Nonlinear Stability Theory Research Articles

Maxwell–Cattaneo double-diffusive convection of Kuvshiniski viscoelastic nanofluid in a Brinkman–Darcy porous medium

Abstract This study examines the stability of double-diffusive convection in a Kuvshiniski viscoelastic nanofluid, in which the fluid is affected by two fields (such as temperature and salinity) that influence its density. The classical Fick’s law, which assumes an immediate response of temperature to the heat flux gradient, is not entirely correct because it suggests an instantaneous reaction at all points, which is not entirely accurate since information propagates at a finite speed. This shortcoming of Fick’s law leads us to consider the Maxwell–Cattaneo effect (MC effect). Thus, our research focuses on Maxwell–Cattaneo double-diffusive convection in a horizontal layer of a porous medium saturated with viscoelastic nanofluid. Here, the fluid’s small dimensions result in its relaxation time being comparable to its thermal diffusion time, necessitating the use of the Maxwell–Cattaneo relationship. The behavior of viscoelastic nanofluids is described by a constitutive equation of the Kuvshiniski kind, and for the porous medium, Brinkman–Darcy model is considered. The nanofluid model includes the effects of Brownian diffusion and thermophoresis, with the assumption that the flux of the nanoparticle volume fraction is zero at the isothermal boundaries. The framework of linear and nonlinear stability theory leads the analysis. By applying linear stability theory with the help of normal mode technique, the conditions for the occurrence of both stationary and oscillatory convective motions are found in terms of a critical thermal Rayleigh number. The Kuvshiniski viscoelastic fluid exhibits Newtonian behavior in a state of stationary convection. We have discussed two cases for oscillatory convection that are when (i) Maxwell–Cattaneo coefficient for temperature (C T ) = 0 and (ii) Maxwell–Cattaneo coefficient for salinity (C C ) = 0. Convective heat and mass transfers are determined using a weakly nonlinear stability analysis. The effects of various factors on oscillatory and stationary states as well as the mass and heat transport are depicted graphically. It is found that with increase in the value of Kuvshiniski parameter F, thermal Rayleigh number Ra also increases for both cases C T = 0 and C C = 0. Ra drops with increasing values of modified diffusivity ratio N A and thermosolutal Lewis number Ls for both stationary as well as oscillatory convection. With increase in the value of Darcy number Da, an interesting pattern can be seen. For stationary convection, Ra increases with Da, but it has reverse effect on oscillatory convection (for both the cases). Streamlines, isotherms, and isohalines are also examined.

Linear and nonlinear stability of double diffusive convection in a micropolar fluid saturated porous layer with magnetic field and throughflow effects

The linear and nonlinear stability of double-diffusive convection in a porous layer saturated with micropolar fluid is examined. A transverse magnetic field is applied to the flow together with vertical throughflow. The normal mode technique is employed for linear stability analysis, whereas the energy method is used for nonlinear stability analysis. The resulting eigenvalue problems corresponding to linear and nonlinear stability theories are solved numerically by employing the bvp4c routine in MATLAB 2022(b). The critical thermal Rayleigh numbers for both linear and nonlinear analyses are computed for the different values of the governing parameters and presented graphically. A comparison is made between linear and nonlinear stability results. It is observed that the flow is more stable whenever a magnetic field is added to the flow, although the subcritical instability region also slightly increases. Increasing the Darcy number, Lewis number, coupling number, and absolute value of the throughflow parameter destabilizes the flow. On the other hand, raising the porosity of the medium and micropolar parameters stabilizes the flow. Furthermore, there is no subcritical gap in the absence of the throughflow effect, which is a good agreement between the linear and nonlinear thresholds.

Thermal convection in a Brinkman–Darcy and Kelvin–Voigt fluid of order one with coupled stresses effect

AbstractOur paper thoroughly examines a model for the thermal convection of specific viscoelastic fluids within a porous medium characterized by Brinkman–Darcy properties. These fluids demonstrate first‐order Kelvin–Voigt behavior as they permeate the medium. This investigation considers the influence of couple stresses on the system. We calculate the critical Rayleigh number required for linear instability theory, focusing on both stationary and oscillatory types of convection. Additionally, we conduct the nonlinear stability theory. In the latter cases, it is demonstrated that oscillatory convection can occur. A detailed investigation is conducted into the characteristics of both stationary and oscillatory convection branches, including the points at which the transition occurs.

