Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 8s (2024) 365 https://internationalpubls.com Dufour Effects on Unstable MHD Viscoelastic Fluid Flow in Porous Vertical Media Current Pressure Gradient S. Kesavarani1, S. Lakshmi Priya2 1Department of Mathematics, SRM Institute of Science and Technology, Kattankulathur, Chengalpattu, Tamil Nadu 603203. G-mail: ks0756@srmist.edu.in 2Department of Mathematics, SRM Institute of Science and Technology, Kattankulathur, Chengalpattu, Tamil Nadu 603203. G-mail: lakshmis3@srmist.edu.in Article History: Received: 04-05-2024 Revised: 26-06-2024 Accepted: 06-07-2024 Abstract This specific investigation analyses Dufour effect in the context of free flow within vertical porous media and unstable MHD (Magnetohydrodynamics). We explore various effects, including Dufour, radiation, and radiation chemical reactions. We convert the flow equations into differential equations using specific parameters and employ the perturbation method with sliding boundary conditions to solve the problem. We investigate differential parameters for the temperature, velocity and concentration distributions, including Grashoff numbers of mass and temperature transfer, as well as the Schmidt number. Additionally, we demonstrate how to analyse the problem using MATLAB. Further research is needed to delve into the effects of the Soret effect. Keywords: MHD; invasion of the media; heat sources; antidote; occupy media; Dufour. 1. Introduction The research focuses on unstable materials in the presence of vertical perforated plates, considering various factors such as thermal explosions, chemical reactions, continuous absorption, and MHD effects. Changes in unstable materials M.L. Ramamohan Reddy vertical perforated plates [1]. In the light of thermal explosion, continuous absorption and chemical reactions, M.C. Raju [3] studied MHD free convection and diffusion boundary layered flow on a permeable vertical surface. Mohanty studied the MHD flow of the viscoelastic fluid through heterogeneous permeable media along with heat source and oscillating suction [4]. Deka [5] found thermal diffraction in a vertically oscillating plate and also turbulent MHD flow different from mass distribution. The unstable MHD double diffusion convective boundary layer passes vertically through the chemical material and dissipates heat. R. A. Mohamed studied radiators [6]. R. Kandasamy [7] investigated the effect of chemical, electricity generation and air movement, and temperature in the light of suction or injection. M. Umamaheswar [8] investigates MHD free convection viscoelastic fluid flow confined by infinitely fine curved perforated plate in a thermal source viscous dispersion, and Joule heating. Murali Gundagani described unstable magnetohydrodynamic free convection along a perforated vertical plate [9]. J. I. Oahimire [10] studied the magnetic flux at the stagnation point of starch with different thermal conductivities and heat sinks. S. Sreenadh [11] studied the effect of cutting and transferring electricity by placing Williamson fluid in an inclined column using peristalsis. B. