Subcritical Instability Region Research Articles

Onset of Darcy–Brinkman convection with thermal anisotropy in an inclined porous layer

In this article, we study the linear instability and the nonlinear stability (through energy functional) analyses of thermal convection in an inclined Darcy–Brinkman porous layer considering uniformly heated horizontal rigid, impermeable walls from below and above. The effects of a uniform internal heat source and anisotropy in effective thermal diffusivity on heat transfer are also considered. Heating the porous layer from below yields the temperature gradient, influencing the buoyancy and making the convection happen. This temperature gradient also impacts the base state. The basic solution for velocity incorporates both hyperbolic and polynomial functions, significantly increasing the complexity of linear and nonlinear analyses. The Chebyshev-tau method, together with the QZ algorithm, is used to solve the linear and nonlinear perturbed system of equations numerically. The region of subcritical instability is obtained by comparing the linear and nonlinear thresholds for the longitudinal and transverse rolls, respectively. We found that perturbations for longitudinal and transverse rolls do not grow after inclination is more than 30.3° and 31.3°, respectively. It has been noted that in transverse roll scenarios, the flow becomes stabilized when the inclination angle, ϕ, is equal to or exceeds 60°, where ϕ plays a leading role in surpassing the impact of internal heating. However, when the inclination angle is ϕ<60°, then internal heating dominates and destabilizes the flow. For the longitudinal rolls, the internal heating dominates the whole range of ϕ, destabilizing the system. Furthermore, it can be seen that the Darcy number (Da) and the anisotropic thermal diffusivity (ξ) delay the onset of convection.

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.

Stability analysis for viscoelastic fluid with thermorheological effects: Linear and nonlinear approaches

This study investigates the stability analysis of Rayleigh-Bénard configuration for a viscoelastic fluid subject to thermorheological effects, using the D2-Chebyshev-τ method. The fluid is modeled as a third-order viscoelastic fluid. This study accentuates how salting the fluid layer affects the thresholds for the onset of instability in a fluid of third order encompassing physically realistic rigid boundaries. The dynamic model incorporates advection-diffusion of temperature and solute concentration and a modified Navier–Stokes equation. We determine instability thresholds for the complex non-Newtonian fluid by analyzing the linear stability of the steady-state conduction solution. Our analysis proves the strong form of the principle of exchange of stabilities, demonstrating that convective motions can only occur through stationary motion. Additionally, a nonlinear stability analysis using the energy method is performed, deriving an unconditional nonlinear stability criterion. The results provide a comprehensive understanding of how variable viscosity and viscoelasticity impact system stability. Both the viscosity parameter and the third-grade fluid parameter exhibit stabilizing effects. Notably, we observe a discrepancy between the linear and global nonlinear stability results, indicating the presence of a subcritical instability region. This study contributes to the understanding of complex fluid dynamics in non-linear mechanical systems, with potential applications in various industrial and natural processes.

On viscous stratified Darcy–Forchheimer flow in a horizontal porous layer with thermal anisotropy and variable permeability

This study analyzes the effect of anisotropy and the internal heat source in a Darcy–Forchheimer porous layer. It is well known that the variations in viscosity can be attributed to the temperature. Therefore, in the present problem, we consider a linear variation in viscosity with temperature for simplicity. We first derived the linear instability theory and then established global stability using the energy functional approach. In the global stability analysis, we show that working with the L2 norm fails to give a sufficient condition for global stability by exhibiting that the associated maximization problem is unbounded in the underlying stability measure space. Then, we show that a conditional stability bound can be achieved by restricting the internal heat source parameter Q with higher-order norms. The eigenvalue problems obtained in linear and nonlinear theories were integrated numerically. The linear and nonlinear instability thresholds are then compared to identify the potential regions of sub-critical instabilities. It is observed that the system is stabilized when the horizontal component of thermal diffusivity dominates and is unstable when the vertical component of thermal diffusivity dominates. We also found that increasing the variable permeability parameter λ destabilized the system. It is observed that increasing viscosity stabilizes the system, and decreasing viscosity encourages the start of convection. It is also interesting that, in the presence of an internal heat source, the region of subcritical instability increases with increasing viscosity effect but reduces with increasing vertical permeability λ.