Nonlinear Stability and Asymptotic Behavior of Periodic Wave Trains in Reaction–Diffusion Systems Against Cub\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C_{\ extrm{ub}}$$\\end{document}-perturbations

We present a nonlinear stability theory for periodic wave trains in reaction–diffusion systems, which relies on pure L∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^\\infty $$\\end{document}-estimates only. Our analysis shows that localization or periodicity requirements on perturbations, as present in the current literature, can be completely lifted. Inspired by previous works considering localized perturbations, we decompose the semigroup generated by the linearization about the wave train and introduce a spatio-temporal phase modulation to capture the most critical dynamics, which is governed by a viscous Burgers’ equation. We then aim to close a nonlinear stability argument by iterative estimates on the corresponding Duhamel formulation, where, hampered by the lack of localization, we must rely on diffusive smoothing to render decay of the semigroup. However, this decay is not strong enough to control all terms in the Duhamel formulation. We address this difficulty by applying the Cole–Hopf transform to eliminate the critical Burgers’-type nonlinearities. Ultimately, we establish nonlinear stability of diffusively spectrally stable wave trains against Cub\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C_{\ extrm{ub}}$$\\end{document}-perturbations. Moreover, we show that the perturbed solution converges to a modulated wave train, whose phase and wavenumber are approximated by solutions to the associated viscous Hamilton–Jacobi and Burgers’ equation, respectively.

Weakly nonlinear stability analysis of non-isothermal parallel flow in a vertical porous annulus

The study examines the stability of parallel airflow within a tall vertical coaxial circular annulus filled with a highly permeable porous medium under stable stratification. The linear stability theory predicts that the least unstable mode can be either axisymmetric or non-axisymmetric, depending on the values of the governing parameters. The curvature parameter, defined as the gap between coaxial circular cylinders, contributes to the stabilization of the flow. Depending on various controlling parameter values, the stably stratified non-isothermal annular parallel flow (SNAPF) can manifest three instability modes: thermal-shear, thermal-buoyant, and mixed or interactive. The instability type can change abruptly with Reynolds number, curvature parameter, and media permeability. A finite-amplitude instability analysis is examined using weakly nonlinear stability theory. The analysis demonstrates both subcritical and supercritical bifurcations near the instability boundary. The range of Reynolds numbers for subcritical bifurcation expands when switching from axisymmetric to non-axisymmetric disturbance. Bifurcation transitions between supercritical and subcritical occur due to changes in the least stable azimuthal mode. The study also investigates the equilibrium/threshold amplitude within the supercritical/subcritical instability regime, and it verifies the existence of subcritical/supercritical bifurcation through nonlinear kinetic energy balance. Furthermore, the finite-amplitude analysis anticipates that the lower critical Rayleigh number increases as the Reynolds number, curvature parameter, or media permeability decreases. An increase in the Reynolds number shifts the point of inflection in the velocity profile toward the inner cylinder, which subsequently leads to a reduction in the size of the core vortex region. In the subcritical instability regime, the radius of the primary vortex cell decreased significantly as the Rayleigh number increased. When the curvature is altered, a structural transition occurs in the isotherms from multicellular to unicellular.

Linear and nonlinear investigations of the impact of chemical reaction on the thermohaline convection in a permeable layer saturated with Casson fluid

In this effort, the consequence of chemical reaction on the thermohaline convection in a permeable layer containing Casson fluid is inspected utilizing both linear and nonlinear stability procedures. To model the momentum equation, the Casson fluid version of Darcy's law is utilized. Applying the linear stability concept, the situation for the start of stationary and oscillatory pattern of convective motion is obtained, whereas the convective heat and mass transferences are determined using nonlinear stability theory. It is found that the oscillatory type of convective flow is probable only if the evaluation of the solutal Rayleigh–Darcy number is less than zero. The effect of the chemical reacting factor, the Casson factor, the Lewis number, and the solutal Rayleigh–Darcy number is to hurry the start of convection toward both stationary and oscillatory styles of convection and also surge the size of the convective cells. The convective heat and mass transfers enhance with increasing the Casson factor, the solutal and thermal Rayleigh–Darcy numbers while both decrease with enhancing the heat capacity quotient. The effect of chemical reacting factor is to increase the mass transmission in the system, while it has no effect on the heat transmission. It is also noted that the convective heat and mass transfers in the system enhance nearly 5.4% and 3.8%, respectively, with a 50% increase in the value of Casson factor.

