https://internationalpubls.com TRANSFER FLOW THROUGH POROUS MEDIUM Y. V. Seshagiri Rao 1, D. Chenna Kesavaiah2*, V. Vishnuvardhan 3, Rakesh Podaralla4, Lavanya Srinathuni5 1 Department of Humanities and Sciences, Guru Nanak Institute of Technology (Autonomous), Ibrahimpatnam, Hyderabad, Telangana-501506, Email: yangalav@gmail.com 2 Department of Basic Sciences & Humanities, Vignan’s Institute of Management and Technology for Women (Autonomous), Kondapur (V), Ghatkesar (M), Medchal-Malkajgiri (Dist), Telangana-501301, India Email: chennakesavaiah@gmail.com 3 Assistant Professor, Department of Humanities and Sciences, Annamacharya University, Rajampet - 516126, Annamayya (Dist), Andhra Pradesh, India, Email: vishnuvardhan36369@gmail.com 4 Assistant Professor, School of Computing and Information Technology, Reva University, Bangalore- 560064, Karnataka, India, Email: rakeshpodaralla@gmail.com 5 Department of H & BS, Visvesvaraya College of Engineering and Technology , M. P. Patelguda (V), Ibrahimpatnam (M), R. R. Dist-501510, Telangana, India, Email: lavanyaram83@gmail.com * Corresponding author: chennakesavaiah@gmail.com Abstract This paper analytically investigates the one-dimensional unsteady laminar magnetohydro- dynamic (MHD) boundary layer flow of a viscous, incompressible fluid past an exponentially accelerated, infinite vertical plate. The effects of a transverse magnetic field, thermal radiation, and flow through a porous medium are considered, with both the plate and the medium assumed to be porous. The fluid is treated as optically thin, and the magnetic Reynolds number is assumed small enough to neglect induced magnetic fields. The governing boundary layer equations are nondimensionalized and solved using a perturbation method. Analytical expressions for transient velocity, temperature, skin friction, and Nusselt number are obtained and graphically analyzed for various physical parameters. Keywords: MHD, Radiation, Pormedium, Convective flow Introduction In recent years, convective heat transfer in porous media has garnered significant attention due to its broad industrial and technological applications, including geothermal energy recovery, oil and gas extraction, thermal insulation using fibrous and granular materials, electronic system cooling, and the development of regenerative heat exchangers. The unique properties of porous media— such as high surface area, complex internal structures, and variable permeability—strongly influence heat and mass transfer processes. Consequently, a thorough understanding of convective heat transfer in these systems is crucial for improving performance and efficiency across various engineering fields [1-14]. The influence of radiation on magnetohydrodynamic (MHD) flow and heat transfer has become increasingly significant in industrial applications, particularly at high operating temperatures where radiative heat transfer cannot be neglected. In engineering systems such as nuclear reactors, gas turbines, and propulsion units for aircraft, missiles, satellites, and space vehicles, effective thermal management is essential. Accurate modeling of radiation effects is critical to achieving optimal design and performance. At elevated temperatures, thermal radiation contributes substantially to overall heat transfer, necessitating its inclusion in predictive models. Consequently, extensive research has focused on transiente- free Received: 23-08-2024 Revised : 11-10-2024 Accepted: 01-11-2024 RADIATION EFFECT ON TRANSIENT FREE CONVECTIVE MHD HEAT Communications on Applied Nonlinear Analysis ISSN:1074-133x Vol31No.7s (2024) 765 Article History: mailto:yangalav@gmail.com mailto:chennakesavaiah@gmail.com mailto:vishnuvardhan36369@gmail.com mailto:rakeshpodaralla@gmail.com mailto:lavanyaram83@gmail.com mailto:chennakesavaiah@gmail.com https://internationalpubls.com There is a substantial interest of the recent researchers in the flows of non-Newtonian fluids. Such motivation in these fluids is mainly because of their use in the industrial and technological applications. Many materials like mud, pasta, personal care products, ice cream, paints, oils, cheese, asphalt etc. are non-Newtonian fluids. Most biological fluids with higher molecular weight components are also non-Newtonian in nature. The usual properties of polymer melts and solutions together with the