches become wider, and the critical wave number kc of the appeared branch becomes small. As the driving frequency ω decreases, the branch tongues become elongated and the critical wave number kc of the appeared branch becomes small.

Dynamics of a thin film of viscoelastic fluid flowing down an inclined or vertical plane

The stability characteristics of a thin film of viscoelastic (Walters’ B′ model) fluid flowing down an inclined or vertical plane are analyzed under the combined influence of gravity and surface tension. A nonlinear free surface evolution equation is obtained by using the momentum-integral method. Normal mode technique and multiple scales method are used to obtain the results of linear and nonlinear stability analysis of this problem. The linear stability analysis gives the critical condition and critical wave number kc which include the viscoelastic parameter Γ, angle of inclination of the plane θ, Reynolds number Re and Weber number We. The weakly nonlinear stability analysis that is based on the second Landau constant J2, reveals the condition for the existence of explosive unstable and supercritical stable zone along with the other two (unconditional stable and subcritical unstable) flow zones of this problem which is 3(1+3Γ)Re−3cotθ−4ReWek2 = 0. It is found that all the four distinct flow zones of this problem exist in Re-k-, θ-k- and Γ-k-plane after the critical value of Rec,θc and Γc, respectively. A novel result of this analysis is that the film flow is stable (unstable) for a negative (positive) value of Γ irrespective of the values of Re and θ, as for example, a solution of polyisobutylene in cetane, compared with the viscous (Γ=0) film flow case. Finally, we scrutinize the effect of Γ on the threshold amplitude and nonlinear wave speed by depicting some numerical examples in supercritical stable as well as subcritical unstable zone of this problem.

Influences of wave resonance on hydrodynamic efficiency and loading of an OWC array under oblique waves

In this paper, a semi-analytical solution was derived from the linear potential flow theory using the matching eigenfunction method to investigate hydrodynamic characteristics of a rectangular oscillating water column (OWC) array under oblique waves. The theoretical results indicate that the sloshing resonance in the alongshore direction varies with the incident wave angle significantly. The presence of the along-shore sloshing resonance decreases the hydrodynamic efficiency and increases the wave reflection. The critical wavenumber kc at which the along-shore sloshing resonance occurs is identified as a key controlling factor of the performance of the OWC array. It is dependent on the chamber length in the along-shore direction. The peak efficiency occurs at the lower frequency for the larger incident wave angle. To our knowledge, this study provides the first theoretical evidence that the peak wave loading of the OWC array is triggered by the along-shore resonance under oblique wave action. Our theory confirms that the optimum coastal protection can be achieved by the peak efficiency of the OWC array in the frequency region of the piston resonance of the OWC.

Effects of adiabaticity of electrons and negative ions on the modulational instability of ion-acoustic waves in electronegative plasma

In the present work, we investigate modulational instability (MI) in an unmagnetised collisionless electronegative plasma consisting of inertialess adiabatic electrons, inertialess adiabatic negative ions and mobile positive ions. For this purpose, a nonlinear Schrodinger equation (NLSE) is derived to study envelope evolution of the ion-acoustic carrier waves for our system by using reductive perturbation technique. In examining the stability/instability of the ion-acoustic envelope wavepackets, the critical wavenumber kc at which MI sets in is found to be shifted to higher values as adiabaticity γ increases. The effects of electron-to-negative ion temperature ratio σn and negative ion density fraction f on the region for propagation of stable wave pulses are also discussed. Within the instability region, the rational solution of NLSE provides the rogue wave (RW) profiles which is localised in space and time. With the increase of adiabaticity, the possibility for the formation of RW decreases at the lower wavenumber which is in good agreement with experimental observations. The change in adiabaticity significantly influences the rogue wave amplitudes. Maximum pulse height for RW is found to be increasing with the increase of adiabaticity. Our theoretical findings may be helpful for basic characteristic study of rogue waves in the laboratory as well as space plasmas.

Modulational instability of electrostatic waves in a magnetized dusty plasma with kappa distributed electrons