Mathematical modeling and optimal control of a deterministic SHATR model of HIV/AIDS with possibility of rehabilitation: a dynamic analysis

In the present work, we developed a deterministic SHATR (Susceptible - HIV infected -AIDS infected - Antiretroviral Treatment - Recovered) compartment model for HIV/AIDS. This model considers the disease outbreak due to a lack of awareness and treatment. The steady states of the proposed model system are obtained and analyzed by using the nonlinear stability theory of differential equations. The basic reproduction number is derived and explored to determine the stability and sensitivity index of some important relative parameters. Further, to know the global behavior of the model one parameter bifurcation study is discussed. Moreover, the optimal control theory has been applied to identify the optimal strategy by taking treatment and awareness for safe intercourse as control parameters. The control problem is solved analytically by using Pontryagin’s maximum principle. Finally, the model is simulated to describe the optimality under various assumptions and the stability of equilibrium points.

A Globally Stable Self-Similar Blowup Profile in Energy Supercritical Yang-Mills Theory

This paper is concerned with the Cauchy problem for an energy-supercritical nonlinear wave equation in odd space dimensions that arises in equivariant Yang-Mills theory. In each dimension, there is a self-similar finite-time blowup solution to this equation known in closed form. It will be proved that this profile is stable in the whole space under small perturbations of the initial data. The blowup analysis is based on a recently developed coordinate system called hyperboloidal similarity coordinates and depends crucially on growth estimates for the free wave evolution, which will be constructed systematically for odd space dimensions in the first part of this paper. This allows to develop a nonlinear stability theory beyond the singularity.

Combined effect of helical force and rotation on the double convection of a binary viscoelastic fluid in a porous medium: Linear and weakly nonlinear analysis

AbstractWe used linear stability theory based on the normal mode decomposition technique and nonlinear stability theory based on the minimum representation of double Fourier series to study the criterion of appearance of the stationary convection and the oscillatory convection; and the rate of heat and mass transfer in a binary viscoelastic fluid mixture in a rotating porous medium under the effect of helical force. We have determined the analytical expression of the Rayleigh number of the system as a function of the dimensionless parameters. Expressions for heat and mass transfer rates are determined as a function of Nusselt and Sherwood number, respectively. The transient behaviors of the Nusselt number and the Sherwood number are studied by solving the finite amplitude equations using the Runge‐Kutta method. Then, the effect of each dimensionless parameter on the system is studied pointed out interesting results. According to the analysis of the different results obtained, it appears that in a porous medium saturated by a binary mixture of viscoelastic fluid in rotation, the Taylor‐Darcy number , the delay time parameter λ2 and the mass Rayleigh‐Darcy number delay the onset of stationary and oscillatory convection. On the other hand, the helical force , the relaxation time parameter λ1, the ratio of diffusivities τ and the number of Vadasz accelerate the onset of stationary and oscillatory convection. In the case of the nonlinear stability study, at steady state, the helical force accelerates the rate of heat and mass transfer. On the other hand, the Taylor‐Darcy number , the diffusivity ratio τ and the mass Rayleigh‐Darcy number retard the rate of heat and mass transfer. In the unsteady state, the helical force , the relaxation time parameter λ1 and the mass Rayleigh‐Darcy number accelerate the rate of heat and mass transfer. This is not the case for the Taylor‐Darcy number and the delay time parameter λ2.

Thermal convection in a higher-gradient Navier–Stokes fluid

We discuss models for flow in a class of generalized Navier–Stokes equations. The work concentrates on producing models for thermal convection, analysing these in detail, and deriving critical Rayleigh and wave numbers for the onset of convective fluid motion. In addition to linear instability theory we present a careful analysis of fully nonlinear stability theory. The theories analysed all possess a bi-Laplacian term in addition to the normal spatial derivative term. The theories discussed are Stokes couple stress theory, dipolar fluid theory, Green–Naghdi theory, Fried–Gurtin–Musesti theory, and a second theory of Fried and Gurtin. We show that the Stokes couple stress theory and the Fried–Gurtin–Musesti theory involve the same partial differential equations while those of Green–Naghdi and dipolar theory are similar. However, we concentrate on boundary conditions which are crucial to understand all five theories and their differences.

