Finite Amplitude Binary Reaction-convection in a Rotating Porous Layer with Thermal Non- Equilibrium Effects

Finite Amplitude Binary Reaction-convection in a Rotating Porous Layer with Thermal Non- Equilibrium Effects

Mahantesh S. Swamy

Department of Mathematics, Government College (Autonomous), Kalaburagi-585 105, India

Abstract – This study explores finite amplitude natural convection within a binary-reacting fluid mixture contained within a rotating porous layer. The primary objectives are to evaluate the conditions necessary for the onset of finite amplitude convective motions and to quantify the associated rates of heat and mass transport. This research incorporates two major physical phenomena: first, the use of the local thermal non-equilibrium (LTNE) model, which is utilized to model the distinct temperature fields maintained by the fluid and solid phases; and second, the effect of the Coriolis force resulting from system rotation. We specifically focus on investigating how the reactive components tendency to precipitate or dissolve significantly influences the characteristics of convection. To determine the qualitative behavior of the finite amplitude solutions and to calculate the quantities of heat and mass transfer, a weakly nonlinear analysis employing the Fourier series method was used. The findings of this current work validate previous research by demonstrating consistency with established results. This investigation ultimately advances the capability to design a control system for convective motions, which is highly relevant for applications in the manufacturing sector.

Keywords – Double-diffusive convection; chemical reaction; modified Darcy model; Rayleigh number; Nusselt number; Sherwood number.

1. INTRODUCTION

   The concurrent presence of gradients from two stratifying agents such as heat and salt, which possess distinct diffusivities within a fluid-saturated porous medium, gives rise to fascinating convective phenomena. These complex behaviors are not observed in single-component fluid systems. Over the past several decades, convection within fluid layers governed by two or more stratifying components has been a central focus of extensive theoretical and experimental research. This includes foundational works summarized in excellent reviews by Turner [1-3], Huppert and Turner [4], and Platten and Legros [5]. The appeal of studying two or multi-component convection stems from its stark difference compared to single- component systems. Notably, in contrast to single-component dynamics, convection can initiate even when the fundamental state is hydrostatically stable-meaning the overall density decreases with height. The concept of double diffusive convection holds critical importance across diverse fields, including the manufacturing of high-quality crystals, liquid gas storage, oceanography, pharmaceutical development, the solidification process of molten alloys, and the dynamics within geothermally heated lakes and magmas. Some of the

   recent works include Akbar et al. [6], Hu et al.[7], Akbal [8] and the references there in.

   In various geological systems, buoyancy forces drive the vertical circulation of a mineral-laden fluid through a porous matrix. During this circulation, the solute concentration in the fluid is often not conserved. This is because the mineral may either dissolve from or precipitate onto the solid porous matrix, as its solubility is dependent on factors like temperature, pressure and the localized rock chemistry. The exact influence of these phase change processes on convective instability remains incompletely understood and has been the subject of limited investigation. Early research into the impact of chemical reactions on the onset of convection in porous media was conducted by Steinberg and Brand[9-10]. Subsequent studies by Gatica et al. [11] and Viljoen et al. [12] explored how exothermic reactions influence the stability of such porous systems. Jupp and Woods [13] also contributed to the field with a study on thermally driven reaction fronts in porous media. The focus has included chemically driven instabilities, such as the work by Malashetty and Gaikwad [14] on binary liquid mixtures with rapid chemical reactions. Specifically, Pritchard and Richardson [15] investigated how temperature- dependent solubility affects the criteria for the onset of convection. The recent important references pertaining to this field are Vaz and Lima [16], Shevchuk et al. [17], Luo et al.

   1. and the references there in.