Nonlinear stability analysis of Rayleigh-Bénard problem for a Navier-Stokes-Voigt fluid

The linear and nonlinear stability analyses of thermosolutal convection in a non-Newtonian Navier-Stokes-Voigt fluid, considering Soret and Ekman damping effects, are conducted analytically. Instability thresholds are determined for thermosolutal convection within a viscoelastic fluid of the Kelvin-Voigt type, wherein a dissolved salt field exists. Two scenarios are examined: one where the fluid layer is heated from the bottom and concurrently salted from the bottom, and the other where the fluid layer is heated from the bottom and concurrently salted from the top. The governing partial differential equations system includes conservation laws of mass, momentum, energy, and salt concentration. Using the energy method, the disturbances to the fluid system are shown to decay exponentially. Analytical expressions are developed for the eigenvalue as a function of Soret, Lewis, Prandtl, Kelvin-Voigt, and Rayleigh friction numbers. The study illustrates the shift from a stationary mode of convection to an oscillatory mode and provides thresholds that indicate these transitions. It is found that the viscoelastic property of the fluid acts as a stabilizing agent for oscillatory mode convection. Rayleigh friction substantially controls the convection threshold. Upon comparing threshold values between linear and nonlinear theories, a subcritical instability region is observed in the heating bottom-salting bottom case (case-1), whereas such a region is absent in the heating bottom-salting top case (case-2).

Effect of variable viscosity, porous walls and mixed thermal boundary condition on the onset of Rayleigh-Bénard convective instability

In this paper, we have studied the criteria for the onset of instability in the Rayleigh-Bénard convection problem, by considering a fluid layer confined between two infinitely extended rough boundaries, which are subjected to mixed-type thermal boundary conditions, physically representing the imperfectly conducting boundaries. The rough boundaries are assumed to be shallow porous layers with different pore size, properties, and permeabilities. Mathematically, it is given by the Saffman type boundary condition. Therefore, we have two generalized forms of boundary conditions, one for the flow field and the other for the thermal field. The two limiting cases of the mixed-type thermal boundary condition correspond to a perfectly conducting boundary and to a perfectly insulating boundary, whereas the two limiting cases of the hydrodynamic boundary condition correspond to a no-slip condition and to a free-surface condition, depending upon the limiting values of the parameters involved in these two generalized boundary conditions. The linear and nonlinear energy stability analyses are performed, and the existence of the region of subcritical instability is checked. Through the principle of exchange of stability analysis, it is found that the instability occurs only in the stationary mode. The adiabatic boundary condition is found to be more restrictive than the isothermal boundary condition. Under given conditions, the instability is observed to be occurring in the infinite wavelength mode for the case of adiabatic boundaries. In the present work, the effect of temperature and pressure dependent viscosity on the stability of the system has been shown and found to be of destabilizing nature. However, the roughness of the boundaries is found to be of stabilizing nature.

Variable gravity effects on penetrative porous convection

The effect of variable gravity on the onset of penetrative convection in a horizontal fluid-saturated porous layer is studied, performing the linear instability analysis and the nonlinear stability analysis (via the weighted energy method) of the thermal conduction solution. The linear and nonlinear critical Rayleigh numbers for the onset of penetrative porous convection are determined and their behaviour with respect to the fundamental physical parameters of the model is studied. In particular, we found that there exists a region of subcritical instabilities and that the increase of the upper bounding plane temperature and the decrease of gravity have a stabilizing effect on the onset of convection.

