Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 2s (2025) 284 https://internationalpubls.com An Analysis of Magnetohydrodynamic Free Convective Flow past an Infinite Vertical Porous Plate *Dr. P. Satish1, Dr. D. Rajani2, Dr. Mohd Ahmed3, Dr. Somarouthu V G V A Prasad4, Dr. G. Manjula5, Dr. P. Bharath Kumar6, Dr. S. Asif Alisha7, 1Associate Professor, Department of Mathematics, Aditya Engineering College, Surampalem, Andhra Pradesh, India, Pincode: 533437.Email Id- satishmaths.7@gmail.com 2Assistant Professor, Department of Mathematics, V R Siddhartha Engineering College, Vijayawada, Andhra Pradesh, India, Pincode: 520007.Email id: rmuppavarapu2009@gmail.com 3Assistant Professor, Department of Freshman Engineering, Sphoorthy Engineering College, Hyderabad, Telangana, India, Pincode: 501510. Email id: ahmed.nitw@sphoorthyengg.ac.in 4Professor, Department of Physics and Electronics, Pithapur Rajah's Government College (A), Kakinada, Andhra Pradesh, India, Pincode: 533001. Email: somarouthu13@yahoo.co.in 5Professor, Department of Physics,SV College of Engineering (Autonomous), Karakambadi, Tirupati, Pincode: 517507. Email id: manjulasvu2@gmail.com 6Assistant Professor, Department of Mathematics, Madanapalle Institute of Technology & Science, Madanapalle, Annamayya (Dist.), Andhra Pradesh, India, Pincode: 517325. Email id: bharathkmr094@gmail.com 7Associate Professor, Department of Mathematics, School of Liberal Arts and Sciences, Mohan Babu University, (Erstwhile Sree Vidyanikethan Engineering College), Sree Sainath Nagar, Rangampeta, Tirupati, Andhra Pradesh, India, Pincode: 517102. Email id: asif.alisha@gmail.com Article History: Received: 13-09-2024 Revised: 21-10-2024 Accepted: 01-11-2024 Abstract: The reason for this study is to find an answer for the issue of magnetohydrodynamic laminar shaky progression of an incompressible electrically leading liquid that is going beyond a limitless vertical permeable plate. This work is being finished with the goal of tracking down an answer for the issue. By looking at the effect that joule warming has on the speed and temperature profiles of a liquid stream that is coupled to a cross over attractive field, the reason for this study is to find out the impact that joule warming has on the progression of the liquid. Certain conditions of MHD stream are explored throughout the exploration. These conditions are demonstrated to be significant. These conditions are addressed mathematically to arrive at the vital outcomes. This is achieved through the use of limited distinction approximations and a PC program. Despite the way that an expansion in the joules warming boundary causes an expansion in the speed and temperature profiles that are consistently positioned close to the plate, these profiles keep on being disseminated away from the plate in a way that is steady. It is feasible to reach the determination from this that the power of the Joules warming that happens near the plate wall and at the standard affects the stream field of the MHD free convective stream. The finishes of this request, which depended on mathematical information, make plainly this peculiarity most certainly happens. Keywords: Magnetohydrodynamic, Incompressible Flow; Vertical Porous Plate, Free Convective Flow past. mailto:satishmaths.7@gmail.com mailto:rmuppavarapu2009@gmail.com mailto:ahmed.nitw@sphoorthyengg.ac.in mailto:somarouthu13@yahoo.co.in mailto:manjulasvu2@gmail.com mailto:bharathkmr094@gmail.com mailto:asif.alisha@gmail.com Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 2s (2025) 285 https://internationalpubls.com 1. Introduction The field of study for electrically leading liquid elements is called molecular hydrodynamics (MHD). These liquids incorporate fluid metals, plasma, and seawater. The terms magneto, and that implies attractive field, hydro, and that implies fluid, and elements, and that implies mechanical properties of liquid, consolidate to frame the name MHD, or magneto hydrodynamics. The investigation of liquid stream and the powers causing it without an electromagnetic field is known as hydrodynamics. Any material that misshapes because of an applied outside force is viewed as a liquid. At the point when a liquid guide streams