      The phenomenon of thermal convection in rotating porous media has been extensively studied. Early work by Friedrich

   2. investigated stability using both linear and nonlinear analyses for a bottom-heated rotating layer. Subsequent investigations incorporated additional complexities: Patil and Vaidyanathan [20] included the influence of variable viscosity, while Palm and Tyvand [21] established a valuable analogy between rotating and anisotropic porous layers. Jou and Liaw

   [22] determined marginal stability conditions for transient heating using the Darcy model. Later research focused on stability criteria. Qin and Kaloni [23] derived sufficient conditions for nonlinear stability in high-porosity media via energy theory. Vadasz [24] performed both linear and weak nonlinear stability analysis, notably finding that, unlike in pure fluids, overstable convection in porous media is not limited by the Prandtl number. Vadasz [25] also provided an excellent overall review. Crucially, studies by Straughan [26-27] utilizing energy stability theory demonstrated key equivalence results. Straughan [26] reported that the global nonlinear stability boundary for Darcy-governed rotating convection matches the linear instability boundary, confirming that

   subcritical instabilities are impossible. Extending this, entire porous layer rotates uniformly about the vertical z -axis

   Straughan [27] later proved this equivalence still holds when applying a thermal non-equilibrium model (LTNE) to a layer rotating about a vertical axis. Further studies have been exhaustively reported in the literature. Some of the recent are Siddheshwar and Krishna [28], Rana et al. [29], Rana and Sharma [30] and the references there in.

   Most prior investigations have relied on the assumption of Local Thermal Equilibrium (LTE) between the fluid and solid phases within the porous medium. This assumption posits that the temperature difference between the two phases at any given point is negligible. However, for numerous practical scenarios involving rapid heat transfer such as the drying and freezing of materials, or microwave heating, the LTE assumption is insufficient. Consequently, the study of Local Thermal Non- Equilibrium (LTNE) effects becomes highly relevant.

   A comprehensive review of research utilizing the LTNE model is provided in the literature [31], with more recent significant contributions found in [32-35]. The concept of a

   two-field energy model was initially proposed by Bories [38] and was adopted by Nield and Bejan [31] and Nield and

   with an angular velocity, = (0, 0, Q) . Due to the assumed

   significance of heat exchange between the solid and fluid components within the porous medium, the Local Thermal Non-Equilibrium (LTNE) model is adopted, requiring separate energy equations for each phase. The effects of variable fluid density are incorporated through the Boussinesq approximation. Regarding the binary reaction, chemical equilibrium is maintained at the boundaries. We invoke a linear relationship to describe the dependence of the equilibrium

   concentration, Ceq (Tf ) = C0 + q (Tf – T0 ) where q = (tiCtiTf )/p>

   is a constant. This relationship is constrained by the physical premise that the solubility of the reactive component increases with temperature ( q > 0 ).The physical phenomena described

   above are governed by a set of coupled partial differential equations, which constitute the mathematical model, including the equations,

   V- q = 0, (1)

   p0 ( aq + q – Vq + 2Qx q ) + µ q + Vp – p g = 0, (2)

   Kuznetsov [39] as the simplest approach for modeling LTNE.

   E I at

   I K f

   Rather than using a single energy equation for a common

   \_ )

   porous media temperature, this model employs two distinct energy equations to describe the fluid and solid phase temperatures separately. These two equations are necessarily

   ((p c) ( E a + q.V) – E k V2 )T

   I 0 f I at I f I

   \_ \_ ) )

   f + h(Tf – Ts ) = 0,

   (3)

   coupled via terms that mathematically account for the heat

   (1- E (

   a 2 )

   \_ )

   transfer occurring between the phases.

   While significant progress has been made in understanding the individual effects of rotation, chemical reaction and thermal

   ) I(p0 c)s at – ks V ITs + h(Ts – Tf ) = 0,

   (E a + q – V – EK V2 + ) – k-C T = 0,

   (4)

   (5)

   non-equilibrium within porous media, a comprehensive study

   I at

   C k IC

   eq ( f )

   that integrates all these complexities is currently lacking. Specifically, the combined influence of the Coriolis force, solute precipitation/dissolution, and Local Thermal Non- Equilibrium (LTNE) on the stability and heat/mass transport of double-diffusive convection has not yet been fully investigated.

   Therefore, the present work aims to fill this critical gap by performing a nonlinear stability analysis of finite amplitude

   \_ )

   p = p0 (1- f3T (Tf – T0 ) + f3C (C – C0 )),

   and the boundary conditions

   w = a2wdz2 = 0 at z = 0, d ,

   Tf = Ts = T0 , C = C0 at z = 0,

   Tf = Ts = Tu , C = Cu at z = d.