A nonlinear stability analysis for magnetized ferrofluid heated from below in the presence of couple stresses for combination of different bounding surfaces

This work aims to inspect the impact of couple stress forces on the convective stability of magnetized ferrofluid for different combinations of bounding surfaces. Both linear and nonlinear analyses are conducted to obtain eigenvalue problems. Normal mode analysis is used for linear analysis, while the energy method is used for nonlinear analysis, and a generalized energy functional is introduced. For solving eigenvalue problems, the Galerkin method is employed. It is found that the Rayleigh numbers for the two analyses did not match, suggesting the existence of a subcritical instability region. Furthermore, it is observed that the subcritical instability region decreased as the magnetic parameter increased, whereas an increase in the couple stress parameter increased the subcritical instability region. For this analysis, three different combinations of bounding surfaces are considered. It is also observed that fluid confined in the rigid–rigid bounding surface is more thermally stable, which is suitable for convection in ferrofluid.

Parametric resonance for pipes conveying fluid in thermal environment

Parametric resonance is a kind of specific fluid-induced vibration for pipes conveying fluid. This paper focuses on revealing the qualitative characteristics of parametric resonances of the pipe with pulsating fluid speed in thermal environment, and compares the differences between the resonance characteristics in subcritical and supercritical regions. According to the generalized Hamilton's principle, the partial-differential-integral governing equation of a straight pipe is established. Under simply-supported boundary conditions, the non-trivial equilibrium configuration is obtained analytically. The governing equation of a supercritical pipe is derived by coordinate substitution on the basis of the new equilibrium configuration. The approximate analytical solution of parametric resonance for a pipe conveying fluid in a thermal environment is obtained by using the direct multi-scale method. The approximate analytical solution is verified to be reliable by using the Runge-Kutta method. The stability bounds of the parameters inducing parametric resonance are investigated via the introduced analytical method. The results show that the influence of temperature increment on sub-harmonic resonance is non-monotonic. In the subcritical region, when the temperature increment increases, the unstable region widens. This means that the system is more prone to parametric resonance, and the response amplitude decreases. In the supercritical region, when the temperature increment increases, the unstable bandwidth decreases, and the response amplitude increases. When viscous damping is increased or the average velocity is decreased, the sub-critical instability region decreases, and the supercritical instability region increases. With the increase in the pulsation velocity, the unstable bandwidth of subcritical and supercritical regions will increase, and parametric resonance is more likely to occur.

Nonlinear instability analysis of internally heated convection in a porous layer with the impact of magnetic field and variable gravity

AbstractThe impact of the magnetic field, heat source, and variable gravity field on the stability of convective phenomena in a porous layer are investigated numerically. Three types of gravity variations, such as linear, parabolic, and cubic functions are considered. For linear and nonlinear theories, the method of normal modes has been employed to solve governing dimensionless equations. It is found that the onset of convection is delayed by increasing the Hartmann number and gravity variation parameter and enhancement of the internal heat source makes the system stable. The subcritical instability region increases as the Hartmann number and gravity variation parameter increase.

Thermorheological effect on Rayleigh–Bénard magnetoconvection in a biviscous Bingham fluid with rough boundary condition on velocity and Robin boundary condition on temperature

AbstractThe thermorheological effect on the onset of Rayleigh–Bénard convection in a biviscous Bingham fluid in the presence of a horizontal magnetic field is investigated considering rough boundary conditions on velocity and Robin boundary conditions on temperature. The viscosity of the electrically conducting fluid is assumed to be sensitive to temperature variation. Linear and global nonlinear stability analyses are performed using the Chebyshev pseudospectral method to determine the existence of instability or otherwise. A general interpretation is made from the results to show the effects of the magnetic field and the variable viscosity on the system's stability. The biviscous Bingham parameter and the Chandrasekhar number are shown to have a delay in the onset of convection, while the effect of temperature sensitivity is to advance the onset. It is found that the results of linear and global nonlinear stability are not in agreement, so the region of subcritical instability exists. Also, the results obtained for Rayleigh–Bénard convection agree pretty well with those of Platten and Legros and Siddheshwar et al. for the limiting cases.

