Weak Nonlinear Theory Research Articles

Riemann-Hilbert problem associated with an integrable generalized mixed nonlinear Schrödinger equation

ABSTRACT The generalized mixed nonlinear Schrödinger (GMNLS) equation is a completely integrable model under certain parameter conditions, which has important applications in nonlinear optics, weak nonlinear dispersive water wave and quantum field theory. In this work, we use the unified transformation method to construct a Riemann-Hilbert problem and discuss the initial-boundary value problems of the GMNLS equation on the half-line. Furthermore, we also obtain that some spectral functions are not independent of each other, but meet a paramount global relation.

The modulation instability of shallow wake flows based on the higher-order generalized cubic-quintic complex Ginzburg–Landau equation

In the context of the parallel flow hypothesis, we derive a higher-order generalized cubic-quintic complex Ginzburg–Landau (GCQ-CGL) equation to describe the amplitude evolution of shallow wake flow from the dimensionless shallow water equations by using multi-scale analysis, perturbation expansion, and weak nonlinear theory. The evolution model includes not only the slowly changing envelope approximation but also the influence of higher-order dissipation, dispersion, and cubic and quintic nonlinear effects. We give the analytical solution of the higher-order GCQ-CGL equation based on the ansatz and coordinate transformation methods, and we discuss the influence of the higher-order dissipation coefficient on the amplitude and frequency of the wake flow by means of three-dimensional diagrams, contour maps, and plane graphs. The subsequent linear stability analysis gives a theoretical basis for the modulation instability (MI) of plane waves, and the linear theory predicts the instability of any amplitude of the main waves. Finally, we focus on the MI of shallow wake flows. Results show that the MI gain function is internally related to the background wave number, disturbance wave number, background amplitude, disturbance expansion parameter, and dissipation coefficient. The area of the MI decreases as the higher-order dissipation coefficient decreases.

Cross-diffusion induced reaction-convection in a Brinkman porous layer

PurposeThis paper aims to investigate the onset of convection, heat and mass transports in a sparse porous layer saturated with chemically reactive binary fluid mixture heated and salted from below under the influence of Soret and Dufour effects.Design/methodology/approachThe Brinkman model is used for momentum equation. Linear stability analysis based on normal mode technique is used to evaluate the onset threshold for stationary and oscillatory convection. In weak-nonlinear theory, the truncated Fourier series method is used. The resulting system of differential equations is solved numerically by using the Runge–Kutta fourth-order method.FindingsBecause of the competition between the processes of thermal, solute diffusions, chemical reaction and cross-diffusions, the onset of instability is via oscillatory mode instead of stationary. The effect of dissolution/precipitation of reactive component and the cross-diffusions on the stability, heat and mass transports is investigated.Originality/valueBy the proper adjustment of underlying parameters, the onset of convection can either be advanced or delayed as per the requirement. Therefore, the present investigation forms a useful tool for regulating the onset of convection.

Methodical fitting for mathematical models of rubber-like materials

A great variety of models can describe the nonlinear response of rubber to uniaxial tension. Yet an in-depth understanding of the successive stages of large extension is still lacking. We show that the response can be broken down in three steps, which we delineate by relying on a simple formatting of the data, the so-called Mooney plot transform. First, the small-to-moderate regime, where the polymeric chains unfold easily and the Mooney plot is almost linear. Second, the strain-hardening regime, where blobs of bundled chains unfold to stiffen the response in correspondence to the ‘upturn’ of the Mooney plot. Third, the limiting-chain regime, with a sharp stiffening occurring as the chains extend towards their limit. We provide strain-energy functions with terms accounting for each stage that (i) give an accurate local and then global fitting of the data; (ii) are consistent with weak nonlinear elasticity theory and (iii) can be interpreted in the framework of statistical mechanics. We apply our method to Treloar's classical experimental data and also to some more recent data. Our method not only provides models that describe the experimental data with a very low quantitative relative error, but also shows that the theory of nonlinear elasticity is much more robust that seemed at first sight.