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 8s (2024) 366 https://internationalpubls.com Mamatha examined the micro-polar fluid through partial absorption cracks in an unstable MHD mixed convective radiative boundary layer [12]. K.V.S. Raj found that free heat and air exchange in a vertical grid is very fast with Newtonian heat [13]. Hot substances such as gas sometimes ionize and are highly charged. Magnetic fields interact with the ionized plasma or liquid, causing changes in heat and friction. Since some liquids can do two jobs, it is interesting to see how the magnetic field is affected by temperature changes when the liquid is an electric current, current, and emitter. Since, the chemical industry increases the reaction rate, more care must be taken when controlling the heat and air changes in the chemical process. V. Rajesh [14] examined the effect of thermal source along with surface quality on the elastoviscous MHD flow in perforated plate. S.F. Ahmed studied the impact of unstable MHD natural convection on mass transfer in perforated plate that is vertical [15]. Mondal [16] studied the electrical and chemical properties of MHD free convection through plates of vertical porous material. Mangali Veerakrishna [17] studied thermal and mass transport of secondary fluid into a weak magnetohydrodynamic oscillating flow between two plates that are vertical with constant change in instruments/sinks and chemicals. Furhad Ali [18] studied heat, air transfer from a freely flowing MHD fluid on a vertical plate to a permeable medium. Misra [19] captured motion of a Biomagnetic viscoelastic fluid on an extended plate. K. Das [20] studied convection without MHD near a moving plate in the visual of thermal radiation. This study, based on this research. Instead of focusing on Newtonian fluids, we study unstable MHD viscoelastic fluid flow in a vertical medium along with pressure gradients. This study is unique in that it explores how the viscoelastic fluid relates to Dufour and the time-dependent fluctuation absorption of the medium in the presence of electrochemistry. 2. Design of the Problem The design problem revolves around the time-dependent oscillating absorption and permeability of infinite vertical permeable plate, immersed in endothermic viscoelastic fluid. This fluid is subject to electric reactions, electric suction along with a change in magnetic-field, while Y* axis remains normal for the X* axis, representing flow direction. The magnetic Reynolds number is considered minimum, and induced magnetic fields are disregarded. Buoyancy forces due to temperature imbalance between the wall and the middle section are the main cause of disturbances in the middle part. The study accounts for Dufour's influence and examines how the plate's temperature changes with time, from t*=0, when the plate and the fluid are at the identical temperature, to t*, when the plate's temperature reaches TW*, and the mode transitions to CW*. The permeability and absorption rate of porous media are represented by the following equations: K * (t *) = K * p (1 + eiw * t *) (1) v * (y *) = (1 + eaw*t*). Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 8s (2024) 367 https://internationalpubls.com Flow Geometry Applying the previously indicated assumption and normal Boussinesq’ s approximation, governing equations and boundary conditions are provided as follows: where t > 0 and 1 are said to be positive constants. πœ•π‘’βˆ— πœ•π‘‘βˆ— = πœ— πœ•2π‘¦βˆ— πœ•π‘¦βˆ—2 + 𝑔𝛽(π‘‡βˆ— βˆ’ π‘‡βˆž βˆ— ) + π‘”π›½βˆ—(π‘βˆ— βˆ’ π‘βˆž βˆ— ) βˆ’ πœŽπ›½0 2 π‘’βˆ— 𝜌 βˆ’ πœ— π‘’βˆ— π‘˜βˆ— βˆ’ π‘˜πœŠ 𝜌 ( πœ•3π‘’βˆ— πœ•π‘‘βˆ—πœ•π‘¦βˆ—2 + 𝑣2 πœ•3π‘’βˆ— πœ•π‘¦βˆ—2 ) (2) πœ•π‘‡βˆ— πœ•π‘‘βˆ— πœŒπ‘π‘ = π‘˜ πœ•2π‘‡βˆ— πœ•π‘¦βˆ—2 βˆ’ πœ•π‘žβˆ— βˆ— πœ•π‘¦2 βˆ’ π‘„βˆ—(π‘‡βˆ— βˆ’ π‘‡βˆž βˆ— ) + 𝑄1 βˆ—(π‘βˆ— βˆ’ π‘βˆž βˆ— ) (3) πœ•π‘βˆ— πœ•π‘‘βˆ— = 𝐷 πœ•2π‘βˆ— πœ•π‘¦βˆ—2 βˆ’ π‘˜π‘Ÿ(π‘βˆ— βˆ’ π‘βˆž βˆ— ) + π·π‘˜π‘‡ π‘‡π‘š πœ•2𝑇 πœ•π‘¦2 (4) 𝑒 = 0, π‘‡βˆ— = 𝑇𝑀 + Ξ΅(𝑇𝑀 βˆ’ π‘‡βˆž)π‘’π‘–π‘€βˆ—π‘‘βˆ—, πΆβˆ— βˆ’ 𝐢𝑀 + Ξ΅(𝐢𝑀 βˆ’ 𝐢∞)π‘’π‘–π‘€βˆ—π‘‘βˆ— at 𝑦 = 0 (5) 𝑒 β†’ 0, π‘‡βˆ— β†’ π‘‡βˆž, πΆβˆ— β†’ 𝐢∞ as 𝑦 β†’ ∞ Introduce non-dimensional equations 𝑦 = π‘£πœŠπ‘‘βˆ— πœ— , 𝑑 = π‘£π‘‘βˆ— 4πœ— , 𝑀 = 4π›Ύπ‘€βˆ— π‘£πœŠ 2 , 𝑒 = π‘’βˆ— π‘£πœŠ , 𝑇 = π‘‡βˆ—βˆ’π‘‡βˆž βˆ— π‘‡π‘€βˆ’π‘‡βˆž , C = Cβˆ—βˆ’C∞ βˆ— Cπ‘€βˆ’C∞ , 𝑠 = πœ—π‘ βˆ— π‘£πœŠ 2 , 𝐾𝑝 = π‘£πœŠ 2π‘˜π‘ 2 πœ—2 , π‘š2 = 𝜎 𝐡𝜊 2πœ— π‘£πœŠ 2𝜌 , π‘ƒπ‘Ÿ = πœ— π‘˜ , 𝑠𝑐 = πœ— 𝐷 , 𝑅𝑐 = π‘£πœŠ 2π‘˜πœŠ πœ—2𝜌 , 𝐺𝑐 = π‘£π‘”π›½βˆ—(𝑐𝑀 βˆ’ π‘βˆž) π‘£πœŠ 3 , πΊπ‘Ÿ = 𝑣𝑔𝛽(𝑇𝑀 βˆ’ π‘‡βˆž) π‘£πœŠ 3 , 𝐹 = 4𝐼1πœ— π‘£πœŠ 2πœŒπ‘π‘ , 𝑠 = π‘„πœ— π‘£πœŠ 2πœŒπ‘π‘ , π‘˜π‘ = π‘˜π‘Ÿπœ— π‘£πœŠ 2 𝑅 = 𝑄1πœ—(𝑐𝑀 βˆ— βˆ’π‘βˆž) π‘£πœŠ 2𝜌(𝑇𝑀 βˆ— βˆ’π‘‡βˆž) , π‘ π‘Ÿ = π·π‘˜π‘‡ π‘‡π‘š ( π‘‡π‘€βˆ’π‘‡βˆž π‘π‘€βˆ’π‘βˆž ) (6) Equation (3), (4), (5) reduces to a non-dimensional form 1 4 πœ•π‘’ πœ•π‘‘ = πœ•2𝑒 πœ•π‘¦2 + πΊπ‘Ÿπ‘‡ + 𝐺𝑐𝐢 βˆ’ (π‘š2 + 1 π‘˜π‘ ) 𝑒 βˆ’ 1 4 𝑅𝑐 { πœ•3𝑒 πœ•π‘‘πœ•π‘¦2 βˆ’ 4(1 + πœ€π‘’π‘–π‘€π‘‘) πœ•3𝑒 πœ•π‘¦3 } (7) 1 4 πœ•π‘‡ πœ•π‘‘ = 1 π‘ƒπ‘Ÿ πœ•2𝑇 πœ•π‘¦2 βˆ’ 𝐻𝑇 + 𝑅𝑐 (8) 1 4 πœ•π‘ πœ•π‘‘ = 1 𝑠𝑐 πœ•2𝑐 πœ•π‘¦2 βˆ’ π‘˜π‘ŸπΆ + π‘†π‘Ÿ πœ•2𝑇 πœ•π‘¦2 (9) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 8s (2024) 368 https://internationalpubls.com 𝑒 = 0, 𝑇 = 1 + Ρ𝑒𝑖𝑀𝑑, 𝑐 = 1 + Ρ𝑒𝑖𝑀𝑑at 𝑦 = 0 (10) 𝑒 β†’ 0, 𝑇 β†’ 0, C β†’ 0 as 𝑦 β†’ ∞ 3. Method of Solution Let us assume the concentration (C), velocity (u) and temperature (T) to be as follows, 𝑒(𝑦, 𝑑) = π‘’πœŠ(𝑦) + Ρ𝑒1(𝑦)𝑒𝑖𝑀𝑑 𝑇( 𝑦, 𝑑) = π‘‡πœŠ(𝑦) + Ρ𝑇1(𝑦)𝑒𝑖𝑀𝑑 C( 𝑦, 𝑑) = C𝜊(𝑦) + Ξ΅C1(𝑦)𝑒𝑖𝑀𝑑 } (11) Misra and Shit [19] and Das and Das [20] use the approach described above to solve a regularly varying flow problem. After substituting (11), we get the harmonic in non-harmonic terms of (7), (8), and (9) π‘…π‘π‘’πœŠ ‴ + π‘’πœŠ β€³ βˆ’ (π‘š2 + 1 π‘˜π‘ ) π‘’πœŠ = βˆ’πΊπ‘Ÿπ‘‡πœŠπΊπ‘C𝜊 (12) 𝑅𝑐𝑒1 ‴ + 𝑒1 β€³ (1 βˆ’ 