in an attractive field, MHD prompts a current. Along these lines, electromagnetic powers follow up on the liquid particles while a leading liquid moves within the sight of a cross over attractive field, changing the smooth movement's calculation. Body powers that follow up on liquid particles a good ways off are remembered for the energy condition that portrays smooth movement in MHD. The powerful impact is created by the applied attractive fields following up on both ionized and electronic molecules. This mass movement makes the electromagnetic field change subsequently. In a liquid, heat is moved by radiation, convection, and conduction. Deciding if the limit layer in a convection issue is violent or laminar is a pivotal step. Which of the circumstances is valid essentially affects surface erosion and convection move rates. Laminar liquid is remembered to move by sliding dainty layers of overlay corresponding to each other. Most of free convective stream examinations utilize mathematical techniques, which have generally huge adaptability in calculation and limit conditions overseeing movement, or exploratory strategies to get the intensity move, temperature dispersions, speed profiles, and disturbance forces. The framework's mechanical movement can consequently be made sense of as far as a solitary directing liquid that has thickness, speed, and strain hydro attractive factors. It is normal practice to ignore the removal current in Amperes regulation at low frequencies. The magneto hydrodynamics guess is then this. At the point when a liquid is upset for quite a few reasons — including unpredictable limit edges, permeable plates, and changes in the actual qualities of smooth movement — fierce stream results. Joule warming is the term used to depict the intensity created when an ongoing goes through a guide. This warming can cause warm strains, which will make the gadget twist since its upper area is smaller than its base part. The mix of the Navier-Stirs up conditions of liquid elements and Maxwell's conditions of electromagnetism, which are differential conditions and might be tackled all the while scientifically or mathematically, makes up the arrangement of conditions that depict MHD. • Mass Transfer A basic process that deals with the movement of materials between a moving fluid and a boundary surface is called convection mass transfer. Mass is transferred from one location to another as a result of the combined actions of concentration gradients and fluid motion. Convection mass transfer allows for the exchange of mass between the fluid and the surface by moving the material either towards or away from the boundary surface. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 2s (2025) 286 https://internationalpubls.com Figure 1: Mass Transfer In many industrial operations, mass transportation is always crucial, such as the removal of contaminants from plant effluent. • Heat Transfer The energy exchange caused by temperature changes between a boundary surface and an adjacent fluid is known as convection heat transfer. Figure 2: Heat Transfer • Free Convection Free convections are characterized by the mobility of fluids that are caused by density gradients that are produced by temperature or concentration gradients that are present in the fluid body. Figure 3: Free Convection Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 2s (2025) 287 https://internationalpubls.com Faraday directed the main concentrate on magnetohydrodynamics by analyzing the way of behaving of current in circuits exposed to fluctuating attractive fields over the long haul. He saw that a voltage was made in a direction opposite to the bearing of the stream and the attractive field in his trial where the leading liquid, mercury, was streaming in a glass tube in an attractive field. A power is applied to a leading liquid toward a path that is opposite to both the attractive and electric fields when an electric field is given like that. In his concentrate on the essentials of MHD, Calvert L. exhibited how an attractive field applies a power opposite to the attractive field on a directing liquid moving, having a tendency to make ordinary speed equivalent to the Eb floats. He guaranteed that the