The effects of the Soret and slip boundary conditions on thermosolutal convection with a Navier–Stokes–Voigt fluid

In this paper, we study the problem of thermosolutal convection in a Navier–Stokes–Voigt fluid when the layer is heated from below and simultaneously salted from above or below. This problem is studied under the effects of Soret and slip boundary conditions. Both linear and nonlinear stability analyses are employed. When the layer is heated from below and salted from above, the boundaries exhibit great concordance, resulting in a very narrow region of probable subcritical instabilities. This proves that linear analysis is reliable enough to forecast the beginning of convective motion. The Chebyshev collocation technique and QZ algorithm have been used to solve systems of linear and nonlinear theories. For thermal convection in a dissolved salt field with a complex viscoelastic fluid of the Navier–Stokes–Voigt type, instability boundaries are computed. When the convection is of the oscillatory type, the Kelvin–Voigt parameter is observed to play a crucial role in functioning as a stabilizing agent. This effect's quantitative size is shown.

Nonlinear magneto convection in an inclined porous layer with artificial neural network prediction

The onset of magnetoconvection in an inclined porous layer is investigated. The effects of physical parameters, such as the Rayleigh number, the inclination angle, and the Hartmann number, are examined. The system becomes more stable by increasing the inclination angle and Hartmann number. It is noted that the transverse rolls are more stable than the longitudinal rolls. A comparison between the linear and nonlinear instability analysis is discussed in more detail. The threshold values for the longitudinal and transverse rolls provide the subcritical instability region. It is noted that the subcritical instability region increases as the inclination angle also increases, whereas it remains unchanged as the Hartmann number increases. Besides, an artificial neural network (ANN) model using the Levenberg‐Marquardt backpropagation algorithm is employed to predict the distribution of critical Rayleigh numbers for both linear and nonlinear stability analyses. The optimal number of neurons in the hidden layer is selected on the basis of coefficient of determination, root mean square error, and root mean relative error. The simulated critical Rayleigh numbers obtained by the numerical study and the predicted critical Rayleigh numbers by the ANN coincides.

Weakly nonlinear stability analysis of salt-finger convection in a longitudinally infinite cavity

This paper is a two-dimensional linear and weakly nonlinear stability analyses of the three-dimensional problem of Chang et al. [“Three-dimensional stability analysis for a salt-finger convecting layer,” J. Fluid Mech. 841, 636–653 (2018)] concerning salt-finger convection, which is seen when there is sideways heating and salting along the vertical walls along with a linear variation of temperature and concentration on the horizontal walls. A two-dimensional linear stability analysis is first carried out in the problem with the knowledge that the result could be different from those of a three-dimensional study. A two-dimensional weakly nonlinear stability analysis, that is, then performed points to the possibility of the occurrence of sub-critical motions. Stability curves are drawn to depict various instability regions. With the help of a detailed stability analysis, the stationary mode is shown to be the preferred one compared to oscillatory. Local nonlinear stability analysis of the system is done in a neighborhood of the critical Rayleigh number to predict a sub-critical instability region. The existence of a stable solution at the onset of a weakly nonlinear convective regime is indicated, allowing one to perform a bifurcation study in the problem. Heat and mass transports are discussed by analyzing the Nusselt number, Nu, and Sherwood number, Sh, respectively. A simple relationship is obtained between the Nusselt number and the Sherwood number exclusively in terms of the Lewis number, Le.

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.

Thermosolutal Convection with a Navier–Stokes–Voigt Fluid

We present a model for convection in a Navier–Stokes–Voigt fluid when the layer is heated from below and simultaneously salted from below, the thermosolutal convection problem. Instability thresholds are calculated for thermal convection with a dissolved salt field in a complex viscoelastic fluid of Navier–Stokes–Voigt type. The Kelvin–Voigt parameter is seen to play a very important role in acting as a stabilizing agent when the convection is of oscillatory type. The quantitative size of this effect is displayed. Nonlinear stability is also discussed, and it is briefly indicated how the global nonlinear stability limit may be increased, although there still remains a region of potential sub-critical instability, especially when the Kelvin–Voigt parameter increases.