𝑅𝑐𝑖𝑀 4 ) βˆ’ (π‘š2 + 1 π‘˜π‘ + 𝑖𝑀 4 ) 𝑒1 = βˆ’πΊπ‘C1 βˆ’ πΊπ‘Ÿπ‘‡1 βˆ’ π‘…π‘π‘’πœŠ ‴ (13) π‘‡πœŠ β€³ βˆ’ π‘ƒπ‘Ÿπ»π‘‡πœŠ = βˆ’π‘…π‘ƒπ‘ŸπΆπœŠ (14) 𝑇1 β€³ βˆ’ (𝐻 + 𝑖𝑀 4 ) π‘ƒπ‘Ÿπ‘‡1 = βˆ’π‘…π‘ƒπ‘Ÿπ‘1 (15) C𝜊 β€³ βˆ’ π‘˜π‘π‘ π‘π‘πœŠ = βˆ’π‘ π‘Ÿπ‘ π‘(1 + 𝑅𝑐)𝐡7𝜐1 2π‘’πœ1𝑦 (16) C1 β€²β€² βˆ’ (π‘˜π‘ + 𝑖𝑀 4 ) 𝑠𝑐𝑐1 = βˆ’π‘ π‘π‘ π‘Ÿ(1 + 𝑅𝑐) 𝐡8 𝜐2 2 π‘’πœ2𝑦 (17) u0 = u1 = 0, T0 = T1 = 1, c0 = c1 = 1 at, y = 0 u0 = u1 β†’ 0, T0 = T1 β†’ 1, c0 = c1 β†’ 1 at, y β†’ ∞ (18) Equations (12) and (13) are third order, but still, we get two boundary conditions. Therefore, our perturbation method for the 𝑅𝑐 (Elastic Parameter) is, 𝑒0 = 𝑒00(𝑦) + 𝑅𝑐𝑒01(𝑦) + 0(𝑅𝑐 2) 𝑒1 = 𝑒10(𝑦) + 𝑅𝑐 𝑒10 (𝑦) + 0(R𝑐 2) } (19) Substituting (19) in (12) and (13) and equating 𝑅𝑐 2 and 𝑅𝑐 Co-efficient We get other following ordinary differential equations Zeroth Order Equations 𝑒00 β€³ βˆ’ (π‘š2 + 1 π‘˜π‘ ) 𝑒00 = βˆ’πΊπ‘π‘0 βˆ’ πΊπ‘Ÿπ‘‡0 (20) 𝑒01 β€³ βˆ’ (π‘š2 + 1 π‘˜π‘ ) 𝑒01 = βˆ’π‘’00 ‴ (21) First Order Equations Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 8s (2024) 369 https://internationalpubls.com 𝑒10 β€³ βˆ’ (π‘š2 + 1 π‘˜π‘ + 𝑖𝑀 4 ) 𝑒10 = βˆ’πΊπ‘π‘1 βˆ’ πΊπ‘Ÿπ‘‡1 (22) 𝑒11 β€³ βˆ’ (π‘š2 + 1 π‘˜π‘ + 𝑖𝑀 4 ) 𝑒11 = βˆ’π‘’00 ‴ βˆ’ 𝑒10 ‴ + 𝑖𝑀 4 𝑒10 β€³ (23) Applying boundary conditions for equations (14) and (15) with the regular perturbation method, we get 𝑇0 = 𝑇00(𝑦) + 𝑅𝑐 𝑇01 (𝑦) 𝑇1 = 𝑇10(𝑦) + 𝑅𝑐 𝑇10 (𝑦) } (24) Substituting (24) in (14) and (15) and equating Rc and constants. We obtain the following ordinary differential equations Zeroth Order Equations 𝑇00 β€³ βˆ’ π‘ƒπ‘Ÿπ»π‘‡00 = βˆ’π‘…π‘ƒπ‘ŸπΆ0 (25) 𝑇01 β€³ βˆ’ π‘ƒπ‘Ÿπ»π‘‡01 = 0 (26) First order Equations 𝑇10 β€³ βˆ’ (𝐻 + 𝑖𝑀 4 ) π‘ƒπ‘Ÿπ‘‡10 = βˆ’π‘…π‘ƒπ‘ŸπΆ1 (27) 𝑇11 β€³ βˆ’ (𝐻 + 𝑖𝑀 4 ) π‘ƒπ‘Ÿπ‘‡11 = 0 (28) Applying boundary conditions for equations (16) and (17) with the regular perturbation method We get C0 = 𝐢00(𝑦) + 𝑅𝑐𝐢01(𝑦) C1 = C10(𝑦) + 𝑅𝑐C10(𝑦) } (29) Substituting (29) in (16) and (17) and equating Rc and constants Zeroth Order Equations C00 β€³ βˆ’ π‘˜π‘π‘ π‘C00 = βˆ’π‘ π‘Ÿπ‘ π‘π΅7β„“1 2𝑒ℓ1𝑦 (30) C01 β€³ βˆ’ π‘˜π‘π‘ π‘C01 = βˆ’π‘ π‘Ÿπ‘ π‘π΅7β„“1 2𝑒ℓ1𝑦 (31) First order Equations C10 β€³ βˆ’ (π‘ π‘π‘˜π‘ + 𝑠𝑐 𝑖𝑀 4 ) C10 = βˆ’π‘ π‘π‘ π‘Ÿπ΅8β„“2 2𝑒ℓ2𝑦 (32) C11 β€³ βˆ’ π‘ π‘π‘˜π‘π‘11(𝑦) βˆ’ 𝑠𝑐 𝑖𝑀 4 C11 = βˆ’π‘ π‘π‘ π‘Ÿπ΅8β„“2 2𝑒ℓ2𝑦 (33) The corresponding boundary conditions are, 𝑒00 = 𝑒01 β†’ 0, 𝑒10 = 𝑒11 β†’ 0, 𝑇00 = 𝑇01 β†’ 0, 𝑇10 = 𝑇11 β†’ 0, 𝐢00 = 𝐢01 β†’ 0, 𝐢10 = 𝐢11 β†’ 0 as 𝑦 = 0 and 𝑒00 = 𝑒01 β†’ 0, 𝑒10 = 𝑒11 β†’ 0, 𝑇00 = 𝑇01 β†’ 0, 𝑇10 = 𝑇11 β†’ 0, 𝑐00 = 𝑐01 β†’ 0, 𝑐10 = 𝑐11 β†’ 0 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 8s (2024) 370 https://internationalpubls.com as 𝑦 = ∞ (34) Solving the above differential equations with help of boundary conditions, we get 𝑒(𝑦, 𝑑) = 𝐡2(π‘’βˆ’βˆšπ‘Ž1𝑦 βˆ’ π‘’βˆšπ‘Ž1𝑦) βˆ’ (πΊπ‘Ÿπ‘‡0 + 𝐺𝑐𝑐0) 𝑦 π‘Ž1 + 𝑅𝑐[𝐡1(π‘’βˆ’βˆšπ‘Ž1𝑦 βˆ’ π‘’βˆšπ‘Ž1𝑦 βˆ’ π‘Ž1 2π‘’π‘Ž1𝑦𝑦)] + πœ€π‘’π‘–π‘€π‘‘ [𝐡3(π‘’βˆ’βˆšπ‘Ž2𝑦 βˆ’ π‘’βˆšπ‘Ž2𝑦) βˆ’ (πΊπ‘Ÿπ‘‡1 + 𝐺𝑐𝑐0) 𝑦 π‘Ž2 + 𝑅𝑐 (𝐡4(π‘’βˆ’βˆšπ‘Ž2𝑦 