smooth movements along the attractive field's bearing is unaffected and that the higher the conductivity, the more grounded this power and the nearer the liquid is drawn by the attractive field (as well as the other way around). The effect of corridor current on MHD stream and intensity move along a permeable level plate with mass exchange was explored by Ramulu et al. He utilized mathematical strategies to track down the response. In an optically slim environment with time- subordinate attractions, Cooker et al. concentrated on the impacts of radiation and gooey dissemination on the issue of precarious magneto hydrodynamics free-flows streaming beyond a boundlessly warmed vertical plate. The discoveries show that while expansions in radiation, attractive field, and Darcy's boundaries are connected to a diminishing in speed, expanding cooling of the plate and Eckert individuals causes an ascent in the speed profile. Jordan J. inspected the impacts of MHD free convection stream over a semi-endless vertical permeable plate, heat radiation, and gooey dissemination. The limit layer conditions in view of the limited distinction plans are tackled utilizing the organization recreation approach. It was found that rising the thick scattering causes the temperature and speed profiles to increment, expanding the attractive boundary causes the temperature profiles to increment and the speed profiles to diminish, and expanding the pull boundary causes the neighborhood skin erosion and Nusselt number to increment. In the circumstance of a power-regulation variance in the wall temperature, Emad et al. explored the impacts of thick dissemination and joule warming on MHD free convection stream by means of a semi-limitless vertical level plate within the sight of consolidated Corridor and Particle slip ebbs and flows. They found that while the attractive field speeds up the created parallel stream, it impedes the digressive stream. At the point when the impacts of gooey dispersal, joule warming, and intensity age are considered, the skin-grinding factor for the extraneous stream and the Nusselt number drops, however the skin-rubbing factor for the unrelated and horizontal streams increments and the nearby Nusselt number abatements. Since Corridor and Particle slip terms have no perceptible effect at little and moderate attractive field values, they were ignored while utilizing Ohm's regulation. Gr 0 within the sight of areas of strength for a field calculated at a point to the plate, Kinyanjui et al. concentrated on unsteady free convection in compressible liquid across a semi-limitless vertical permeable plate with impacts from particle slip flows and Corridor. On the convectively cooled or convectively warmed plate restricted to Laminar limit layer, the effects of changed Grashof number, pull speed, the point of tendency, time, Corridor current, particle slip current, Eckert number, Schmidt number, and intensity source boundary were explored. He found that rising the temperature profile results from an expansion in the Eckert number Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 2s (2025) 288 https://internationalpubls.com Ec, expanding the mass dissemination boundary Sc causes an expansion in the focus profile, and expanding the point of tendency builds the essential speed profiles yet diminishes the optional profiles. In his examination of the constrained convection stream of ionized gases close to isothermal permeable surfaces, Duwairi G. inspected the impacts of thick and joule warming. He utilized mathematical examination to look at his situations and found that the thick scattering impact lessens the pace of intensity move in both pull and infusion situations. Consequently, we concentrate on the MHD free convective progression of an incompressible directing liquid within the sight of a uniform cross over attractive field past a boundless vertical permeable plate that is warmed to joules. 