Nonlinear convective analysis of a rotating Oldroyd-B nanofluid layer under thermal non-equilibrium utilizing Al2O3-EG colloidal suspension

In this paper, we examine a layer of Oldroyd-B nanofluid for linear and nonlinear regimes under local thermal non-equilibrium conditions for the classical Rayleigh-Benard problem. The free-free boundary condition has been implemented with the flux for nanoparticle concentration being zero at edges. The Oberbeck-Boussinesq approximation holds good and for the rotational effect Coriolis term is included in the momentum equation. A two-temperature model explains the effect of local thermal non-equilibrium among the particle and fluid phases. The criteria for onset of stationary convection has been derived as a function of the non-dimensionalized parameters involved including the Taylor number. The assumed boundary conditions negate the possibility of overstability due to the absence of opposing forces responsible for it. The thermal Nusselt number has been obtained utilizing a weak nonlinear theory in terms of various pertinent parameters in the steady and transient mode, and has been depicted graphically. The main findings signify that the rotation has a stabilizing effect on the system. The stress relaxation parameter $ \lambda_1$ inhibits whereas the strain retardation parameter $ \lambda_2$ exhibits heat transfer utilizing Al2O3 nanofluids.

Mechanism for the Formation of Equatorial Superrotation in Forced Shallow-Water Turbulence with Newtonian Cooling

Abstract Forced shallow-water turbulence on a rotating sphere with Newtonian cooling is examined with the aim of elucidating the mechanism of the robust formation of equatorial superrotation reported by R. K. Scott and L. M. Polvani. It is shown that the Newtonian cooling term distorts the structure of the Hough modes. This distortion can be visualized as either the westward or eastward tilting of the equiphase line with increasing absolute value of latitude; the structural change of the Hough modes leads to the acceleration of the zonal-mean flow. A statistical analysis based on a weak-nonlinear theory predicts that stochastically excited Hough modes generate a prograde equatorial jet, the profile of which is quantitatively consistent with that of the ensemble-averaged zonal-mean flow obtained in nonlinear time evolutions. The predicted prograde equatorial jet originates mainly from the acceleration produced by Rossby modes, the equiphase line of which is tilted westward by the Newtonian cooling term. This tilt of the equiphase line of the Hough modes is clarified and a comparison between the acceleration mechanism presented in the present paper and that in other numerical studies in which equatorial superrotation emerges is made.

Thermal instability of a nanofluid layer under local thermal non-equilibrium

In this paper, we study the effect of local thermal non-equilibrium on the linear thermal instability in a horizontal layer of a Newtonian nanofluid. The nanofluid layer incorporates the effect of Brownian motion along with thermophoresis. A two-temperature model has been used for the effect of local thermal non-equilibrium among the particle and fluid phases. The linear stability is based on normal mode technique and for nonlinear analysis, a minimal representation of the truncated Fourier series analysis involving only two terms has been used. We observe that for linear instability, the value of Rayleigh number can be increased by a substantial amount on considering a bottom heavy suspension of nano particles. The effect of various parameters on Rayleigh number has been presented graphically. A weak nonlinear theory based on the truncated representation of Fourier series method has been used to find the concentration and the thermal Nusselt numbers. The behavior of the concentration and thermal Nusselt numbers is also investigated by solving the finite amplitude equations using a numerical method.

Rayleigh–Bénard Convection in a Nanofluid Layer Using a Thermal Nonequilibrium Model

This paper studies the effect of local thermal nonequilibrium (LTNE) on the thermal instability in a horizontal layer of a Newtonian nanofluid. The nanofluid layer incorporates the effect of Brownian motion along with thermophoresis. A two temperature model has been used for the effect of LTNE among the particle and fluid phases. The boundary condition involved assumes that the nano-concentration flux is zero thereat, including the effect of thermophoresis. The linear stability is based on normal mode technique and for nonlinear analysis, a minimal representation of the truncated Fourier series analysis involving only two terms has been used. The effect of various parameters on Rayleigh number has been presented graphically. A weak nonlinear theory based on the truncated representation of Fourier series method has been used to obtain the thermal Nusselt number, whose variation with respect to various parameters has been depicted graphically.