desirable attributes of many polymeric solids, have given rise to the world-wide industry of polymer processing. The non- Newtonian fluids in particular have key importance in geophysics, chemical and nuclear industries, material processing, oil reservoir engineering, bioengineering and many others. Rheological properties of all the non-Newtonian fluids cannot be predicted using single constitutive equation (unlike the case of viscous fluids). Therefore many models of non- Newtonian fluids are based either on ―natural‖ modifications of established macroscopic theories or molecular considerations. The additional rheological parameters in the constitutive equations of non-Newtonian fluids are the main culprit for the lack of analytical solutions. The resulting equations are more complex and higher order than the Navier-Stokes equations [33-43]. The objective of this study is to derive an analytical solution for the one-dimensional, unsteady, laminar boundary layer flow of a viscous, incompressible fluid past an infinitely long vertical plate that is exponentially accelerated. The analysis considers the influences of the following physical effects:  Transverse magnetic field (representing the MHD aspect),  Thermal radiation (affecting energy transport),  Porous medium (affecting momentum and heat transfer resistance). Formulation of the Problem An unsteady one – dimensional laminar free convection flow of a viscous incompressible fluid past an infinite vertical porous plate through a porous medium with variable temperature is considered. The x − axis is being taken vertically upwards along the vertical plate and y − axis to be normal to the plate. The physical model and coordinate system of the flow problem is shown in figure (1). Figure (1): Physical model and coordinate system convective MHD flows, with special emphasis on the role of radiative heat transfer in shaping the fluid dynamics and thermal behavior of such systems [15-32]. Communications on Applied Nonlinear Analysis ISSN:1074-133x Vol31No.7s (2024) 766 Initially, it is assumed that the plate and fluid are at the same temperature T in the stationary condition. At 0t  , the plate is exponentially accelerated with a velocity  0 expu u a t   in its own plane and the plate temperature is raised linearly with time t . A uniform magnetic field is applied in the direction perpendicular to the plate. The fluid is assumed to be slightly conducting, so that the magmatic Reynolds number is much less than unity and hence the induced magnetic field is negligible in comparison with the applied magnetic field. The fluid considered here is a gray, absorbing/emitting radiation but a non scattering medium. The viscous dissipation is also assumed to be negligible in the energy equation as the motion is due to free convection only. It is also assumed that all the fluid properties are constant except for the density in the buoyancy term, which is given by the usual Boussinesq’s approximation. Under these assumptions the governing boundary layer equations are   22 0 2 Bu u u u v g T T u t y y k                             (1) 2 2 r p qT T T C v t y y y                      (2) with the following initial and boundary conditions:    0 0, , 0 exp , 0 0, w u T T y t u u a t T T T T At at y u T T as y                            (3) where 1 0u A   The local radiant for the case of an optically thin gray gas is expressed by  4 44rq a T T y         (4) We assume that the temperature differences within the flow are sufficiently small such that 4T  may be expressed as a linear function of the temperature. This is accomplished by expanding 4T  in a Taylor series about 4T  and neglecting higher order terms; thus  4 3 44 3T T T T        (5) Communications on Applied Nonlinear Analysis ISSN:1074-133x Vol31No.7s (2024) https://internationalpubls.com 767   2 3 2 16p T T C a T T T t y                 (6) In order to write the governing equations, initial and the boundary conditions the following non-dimensional quantities are introduced.   