2. LITERATURE REVIEW Abo-Eldahab and El Aziz (2005) In this review, an examination was done to decide the effect that thick dissemination and Joule warming have on MHD free convection from an upward plate that encounters power-regulation variances in surface temperature. Their discoveries shed light on the impact that temperature boundaries have on the stream elements, considering the presence of Lobby and particle slip ebbs and flows. Their research was conducted to investigate this influence. As a result of the research, our comprehension of the mechanisms that are responsible for heat transmission in MHD flows has been enhanced since it takes into consideration the complexity of the situation. This is especially true in circumstances in which the temperatures of the surface are not uniform. Ahmed and Sarmah (2009) Inside the extent of this examination, we explored a transient radiative MHD free and constrained convection stream with mass exchange that navigated a limitless vertical plate that was set up for eternity. The plate was established in its position. The transient behaviour of MHD flows that are subjected to radiative heat transfer, mass transfer, and fluid motion that is generated by external forces is the core focus of their research. This is because these flows are subjected to these conditions. An important addition to the characterization of time-dependent MHD convective flows is made by the research through the investigation of these transitory occurrences. On the other hand, this provides crucial insights into the dynamics of fluids that are unstable. Ahmed and Sengupta (2011) We explored the effects of Thermo-Dispersion and Dissemination Thermo on a three-layered MHD convection stream that was communicated through an endless vertical permeable plate while heat radiation was available. This research endeavours to understand the complex relationship that exists between thermal radiation, diffusion-thermo effects, and thermal diffusion in the context of MHD flows that occur through porous media. Consequently, the findings contribute to the optimisation of processes that include MHD convection in porous settings, since they provide a thorough understanding of heat and mass transfer phenomena in porous media under the effect of thermal radiation. Ahmed, Sarma, and Deka (2012) Inside the setting of this examination, the emphasis was on the MHD convective mass exchange stream that happened beyond an endless vertical permeable plate while a consistent intensity and synthetic response was occurring within the sight of an intensity source. This research explores the combined effects of mass transfer, heat transfer, and chemical reactions on MHD flows across porous media. The focus of their investigation is on MHD flows that occur across porous medium. The research makes a contribution to a better understanding of the complex transport processes that take place in porous settings by taking into consideration the Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 2s (2025) 289 https://internationalpubls.com interactions that take place across several dimensions. This has repercussions for a wide range of technical applications, including chemical reactors and heat exchangers, among others. Ashgar, Mohyuddin, (2003) An investigation into the flow of a non-Newtonian fluid that was brought about by the vibrations of a porous plate was carried out as part of academic study. They are conducting research that looks into the behaviour of fluids that have viscosity characteristics that are not linear. During the course of their work, they were able to get valuable insights into the flow characteristics that are brought about by the oscillations of a porous medium. By conducting an analysis of the dynamics of non-Newtonian fluids, the research results in a contribution to the existing body of information regarding the behaviour of fluids in porous environments. The consequences of this insight can be seen in a wide range of geological applications and industrial activities. Calvert and Santos (2002) doing explore on the essential standards of MHD determined to decide the fundamental systems that direct the connection between attractive fields and smooth movement as the essential focal point of the examination. This research is being conducted with the intention of shedding light on the fundamental principles of magnetohydrodynamics, thereby laying the groundwork for further breakthroughs in the theory and applications of magnetohydrodynamics (MHD). Through this