The modulational instability of dust-ion acoustic wave (DIAW) and dust-ion cyclotron wave (DICW) is investigated in a magnetized plasma containing static dust particles. The static dust particles can be positively or negatively charged, while ions are taken to be dynamic and inertialess electrons follow the kappa distribution in a magnetized dusty plasma. The nonlinear Schrödinger equation is derived to study the amplitude modulation of obliquely propagating electrostatic waves in a magnetized dusty plasma using the Krylov-Bogoliubov-Mitropolsky method. The dispersive and nonlinear coefficients, i.e., P and Q of nonlinear Schrödinger equation, are obtained which depend on the dust charge concentration, the magnetic field intensity, dust charge polarity, angle of wave propagation, and spectral index kappa for the nonthermal electrons. The modulationally stable and unstable regions of DIAW and DICW are investigated numerically, and the illustration of contour plots of product PQ with wave propagation angle θ and critical wave number kc is also presented. It is found that DIAW and DICW become modulationally unstable at low values of wave number k for negatively charged dust particles in comparison with positively charged dust particles or without dust particles case in a magnetized plasma. The stable region for DIAW, whose wave frequency now depends also on wave propagation angle in a magnetized plasma, is found to be increased in comparison with the unmagnetized plasma case. The observations and existence of positively and negatively charged dust particles in different regions of space plasmas and in laboratory experiments are also pointed out.

Ion-acoustic dark solitons collision in an ultracold neutral plasma

Dark solitons collision in an ultracold neutral plasma in the framework of the phenomenological generalized hydrodynamic model is reported. The derivative expansion method is used to derive a nonlinear Schrödinger equation for describing the low frequency modulation. The critical wavenumber kc, which indicates where the modulational instability sets in, is determined precisely. Also, the modulationally stable and unstable regions are obtained for a wide range of the wavenumber k and the ions effective temperature ratio . In the stable regions, the collisions between two-dark solitons are investigated using the extended Poincaré–Lighthill–Kuo method and their corresponding phase shifts are derived. The dependence of both and k on the phase shift is examined.

Ion-acoustic super rogue waves in ultracold neutral plasmas with nonthermal electrons

The ion-acoustic rogue waves in ultracold neutral plasmas consisting of ion fluid and nonthermal electrons are reported. A reductive perturbation method is used to obtain a nonlinear Schrödinger equation for describing the system and the modulation instability of the ion-acoustic wave is analyzed. The critical wave number kc, which indicates where the modulational instability sets in, has been determined. Moreover, the possible region for the ion-acoustic rogue waves to exist is defined precisely. The effects of the nonthermal parameter β and the ions effective temperature ratio σ∗ on the critical wave number kc are studied. It is found that there are two critical wave numbers in our plasma system. For low wave number, increasing β would lead to cringe kc until β approaches to its critical value βc, then further increase of β beyond βc would enhance the values of kc. For large wave numbers, the increase of β would lead to a decrease of kc. However, increasing σ∗ would lead to the reduction of kc for all values of the wave number. The dependence of the rogue waves profile on the plasma parameters is numerically examined. It is found that the rogue wave amplitudes have complex behavior with increasing β. Furthermore, the enhancement of σ∗ and the carrier wave number k reduces the rogue wave amplitude. It is noticed that near to the critical wave number, the rogue wave amplitude becomes high, but it shrinks whenever we stepped away from kc. The implications of our results in laboratory ultracold neutral plasma experiments are briefly discussed.

Analysis on the onset of buoyancy-driven convection in a fluid-saturated porous medium heated uniformly from below

The onset of convective motion in an initially quiescent, horizontal isotropic porous layer is analyzed theoretically. Using linear theory, the stability equations are obtained with and without the quasi-steady state approximation (QSSA). They are solved by expanding the disturbances as a series of orthogonal functions. Initial value problem analyses also are conducted, and their results are compared with the eigenanalysis and the QSSA. For the large Darcy–Rayleigh number RaD, the critical time of the onset of convection is found to be τc=11.74RaD−1 with the critical wavenumber kc=0.29RaD1/2. Based on linear analyses, numerical simulations are conducted to consider nonlinear effects. The nonlinear analysis using the Fourier spectral method and the finite volume method reproduces the linear stability limit obtained theoretically. From the present nonlinear analysis, it is shown that manifest convection is observed experimentally from τm=51.42RaD−1.

On short wave stability and sufficient conditions for stability in the extended Rayleigh problem of hydrodynamic stability

We consider the extended Rayleigh problem of hydrodynamic stability dealing with the stability of inviscid homogeneous shear flows in sea straits of arbitrary cross section. We prove a short wave stability result, namely, if k > 0 is the wave number of a normal mode then k > kc (for some critical wave number kc) implies the stability of the mode for a class of basic flows. Furthermore, if \( K(z) = \frac{{ - (U''_0 - T_0 U'_0 )}} {{U_0 - U_{0s} }} \), where U0 is the basic velocity, T0 (a constant) the topography and prime denotes differentiation with respect to vertical coordinate z then we prove that a sufficient condition for the stability of basic flow is \( 0 < K(z) \leqslant \left( {\frac{{\pi ^2 }} {{D^2 }} + \frac{{T_0^2 }} {4}} \right) \), where the flow domain is 0 ≤ z ≤ D.

