A study of continuous double diffusive convection of binary liquid in an alloy form by employing Lapwood-Brinkman model with cross diffusion effect.

In this paper we study the effect of DuFour and Soret on linear double diffusive convection of binary liquid in a two dimensional rectangular channel filled with anisotropy porous material using Lapwood-brinkman model with cross diffusion effect. Darcy model without a time derivative is used in the momentum equation. Linear stability assessment is used to investigate the walls of channels that are heated and salted from below due to Soret and DuFour effects. The anisotropic parameters, the solute Rayleigh number, and the impacts of Soret and DuFour on the onset of convection are examined and same is plotted graphically.


Introduction:
Diffusive double convection is a fluid flow phenomena in which two different components of a fluid go through separate molecular diffusion processes, resulting in distinct density fluctuations.Temperature and concentration of a solute, such as salt, are often involved in the two components.There are numerous scientific and engineering uses for the study of double diffusive convection., including oceanography, astronomy, geophysics, and industrial processes requiring heat and mass movement.Researchers use mathematical models, numerical simulations, and experimental experiments to help them understand how fluids behave under certain circumstances.The importance of twofold diffusive convection in fluids with pores has lately increased because of its multiple scientific applications, particularly in saltwater hydrothermal areas.Instabilities can only occur when two diffusing abilities present in a system and one of them is causing instability.Whenever both heat and mass transfers occur simultaneously in a liquid that is moving, the relationship between the fluxes and driving abilities gets more complicated It was recently found that composition gradients, in addition to the temperature gradients, can also generate an energy flux.The diffusion-thermoeffect, commonly referred to as the DuFour effect, is a thermal flux caused by a composition gradient.Temperature gradients, on the other hand, can cause mass fluxes, a phenomenon referred as the Soret or diffusion of heat effect.Once cross-diffusion components are taken into account in transport equations, the circumstance changes substantially.Cross-diffusion effects allow every component of the gradient to have a significant influence on the motion of the others.Several research efforts have been undertaken to determine the impact of cross diffusions on the onset of twofold diffusive convection in media with pores.A stationary point in a porous media, where mass and heat pass through natural convection, with Soret and DuFour effects.was recently studied by Adrain Postelnicu, (2010) [1].Nonlinear as well as linear double diffusive convection in an anisotropic porous layer with the Soret effect was studied by Gaikwad et.al.(2009)[2].Soret and DuFour impacts on non-Darcy MHD free convective heat and mass transport on a vertical surface placed in a porous medium.was studied by Ahmed and Afify (2007) [3].The phenomenon of natural convection across an upward surface in a medium that is porous with variable surface temperature using Soret and  [10].The effect of Soret/DuFour and energy optimization on double diffusive convection in a novel form of wavy walled I shaped domain was studied by Tahar Tayebi et.al (2023) [11].With a binary Maxwell fluid with cross-diffusion effects, on double diffusive convection was studied by M.S. Malashetty et.al (2011) [12].Anisotropic permeable layer with twofold diffusion and an inside heat source was studied by Bhadauria (2012) [13].In a horizontal chamber impact of Soret and DuFour on twofold diffusive convection was studied by J Wang et.al (2014) [14] 1.1 Mathematical formulation: In a two-dimensional porous rectangular channel with height "d" and breadth "a,".we examine the free convection of a binary liquid with the channel wall being salted and heated from below.Between the borders, gradients of constant concentration (△S) and temperature (△T) are maintained.Both of the diffusion terms can be found in the thermal and concentrations formulae.while the momentum equation uses the Darcy model without a time derivative.Since the channel is a rectangle, we consider vertical direction as z-axis and horizontal direction as x-axis.Diagram 1 shows vertical wall from

Basic Governing Equations are;
Equation of Continuity; Momentam Equation; Energe Equation; Concentration Equation; Equation of State; Where, ρ and μ denote density and the coefficient of viscosity, H is the porosity, k, the thermal conductivity, 11 K the diffusivity of thermal, 12 K the solute component due to cross-diffusion, Where; ( Using the equation ( 12) in ( 6) -( 11) we get Stream function \ is introduced as the flow is two dimensional are Where, p represents pressure, I represents scaled temperatures of the fluid segment respectively.By using these non-dimensional variables from the equations ( 7) - (11) and eliminating the pressure and density, and deleting the asterisk, we get the following equations:

NUMERICAL SOLUTION AND LINEAR STABILITY ANALYSIS;
The linear version of equations ( 16) -(18) are Boundary conditions used as Equations ( 19) to (21) describe the onset of convection, and the solution for I \ , , and S can be expanded in the Fourier series Where, Ci , Di , Fi , Gi , Hi ,and Ii , are the functions x-only, and σ is the Marginal stability.
The boundary conditions are satisfied if Ci = Fi = Hi = 0 for all the value of x.When ψ, φ, and S are replaced by a single term and marginal stability is assumed (σ = 0), the equations ( 23 Using the equation (26) in equations ( 23) to (25), we get Where,  The boundary condition on G which can be obtained from equation ( 27) are 0 2 The solution of auxiliary equation ( 31) is .) ( Where, Where, We came across that a corresponding rise in Sr parameter leads to a decreasing in the value of C Ra .However, we find that strengthening small Sr numbers, strengthens the C Ra .This is due to the fact that, with minimal Sr numbers, the heavier element shift to the warmer zone, preventing the temperature-induced density gradient. DuFour effects was studied byHassan (2009) ][4].The consequence of twofold diffusion and cross diffusion on the Casson liquid with a Lorentz force in a porous medium was studied by Asogwa kk.et.al (2022) [5].Double diffusive convection in a rotating vertical porous cylinder with vertical flow through is affected by the cross diffusion effect.Was studied by noreldin O. et.al (2023) [6].Noureen et.al (2022) [7] studied the double-diffusive convection in flow of viscous fluid is investigated inside a horizontal channel.Under local thermal nonequilibrium the effect of cross diffusion on natural convection of binary liquid in a rectangular enclosures was studied by KM Lakshmi et.al (2022) [8].M.S Aghigi et.al (2022) [9] studied the Double-diffusive natural convection of Casson fluids in an enclosure.Channel constructed by electrically conducting and non-conducting walls, the effect of Soret and DuFour on MHD double-diffusive mixed convection was studied by B Shilpa et.al (2022)

Diagram 1 ;
Flow configuration of binary liquid.

21K
the temperature component due to cross-diffusion, 22 K the diffusivity of mass, T E the thermal expansion coefficient, S E the solute expansion co-efficient, c P effective Brinkman Viscosity, f P viscosity of the fluid.Since the Darcy Prandtl number is high, we ignore the inertia term and use the Boussinesq approximation to arrive at the basic governing equation in component-wise are.
State; Initially fluid is inactive, hence we have 0 u satisfy the boundary condition (22) on horizontal boundary.Boundary conditions in terms of G D, and I are the equations (28) to (30), we get six order differential equation in the form\

2 D
c i ' represents random constants and s m i ' represents roots of the auxiliary equation (31).Since the auxiliary equation involves cubic in equations are arising from equations (31) -(33) and we get a non-trivial answer for this system of equations are International Conference on Advanced Materials and Fluid Mechanics Journal of Physics: Conference Series 2748 side of above equation (34) is also consider as C Ra f with depending on [ , s Ra , W , Du and Sr. Hence equation (34) are written as 0 C Ra f Newton-Raphson method is used to calculate C Ra for various values of [ , W , Du and Sr .

1
diagrams that strengthening the values of Du parameter strengthens the C Ra , which indicate that increase in the value of Du makes the whole system is stable.Diagrams 6-8 indicate the correlation of C Ra with S Ra for various ranges of Sr parameter and for a particular value of another'

Diagram 9
indicates the variability of C Ra with S Ra for multiple values of the Sr parameter and the set values of the remaining parameter indicates that for small value of S Ra , even we increase in the value of W there is a stable mode in C Ra .While for large values of S Ra , if we increase in the magnitude of W , decreases the value of C Ra .Thus increases the value of W destabilizing the system.