research, a theoretical framework has been developed for understanding and forecasting the effects of magnetic fields on fluid flows. This framework has applications in a wide range of domains, including engineering and astronomy. To achieve this goal, it is necessary to provide an explanation of the fundamental principles that underlie magnetically influenced fluid dynamics (MHD). Chaudhary and Jain (2008) explored the progression of magnetohydrodynamic transient convection past an upward surface that was lowered in a permeable material with a temperature that was continually evolving. Their investigation is centred on the observation of transient convection events that take place in porous media that are subjected to varying temperature conditions. The influence that magnetic fields have on the behaviour of fluid flow is also taken into consideration by these researchers. Through the examination of transient convection flow over vertical surfaces in porous media, this study makes a contribution to the understanding of heat and mass transfer mechanisms that occur in porous environments. The extraction of geothermal energy and the treatment of groundwater are both affected by this understanding if it is taken into consideration. Cooker and Mubo-Pepple (2003) The progression of precarious MHD free-convection through a permeable medium with time-autonomous pull was investigated to decide the effect of radiation and thick scattering on the stream as it went through an endlessly warmed vertical plate. Within the framework of unstable free-convection flows, the study that they conduct studies the complex interactions that take place between radiation, magnetic fields, and viscous dissipation. The research makes a contribution to a better understanding of the mechanisms that govern heat transmission in porous media when magnetic fields are present. This is accomplished by taking into account the aforementioned factors. The ramifications of this are significant for applications in thermal management systems and the extraction of geothermal energy. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 2s (2025) 290 https://internationalpubls.com 3. MATERIALS AND METHODS A. Specific Equations Governing Fluid Flow The free convection magneto hydrodynamic liquid move through an endless vertical permeable plate that is exposed to a uniform attractive field and a consistent pull speed is thought about. B. Momentum Equation Newton's second rule of movement, which expresses that the all-out body power and surface powers following up on a framework are equivalent to the time pace of progress of the framework's force, is the wellspring of the situation of energy. This condition is gotten from Newton's second rule of movement. Due to this law of movement, the connection between the two powers was appropriately settled. This assertion fills in as the establishment for the situation that makes sense of energy beginning starting here. At the point when vector documentation is used, the condition of movement that thinks about just the body force that is achieved by gravity and electromagnetic power might be addressed as; 𝜕𝑞 𝜕𝑡 + (𝑞. ∇)𝑞 = − 1 𝑝 ∇𝑝 + 𝜗∇2𝑞 + 𝐹 (1) In the shape of components and in the x-direction Figure 4: Configuration of the flow ( 𝜕𝑢 𝜕𝑡 + 𝜕𝑢 𝜕𝑥 + 𝑣 𝜕𝑢 𝜕𝑦 ) = − 𝜕𝑝 𝜕𝑥 + 𝜇 ( 𝜕2𝑢 𝜕𝑥2 + 𝜕2𝑢 𝜕𝑦2 ) + 𝑝𝑔 + 𝐽 × 𝛽 (2) Whare for the explanation that to compute the volumetric thickness of the outside force, we are considering both the gravitational power, meant by g, and the electromagnetic power. A determination of the pressure gradient is made by evaluating the momentum equation near the boundary layer's edge, where the pressure gradient is found. The phrase for pressure in the x direction. C. Method of Solution In our investigation, the equations that govern fluid flow are nonlinear; hence, it is challenging to acquire the exact solutions to these equations. For the purpose of solving these differential equations, Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 2s (2025) 291 https://internationalpubls.com we therefore utilised an approach that was both quick and reliable. A number of fundamental requirements, including consistency, stability, and convergence, were satisfied by the