Weak Nonlinear Double-Diffusive Magnetoconvection in a Newtonian Liquid under Temperature Modulation

The present paper deals with a weak nonlinear theory of double-diffusive magnetoconvection in an electrically conducting Newtonian liquid, confined between two horizontal surfaces, under a constant vertical magnetic field, and subjected to imposed time-periodic thermal boundaries. The temperature of both walls is varied time periodic in this case. The disturbances are expanded in terms of power series of amplitude of convection, which is assumed to be small. Using nonautonomous Ginzburg-Landau equation, the Nusselt and Sherwood numbers obtained analytically and studied heat and mass transport in the system. Effect of various parameters on the heat and mass transport is discussed extensively. It is found that the effect of magnetic field is to stabilize the system. Further, it is also notified that the heat and mass transport can be controlled by suitably adjusting the external parameters of the system.

Onset of Double-Diffusive Reaction–Convection in an Anisotropic Rotating Porous Layer

The linear and non-linear stability of a rotating double-diffusive reaction–convection in a horizontal anisotropic porous layer subjected to chemical equilibrium on the boundaries is investigated considering a Darcy model that includes the Coriolis term. The effect of Taylor number, mechanical, and thermal anisotropy parameters, reaction rate, solute Rayleigh number, Lewis number, and normalized porosity on the stability of the system is investigated. We find that the Taylor number has a stabilizing effect, chemical reaction may be stabilizing or destabilizing and that the anisotropic parameters have significant influence on the stability criterion. The effect of various parameters on the stationary, oscillatory, and finite-amplitude convection is shown graphically. A weak nonlinear theory based on the truncated representation of Fourier series method is used to find the finite amplitude Rayleigh number and heat and mass transfer.

Effect of rotation on the onset of double diffusive convection in a Darcy porous medium saturated with a couple stress fluid

The effect of rotation on the onset of double diffusive convection in a horizontal couple stress fluid-saturated porous layer, which is heated and salted from below, is studied analytically using both linear and weak nonlinear stability analyses. The extended Darcy model, which includes the time derivative and Coriolis terms, has been employed in the momentum equation. The onset criterion for stationary, oscillatory and finite amplitude convection is derived analytically. The effect of Taylor number, couple stress parameter, solute Rayleigh number, Lewis number, Darcy–Prandtl number, and normalized porosity on the stationary, oscillatory, and finite amplitude convection is shown graphically. It is found that the rotation, couple stress parameter and solute Rayleigh number have stabilizing effect on the stationary, oscillatory, and finite amplitude convection. The Lewis number has a stabilizing effect in the case of stationary and finite amplitude modes, with a destabilizing effect in the case of oscillatory convection. The Darcy–Prandtl number and normalized porosity advances the onset of oscillatory convection. A weak nonlinear theory based on the truncated representation of Fourier series method is used to find the finite amplitude Rayleigh number and heat and mass transfer. The transient behavior of the Nusselt number and Sherwood number is investigated by solving the finite amplitude equations using Runge–Kutta method.

Nonlinear Two-Dimensional Convection in a Nanofluid Saturated Porous Medium

In this article, we study the linear and nonlinear thermal instability in a horizontal porous medium saturated by a nanofluid. For this, the momentum equation with Brinkman model has been used. Also, it incorporates the effect of Brownian motion along with thermophoresis. The linear stability is based on normal mode technique, and for nonlinear analysis, the truncated Fourier series involving only two terms has been used. The expression of Rayleigh number for linear theory has been derived, and the effects of various parameters on Rayleigh number have been presented graphically. Weak nonlinear theory is used to find the concentration and the thermal Nusselt numbers. The behavior of the concentration and thermal Nusselt numbers is investigated and depicted graphically, by solving the finite amplitude equations using a numerical method.