   (6)

   (7)

   double-diffusive convection in a binary-reacting fluid mixture saturated in a rotating porous layer under the governing framework of the LTNE model. The primary objectives are to

   (i) determine the conditions required for the occurrence of finite amplitude convective motions, (ii) quantify the resulting heat and mass transfer rates, and (iii) understand how the parameters associated with rotation, chemical reaction, and LTNE interact to control the onset and magnitude of convection.

   Here h denotes the coefficient of inter-phase heat transfer, k- , the lumped effective reaction rate. There is no motion and inter-phase heat transfer in the basic state, and all the unknown variables of above equations are homogeneous in the horizontal direction. To analyze the systems stability the finite amplitude perturbations are enforced on to the static state. For the ease of analysis, the focus is made on two- dimensional rolls (perturbations). Thus, all physical quantities become free from the y-coordinate. Then, it is convenient to

2. Formulation of the Problem

   introduce the stream function

   \jf ,

   defined by u = a\jf az

   The present study analyzes a long, horizontal layer of

   and w = – a\jf

   ax . Finally, up on rescaling the length, time,

   porous medium saturated with a binary-reacting fluid mixture. velocity, temperatures, and concentration with division by d,

   The system is confined between two surfaces located at z = 0

   d 2 K , K d , tiT and

   f

   f

   tiC

   respectively the Eqs. (1)-(6)

   and z = d . Both boundaries are assumed to be stress-free, impermeable, and isothermal. The layer is subjected to a

   constant gravitational field, g = (0, 0, – g ) , acting in the

   take dimensionless form

   ( 1 a ) a0) aT aC

   I + 1IV2\jf – Ta z + Ra f – Ra = 0,

   (8)

   negative z -direction. To establish the driving forces for convection, the lower boundary is uniformly heated and salted

   T C

   \_ PrD at ) az ax ax

   to create a destabilizing thermal gradient and a stabilizing

   aTf

   a (\jf , Tf )

   a\jf

   solute gradient relative to the upper surface. Furthermore, the

   -V2T

   – + – H (T – T ) = 0,

   (9)

   at f

   a ( x, z ) ax s f

   a aTs – V2T – y H (T – T ) = 0,

   (10)

   amplitudes A11 , B11 ,… using the dynamics of the system. The

   at s f s

   following eighth order Lorenz model is obtained on using Eqs. (18)-(22) into the Eqs. (8)-(12)

   ( f )

   aC a (\jf , C ) 1 2 a\jf

   at – a ( x, z

   – V C + – x T – C = 0,

   Le ax

   (11)

   dA11 (t ) = PrD (-8 2 A (t ) +

   Ta (n E

   (t )) – Ra (aB

   (t )) + Ra (aD

   (t )))

   )

   Ta az 2

   a2\jf

   1 a ( a0) ) a0)

   dt 8 2

   11 11

   T 11

   C 11

   (23)

   I z I + z + = 0.

   (12)

   aB (t)

   PrD at \_ az ) az

   11 = -8 2 B

   (t) – aA

   (t ) – 2n aA

   (t ) B

   (t ) + H (C

   (t) – B (t) )

   Here

   1. z

   at

   represents z-component of vorticity vector

   11 11 11 02 11 11

   (24)

   Pr = d 2Ev ( p c) Kk ,

   the Darcy-Prandtl number,

   aB (t) n a

   D 0 f f

   02 = -4n 2 B (t) + A (t ) B (t ) + H (C (t) – B

   (t) ) (25)

   Ta = (2K QEv )2 ,

   the Taylor number,

   at 02 2 11 11 02 02

   Ra = f3 gtiTdK ( p c )

   E k v , the thermal Rayleigh number,

   aC11 (t) =- 1 (8 2C

   (t) – y H ( B

   (t) – C

   (t) ) )

   (26)

   T T 0 f f

   at a

   11 11 11

   f

   RaC = f3C gtiCdK ( p0 c) E k fv , the solutal Rayleigh number,

   dD (t) 8 2

   H = d 2h E k , non-dimensional inter-phase heat transfer

   11 = –

   D (t) – aA (t ) – 2n aA (t ) D

   (t ) + x (B (t) – D (t))

   f

   coefficient,

   a = K f

   Ks ,

   the diffusivity ratio,

   dt Le

   11 11 11 02 11 11

   (28)

   y = E k f (1- E )ks ,

   the porosity modified conductivity ratio,

   dD (t) 4n 2

   02 =- D

   (t) + n a A (t ) D (t ) + x ( B (t) – D

   (t))