finite difference approach that was utilised in the process of solving these differential equations. A strategy is supposed to be joined assuming the mathematical arrangement moves toward the exact response as the quantity of network focuses that are thought about increments or the step size diminishes. In the event that the impact of any single fixed adjust mistake is limited, the technique being referred to is viewed as steady. All in all, a strategy is thought of as reliable on the off chance that the truncation mistake increments until it approaches zero as the step size diminishes. The presence of the mathematical blunder is because of the way that in most of calculations, we can't exactly figure the distinction arrangement because of the presence of adjust mistakes. Truly, there are circumstances in which the real response could be to some degree unique in relation to the distinction arrangement. In the occasion on the off chance that the impact of the adjust blunder keeps on being limited as the cross section directs pattern toward limitlessness with fixed sizes, then the distinction procedure is supposed to be steady. In this study we use addendums to demonstrate spatial focuses and superscript to show time Expect that the variable addressing the cross section point at time n is addressed by. The forward distinction for the main request subsidiaries as for time t is given by: ∅(𝑗,𝑖) 𝑓𝑛 = ∅(𝑗,𝑖) 𝑛+1−∅(𝑗,𝑖) 𝑛 ∇𝑡 + 𝐻𝑜𝑡 (3) 𝑢(𝑗,𝑖) 𝑛+1−𝑢(𝑗,𝑖) 𝑛 ∇𝑡 + 𝑢(𝑗,𝑖) 𝑛 ( 𝑢(𝑗,𝑛+1) 𝑛 −𝑢(𝑗,𝑖−1) 𝑛 2∆x ) + 𝑉(𝑗,𝑖) 𝑛 ( 𝑢(𝑗+1,𝑖) 𝑛 −𝑢(𝑗−1,𝑖) 𝑛 2∆𝑦 ) = ( 𝑢(𝑗,𝑖+1) 𝑛 −2𝑢(𝑗,𝑖) 𝑛 +𝑢(𝑗,𝑖−1) 𝑛 (∆𝑥)2 ) + ( 𝑢(𝑗+1,𝑖) 𝑛 −2𝑢(𝑗,𝑖) 𝑛 +𝑢(𝑗−1,𝑖) 𝑛 (∆𝑦)2 ) + 𝐺𝑟𝜃(𝑗,𝑖) + 𝑀2𝑢(𝑗,𝑖) 𝑛 (4) This is the form that the initial conditions adopt: 𝐴𝑡 𝑦 = 0, 𝑢(0,𝑖) 0 = 1, 𝑣(0,𝑖) 0 = 0, 𝜃(0,𝑖) 0 = 1, 𝐵(0,𝑖) 0 = 0 (5) 𝐴𝑡 𝑦 = 0, 𝑢(𝑗,𝑖) 0 = 1, 𝑣(𝑗,𝑖) 0 = 0, 𝜃(𝑗,𝑖) 0 = 1, 𝐵(𝑗,𝑖) 0 = 0 (6) For every i and j greater than zero, the boundary condition is expressed as; 𝐴𝑡 𝑦 = 0, 𝑢(0,𝑖) 𝑛 = 1, 𝑣(0,𝑖) 𝑛 = 0, 𝜃(0,𝑖) 𝑛 = 1, 𝐵(0,𝑖) 𝑛 = 1 (7) 𝐴𝑡 𝑥 = 0, 𝑢(𝑗,0) 𝑛 = 1, 𝑣(𝑗,0) 𝑛 = 0, 𝜃(𝑗,0) 𝑛 = 0, 𝐵(𝑗,0) 𝑛 = 0 (8) For All n According to the energy equation with time t, the beginning and boundary conditions for the non- dimensional form are as follows: For 𝑡 ≤ 0, 𝑢(𝑥, 𝑦, 0) = 0, 𝑣(𝑥, 𝑦, 0) = 0, 𝐵 (𝑥, 𝑦, 0) = 0 (9) For 𝑡 > 0, 𝑢(0, 𝑦, 𝑡) = 1, 𝑣(0, 𝑦, 𝑡) = 1, 𝐵 (0, 𝑦, 𝑡) = 1 (10) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 2s (2025) 292 https://internationalpubls.com For 𝑡 ≥ 0, 𝑢(∞, 𝑦, 𝑡) = 0, 𝑣(∞, 𝑦, 𝑡) = 0, 𝐵 (∞, 𝑦, 𝑡) = 0 (11) The computations are carried out by employing relatively reduced values of t. As part of our investigation, we established the value of αt to be 0.0012 and αx to be equal to y equal to 0.1. When we fixed it, it corresponds to the equation y=∞. There are equations that are used to compute the velocities. This technique is done until the value of n is equal to 400, which is 𝑡 = 0.5 𝑓𝑜𝑟 𝑖 = 1, 𝑥 = 0.1. During the course of our computations, we have determined that the Prandtl number is 0.71, which is the magnetic parameter that correlates to air and indicates a powerful magnetic field. In this particular instance, we take into consideration the situation in which the convective cooling of the plate causes. Additionally, in order to guarantee the stability and convergence of the finite difference approach, the Java computer programme is executed with smaller values of ⁄t, specifically ⁄t = 0.007, 0.005, and 0.0015. D. Energy Equation 𝜕𝜃 𝜕𝑡 + 𝜕𝜃 𝜕𝑥 + 𝑣 𝜕𝑣 𝜕𝑦 = 1 𝑝𝑟 ( 𝜕2𝜃 𝜕𝑥2 + 𝜕2𝜃 𝜕𝑦2) + 𝑅 ( 𝐵0 2 (∆𝑥)2) (12) When it is solved using the method of finite differences, the result is; 𝑢(𝑗,𝑖) 𝑛+1−𝑢(𝑗,𝑖) 𝑛 ∆𝑡 + 𝑢(𝑗,𝑖) 𝑛 ( 𝜃(𝑗,𝑛+1) 𝑛 −𝜃(𝑗,𝑖−1) 𝑛 2∆𝑥 ) = 1 𝑝𝑟 ( 𝜃(𝑗,𝑖+1) 𝑛 −2𝜃(𝑗,𝑖) 𝑛 +𝜃(𝑗,𝑖−1) 𝑛 (∆𝑥)2 ) + 1 𝑝𝑟 ( 𝜃(𝑗+1,𝑖) 𝑛 −2𝜃(𝑗,𝑖) 𝑛 +𝜃(𝑗−1,𝑖) 𝑛 (∆𝑦)2 ) +𝑅 ( 𝐵(𝑗,𝑛+1) 𝑛 −𝐵(𝑗,𝑖−1) 𝑛 2∆𝑥 ) 2 (13) 4. RESULTS AND DISCUSSION After defining the initial and boundary conditions in accordance with the provided equations, we solved equations using the finite difference approach and then ran them through a Java computer programme in order to acquire numerical results for this study. These findings were presented in tables and graphs. Following that, a discussion of the outcomes follows. By alluding to Table 1 and Figure 5, we can see that