Adaptive control for unknown HOFA nonlinear systems without overparametrization

SummaryThe problem of adaptive control is investigated for unknown mixed high‐order fully actuated (HOFA) nonlinear systems. To deal with the parametric uncertainties, the tuning function is utilized to design an adaptive update law without parameter overestimation. Based on the full‐actuation feature of HOFA nonlinear systems and Lyapunov stability theory, an adaptive controller is constructed and its stability is analyzed. Finally, the effectiveness of the control strategy is verified by simulation results.

Stability Analysis of Double Diffusive Convection in Local Thermal Non-equilibrium Porous Medium with Internal Heat Source and Reaction Effects

Abstract The internal heat source and reaction effects on the onset of thermosolutal convection in a local thermal non-equilibrium porous medium are examined, where the temperature of the fluid and the solid skeleton may differ. The linear instability and nonlinear stability theories of Darcy–Brinkman type with fixed boundary condition are carried out where the layer is heated and salted from below. The D 2 {D^{2}} Chebyshev tau technique is used to calculate the associated system of equations subject to the boundary conditions for both theories. Three different types of internal heat source function are considered, the first type increases across the layer, while the second decreases, and the third type heats and cools in a nonuniform way. The effect of different parameters on the Rayleigh number is depicted graphically. Moreover, the results detect that utilizing the internal heat source, reaction, and non-equilibrium have pronounced effects in determining the convection stability and instability thresholds.

Linear and nonlinear stability analyses of penetrative convection in porous media with a gravity field effect

We examine resonant flow of fluid in porous media over a variable gravity. A heat source/sink which varies with height has been investigated. In this case, it is shown that there are three distinct sub-layers that can cause convective overturning instability. The possibility of resonance between the motion in these layers is discussed. A specific area that has a very rapid increase in Rayleigh number is noticed. In addition to a linear theory, the nonlinear stability theory is studied. To verify the reliability of the linear instability limits, a three dimensional approximation is used.

Nonlinear stability of Prats flow in Brinkman porous medium with Soret effect

The study aims at establishing the nonlinear stability theory for a uniform flow along a Brinkman porous layer with two impermeable and isothermal horizontal boundaries. Convection in the flow is initiated by the integrated effect of the buoyant forces in thermal and solutal fields. The cross-diffusion effect is also considered as the contributing factor towards the convective instability in the medium. The equations associated with the nonlinear analysis are derived with the help of the Energy method. The results obtained from the linear stability analysis are compared with those of nonlinear stability analysis and the conditions under which the two stability boundaries are widely separated, are discussed elaborately. It is observed that the stability boundary obtained using the nonlinear analysis is solely affected by the concentration gradient (Sa) and the Soret parameter (Sr). However, the concentration gradient is found to have a destabilizing effect on both linear as well as nonlinear stability boundaries. It is also observed that the oscillatory instability is seen only in the case of oblique rolls, whereas, the longitudinal rolls always occur in the stationary mode. For Sa<0, the basic state flow gets stabilized with the increasing values of the Soret parameter. The flow tends to stabilize with the increasing magnitude of throughflow in both forward as well as backward directions. However, an opposite trend is noticed for La>1 and Sa<0, for the smaller magnitude of throughflow (|Pe|<20).

On the stability of thermosolutal rotating porous convection subject to a lack of thermal equilibrium

This paper investigates the linear and weakly nonlinear stability of thermosolutal rotating porous convection subject to a lack of thermal equilibrium. Being an example of a triply diffusive fluid system in porous media with multiparameters, some novel results on the linear instability are delineated for a few isolated cases. The otherwise stabilizing factors such as stabilizing solute concentration and rotation are shown to destabilize the system under certain parameters space. Besides, the existence of a completely detached closed convex oscillatory neutral curve from that of the stationary neutral curve is uncovered indicating the requirement of three critical thermal Darcy-Rayleigh numbers to identify the linear instability criteria instead of the usual single critical value. The multivalued nature of the stability boundaries for the case of completely detached oscillatory neutral curves is displayed. The co-dimension-2 points are found through a stability map with the demarcation of stationary and oscillatory convective regions in a plane of Darcy-Taylor and scaled Vadasz numbers. Weakly nonlinear stability theory is carried out for a stationary case by using a modified multi-scale method and thereby the complex Ginzburg-Landau amplitude equation is derived identifying the pitchfork bifurcation. Moreover, heat and mass transport are quantified in terms of average thermal and solute Nusselt numbers, respectively.