   (29)

   Le = K f KC ,

   the Lewis number and x = d 2k- EK , the

   dt Le 02

   2 11 11 02 02

   f

   Damkohler number (non-dimensional reaction-rate).

   dE11 (t ) =- Pr (E

   (t ) + n

   Ta A

   (t ))

   (30)

3. Non-linear Stability Analysis

   We now analyze the onset of finite amplitude convection,

   dt D

   11 11

   amplitude of convective motions, quantity of heat and mass transports. The methodology we adopted is based on double Fourier series representation of stream function, temperature, concentration and vorticity component in the form

   1. Steady Finite Amplitude Convection

      Tc

      The derivation of onset threshold for steady finite amplitude convection, namely, RaF , the critical finite amplitude

      00 00

      Rayleigh number is the major point of emphasis. This is

      \jf = II Amn (t )sin (max)sin (nn z ),

      m =1 n =1

      (13)

      achieved by first evaluating the finite amplitudes

      A11 , B11

      ,…,

      II

      00 00

      Tf = Bmn (t )cos(max)sin (nn z ),

      m =0 n=1

      00 00

      Ts = IICmn (t )cos (max)sin (nn z ),

      m =0 n =1

      00 00

      (14)

      (15)

      from the system of homogeneous non-linear algebraic equations which arise from the Eqs. (23)-(30), upon setting d dt (-) = 0 . Since there are nine unknowns with only eight

      11

      equations, it is required to express, in terms the first unknown A11 , all the remaining amplitudes. This yields the following relation, where x = A2 ,

      C = II Dmn

      (t )cos (max)sin (nn z ),

      (16)

      ( a x2 + b x + c )

      )

      11

      I 1 1 1 I A

      = 0.

      (31)

      m=0 n =1

      00 00

      \_ d1

      0)z = II Emn (t )sin (max)cos(nn z ),

      m=1 n=0

      (17)

      The expressions of the coefficients omitted for the sake of conciseness. There exist two possibilities from Eq. (31), either

      It is observed that in the laboratory setups and practical

      A11 = 0

      or (a x2 + b x + c ) d

      = 0 . The first one corresponds

      1 1 1 1

      contexts the flows are dominated by few spatial harmonics. Thus, one can truncate the above series as

      to the non-convective case, i.e., a purely conduction state. On the other hand the later possibility yields the expression for

      \jf = A11 (t) sin(ax) sin(n z),

      (18)

      convective amplitude A11 , in the form of the non-negative

      T = B (t) cos(ax) sin(n z) + B

      (t) sin(2n z),

      (19)

      double root of quadratic equation

      f 11 02

      1 1 1

      T = C (t) cos(ax) sin(n z) + C

      (t) sin(2n z),

      (20)

      a x2 + b x + c = 0.

      (32)

      s 11 02

      C = D11 (t) cos(ax) sin(n z) + D02 (t) sin(2n z),

      (21)

      The condition for existence of double root provides the quadratic equation in RaF .

      T

      0)

      = E

      (t) sin(ax) cos(n z) + E

      (t) sin(2n x).

      2

      z 11 20

      a2 (RaT )

      + b2 (RaT ) + c2 = 0.

      (33)

      (22)

      F

      F

      This gives the initial values for solving a full non-linear convection problem. Our first intention is to find the finite

      The real, positive, minimum root of this equation represents the

      ( Conduction + Convection

      onset threshold of finite amplitude convection, i.e.,

      RaF .

      Nu = Heat transport via i

      l

      Conduction

      Tc

      Further, up on using this for the computation of a non-negative

      ( 2n ac (aT

      az )dx )

      (36)

      double root of Eq. (32) one can obtain the value of the amplitude of steady convective motions, namely, A .

      = 1+ I I0 f I

      I 2n ac I

      Consequently, the values of all other unknowns B , B

      11

      ,…, are

      \_ I0

      (aTfb

      az ) dx )

      z =0

      obtained.