when the worth of Gr is equivalent to nothing, the essential speed profile encounters a fast expansion because of an expansion in the joule's boundary R nearby the plate, while the speed profile dispersion stays steady while creating some distance from the plate. Assuming we examine Table 2 and Figure 6, we can see that the seco is dependably no close to the plate until it arrives at a specific separation away from the plate. From that point forward, it Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 2s (2025) 293 https://internationalpubls.com begins to increment consistently with an expansion in the joule's boundary R, and afterward it gets back to a steady circulation far away from the plate. Table 1: principal velocity profile Velocity Profiles (U) 0 10 20 30 40 50 60 I 0 0.00005 0.00005 0.00005 0.00005 0.00005 II 0 0.0001 0.0001 0.0001 0.0001 0.0001 III 0 0.00015 0.00015 0.00015 0.00015 0.00015 Figure 5: The principal velocity profile Table 2: Profile of secondary velocity Velocity Profiles (V) 0 10 20 30 40 50 60 70 I 0 0 0.5 0.8 0.9 0.9 0.9 II 0 1 2 3 4 4 4 III 0 1 7 8 9 9 9 Figure 6: The secondary velocity profile Table 3: Distribution of temperatures Temperature Profiles 0 10 20 30 40 50 60 I 0 0.2 0.4 0.4 0.4 0.4 II 0 1 1.5 1.8 1.9 1.9 III 0 2 3 3.5 3.6 3.7 0 0.00005 0.0001 0.00015 0.0002 0 10 20 30 40 50 60 Velocity Profiles (U) I II III 0 2 4 6 8 10 0 10 20 30 40 50 60 70 Velocity Profiles (V) I II III Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 2s (2025) 294 https://internationalpubls.com Figure 7: Distribution of temperatures In light of the information displayed in Table 3 and Figure 7, we can see that the pace of temperature profiles dispersion is zero when the plate is extremely near it. In any case, at a specific distance, the pace of dissemination increments. After a specific separation from the plate, an expansion in the Joules boundary R causes a uniform expansion in the optional speed and temperature profiles. Notwithstanding, as the distance will in general get away from the plate, the bends show consistent speed and temperature profile disseminations, as exhibited in the bends that are situated previously. On account of essential speed profiles, the speed increments quick with an expansion in the Joule boundary, which is very near the plate, and shows consistent profiles circulation far away from the plate, as found in the figure. 5. CONCLUSION An examination of the effect of various Joule boundaries on the speeds and temperature dissemination of a temperamental free convection incompressible liquid stream that is going through an endless vertical permeable level plate has been done. Our work was restricted to the laminar limit layer, and in each and every situation that was thought about, the speed was separated into two parts, yet the temperature was just separated into one part. At the point when the temperature of the plate is higher than the temperature of the liquid in the free stream zone, the outcomes for Gr are equivalent to nothing. This demonstrates that intensity will be moved from the plate to the liquid, which is a term that alludes to the cooling of the plate by free convection flows. Our examination uncovered that an expansion in the quantity of Joule boundaries brings about an expansion in speed, and that temperature is dispersed consistently close to the plate. In any case, when the plate is taken out from the situation, the speed and temperature profiles keep on being disseminated in a steady way. It is obvious from this that the general intensity move in MHD free convective liquid stream is reliant upon the strength of Joule warming anases, wherein an ascent in the Joule warming boundary brings about a diminishing in the warming boundary. Moreover, it is seen that the stream field with the most minimal typical temperature is created when the worth of R is equivalent to nothing. REFERENCES [1] Abo-Eldahab, M. E., & El Aziz, A. M. (2005). Viscous dissipation and Joule heating effects on MHD free convection from a vertical plate with power-law variation in surface temperature in the presence of Hall and ion-slip currents. Applied Mathematical Modeling, 29(6), 579-595. 