0 3 0 2 2 3 0 0 2 2 2 0 0 0 , , , , 16 , , , , ,Pr w o w w p g T Ty v t u T Tu Y U t T Gr u uT T CB k u a T a M k R a u u u k                                          (7) In view of (7) the equations (1) and (6) are reduced to the following non-dimensional form 2 2 1U U U GrT M U U t Y Y k             (8) 2 2 1 Pr Pr T T T R T t Y Y           (9) with following initial and boundary conditions:   0, 0 , 0 exp , 0, 0 0, 0 U T Y t U at T t t at Y U T as Y            (10) where Gr is the thermal Grashof number, Pr is the fluid Prandtl number, R is the radiation parameter, M is the magnetic parameter, a is the accelerating parameter, a dimensionless accelerating parameter, a absorption coefficient, pc specific heat at constant pressure, 0B transverse magnetic field strength, g acceleration due to gravity,  thermal conductivity of the fluid, k permeability parameter, k dimensionless permeability parameter, rq radiative heat flux in the y  direction, t time, t dimensionless time, T  temperature, T dimensionless temperature, wT  is the temperature of the plate, T  is the temperature of the fluid far away from the plate, u is the x component of the velocity, 0u velocity of the plate, U is the dimensionless velocity, V  is the y  component of velocity, y is the coordinate axis normal to the plate, Y is the dimensionless coordinate axis normal to the plate,  is the volumetric By using equation (4) and (5), equation (2) gives Communications on Applied Nonlinear Analysis ISSN:1074-133x Vol31No.7s (2024) https://internationalpubls.com 768 coefficient of thermal expansion,  is the suction parameter,  is the kinematic viscosity,  is the fluid density,  is the electrical conductivity of fluid. Method of Solution Equation (8) - (9) are coupled, non – linear partial differential equations and these cannot be solved in closed – form using the initial and boundary conditions (10). However, these equations can be reduced to a set of ordinary differential equations, which can be solved analytically. This can be done by representing the velocity, temperature and concentration of the fluid in the neighbourhood of the fluid in the neighbourhood of the plate as             0 1 0 1 , , at at U y t U y U y e T y t T y T y e     (11) Substitute equation (11) in to the equations (8) and (9) the set of ordinary differential equations are the following form 0 0 3 0 0U U U GrT      (12) 1 4 1 1U U GrT    (13) 0 1 0 0 0T T RT    (14) 1 2 1 0T T   (15) The exact solution for the fluid velocity  ,U y t , fluid temperature  ,y t are obtained and expressed from equations from (12) - (15) in the following form:   2 4 1 2, m y m yU y t K e K e    2, m yT y t t e Skin friction 2 1 4 2 0y U m K m K y           Nusselt number Communications on Applied Nonlinear Analysis ISSN:1074-133x Vol31No.7s (2024) https://internationalpubls.com 769 2 0y T Nu t m y         Appendix       22 2 3 2 4 1 2 2 2 3 2 1 1 2 3 4 3 4Pr Pr 4 , , 2 2 1 , Pr, Pr Pr , ,at R Grt m m K m m K e K R at M at k                                                        Results and Discussion In order to better understand the physical characteristics of the problem, numerical simulations have been conducted to evaluate the velocity profiles, temperature distribution, skin friction coefficient, and Nusselt number for a range of parameter values of magnetic field parameter  M , Grashof number  Gr , accelerating parameter  a , suction parameter   , permeability parameter  K , radiation parameter  R and time  t are presented graphically in figures (2) – (13). The transient velocity profiles for different values of Grashof number  Gr are shown in figure (2). The Grashof number signifies the relative effect of the buoyancy force to the hydrodynamic viscous force. The positive values of Grashof number correspond to cooling of the plate and the negative values of Grashof number correspond to heating of the plate by free convection. As expected, it is found that an increase in the Grashof number lead to increase in the velocity due to enhancement in the buoyancy force. The transient velocity profiles for different values of magnetic parameter  M are depicted in figure (3). It is observed form this figure that an increase in magnetic field leads to decrease in the velocity profiles for both the cases of cooling  2Gr  and heating  2Gr   of the porous plate. It is because that the application of transverse magnetic field will result a resistive type force (Lorentz force) similar to drag force which tends to resist the fluid flow and thus reducing its velocity. The effects of suction parameter on velocity profiles illustrated in figure (4), it is found here that velocity decreases with increase of the suction parameter