βˆ’ π‘’βˆšπ‘Ž2𝑦) βˆ’ 𝐡1π‘Ž1 2π‘’π‘Ž1𝑦𝑦 βˆ’ 𝐡2π‘Ž2 2π‘’π‘Ž2𝑦𝑦 + 𝑖𝑀 4 𝐡2π‘Ž2π‘’π‘Ž2𝑦𝑦)] (35) 𝑇( 𝑦, 𝑑) = 𝐡5 (π‘’βˆ’βˆšπ‘1𝑦 βˆ’ π‘’βˆšπ‘1𝑦) βˆ’ π‘…π‘ƒπ‘Ÿ 𝛾1 𝑐0𝑦 + πœ€π‘’π‘–π‘€π‘‘ (𝐡8 (π‘’βˆ’βˆšπ‘1𝑦 βˆ’ π‘’βˆšπ‘1𝑦)) (36) C(𝑦, 𝑑) = 𝐡11(π‘’π‘˜π‘π‘ π‘π‘¦ βˆ’ 1) βˆ’ π‘ π‘Ÿπ‘ π‘π΅7β„“1𝑒ℓ1𝑦𝑦 + πœ€π‘’π‘–π‘€π‘‘ [𝐡10 (𝑒(π‘˜π‘+ 𝑖𝑀 4 )𝑠𝑐𝑦 βˆ’ 1) βˆ’ π‘ π‘Ÿπ‘ π‘π΅8𝛾2 2𝑒𝛾2𝑦𝑦] (37) 4. Result and Discussion The work provides a series of findings related to various parameters and their impact on the velocity, temperature, concentration, and other factors in the studied system. Here's an enhanced explanation of each set of findings. Figure 1 Magnetic Parameters: Several magnetic parameters, namely Grashof number and modified Grashof number, were examined. The results of these investigations are presented in multiple tables. Magnetic Field Influence on Velocity: The figure above illustrates how the velocity profile changes concerning different magnetic field strengths, represented by M values ranging from 18 to 21. The velocity distribution displays variations based on the magnetic load (M). As M increases, the velocity decreases. This phenomenon occurs because the Lorentz force acts as a hindrance when a magnetic field is introduced into the system. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 8s (2024) 371 https://internationalpubls.com Figure 2 Figure 3 Buoyancy and Velocity Distribution: The two graphs above depict the differences in velocity distribution associated with varying Grashof numbers (Gr) and modified Grashof numbers (Gc). Due to significant buoyancy forces, the water velocity increases, and evidently speed increases with rising values of Gr and Gc. Figure 4 Figure 5 Viscoelastic Parameter Effect: The figure above illustrates the relationship between viscoelastic parameters (Rc), with values ranging from 0.10 to 0.25. It has been shown that speed decreases when the viscoelastic parameter (Rc) rises. Oscillation Frequency Impact on Velocity: The figure above showcases the variation in velocity distribution concerning different oscillation frequencies (W) with values of 0.1, 1, 2, and 3. The effect of oscillation frequency (W) on velocity becomes more pronounced as W increases. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 8s (2024) 372 https://internationalpubls.com Figure 6 Figure 7 Prandtl Number Influence: The relationship between velocity profile and the Prandtl number (Pr) is displayed above, with Pr values ranging from 0.5 to 2. A rapid increase in Prandtl number is associated when the fluid's velocity drops. This is attributed to the reduction in thermal conductivity as the Prandtl number rises. Chemical Concentration Distribution: The diagram above depicts the distribution of chemical concentrations (Kc) with values of 3, 4, 5, and 6. Remarkably, there appears to be no significant effect on speed. However, the velocity of the particular flow does decrease with the increasing chemical resistance (Kc) for