Analysis of the effect of variable gravity on stationary convection in Rayleigh-Bénard convection for free-free, rigid-rigid and rigid-free boundary conditions

Introduction

The Rayleigh-Bénard problem, which describes the onset of natural convection in a fluid layer heated from below, is a fundamental problem in fluid dynamics, particularly in the study of heat transfer and stability. This problem involves the spontaneous conversion of potential energy (due to the thermal gradient) into kinetic energy (due to fluid motion), leading to convection when the temperature gradient exceeds a critical threshold. The stability of such convection is influenced by the interaction of buoyancy forces, viscosity, and thermal gradients.

The phenomenon was first experimentally demonstrated by Bénard [1] , but it was Rayleigh [2] who laid down the theoretical framework for the onset of instability. Rayleigh formulated the Rayleigh number, a dimensionless number that characterizes the convection process. According to Rayleigh's criterion, convection becomes unstable when the Rayleigh number exceeds a critical value, which depends on the fluid properties (like viscosity and thermal conductivity), the temperature gradient, and the physical dimensions of the system. This threshold marks the transition from a stationary, stable state to a convective regime. Subsequent studies, including those by Jeffreys [3] , Pellew and Southwell [4] , and Chandrasekhar [5] , further refined the understanding of Rayleigh-Bénard convection, introducing the principle of the exchange of stabilities and exploring various effects such as rotation, magnetic fields, and the impact of different boundary conditions. For instance, the work by Gupta et al. [6] and Banerjee et al. [7] examined the influence of rotation and magnetic fields on the Rayleigh-Bénard problem, and their work provided upper bounds and sufficient conditions for the stability of the system under these conditions. Bodenschatz et al. [8] work provides a comprehensive summary of results on Rayleigh-Bénard convection, particularly focusing on compressed gases and gas mixtures. These systems are of interest because they exhibit different behaviors compared to conventional fluids, especially when the Prandtl number is near one or smaller.