for both cases of cooling  2Gr  and heating  2Gr   of the porous plate. The effect of permeability parameter  K and time  t on velocity profiles are depicted in figure (5). It can be seen that the velocity increases Communications on Applied Nonlinear Analysis ISSN:1074-133x Vol31No.7s (2024) https://internationalpubls.com 770 https://internationalpubls.com with increase of permeability parameter and time. The transient velocity profiles for different values of the accelerating parameter and the radiation parameter are presented in figure (6). It is observed that the velocity decreases with an increase in the radiation parameter, whereas it increases with an increase in the accelerating parameter. Figure (7) displays the velocity profiles for different values of Prandtl number (Pr) , it is clear that the velocity decreases with increasing values of Prandtl number. Effects of radiation parameter (R) , suction parameter (a) , Prandtl number (Pr) and time (t ) on temperature profiles are shown in figure (8), (9), (10) and (11) respectively. It is observed form these figures that temperature decrease with increased values of radiation parameter, suction parameter and Prandtl number, but increases with increased values of time. Figure (12) shows the variation of the Nusselt number for different values of the suction parameter. It is observed that the rate of heat transfer decreases as the suction parameter increases. Figure (13) depicts the influence of the Grashof number on skin friction. It is evident from the figure that, for the case of cooling of the porous plate, an increase in the Grashof number leads to an increase in skin friction. This trend can be attributed to the enhancement of buoyancy forces which augment Reference 1. D. Chenna Kesavaiah, B Venkateswarlu, N. Nagendra and O.D. Makinde (2024): Magneto- Compound Reaction of Convective Flow via a Porous Inclined Plate with Heat Energy Absorption, Journal of Nonlinear Modeling and Analysis, Vol. 6 (1), pp. 88–106 2. Damala Chenna Kesavaiah, Vellanki Nagaraju, Bhumarapu Venkateswarlu (2023): Investigating the Influence of Chemical Reaction on MHD-Casson Nanofluid Flow via a Porous Stretching Sheet with Suction/Injection, Science, Engineering and Technology, Vol.3 (2), pp. 47-62 3. D. Chenna Kesavaiah, Mohd Ahmed, G. Chandu, Dr. M. Vijaya Bhaskar Reddy Y. V. Seshagiri Rao, Dr. Nookala Venu (2023): Heat and mass transfer of unsteady hydromagnetic free convection flow in porous medium past a vertical plate with chemical reaction, Eur. Chem. Bull. Vol.,12 (9), pp. 502-521 4. P. Krishna Jyothi, D. Chenna Kesavaiah, G. Ravindranath Reddy, M. Chitra, Y. V. Seshagiri Rao, Dr. Nookala Venu (2023): Chemical reaction, radiation absorption and Hall effects on unsteady flow past an isothermal vertical plate in a rotating fluid with variable mass diffusion with heat source, Eur. Chem. Bull. Vol. 12 (11), pp.581-599 5. Ch. Shashi Kumar, K. Ramesh Babu, M. Naresh, D. Chenna Kesavaiah, Dr. Nookala Venu (2023): Chemical reaction and Hall effects on unsteady flow past an isothermal vertical plate in a rotating fluid with variable mass diffusion, Eur. Chem. Bull. Vol. 12 (8), pp. 4991-5010 6. G. Balreddy, Y. V. Seshagiri Rao, D. Chenna Kesavaiah, Lavanya Srinathuni (2023): Effects of hall current and rotation, heat generation on MHD free convection heat and mass transfer flow past an accelerated vertical plate, Journal of Computational Analysis and Applications, Vol. 31 (4), pp. 775-789 7. K. Venugopal Reddy, B. Venkateswarlu, D. Chenna Kesavaiah, N. Nagendra (2023): Electro-Osmotic Flow of MHD Jeffrey Fluid in a Rotating Microchannel by Peristalsis: Thermal Analysis, Science, Engineering and Technology, Vol. 3, No. 1, pp. 50-66 Communications on Applied Nonlinear Analysis ISSN:1074-133x Vol31No.7s (2024) 771 https://internationalpubls.com 8. Anita Tuljappa, D. Chenna Kesavaiah, M. Karuna Prasad, Dr. V. Bharath Kumar (2023): Radiation absorption and chemical reaction effects on MHD free convection flow heat and mass transfer past an accelerated vertical plate, Eur. Chem. Bull. Vol. 12 (1), pp. 618-632 9. P. Ramesh Babu, D. Chenna Kesavaiah, Y. V. Seshagiri Rao (2022): Chemical reaction and hall effects on unsteady flow past an isothermal vertical plate in a rotating fluid with variable mass diffusion with heat source, Eur. Chem. Bull. Vol. 11 (11), pp. 1432–1446 