The onset of double diffusive reaction-convection in an anisotropic porous layer

The onset of double diffusive reaction-convection in a horizontal anisotropic porous layer saturated with binary mixture, which is heated and salted from below and subjected to chemical equilibrium on the boundaries, is studied analytically using both linear and nonlinear stability analyses. A linear stability analysis is performed to investigate how the dissolution or precipitation of reactive component affects the onset of convection. The Darcy model is employed as momentum equation. The effect of mechanical and thermal anisotropy parameters, reaction rate, Lewis number, solute Rayleigh number, and normalized porosity on the stability of the system is investigated. We find that the chemical reaction may be stabilizing or destabilizing and that the anisotropic parameters have significant influence on the stability criterion. The effect of various parameters on the stationary, oscillatory, and finite amplitude convection is shown graphically. A weak nonlinear theory based on the truncated representation of Fourier series method is used to find the finite amplitude Rayleigh number and heat and mass transfer.

Linear and nonlinear light propagations in a Doppler-broadened medium via electromagnetically induced transparency

We present a systematic theoretical study to deal with linear and nonlinear light propagations in a Doppler-broadened three-level $\ensuremath{\Lambda}$ system via electromagnetically induced transparency (EIT), with incoherent population exchange between two lower energy levels taken into account. Through a careful analysis of base state and linear excitation, we show that the EIT condition of the system is given by $|{\ensuremath{\Omega}}_{c}|{}^{2}{\ensuremath{\gamma}}_{31}\ensuremath{\gg}2{\ensuremath{\gamma}}_{21}\ensuremath{\Delta}{\ensuremath{\omega}}_{D}^{2}$, where ${\ensuremath{\Omega}}_{c}$ is half the Rabi frequency of the control field, $\ensuremath{\Delta}{\ensuremath{\omega}}_{D}$ is the Doppler width, and ${\ensuremath{\gamma}}_{\mathit{jl}}$ is the decay rate of the coherence between states $|j\ensuremath{\rangle}$ and $|l\ensuremath{\rangle}$. Under this condition, the effect of incoherent population exchange is insignificant, while dephasing dominates the decoherence of the system. This condition also ensures the validity of the weak nonlinear perturbation theory used in this work for solving the Maxwell-Bloch equations with inhomogeneous broadening. We then investigate the nonlinear propagation of the probe field and show that it is possible to form temporal optical solitons in the Doppler-broadened medium. Such solitons have ultraslow propagating velocity and can be generated in very low light power. The possibility of realizing ($1+1$)-dimensional and ($2+1$)-dimensional spatial optical solitons in the adiabatic regime of the system is also discussed.

Thermal instability in a rotating anisotropic porous layer saturated by a viscoelastic fluid

Linear and non-linear thermal instability in a rotating anisotropic porous medium, saturated with viscoelastic fluid, has been investigated for free–free surfaces. The linear theory is being related to the normal mode method and non-linear analysis is based on minimal representation of the truncated Fourier series analysis containing only two terms. The extended Darcy model, which includes the time derivative and Coriolis terms has been employed in the momentum equation. The criteria for both stationary and oscillatory convection is derived analytically. The rotation inhibits the onset of convection in both stationary and oscillatory modes. A weak non-linear theory based on the truncated representation of Fourier series method is used to find the thermal Nusselt number. The transient behaviour of the Nusselt number is also investigated by solving the finite amplitude equations using a numerical method. The results obtained during the analysis have been presented graphically.