0 1 2 3 4 0 10 20 30 40 50 60 Temperature Profiles I II III Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 2s (2025) 295 https://internationalpubls.com [2] Ahmed, N., & Sarmah, H. K. (2009). A transient radiative MHD free and forced convection flow with mass transfer past an impulsively fixed infinite vertical plate. International Journal of Applied Mathematics and Mechanics, 5, 87- 98. [3] Ahmed, N., & Sengupta, S. (2011). Thermo-Diffusion and Diffusion-Thermo effects on a three-dimensional MHD convection flow past an infinite vertical porous plate with thermal radiation. Magnetohydrodynamics, 47, 41-60. [4] Ahmed, N., Sarma, D., & Deka, H. (2012). MHD convective mass transfer flow of an electrically conducting incompressible viscous past an infinite vertical porous plate with constant heat and chemical reaction in presence of a heat source. Applied Mathematics Sciences, 6, 1011-1020. [5] Ashgar, S., Mohyuddin, M. R., Hayat, T., & Siddiqui, A. M. (2003). The flow of non-Newtonian fluid induced due to the oscillations of a porous plate. Mathematical Problems in Engineering, 2, 133-143. [6] Calvert, L., & Santos, C. (2002). Research article on the fundamental principle of MHD. Applied Mathematics and Computation, 108, 11-21. [7] Chaudhary, R. C., & Jain, A. (2008). Magnetohydrodynamic transient convection flow past a vertical surface embedded in a porous medium with oscillating temperature. Turkish Journal of Engineering & Environmental Sciences, 32, 13-22. [8] Cooker, I. C., & Mubo-Pepple, O. (2003). Influence of viscous dissipation and radiation on unsteady MHD free- convection flow past an infinite heated vertical plate in a porous medium with time independent suction. International Journal of Heat and Mass Transfer, 46(13), 2305-2311. [9] Das, S. S., Maity, M., & Das, J. K. (2012). Unsteady hydromagnetic convective flow past an infinite vertical porous flat plate in a porous medium. International Journal of Energy and Environment, 1(3), 109-118. [10] Das, S. S., Sahoo, S. K., & Dash, G. C. (2006). Numerical solution of mass transfer effects on unsteady flow past an accelerated vertical porous plate with suction. Bulletin of the Malaysian Mathematical Sciences Society, 29(1), 33- 42. [11] Duwahiri, G. C., & Das, D. P. (2005). Viscous and Joule heating effect on forced convection flow of ionized gases along a plane wall with periodic suction and heat source. Simulation and Control B, 27(2), 47-55. [12] Jha, A. K., & Sharma, R. (2011). Mass transfer with chemical reaction effects on MHD free convective flow past an accelerated vertical plate. In International Conference on Fluid Dynamics and its Applications (pp. 648-655). [13] Jordan, J. (2007). Network simulation method to radiation and viscous dissipation effect on MHD unsteady free convection over vertical porous plate. Applied Mathematics Modeling, 31(9), 2019-2033. [14] Kandasamy, R. K. P., & Prabhu, K. K. S. (2005). Chemical reaction, heat and mass transfer on MHD flow over a vertical stretching surface with heat source and thermal stratification effects. International Journal of Heat and Mass Transfer, 48, 4557-4561. [15] Ramulu, G. S., & Ghosh, S. K. (2007). Effect of Hall current on MHD flow and heat transfer along a porous flat plate with mass transfer. International Journal for Engineering Science, 24(7), 1183-1193. [16] Raptis, A., Perdikis, A. C., & Takhar, H. S. (2004). Effect of thermal radiation on MHD flow. Applied Mathematics and Computation, 153, 645-649. [17] Rubin, H., & Atkinson, J. (2001). Environmental Fluid Mechanics. Marcel Dekker, Inc. [18] Sarma, D., & Deka, H. (2012). Effect of thermal radiation on a three-dimensional free convective flow with mass transfer past a porous vertical plate. International Journal of Mathematical Sciences & Engineering Applications, 6, 135-151. [19] Singh, A. K., Singh, A. K., & Singh, N. P. (2003). Heat and mass transfer in MHD flow of a viscous fluid past a vertical plate under oscillatory suction velocity. Indian Journal of Pure and Applied Mathematics, 34(3), 429-442. [20] Singh, P., & Gupta, C. B. (2005). MHD free convective flow of a viscous fluid through a porous medium bounded by an oscillating porous plate in slip flow regime with mass transfer. Indian Journal of Theoretical Physics, 53(2), 111-120.