STABILITY ANALYSIS FOR ROTATING DOUBLE-DIFFUSIVE CONVECTION IN THE PRESENCE OF VARIABLE GRAVITY AND REACTION EFFECTS: DARCY MODEL

The linear instability and nonlinear stability theories are performed to investigated the variable gravity, rotation and reaction effects on the thermosolutal convection in a porous medium of Darcy type. Two statuses are analyzed, the layer is heated from below and simultaneously salted from above or below. Three different types of gravity variation functions are presented: linear, parabolic, and exponential. The associated system of equations with the corresponding boundary conditions for both theories are calculated by using the $D^2$ Chebyshev tau method. The effect of various parameters on the Rayleigh number is shown graphically. Furthermore, the results reveal that employing the effects of gravity variation, rotation and reaction has a pronounced in determining the convection instability threshold.

On closures for reduced order models—A spectrum of first-principle to machine-learned avenues

For over a century, reduced order models (ROMs) have been a fundamental discipline of theoretical fluid mechanics. Early examples include Galerkin models inspired by the Orr–Sommerfeld stability equation and numerous vortex models, of which the von Kármán vortex street is one of the most prominent. Subsequent ROMs typically relied on first principles, like mathematical Galerkin models, weakly nonlinear stability theory, and two- and three-dimensional vortex models. Aubry et al. [J. Fluid Mech. 192, 115–173 (1988)] pioneered the data-driven proper orthogonal decomposition (POD) modeling. In early POD modeling, available data were used to build an optimal basis, which was then utilized in a classical Galerkin procedure to construct the ROM, but data have made a profound impact on ROMs beyond the Galerkin expansion. In this paper, we take a modest step and illustrate the impact of data-driven modeling on one significant ROM area. Specifically, we focus on ROM closures, which are correction terms that are added to the classical ROMs in order to model the effect of the discarded ROM modes in under-resolved simulations. Through simple examples, we illustrate the main modeling principles used to construct the classical ROMs, motivate and introduce modern ROM closures, and show how data-driven modeling, artificial intelligence, and machine learning have changed the standard ROM methodology over the last two decades. Finally, we outline our vision on how the state-of-the-art data-driven modeling can continue to reshape the field of reduced order modeling.

Stability of Darcy thermosolutal convection in bidispersive porous medium with reaction

AbstractAnalysis of fluid flow in a bidispersive medium with reactions is explored. The Darcy model is used in the momentum equation with the density being a linear function in temperature and salt concentration. Two cases are regarded which are heated below and salted above system and heated and salted below system. It is presumed that the equilibrium solute concentration follows a linear relationship to temperature. A linear instability and nonlinear stability theories are conducted, and from the outputs of the analysis, the Chebyshev collocation approach is used to solve the resulting eigenvalue systems. The effects of the chemical reaction and other parameters on temperature Rayleigh number are described graphically.

Critical stable sectional area of downstream surge tank of hydropower plant with sloping ceiling tailrace tunnel

AbstractThis paper investigates the critical stable sectional area (CSSA) of downstream surge tank (DST) of hydropower plant with sloping ceiling tailrace tunnel (SCTT). Firstly, two types of nonlinear mathematical model of hydropower plant with SCTT during transient process are established. Then, the criterion of operation stability of hydropower plant is obtained based on nonlinear stability theory. Formulae for CSSAs of DST under two types of mathematical model are derived. Finally, the similarities, differences, and relationships of different CSSAs are clarified. The effect mechanisms of the assumption of steady power output of hydro‐turbine and the SCTT on CSSA of DST are revealed. The recommended use conditions of different CSSAs are explained. The results indicate that, under the assumption of steady power output of hydro‐turbine, the formula for CSSA of DST of hydropower plant with SCTT is analytical and has a simple expression. Under the complete model, the formula for CSSA of DST of hydropower plant with SCTT is not analytical and has a complicated expression. The former formula is suggested to be given the priority of use because of good precision and concise expression. SCTT can improve the stability of hydropower plant and decrease the CSSA of DST.

Conditional and unconditional stability for double diffusive convection when the viscosity has a maximum

We here analyse two models of double-diffusive convection in fluid layer when viscosity depends on temperature quadratically. However, to a linearized instability analysis, conditional and global (unconditional) nonlinear stability theories are applied. For the first model, we establish an unconditional nonlinear energy stability. Moreover, in the second model the standard energy method does not yield unconditional stability so a conditional energy analysis is employed to achieve nonlinear results. In addition, the nonlinear stability bounds is found to be independent of the salt field and a presentation of the region of possible subcritical instabilities is given.