10. G. Balreddy, D. Chenna Kesavaiah, Y. V. Seshagiri Rao (2022): Analytical solution for transient free convection MHD flow through a porous medium between two vertical plates with heat source, Eur. Chem. Bull. Vol. 11(10), 653 –661 11. D. Chenna Kesavaiah, G. Rami Reddy, Y. V. Seshagiri Rao (2022): Impact of thermal diffusion and radiation effects on MHD flow of Walter’s liquid model-b fluid with heat generation in the presence of chemical reaction, International Journal of Food and Nutritional Sciences, Vol. 11, (12), pp. 339- 359 12. D. Chenna Kesavaiah, G. Rami Reddy, G. Maruthi Prasada Rao (2022): Effect of viscous dissipation term in energy equation on MHD free convection flow past an exponentially accelerated vertical plate with variable temperature and heat source, International Journal of Food and Nutritional Sciences, Vol. 11,(12), pp. 165- 183 13. Dr. Pamita, D. Chenna Kesavaiah, Dr. S. Ramakrishna (2022): Chemical reaction and Radiation effects on magnetohydrodynamic convective flow in porous medium with heat generation, International Journal of Food and Nutritional Sciences, Vol. 11,(S. Iss 3), pp. 4715- 4733 14. K. Ramesh Babu, D. Chenna Kesavaiah, B. Devika, Dr. Nookala Venu (2022): Radiation effect on MHD free convective heat absorbing Newtonian fluid with variable temperature, Neuro Quantology, Vol. 20 (20), pp. 1591-1599 15. D. Chenna Kesavaiah, Mohd Ahmed, K. Venugopal Reddy, Dr. Nookala Venu (2022): Heat and mass transfer effects over isothermal infinite vertical plate of Newtonian fluid with chemical reaction, NeuroQuantology, Vol. 20 (20), pp. 957-967 16. D. Chenna Kesavaiah, K. Ramakrishna Reddy, Ch. Shashi Kumar, M. Karuna Prasad (2022): Influence of joule heating and mass transfer effects on MHD mixed convection flow of chemically reacting fluid on a vertical surface, NeuroQuantology, Vol. 20 (20), pp. 786-803 17. G. Bal Reddy, D. Chenna Kesavaiah, G. Bhaskar Reddy, Dr. Nookala Venu (2022): A note on heat transfer of MHD Jeffrey fluid over a stretching vertical surface through porous plate, NEUROQUANTOLOGY, Vol. 20 (15), pp. 3472-3486 18. D. Chenna Kesavaiah, P. Govinda Chowdary, Ashfar Ahmed, B. Devika (2022): Radiation and mass transfer effects on MHD mixed convective flow from a vertical surface with heat source and chemical reaction, NEUROQUANTOLOGY, Vol.20 (11), pp. 821-835 19. Y. Haranth and A. Sudhakaraiah (2015): Viscosity and Soret effects on unsteady hydromagnetic gas flow along an inclined plane, International Journal of Science and Research, vol. 4 (2), pp. 2650-2654 20. D. Chenna Kesavaiah, P. Govinda Chowdary, G. Rami Reddy, Dr. Nookala Venu (2022): Radiation, radiation absorption, chemical reaction and hall effects on unsteady flow past an Communications on Applied Nonlinear Analysis ISSN:1074-133x Vol31No.7s (2024) 772 https://internationalpubls.com isothermal vertical plate in a rotating fluid with variable mass diffusion with heat source, NEUROQUANTOLOGY, Vol. 20 (11), pp. 800-815 21. M. Rajaiah and A. Sudhakaraiah (2015): Unsteady MHD free convection flow past an accelerated vertical plate with chemical reaction and Ohmic heating, International Journal of Science and Research, vol. 4 (2), pp. 1503-1510 22. D. Chenna Kesavaiah, M. Karuna Prasad, G. Bhaskar Reddy, Dr. Nookala Venu (2022): Chemical reaction, heat and mass transfer effects on MHD peristaltic transport in a vertical channel through space porosity and wall properties, NEUROQUANTOLOGY, Vol. 20 (11), pp. 781-794 23. D. Chenna Kesavaiah, G. Bhaskar Reddy, Anindhya Kiran, Dr. Nookala Venu (2022): MHD effect on boundary layer flow of an unsteady incompressible micropolar fluid over a stretching surface, NEUROQUANTOLOGY, Vol. 20 (8), pp. 9442-9452 24. M. Rajaiah, A. Sudhakaraiah, P. Venkatalakshmi and M. Sivaiah (2014): Unsteady MHD free convective fluid flow past a vertical porous plate with Ohmic heating In the presence of suction or injection, International Journal of Mathematics and Computer Research, Vol. 2 (5), pp. 428-453 25. D. Chenna Kesavaiah, P. Govinda Chowdary, M. Chitra, Dr. Nookala Venu (2022): Chemical reaction and MHD effects on free convection flow of a viscoelastic dusty gas through a semi infinite plate moving with radiative heat transfer, NEUROQUANTOLOGY, Vol. 20 (8), pp. 9425-9434 26. M. Rajaiah, A. Sudhakaraih, S. V. K. Varma and P. Venkatalakshmi (2015): Chemical and