Tagare et al. [9] research investigates the linear and weakly nonlinear properties of Rayleigh-Bénard convection in rotating fluids and explores the effects of the Coriolis force on gravity-driven convection. Their work is particularly important in understanding how rotation influences convection, especially for idealized stress-free boundary conditions, which are a simplifying assumption often made in theoretical studies. All these researchers, in their study of Rayleigh-Bénard convection problem, consider Earth's gravitational field as constant. A crucial simplification in many studies of Rayleigh-Bénard convection is the assumption of constant gravitational acceleration. However, this assumption may not always be appropriate, particularly in geophysical or astrophysical contexts, where gravity can vary with height. For example, in the Earth's atmosphere, ocean currents, or in planetary mantles, the gravitational field changes with altitude, and this can influence the convection process. Several studies have begun to explore the effect of variable gravity on the onset of instability in Rayleigh-Bénard convection. Gresho and Sani [10] were among the first to investigate the impact of timedependent gravity on fluid stability. Later, researchers like Pradhan and Samal [11] did analytical investigation of the instability, sufficient condition for stability and a circle theorem for the growth rate of a layer of fluid under the action of a gravitational field varying with height, was derived. Straughan [12] showed that gravity variation could either stabilize or destabilize the convection depending on the direction and the magnitude of the change in gravity. Pradhan et al. [13] showed that increasing the vertical gravitational field accelerates the onset of convection. Rionero and Straughan [14] numerically investigated convection in a porous layer under a depth-dependent gravitational field, highlighting how gravity variations influence the thermal instability and convection patterns in porous media. Several studies [15,16] have explored the effects of timedependent gravity on thermal instability. Time-varying gravitational fields can occur in systems undergoing acceleration or deceleration, such as in rotating fluids or in systems under the influence of tidal forces or acceleration due to external influences. The impact of such time-dependent gravity is significant because it can modulate the strength of buoyancy forces that drive convection, leading to changes in the onset of instability and the nature of convection patterns. Suma and Indira [17] study examined the effect of throughflow (fluid movement due to pressure gradients) and variable gravity on thermal convection in porous media using the regular perturbation technique. Chand et al. [18] investigated the impact of variable gravity on thermal instability in a nanofluid in a porous medium. They showed that the gravity parameter (which characterizes the strength of gravity variation) has a significant effect on fluid velocity. For porous medium the effect of variable gravity for different kind of fluids was studied by different researchers [19][20][21][22][23] . The study by Han et al. [24] investigates the dynamic transition in Rayleigh-Bénard convection under the influence of internal heating and a varying gravity field. This work adds an important layer to our understanding of convection by considering both internal heating and variable gravity, which are common in many natural and engineered systems. Yadav [25] used Galerkinweighted residual technique, to study the influence of varying downward gravity field and constant throughflow on the onset of Darcy-Brinkman convection in a porous medium layer. Shekar et al. [26] studied the effects of variable gravity on rotating convection in a sparsely packed porous layer. Using the MATLAB-based bvp4c method, they showed that the gravity variation parameter and the Taylor number (which characterizes the effects of rotation on the convection) have an impact on the onset of convection. Specifically, they found that gravity variations and the Taylor number together could postpone the onset of convection, whereas the Darcy number (which relates to the permeability of the porous medium) could accelerate it. Surya and Gupta [27] investigated the instability of a liquid layer heated from below under the influence of a variable gravitational field. They considered thermally conducting permeable boundaries. Their results show that the onset of convection is significantly affected by the variation in gravity. Pant and Algehyne [28] numerically studied ferromagnetic convection influenced by variable gravity and viscosity. They hypothesized six different types of gravitational field variations to explore how gravity, in combination with magnetic effects, can influence convection patterns. Francis et al. [29] conducted a comparative study on the effect of gravity modulation on Rayleigh-Bénard convection with different boundary conditions. They found that gravity modulation has a stabilizing effect on the system, particularly when the boundaries are configured in certain ways. The effect of variable gravity field, using nonlinear stability analysis, on the onset of penetrative convection was analysed by Arone et al. [30] and conditions for the onset of steady convection were also obtained. While much has been done to understand the role of gravity in Rayleigh-Bénard convection, the effect of varying gravity on convection with free-free, rigid-rigid and rigid-free boundary conditions; particularly when the gravity field varies in specific ways, such as the six distinct gravitational field variations hypothesized by Pant and Algehyne [28] , has not yet been thoroughly studied. This gap in the literature motivates further investigation into how variable gravity might influence the onset of instability and the resulting convective patterns, especially in systems with free-free, rigid-rigid and rigid-free boundary conditions, where the fluid boundaries are not constrained.

Physical Problem and Governing Equations

Here, we study finitely heat conducting viscous Boussinesq fluid, which is confined statically, between two horizontal boundaries 𝑧 = 0 and 𝑧 = 𝑑, of infinite horizontal extension. At lower boundary 𝑧 = 0 temperature is taken to be 𝑇 0 and 𝑇 1 (< 𝑇 0 ) is temperature at upper boundary 𝑧 = 𝑑 (see Figure 1). Variable gravity acting on entire system is assumed from Pant and Algehyne [28] of the form 𝑔 ⃗ = (0,0, -𝑔(1 + 𝛿𝑓(𝑧))) where 𝑓(𝑧) is changeable gravity function and 𝛿 is a gravity variation parameter (𝛿 ∈ (0,1) ). It is further assumed that the cross-diffusion effects of the stratifying agencies may be neglected. Following the linear stability theory [5] , making use of the initial state solutions, linearized perturbation equations and normal mode analysis, we get the following non-dimensional governing equations and boundary conditions