      11 02

      Sh = Mass transport via ( Conduction + Convection

   2. Steady Finite Amplitude Convection

      It is impossible to tackle the above non-linear system of differential equations (23)-(30), representing unsteady finite amplitude convection, by the analytical approach. Thus a

      i l

      I I

      2n a

      \_ 0

      c (aC az ) dx I

      b

      )

      = 1 + I 0 I

      ( I 2n ac (aCaz ) dx )

      z =0

      Conduction

      (37)

      powerful numerical technique is to be employed to do so. However in the realm of unsteady finite amplitude analysis, one can derive several important qualitative predictions. Let us calculate the divergence of velocity field represented by the

      above system in the form

      Up on using the basic-conduction state temperature and concentration distributions Tfb (z) = T0 – tiT ( zd ) and

      Cb (z) = C0 – tiC ( zd ) along with the Eqs. (19) and (21) in

      (36) and (37) respectively, one can obtain

      E11

      a ( dA ) ( (

      Hy )

      ( 1 1 ))

      Nu = 1+ 2n (-B02 )

      (38)

      I I I =- I 2 IPr + H + x + I + (8 2 + 4n 2 )I1+ +

      II.

      A= A11 aA \_ dt ) \_ \_ a )

      D

      \_ Le

      a ))

      Sh = 1+ 2n (-D )

      (39)

      (34)

      The negative divergence of velocity field signifies the

      02

      In the above expressions the second terms, viz.,

      2n (-B02 )

      contraction of volume in the phase space. Consequently, in the phase space the trajectories tend to a set of measure zero or in particular to a fixed point. The Eq. (34) demonstrates that if

      and 2n (-D02 ) signify the contribution of finite amplitude convective motions towards the heat and mass transports

      V (0) is the volume of the region in the phase space occupied

      respectively. Since both

      B02

      and

      D02

      decrease from zero to

      by the initial points of the trajectories at t = 0 , then after the

      T

      time t seconds the volume occupied by the final points of the trajectories is

      negative values as the thermal Rayleigh number is elevated above its critical value, it is observed that at the onset of convection both Nu and Sh take their conduction state value

      ( ( (

      H y )

      ( 1 1 )) )

      viz., one and then increase with

      RaF . Further in the limiting

      V (t ) = V (0)exp I- I 2 Pr + H + x + + (8 2 + 4n 2 ) 1 + +

      t I. F

      I D a I

      I Le

      a II

      case as

      RaT

      • 00 , the rate of heat and mass transfer attain

      \_ \_ \_

      ) \_ )) )

      (35)

      independence from

      T

      RaF . Its prime reason is our choice of

      Thus, there is an exponential decay in the volume with time. This dissipation is enhanced by the governing parameters like

      truncated Fourier series expansion with only two terms (see Eqs. (18)-(22)).

      PrD , x, H

      and y . Further as the above system (23)-(30)

      preserves the dissipative nature of the original equations (8)- (12), one can expect that this system continue to hold many properties of the full non-linear problem. The above system of non-linear differential equations is solved numerically by Runge-Kutta-Gill method to obtain the values of the amplitudes A11 , B11 ,…, as the functions of time t. Among

      these B02 (t ) and .. are used for the computation of unsteady

      heat and mass transfer rates (the derivation of formulae is shown in the following Section).

   3. Heat and Mass Transports

   The estimation of heat and mass transports is of great emphasis in the study of convection. The onset of convection significantly influences the heat and mass transfer rates. Before the convection sets in the heat and mass transfer take place only through the conduction mode. The non-dimensional coefficient of heat and mass transfer due to the onset of convection are denoted by the Nusselt and Sherwood numbers respectively. They are defined as

4. Results and Discussion

   The combined influence of the Coriolis force, solute precipitation/dissolution and LTNE on the double-diffusive convection is investigated. This work mainly focused on a nonlinear stability analysis of finite amplitude double- diffusive convection in a binary-reacting fluid mixture saturated in a rotating porous layer under the governing framework of the LTNE model. As a result of this study we determined the conditions for the occurrence of finite amplitude convective motions, quantified the resulting heat and mass transfer rates, and understood how the parameters associated with rotation, chemical reaction and LTNE interact to control the onset and magnitude of convection.

   The onset threshold in the form of critical Rayleigh number for finite amplitude convective motions is computed as a function of various governing parameters. The impact of LTNE and chemical reaction on the stability boundary is revealed in Figures 1 (a-b). These figures show the

   T

   Tc

   dependence of RaF on interphase heat transfer coefficient. The influence of interphase heat transfer coefficient is to elevate the value of RaF .