the given problem. Figure 8 Figure 9 Electrical Absorption Effect: The diagram above demonstrates the variation in temperature due to changes in the electrical absorption coefficient (R) with values of 0, 0.2, 0.6, and 1. There is a raise in the electrical absorption coefficient leads to a drop in temperature in the temperature profile. Prandtl Number and Temperature: The figure above presents the temperature at different Prandtl numbers (Pr) ranging from 1 to 4. As the Prandtl number raises, the influence on temperature drops for the respective flow pattern. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 8s (2024) 373 https://internationalpubls.com 5. Conclusion We investigated the dynamics of a viscoelastic fluid flow along a vertical porous plate that is unstable in Magnetohydrodynamics (MHD). Many contributing elements, such as a pressure gradient, a magnetic field, a permeable material with time-dependent oscillatory permeability and suction, and the Dufour effect, were taken into account when conducting this research. These are the main conclusions that need to be discussed: (i) Enhancement of Concentration Profile by Dufour Effect: One significant observation from our study is the impact of the Dufour effect. It is noteworthy that this effect has a somewhat enhancing influence on the concentration profile of the fluid. The Dufour effect is a heat-mass transfer phenomenon, and its presence in this system plays a role in shaping the concentration distribution of the fluid. This finding underscores the importance of considering heat- mass transfer effects in similar systems. (ii) Magnetic Field Slows Fluid Flow: The effect of a magnetic field on the flow dynamics was observed to be significant. It was noted that the magnetic field hinders the fluid, flow. The decrease in speed of the fluid is caused by the interaction between magnetic field and fluid's ability to conduct electricity. Gaining a comprehensive understanding of this impact is essential for optimizing procedures that need precise control or adjustment of fluid flow. (iii) Reduction of Flow at Boundary Layer Due to Species Conductivity: Another important finding pertains to the flow characteristics at boundary layer of the fluid. The study revealed that the flow at the boundary layer is reduced. This reduction can be attributed to the presence of heavier species in the fluid, which generally have lower conductivities. The relationship between species properties, conductivities, and flow behaviour at the boundary layer is a crucial insight, particularly in applications where species differentiation plays a vital role. To summarize, our study of the flow of unstable magnetohydrodynamic viscoelastic fluid in a complicated system containing3 a vertical permeable plate, a pressure gradient, a magnetic field, a permeable medium with time-dependent permeability, and the Dufour effect has resulted in these significant discoveries. These findings enhance our comprehension of the intricate interactions between different elements in these systems, and have practical implications for domains including fluid dynamics, heat-mass transfer, and materials processing. References: [1] L. Ramamohan Reddy, M.C. Raju and G.S.S Raju (2016), β€œUnsteady MHD Free convection flow characteristics of a viscoelastic fluid past a vertical porous plate” International Journal of Applied Science and Engineering. 