The Effect of Magnetic Field on the Stability of Double-Diffusive Convection in a Porous Layer with Horizontal Mass Throughflow

The onset of double-diffusive convection in an electrically conducting fluid-saturated porous layer is studied. The convective flow in the porous medium is induced by horizontal temperature and concentration gradients with net horizontal mass throughflow. The effect of magnetic field on the instability of convection is taken into consideration. The stability of the steady-state solution is investigated in two different approaches, namely the linear instability analysis and nonlinear stability analysis. The nonlinear stability analysis is performed by constructing the energy functional. The eigenvalue problems which are derived from the stability analyses are numerically integrated using the shooting and Runge–Kutta methods. The variation in the critical thermal Rayleigh number against each flow governing parameter is shown graphically. It is observed that Hartman number $$Ha^{2}$$ delays the onset of convection to commence and helps to reduce the region of subcritical instabilities. When the solute is concentrated at lower boundary of the porous layer, the onset of convection is in the form of stationary modes, but it switches to oscillatory mode of convection when the solute is concentrated at upper boundary. Interestingly, Hartman number $$Ha^{2}$$ plays an important role in delaying this transition from stationary mode to oscillatory mode of convection.

Effect of anisotropic permeability on double‐diffusive bidisperse porous medium

AbstractThe model of thermosolutal convection in a fluid‐saturated bidisperse porous medium of Darcy type is studied in this paper. The permeability is allowed to be horizontally isotropic for both the macro‐ and microphases. The linear instability and nonlinear stability are analyzed by taking the Soret effect into account. Furthermore, the effect of anisotropy parameter, Soret coefficient, and other physical parameters on the stability of the system are investigated. It is shown that the linear instability boundaries and the energy stability boundaries do not coincide when the layer is heated and salted from below, where a region of potential subcritical instability occurs. The results reveal that the horizontal to vertical permeability ratio plays a crucial role in the stability of the system. It is also observed that for large values of the salt Rayleigh number, the onset of thermal convection is more likely to be via oscillatory convection rather than stationary convection. Furthermore, the onset of stationary convection is significantly influenced by the presence of the Soret coefficient.

Linear and nonlinear thermosolutal instabilities in an inclined porous layer

We investigate the double-diffusive instability in an inclined porous layer with a concentration-based internal heat source by conducting linear instability and nonlinear energy analyses. The effects of different dimensionless parameters, such as the thermal ( Ra T ) and solutal ( Ra S ) Rayleigh numbers, the angle of inclination ( ϕ ), the Lewis number ( Le ) and the concentration-based internal heat source ( Q ) are examined. A comparison between the linear and nonlinear thresholds for the longitudinal and transverse rolls provides the region of subcritical instability. We found that the system becomes more unstable when the thermal diffusivity is greater than the solute and the internal heat source strength increases. It is observed that the system is stabilized by increasing the angle of inclination. While the longitudinal roll remains stationary without the region of subcritical instability, as the angle of inclination increases, the transverse roll switches from stationary-oscillatory-stationary mode. Our numerical results show that for Ra S < 0, for all Q values, the subcritical instability only exists for transverse rolls. For Ra S ≥ 0, however, the subcritical instability appears only for Q = 0 and Q ≥ 0, respectively, for longitudinal and transverse rolls.

Mono‐diffusive Hadley‐Prats flow in a horizontal porous layer subject to variable gravity and internal heat generation

AbstractAnalysis of internal heated and gravity effect on the onset of Hadley‐Prats flow in a horizontal porous layer with inclined temperature gradients is investigated using the linear and nonlinear instability analysis. The transformed eigenvalue problem is evaluated numerically to find the eigenvalue, which is treated as a vertical thermal Rayleigh number (Rz). It is evaluated by applying shooting and Runge‐Kutta method. Also, the critical Rz is investigated for different parameters governing the flow. A theoretical study is made to understand the influence of gravity field on the mechanism of mono‐diffusive instability of Hadley‐Prats convection in a fluid saturated horizontal porous layer. Nonlinear stability is evaluated by using energy functional. The comparison between linear and nonlinear instability results are presented and it is noted that linear theory of instability may not be useful to capture the complete picture of stability and instabilities may arise before one attains the linear stability threshold. This subcritical instability region is identified between the linear and energy thresholds in the parameter space of the problem considered.