   Figure 1. Finite amplitude Rayleigh number versus H for different values of (a) y and (b) x .

   Thus, due to the LTNE effect the onset of finite amplitude on the double-diffusive system. Higher values of solutal

   convection is significantly delayed. This stabilizing effect of H is confined to specific moderate range. However, when H is very small, i.e., the difference between fluid and solid temperature is almost zero, the curves become flat. This reveals that the onset of convection is independent of H and hence corresponds to the LTE case. On the other hand when H is very large, i.e., fluid and solid temperature fields are not comparable, the impact of solid phae temperature field becomes insignificant and thus once again the curves become horizontal and approach those of LTE case.

   Furthermore, Fig. 1 (a) reports that increasing the porosity-modified conductivity ratio ( y ) from 0.01 to 1 leads

   Tc

   to a consistent decrease in RaF , suggesting that a more

   thermally conductive solid phase relative to the fluid phase destabilizes the system. Figure 1(b) illustrates that increasing the chemical reaction parameter ( x ) results in a decrease in

   Tc

   RaF . This indicates that the reaction

   (precipitation/dissolution) acts as a destabilizing agent, advancing the onset of finite amplitude convective motions.

   Tc

   The influence of the Coriolis force is depicted in Figure 2. Increasing the Taylor number leads to a sharp increase in Ra F , confirming that rotation has a strong stabilizing effect

   Rayleigh number RaC elevate the stability threshold (see Fig. 2a), while an increase in the Lewis number-representing a decrease in solutal diffusivity relative to thermal diffusivity- further stabilizes the system against finite amplitude perturbations (see Fig. 2b).

   Figure 3 display the steady-state heat and mass transfer rates are quantified by the Nusselt number (Nu) and Sherwood number (Sh) as a function of the Rayleigh number ratio

   T Tc

   ( Ra / RaF ). In all cases shown in Figure 3, the broken curves

   (Sh) lie above the solid curves (Nu), indicating that mass transfer is consistently more efficient than heat transfer in this binary-reacting mixture. Increasing H or y leads to an

   increase in both Nu and Sh, as a stronger thermal coupling and higher solid conductivity facilitate more vigorous convection. An increase in the Taylor number suppresses the transport rates (decreasing Nu and Sh), while an increase in the chemical reaction parameter ( x ) enhances them. Higher

   values of Le significantly enhance the Sherwood number while having a more moderate effect on the Nusselt number, highlighting the solutal sensitivity of the system.

   Figure 2. Finite amplitude Rayleigh number versus Ta for different values of (a) RaC

   and (b) Le .

   Tc

   Figure 3. Nu and Sh versus Ra F

   for different values of (a) H, (b) y , (c) Ta, (d) x , (e) RaC , (f) Le

5. Conclusions

This study investigated the nonlinear stability of double- diffusive convection in a rotating porous layer saturated with a binary-reacting fluid mixture under the framework of Local Thermal Non-Equilibrium. The primary conclusions derived from the weakly nonlinear analyses are as follows:

Tc

• The finite amplitude Rayleigh number ( RaF ) is significantly influenced by the thermal coupling

  between the fluid and solid phases. An increase in the interphase heat transfer coefficient H delays the onset of convection, demonstrating a stabilizing effect. Conversely, a higher conductivity ratio ( y ) and an

  increase in the chemical reaction parameter ( x ) are found to destabilize the system, advancing the onset of convective motions.

  Tc

  Figure 4. Nu and Sh versus time for different values of (a) RaF , (b) Ta, (c) H, (d) y .

• The Coriolis force, represented by the Taylor number, acts as a strong stabilizing agent. Increasing Ta leads to a substantial increase in the critical Rayleigh number, effectively suppressing the transition to convective flow.

• The heat and mass transport rates, quantified by the Nusselt (Nu) and Sherwood (Sh) numbers, increase with

  the thermal Rayleigh number. Notably, the Sherwood number is consistently higher than the Nusselt number across all parameter ranges, indicating that mass transfer is the dominant transport mechanism in this reactive system.

  Figure 4. Nu and Sh versus time for different values of (e) RaC , (f) Le, (g) x , (h) a .