𝑤 = 0 = 𝜃 = 𝐷 2 𝑤 at 𝑧 = 0 and 𝑧 = 1 (Lower and upper boundaries are free),

𝑤 = 0 = 𝜃 = 𝐷𝑤 at 𝑧 = 0 and 𝑧 = 1 (Lower and upper boundaries are rigid),

𝑤 = 0 = 𝜃 = 𝐷𝑤 at 𝑧 = 0 (Lower boundary is rigid), ( 5)

where 𝑎, 𝑧, 𝐷 = 𝑑 𝑑𝑧 , 𝜎 > 0, 𝑅 > 0, 𝑝 = 𝑝 𝑟 + 𝑖 𝑝 𝑖 , 𝑤 and 𝜃 represents the wave number, vertical coordinate, differentiation with respect to 𝑧, Prandtl number, thermal Rayleigh number, complex growth rate (complex number) where 𝑝 𝑟 and 𝑝 𝑖 are real constants, vertical velocity and temperature, respectively.

Mathematical Analysis

We consider two categories of gravitational field variation, here, namely • Decreasing gravitational field variation • Increasing gravitational field variation We now solve the above equations ( 1)-( 2) using Galerkin Method for different combinations of boundary conditions (3)-( 6). We use Galerkin type weighted residual method to obtain Rayleigh numbers for free-free, rigid-rigid and rigidfree boundary conditions.

Stability Analysis when Lower and Upper boundaries are free

We consider the trial functions for 𝑤 and 𝜃, satisfying the boundary condition (3), as given below 𝑤 = 𝑤 𝑜 (𝑧 4 -2𝑧 3 + 𝑧) and 𝜃 = 𝜃 𝑜 𝑧(𝑧 -1).

Now, let us consider the case (iv) i.e. 𝑓(𝑧) = 𝑧, for increasing gravitational field variation, and obtain the Rayleigh Number for free-free boundary condition. Substituting equation (7) in equations ( 1)-( 2), integrating each equation within the limits 0 to 1 and using Galerkin type weighted residual method, we get following systems of equations

] 𝑤 𝑜 + 17𝑅𝑎 2 (2 + 𝛿)𝜃 𝑜 = 0,

and 17𝑤 𝑜 + 14(10 + 𝑎 2 + 𝑝)𝜃 𝑜 = 0,

For non-trivial solution of simultaneous system of equations ( 8) and ( 9), the determinant of coefficients of 𝑤 𝑜 and 𝜃 𝑜 must vanish, mathematically . when 𝑓(𝑧) = log (1 + 𝑧).

Following the same, above mentioned, steps, Rayleigh number for remaining five cases of gravitational field variation can be obtained and are given below

𝑅 𝑓𝑓 = , when 𝑓(𝑧) = 𝑒 𝑧 . ( 14)

Stationary Convection

Condition for stationary convection at the marginal state, is obtained by putting 𝑝 = 0 in the equations ( 10)-( 15), which on simplification yields Rayleigh Numbers (stationary convection) for six cases of gravitational field variation, given by

, when 𝑓(𝑧) = 𝑧. (16) ( 11)

17𝑎 2 (51+1260𝛿(1120-412𝑒))

In order to obtain the critical wave number (𝑎 𝑐 ), we differentiate 𝑅 𝑠 𝑓𝑓 with respect to square of wave number (𝑎 2 )

and resultant is equated to zero, we get the value of 𝑎 𝑐 . For all six cases of gravitational field variation, when both boundaries are free, we get 𝑎 𝑐 = 2.226972.

Stability Analysis when Lower and Upper boundaries are rigid

We consider the trial functions for 𝑤 and 𝜃, satisfying the boundary condition (4), as given below 𝑤 = 𝑤 𝑜 (𝑧 4 -2𝑧 3 + 𝑧 2 ) and𝜃 = 𝜃 𝑜 𝑧(𝑧 -1).