Modulation instability of low-frequency electrostatic ion waves in magnetized electron-positron-ion plasma

The nonlinear amplitude modulation of low-frequency electrostatic ion waves propagating in collisionless magnetized electron-positron-ion plasma are studied. The Krylov–Bogoliubov–Mitropolsky perturbation method is employed to derive the nonlinear Schrödinger equation. Modulation instability of both ion-acoustic and ion-cyclotronlike modes is examined. We found that the ion-acoustic mode, which propagates below the ion-cyclotron frequency, is stable if the strength of the external magnetic field is small. However, as the strength of the magnetic field increases, this mode becomes modulationally unstable for a range of wave numbers and angles of propagation. This range increases as the strength of the magnetic field and/or positron density increases. For the ion-cyclotronlike mode, which propagates above the ion-cyclotron frequency, a number of stability/instability regions appear in the (kc,θ) plane even for a very small value of the magnetic field. It is found that, for both modes, critical wave number kc separating the stability and instability regions, shifts towards higher values as the density of the positron increases.

Influence of a mean shear on the spectrum of passive scalar fluctuations

AbstractThe form of the spectrum of passive scalar fluctuations in the inertial–convective range is investigated. In the presence of a mean shear, there exists a critical wave number kc. At wave numbers greater than kc, the mean shear does not influence the spectrum of scalar fluctuations and the Obukhov–Corrsin prediction holds. At wave numbers smaller than kc, the mean shear dominates and the spectrum is less steep. These predictions are based on the study of Lagrangian dynamics of fluid particles and account for the experimental observations. (© 2008 WILEY‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

Inertial effect on the stability of viscoelastic cone-and-plate flow

The stability of axially symmetric cone-and-plate flow of an Oldroyd-B fluid at non-zero Reynolds number is analysed. We show that stability is controlled by two parameters: [Escr ]1≡DeWe and [Escr ]2≡Re/We, where De, We, and Re are the Deborah, Weissenberg and Reynolds numbers respectively. The linear stability problem is solved by a perturbation method for [Escr ]2 small and by a Galerkin method when [Escr ]2=O(1). Our results show that for all values of the retardation parameter β and for all values of [Escr ]2 considered the base viscometric flow is stable if [Escr ]1 is sufficiently small. As [Escr ]1 increases past a critical value the flow becomes unstable as a pair of complex-conjugate eigenvalues crosses the imaginary axis into the right half-plane. The critical value of [Escr ]1 decreases as [Escr ]2 increases indicating that increasing inertia destabilizes the flow. For the range of values considered the critical wavenumber kc first decreases and then increases as [Escr ]2 increases. The wave speed on the other hand decreases monotonically with [Escr ]2. The critical mode at the onset of instability corresponds to a travelling wave propagating inward towards the apex of the cone with infinitely many logarithmically spaced toroidal roll cells.

Analyticity of Ginzburg-Landau Modes

Ginzburg-Landau formalism applies for dissipative systems defined on cylindrical domains which are close to the threshold of instability and for which the unstable Fourier modes belong to non-zero wave numbers. In these situations the real part of the curve of critical eigenvalues as function drawn over the wave numbers k is positive of height O(ϵ2) and of width O(ϵ). Here it is shown that the set of solutions which can be described by the Ginzburg-Landau formalism is attractive. To do this we demonstrate that in Fourier space peaks appear at integer multiples of the critical wave number kc. These peaks called Ginzburg-Landau modes concentrate in time like e−|k−mkc|√t for 0 ≤ t ≤ O(1/ϵ2) and m ∈ Z. The inverse Fourier transform of such a Ginzburg-Landau mode is an analytic Function in a strip of width √t. This result extends a former work of W. Eckhaus.

A proof of Howard's conjecture in homogeneous parallel shear flows—II: Limitations of Fjortoft's necessary instability criterion

The present paper on the linear instability of nonviscous homogeneous parallel shear flows mathematically demonstrates the correctness of Howard's [4] prediction, for a class of velocity distributions specified by a monotone functionU of the altitudey and a single point of inflexion in the domain of flow, by showing not only the existence of a critical wave numberkc>0 but also deriving an explicit expression for it, beyond which for all wave numbers the manifesting perturbations attain stability. An exciting conclusion to which the above result leads to is that the necessary instability criterion of Fjortoft has the seeds of its own destruction in the entire range of wave numbersk>kc—a result which is not at all evident either from the criterion itself or from its derivation and has thus remained undiscovered ever since Fjortoft enunciated [3].