• While rotation (Ta) reduces transport efficiency, increasing the interphase heat transfer (H) and the conductivity ratio ( y ) enhances both heat and mass

  transfer rates once convection is established. Furthermore, the Lewis number has a more pronounced effect on the Sherwood number compared to the Nusselt number.

• Transient analysis reveals that the system reaches a steady state following initial oscillatory behavior. The frequency of these oscillations is particularly sensitive to the solutal Rayleigh number ( RaC ), the Lewis number, and the Darcy-Prandtl number, though PrD does not significantly impact the final steady-state transport values.

References

1. J. S. Turner, Buoyancy Effects in Fluids. Cambridge University Press, London, 1973.

2. J. S. Turner, "Double diffusive phenomena," Annu. Rev. Fluid Mech., vol. 6, pp. 37-56, 1974.

3. J. S. Turner, "Multicomponent convection," Annu. Rev. Fluid. Mech., vol. 17, pp. 11-44, 1985.

4. H. E. Huppert and J. S. Turner, "Double diffusive convection," J. Fluid Mech., vol. 106, pp. 299-329, 1981.

5. J. K. Platten and J. C. Legros, Convection in Liquids. Springer, Berlin, 1984.

6. R. Akbar, A. Arshad, and M. Farooq, "Numerical Investigation of Double-Diffusive Convection in an Irregular Porous Cavity Subjected to Inclined Magnetic Field Using Finite Element Method," Mathematics, vol. 12, no. 6, 808, 2023.

7. Z. Hu, X. Xu, and M. Yang, "Double-diffusive convection with gravitationally unstable temperature and concentration gradients in homogeneous and heterogeneous porous media," Journal of Fluid Mechanics, vol. 998, A1, 2024.

8. Z. Akbal, "Investigating double-diffusive natural convection in a sloped dual-layered homogenous porous-fluid square cavity," International Journal of Numerical Methods for Heat & Fluid Flow, vol. 35, no. 5, pp. 1837-1859, 2025.

9. V. Steinberg and H. Brand, "Convective instabilities of binary mixtures with fast chemical reaction in a porous medium," J. Chem. Phys., vol. 78, no. 5, pp. 2655-2660, 1983.

10. V. Steinberg and H. Brand, "Amplitude equations for the onset of convection in a reactive mixture in a porous medium," J. Chem. Phys., vol. 80, no. 1, pp. 431- 435, 1984.

11. J. E. Gatica, H. J. Viljoen, and V. Hlavacek, "Interaction between chemical reaction and natural convection in porous media," Chem. Eng. Sci., vol. 44, no. 9, pp. 1853-1870, 1989.

12. H. J. Viljoen, J. E. Gatica, and V. Hlavacek, "Bifurcation analysis of chemically driven convection," Chem. Eng. Sci., vol. 45, no. 2, pp. 503-517, 1990.

13. T. E. Jupp and A. W. Woods, "Thermally-driven reaction fronts in porous media," J. Fluid Mech., vol. 484, pp. 329-346, 2003.

14. M. S. Malashetty and S. N. Gaikwad, "Onset of convective instabilities in a binary liquid mixture with fast chemical reactions in a porous medium," Heat Mass Transfer, vol. 39, pp. 415-420, 2003.

15. D. Pritchard and C. N. Richardson, "The effect of temperature-dependent solubility on the onset of thermosolutal convection in a horizontal porous layer," J. Fluid Mech., vol. 571, pp. 59-95, 2007.

16. F. E. Vaz and G. M. de Lima, "Convective mixing in porous media: a review of Darcy, pore-scale and Hele- Shaw studies," Journal of Fluid Mechanics, vol. 978, A1, 2023.

17. V. Shevchuk, A. N. Cherepanov, and N. A. Cherepanov, "Instability Problems and Density-Driven Convction in

    Saturated Porous Media Linking to Hydrogeology: A Review," Geosciences, vol. 13, no. 2, 36, 2023.

18. C. Luo, Z. Liu, and Y. Chen, "State of the Art on Fe Precipitation in Porous Media: Hydrogeochemical Processes and Evolving Parameters," Hydrogeology Journal, vol. 32, no. 4, 690, 2024.