14,2:69-85. [2] F.S. Ibrahim, A.M. Elaiw, A.A. Bakr. Effect of the chemical reaction and radiation absorption on the unsteady free convection flow past a semiinfinite vertical permeable moving plate with heat source and suction. Communication in nonlinear science and numerical simulation. 13 (2008)1056-1066. www.elsivier.com/locate/cnsns [3] M.C. Raju, N. Ananda Reddy, S.V.K. Varma (2014), β€œAnalytical study of MHD free convective, dissipative boundary layer flow past a permeable vertical surface in the presence of thermal radiation, chemical reaction and constant suction,” Am Shams Engineering Journal 5, 1361-1369. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 8s (2024) 374 https://internationalpubls.com [4] B. Mohanty, S.R. Mishra and H.B. Pattnaik (2014), β€œMHD Flow of a Visco-elastic Fluid through Non- homogeneous Porous Medium with Oscillatory Suction and Heat Source/sink,” International Journal of Mathematical Education. ISSN 0973-6948, Volume 4, Number 1, pp. 21-34. [5] R.K. Deka and B.C. Neog (2009), β€œUnsteady MHD Flow Past a Vertical Oscillating Plate with Thermal Radiation and Variable Mass Diffusion,” Chamchuri Journal of Mathematics volume 1 number 2, 79-92, http://www.math.sc.chula.ac.th/cjm [6] R.A. Mohamed Abdel-Nasser A. Osman S.M. Abo-Dahab (2013), β€œUnsteady MHD double-diffusive convection boundary-layer flow past a radiate hot vertical surface in porous media in the presence of chemical reaction and heat sink,” Meccanica 48:931-942 DOI 10.1007/s11012-012-9644-0. [7] R. Kandasamya, K. Periasamy b, K.K. Sivagnana Prabhu (2005), β€œEffects of chemical reaction, heat and mass transfer along a wedge with heat source and concentration in the presence of suction or injection,” International Journal of Heat and Mass Transfer 48, 1388-1394. [8] Murali Gundagani, Sivaiah Sheri, Ajit Paul, and M.C.K. Reddy (2013), β€œUnsteady Magnetohydrodynamic Free Convective Flow Past a Vertical Porous Plate,” International Journal of Applied Science and Engineering. 11, 3: 267-275. [9] M. Umamaheswar, S.V.K. Varma and M.C. Raju (2013), β€œUnsteady MHD Free Convective Visco-Elastic Fluid Flow Bounded by an Infinite Inclined Porous Plate in the Presence of Heat Source, Viscous Dissipation and Ohmic Heating,” International Journal of Advanced Science and Technology Vol.61. pp.39-52 http://dx.doi.org/10.14257/ljast.2013.61.05. [10] S. Sreenadh, P. Govardhan, and Y.V.K. Ravi Kumar (2014), β€œEffects of Slip and Heat Transfer on the Peristaltic Pumping of a Williamson Fluid in an Inclined Channel,” International Journal of Applied Science and Engineering. 