Following the similar steps mentioned above, in section 3.1, we obtained Rayleigh Number for six different cases of gravitational field variation for rigid-rigid boundary condition and are given below

, when 𝑓(𝑧) = log (1 + 𝑧). (28)

Stationary Convection

For stationary convection, Rayleigh Number (𝑅 𝑠 𝑟𝑟 ) for different cases of gravitational field variation are obtained by putting 𝑝 = 0 in equations ( 23)-( 28), given by

In order to obtain the critical wave number (𝑎 𝑐 ), we differentiate 𝑅 𝑠 𝑟𝑟 with respect to square of wave number (𝑎 2 ) and resultant is equated to zero, we get the value of 𝑎 𝑐 . For all six cases of gravitational field variation, when both boundaries are rigid, we get 𝑎 𝑐 = 3.116524.

Stability Analysis when Lower boundary is rigid and Upper boundary is free

We consider the trial functions for 𝑤 and 𝜃, satisfying the boundary condition (4), as given below 𝑤 = 𝑤 𝑜 (2𝑧 4 -5𝑧 3 + 3𝑧 2 ) and 𝜃 = 𝜃 𝑜 𝑧(𝑧 -1).

Following the similar steps mentioned above, in section 3. , when 𝑓(𝑧) = -𝑧 2 .

(37) Figures 234are the graphs between 𝑅 𝑠 and 𝑎, for the decreasing gravitational field variation 𝑓(𝑧) = 𝑧 2 -2𝑧, 𝑓(𝑧) = -𝑧 2 and 𝑓(𝑧) = -𝑧 respectively, for different values of 𝛿 = 0.3, 0.6, 0.9, for free-free, rigid-rigid and rigid-free boundary conditions. It is seen that as the value of 𝛿 increases, value of Rayleigh number increases, which clearly shows the stabilizing effect of 𝛿 on stationary convection for decreasing gravitational field variation. This happens because, an increase in the value of δ decreases the gravitational force for decreasing gravitational field variation. Hence, we can say that decreasing gravitational force postpone the onset of convection, resulting in enhancement of the stability of system.

Figures 567are the graphs between 𝑅 𝑠 and 𝑎 for the increasing gravitational field variation 𝑓(𝑧) = 𝑧, 𝑓(𝑧) = 𝑒 𝑧 and 𝑓(𝑧) = log (1 + 𝑧) respectively. It is seen that as the value of 𝛿 increases the value of 𝑅 𝑠 decreases. Hence, in these cases, gravitational forces prepone the onset of convection by helping in early moments of fluid molecules, resulting in destabilizing the system. This is due to the reason that for increasing gravitational field variation, as the value of 𝛿 increases gravitational forces also increases.

Conclusion

One of the main findings of this study is that the stability of the convection system is highly dependent on both the type of gravitational field variation and the boundary conditions.

Here's what we reveal: a) Effect of Decreasing Gravitational Field (Stabilizing):

When the gravitational field decreases with height (f(z) = z 2 -2z, f(z) = -z 2 and f(z) = -z), the system becomes more stable, particularly when rigid-rigid boundary conditions are applied. This suggests that decreasing gravity tends to suppress convection and stabilize the fluid motion. b) Effect of Increasing Gravitational Field (Destabilizing): Conversely, when gravity increases with height (f(z) = z, f(z) = e z and f(z) = log (1 + z)), the system becomes more unstable. This destabilizing effect is most pronounced under free-free boundary conditions, where the fluid is more susceptible to instabilities.

c) Comparing Boundary Conditions:

• Rigid-Rigid Boundaries: These boundary conditions generally lead to the most stable convection systems, especially when the gravitational field decreases with height. The rigidity of the boundaries seems to limit the fluid motion and prevent large convection cells from forming. • Rigid-Free Boundaries: These are more unstable than rigid-rigid, but less so than free-free boundaries. The rigidity of one surface helps control fluid motion, but the free surface still allows some convection. • Free-Free Boundaries: These are the least stable, particularly when the gravitational field increases. The absence of boundary constraints on both sides of the fluid allows for greater fluid motion, leading to more pronounced convection instabilities.

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