19. R. Friedrich, "Einflub der Prandtl-Zahl auf die Zellularkonvektion in einem rotierendent mit Fluid gesattigten porosen Medium," Z. Angew. Math. Mech., vol. 63, pp. 246-249, 1983.

20. P. R. Patil and G. Vaidyanathan, "On setting up of convection currents in a rotating porous medium under the influence of variable viscosity," Int. J. Engng. Sci., vol. 21, pp. 123-130, 1983.

21. E. Palm and A. Tyvand, "Thermal convection in a rotating porous layer," Z. Andrew. Math. Phys., vol. 35,

    pp. 122-123, 1984.

22. J. J. Jou and J. S. Liaw, "Thermal convection in a porous medium subject to transient heating and rotation," Int. J. Heat Mass Transfer, vol. 30, pp. 208-211, 1987.

23. Y. Qin and P. N. Kaloni, "Nonlinear stability problem of a rotating porous layer," Quart. Appl. Maths., vol. 53, no. 1, pp. 129-142, 1995.

24. P. Vadasz, "Coriolis effect on gravity-driven convection in a rotating porous layer heated from below," J. Fluid Mech., vol. 376, pp. 351-375, 1998.

25. P. Vadasz, "Flow and thermal convection in rotating porous media," in Handbook of Porous Media, K. Vafai, Ed. Marcel Dekker, Inc, New York, pp. 395-440, 2000.

26. B. Straughan, "A sharp nonlinear stability threshold in rotating porous convection," Proc. Roy. Soc. Lond. A, vol. 457, pp. 87-93, 2001.

27. B. Straughan, "Global non-linear stability in porous convection with a thermal non-equilibrium model," Proc. Roy. Soc. Lond. A, vol. 462, pp. 409-418, 2006.

28. P. G. Siddheshwar and M. V. Krishna, "Effect of maximum density and internal heating on the stability of rotating fluid saturated porous layer using LTNE model," Heliyon, vol. 8, no. 6, e09721, 2022.

29. R. K. Rana, A. Kumar, and V. Sharma, "Thermal convection in a rotating porous layer with generalized heat conduction law and prescribed heat flux," Physics of Fluids, vol. 35, no. 3, 034102, 2023.

30. R. K. Rana and A. Sharma, "The Effect of Rotation on Double Diffusive Convection: Perspectives from Linear Stability Analysis," Journal of Physical Oceanography, vol. 54, no. 3, pp. 857-873, 2024.

31. D. A. Nield and A. Bejan, Convection in Porous Media, 3rd ed. Springer, Berlin, 2006.

32. P. Vadasz, Emerging Topics in Heat and Mass Transfer in Porous Media. Springer, Berlin, 2008.

33. M. S. Malashetty, M. S. Swamy, and R. Heera, "Double diffusive convection in a porous layer using a thermal non-equilibrium model," Int. J. Thermal Sci., vol. 47, pp. 1131-1147, 2008.

34. M. S. Malashetty, M. S. Swamy, and R. Heera, "The onset of convection in a binary viscoelastic fluid saturated porous layer," Z. Angew. Math. Mech., vol. 89,

    pp. 356-369, 2009.

35. A. Postelnicu, "The effect of a horizontal pressure gradient on the onset of a Darcy-Benard convection in a thermal non-equilibrium condition," Int. J. Heat Mass Transf., vol. 53, pp. 68-75, 2010.

36. A. A. Khan, S. Hussain, and M. Y. Malik, "Local thermal non-equilibrium (LTNE) model for developed flow in porous media with spatially-varying Biot number," International Journal of Heat and Mass Transfer, vol. 164, 120538, 2021.

37. S. Pati, A. Kumar, and S. Pati, "Local Thermal Non- Equilibrium Models in Porous Media: A Comparative Study of Conduction Effects," Journal of Porous Media, vol. 28, no. 5, 2025.

38. S. A. Bories, "Natural convection in porous media," in Advances in Transport Phenomena in Porous Media, J. Bear and M. Y. Corapcioglu, Eds. Martinus Nijhoff, The Netherlands, pp. 77-141, 1987.

39. D. A. Nield and A. V. Kuznetsov, "The interaction of thermal non-equilibrium and heterogeneous conductivity effects in forced convection in layered porous channels," Int. J. Heat Mass Transf., vol. 44, pp. 4369-4373, 2001.