Soliton Formation in a Self-Gravitating Gas

Nonlinear wave modulation associated with the gravitational stability of an infinite homogeneous gas is investigated by the reductive perturbation method. It is shown that the weakly nonlinear wave with the carrier wave number more than the Jeans wave number kJ is governed by a nonlinear Schrödinger (NLS) equation. That NLS equation changes its type from modulationally unstable one to stable one across a critical wave number kc (> kJ). Further it is shown that the weakly nonlinear wave near the marginal state of instability, i.e., near kJ, obeys an unstable NLS equation. From these results, it is conjectured that the nonlinearity may lead to various types of envelope soliton formations in a self-gravitating medium.

Soliton Formation in a Self-Gravitating Gas

Nonlinear wave modulation associated with the gravitational stability of an infinite homogene­ ous gas is investigated by tbe reductive perturbation method. It is shown that the weakly nonlinear wave with the carrier wave number more than the Jeans wave number k, is governed by a nonlinear Schrodinger (NLS) equation. That NLS equation changes its type from modulationally unstable one to stable one across a critical wave number kc ( > k, ). Further it is shown that the weakly nonlinear wave near the marginal state of instability, i. e., near k1 , obeys an unstable NLS equation. From these results, it is conjectured that tbe nonlinearity may lead to various types of envelope soliton formations in a self-gravitating medium.

Transition from hexagons to rolls in convection in fluids under non-Boussinesq conditions

The transition between hexagons and rolls in convective patterns has been studied. The transition thresholds and changes in the Nusselt number are discussed theoretically in terms of calculations made by Busse (1967a) and with amplitude equations. Experiments have been done in a shallow layer of pure water under non-Boussinesq conditions using complementary techniques: shadowgraph (qualitative), optical (based on the deflections of a laser beam) and calorimetric. The experimental values of the critical Rayleigh number Rc and the critical wavenumber kc are in agreement with theoretical ones. However, theory and experiments show some discrepancies in the slopes of the non-dimensional convective heat flow curve and in the thresholds of the hysteretic hexagons-rolls transition. These discrepancies are discussed in terms of lateral effects and of the presence of defects in the pattern.

Primordial fluctuations and non-linear structure

We explore several aspects of non-linear gravitational instability using two numerical experiments, each of which employs a series of 3D cosmological N-body simulations. In the first experiment we truncate all of the initial high-frequency power above some critical wavenumber kc; in the second experiment, we replace initial Fourier components with wavenumbers |$k\gt{k}_\text{c}$| by waves taken from an independent realization with the same power spectrum but with unrelated phases. We evolve these initial conditions for different values of kc to see how the progressive elimination or substitution of initial high-frequency components affects the final, non-linear structure. We find that such structure is determined mainly by initial waves with |$k\lesssim{k}_{1}\sim R_{1}^{-1},\,\text{where}\,{R}_{1}$| is the comoving scale on which rms fluctuations are just becoming non-linear. Changes in high-frequency components of the initial conditions do not disturb the evolution of longer waves, and even on small scales the influence of waves with |$k\gt{k}_{1}$| is erased by gravitational collapse. Since fluctuations with |${k}\ll {k}_{1}$| are weak, we conclude that the appearance of non-linear structure in the universe is determined by fluctuations from a fairly small range in wavenumber. Our first experiment also confirms that a truncated initial power spectrum leads to coherent, large-scale filamentary structures when |${k}_\text{c}\sim {k}_{1}.$|

Effect of ion streaming on the onset of the modulational instability

The effect of ion streaming on the modulational instability (MI) of ion-acoustic waves in a collisionless unmagnetized plasma is studied. The nonlinear Schrödinger equation is derived by using the Krylov–Bogoliubov–Mitropolsky (KBM) perturbation technique. It is found that the value of the critical wavenumber kc is highly dependent upon the ion streaming velocity V0 and the system breaks into a number of stable and unstable regions. In the weak relativistic limit of V0, instability sets in for kc &amp;lt;1.47. It is reconfirmed that in the absence of ion streaming the modulational instability of ion-acoustic waves sets in at a critical wavenumber kc =1.47. UFaipxr

The modulation of weakly non-linear ion acoustic plasma waves near the marginal state of instability

The modulation of a one-dimensional weakly non-linear quasimonochromatic ion acoustic plasma wave (the carrier) is considered. It is well known that the carrier is modulationally unstable for wavenumbers k larger than a critical wavenumber kc, and that when k is not near kc the modulations of the carrier are governed by a non-linear Schrodinger (NS) equation. The author shows that when k is near kc, and under certain assumptions, the modulations are governed by a modified form of the NS equation that involves higher-order non-linearities, and that the correct critical wavenumber for marginal modulational instability is slightly different from kc.