The Effect of Rotation on the Onset of Double Diffusive Convection in a Horizontal Anisotropic Porous Layer

The effect of rotation and anisotropy on the onset of double diffusive convection in a horizontal porous layer is investigated using a linear theory and a weak nonlinear theory. The linear theory is based on the usual normal mode technique and the nonlinear theory on the truncated Fourier series analysis. Darcy model extended to include time derivative and Coriolis terms with anisotropic permeability is used to describe the flow through porous media. The effect of rotation, mechanical and thermal anisotropy parameters, and the Prandtl number on the stationary and overstable convection is discussed. It is found that the effect of mechanical anisotropy is to allow the onset of oscillatory convection instead of stationary. It is also found that the existence of overstable motions in case of rotating porous medium is not restricted to a particular range of Prandtl number as compared to the pure viscous fluid case. The finite amplitude analysis is performed to find the thermal and solute Nusselt numbers. The effect of various parameters on heat and mass transfer is also investigated.

Thermal convection in a rotating porous layer using a thermal nonequilibrium model

Linear stability of a rotating fluid-saturated porous layer heated from below and cooled from above is studied when the fluid and solid phases are not in local thermal equilibrium. The extended Darcy model, which includes the time derivative and Coriolis terms, is employed as a momentum equation. A two-field model that represents the fluid and solid phase temperature fields separately is used for energy equation. The onset criterion for both stationary and oscillatory convection is derived analytically. It is found that a small interphase heat transfer coefficient has significant effect on the stability of the system. There is a competition between the processes of rotation and thermal diffusion that causes the convection to set in through the oscillatory mode rather than the stationary one. The rotation inhibits the onset of convection in both stationary and oscillatory mode. In addition, the effect of porosity modified conductivity ratio, Darcy-Prandtl number, and the ratio of diffusivities on the stability of the system is investigated. A weak nonlinear theory based on the truncated representation of Fourier series method is used to find the Nusselt number. The effect of thermal nonequilibrium on heat transfer is brought out. The transient behavior of the Nusselt number is also investigated by solving the finite amplitude equations using Runge-Kutta method.

The effect of rotation on the onset of convection in a horizontal anisotropic porous layer

The effect of rotation and anisotropy on the onset of convection in a horizontal porous layer is investigated using a linear theory and a weak nonlinear theory. The linear theory is based on the usual normal mode technique and the nonlinear theory on the truncated Fourier series analysis. Darcy model extended to include time derivative and Coriolis terms with anisotropic permeability is used to describe the flow through porous media. A modified energy equation including the thermal anisotropy is used. The effect of rotation, mechanical and thermal anisotropy parameters and the Prandtl number on the stationary and overstable convection is discussed. It is found that the effect of mechanical anisotropy is to allow the onset of oscillatory convection instead of stationary. It is also found that the existence of overstable motions in case of rotating porous medium is not restricted to a particular range of Prandtl number as compared to the pure viscous fluid case. The steady finite amplitude analysis is performed using truncated Fourier series to find the Nusselt number. The effect of various parameters on heat transfer is investigated.

Secondary instabilities of linearly heated falling films

Secondary instabilities of linearly heated falling films are studied through three steps.Firstly,the analysis of the primary linear instability on Miladinova’s long wave equation of the linearly heated film is performed.Secondly,the similar Landau equation is de-rived through weak nonlinear theory,and a two-dimensional nonlinear saturation solution of primary instability is obtained within the weak nonlinear domain.Thirdly,the secondary(three-dimensional)instability of the two-dimensional wave is studied by the Floquet theorem.Our secondary instability analysis shows that the Marangoni number has destabilization effect on the secondary instability.

The physics of evaporative instability in bilayer systems: Weak nonlinear theory

A weak nonlinear method of dominant balance is used to describe the fluid dynamics in a bilayer system in the vicinity of phase-change instability. The calculations are performed for a case where gravity and Marangoni effects are ignored. They give us an estimate of the enhancement of heat transfer and, equally importantly, lend credence to physical arguments on the effect of vapor dynamics on the instability.