12, 2: 143-155. [11] J. I. Oahimire and B.I. Olajuwon (2013), β€œHydromagnetic Flow Near a Stagnation Point on a Stretching Sheet with Variable Thermal Conductivity and Heat Source/Sink,” International Journal of Applied Science and Engineering. 11, 3: 331-341. [12] B. Mamatha, S.V.K. Varma, and M.C. Raju (2015), β€œUnsteady MHD Mixed Convection, Radiative Boundary Layer Flow of a Micro Polar Fluid Past a Semi-infinite Vertical Porous Plate with Suction,” International Journal of Applied Science and Engineering, 13, 2: 133-146. [13] K.V.S. Raju, T. Sudhakar Reddy, M.C. Raju & S. Venkata Ramana (2013), β€œFree convective heat and mass transfer transient flow past an exponentially accelerated vertical plate with Newtonian heating in the presence of radiation,” International Journal of Mathematics and Computer Applications Research (IJMCAR) ISSN 2249-6955 Vol.3, Issue 2, 215226o. [14] V. Rajesh (2011), β€œHeat source and mass transfer effects on MHD flow of an elastic-viscous fluid through a porous medium,” Annals of engineering Hunedoara International journal of engineering volume IX. number 2. (ISSN 1584-2665). [15] S.F. Ahmmed, M.K. Das, L.E. Ali (2015), β€œAnalytical Study on Unsteady MHD Free Convection and Mass Transfer Flow Past a Vertical Porous Plate,” Published online March 28,2015(http://www.sciencepublishinggroup.com/j/ajam)doi:0.11648/j.ajam.20150302.16 ISSN: 2330-0043 (Print); ISSN: 2330-006X (Online) American Journal of Applied Mathematics; 3(2): 64-74. [16] Mangali Veera Krishna, Kamboji Jyothi, & Ali J. Chamkha (2018), β€œHeat and mass transfer on the unsteady, magnetohydrodynamic, oscillatory flow of second-grade fluid through a porous medium between two vertical plates, under the influence of fluctuating heat source/sink, and chemical reaction,” International Journal of Fluid Mechanics Research, 45(5):459-477. [17] S. Mondal, S. Parvin, S.F. Ahmmed, β€œEffects of Radiation and Chemical Reaction on MHD Free Convection Flow past a Vertical Plate in the Porous Medium,” American Journal of Engineering Research (AJER) e-ISSN: 2320- 0847 p-ISSN: 2320-0936 Volume-03, Issue-12, pp-15-22 www.ajer.org. [18] Farhad Ali, Ilyas Khan, Sharidan Shafie, and Norzieha Musthapa (2013), β€œHeat and Mass Transfer with Free Convection MHD Flow Past a Vertical Plate Embedded in a Porous Medium,” Hindawi Publishing Corporation Mathematical Problems in Engineering Volume, Article ID 346281,13pageshttp://dx.doi.org/10.1155/2013/346281. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 8s (2024) 375 https://internationalpubls.com [19] Misra. J.C. and Shit. G.C. 2009. Biomagnetic viscoelastic fluid flow over a stretching sheet. Applied Mathematics and Computation, 210, 2: 350-361. [20] Das. K and Das. R. 2009. MHD free convection flow near a moving vertical plate in the presence of thermal radiation: an analytical solution. Moldavian Journal of the physical sciences, 8, 3-4: 358-365.