Soret effect on MHD free convective flow past an accelerated vertical plate in presence of inclined magnetic field through porous medium. i-manager’s Journal on Mathematics, Vol. 4(1), pp. 32-39 27. Chenna Kesavaiah DAMALA, Venkateswarlu BHUMARAPU, Oluwole Daniel MAKINDE (2021): Radiative MHD Walter’s Liquid-B Flow Past a Semi-Infinite Vertical Plate in the Presence of Viscous Dissipation with a Heat Source, Engineering Transactions, Vol. 69(4), pp. 373–401 28. G. Rami Reddy, D. Chenna Kesavaiah, Venkata Ramana Musala and G. Bkaskara Reddy (2021): Hall Effect on MHD Flow of a Visco-Elastic Fluid through Porous Medium Over an Infinite Vertical Porous Plate with Heat Source, Indian Journal of Natural Sciences, Vol. 12 (68), pp. 34975-34987 29. G. Rami Reddy, Venkata Ramana Musala, G Bkaskara Reddy and D Chenna Kesavaiah (2021):Applications of Group Theory in Molecular Systems Biology, Indian Journal of Natural Sciences, Vol. 12 (68), pp. 34867-34875 30. D. Chenna Kesavaiah, T. Ramakrishna Goud, Nookala Venu, Y. V. Seshagiri Rao (2021): MHD effect on convective flow of dusty viscous fluid with fraction in a porous medium and heat generation, Journal of Mathematical Control Science and Applications, Vol. 7 (2), pp. 393-404 31. D. Chenna Kesavaiah and B. Venkateswarlu (2020): Chemical reaction and radiation absorption effects on convective flows past a porous vertical wavy channel with travelling thermal waves, International Journal of Fluid Mechanics Research, Vol. 47 (2), pp. 153-169 32. D. Chenna Kesavaiah, T. Ramakrishna Goud, Y. V. Seshagiri Rao, Nookala Venu (2019): Radiation effect to MHD oscillatory flow in a channel filled through a porous 773 Communications on Applied Nonlinear Analysis ISSN:1074-133x Vol31No.7s (2024) medium with heat generation, Journal of Mathematical Control Science and Applications, Vol. 5 (2), pp. 71-80 33. B. Mallikarjuna Reddy, D. Chenna Kesavaiah and G. V. Ramana Reddy (2019): Radiation and Diffusion Thermo Effects of Visco-Elastic Fluid Past a Porous Surface in the Presence of Magnetic Field and Chemical Reaction with Heat Source, Asian Journal of Applied Sciences, Vol. 7 (5), pp. 597-607 34. D. Chenna Kesavaiah, T. Ramakrishna Goud, Nookala Venu, Y. V. Seshagiri Rao (2017): Analytical study on induced magnetic field with radiating fluid over a porous vertical plate with heat generation, Journal of Mathematical Control Science and Applications, Vol. 3 (2), pp. 113-126 35. R. C. Chaudhary and A. Jain (2009): MHD heat and mass diffusion flow by natural convection past a surface embedded in a porous medium, Theoretical and Applied Mechanics, Vol. 36 (1), pp. 1-27 36. V. Rajesh, M. Mallesh and O A B’eg (2015): Transient MHD free convection flow and heat transfer of nanofluid past an impulsively started vertical porous plate in the presence of viscous dissipation, Procedia Materials Science, Vol. 10, pp. 80-89 37. Ashish Paul (2017): Transient free convective MHD flow past an exponentially accelerated vertical porous plate with variable temperature through a porous medium, Hindawi, International Journal of Engineering Mathematics, pp. 1-9 38. S. Mukhopadhyay (2013): MHD boundary layer flow and heat transfer over an exponentially stretching sheet embedded in a thermally stratified medium,. Alexandria Eng Journal, Vol. 52, pp. 259–265 39. M. Hussain, M. Ashraf, S. Nadeem, M. Khan (2013): Radiation effects on the thermal boundary layer flow of a micropolar fluid towards a permeable stretching sheet,. J Fran Institute, Vol. 350, pp. 194–210 40. M. M. Rashidi, S. Abelman, N. F. Mehr (2013): Entropy generation in steady MHD flow due to a rotating disk in a nanofluid,. Int. J. Heat Mass Transfer 62, pp. 515–525. 41. G. Balreddy, Y. V. Seshagiri Rao, D. Chenna Kesavaiah, Lavanya Srinathuni (2024): Radiation Absorption and Chemical Reaction Effects on MHD Flow Through Porous Medium Past an Exponentially Accelerated Inclined Plate with Variable Temperature, Nanotechnology Perceptions, Vol. 20 (3), pp. 346–362 42. P. Ramesh Banu, G. Balreddy, D. Chenna Kesavaiah, Lavanya Srinathuni (2024): Variable temperature, radiation absorption and chemical reaction effects on unsteady MHD flow through porous medium past an oscillating inclined plate, Journal of Computational Analysis and Applications, Vol. 33 (2), pp. 925-941 43. D. Chenna Kesavaiah, Ch. Shashi Kumar, M. Chitra, Vuppala Lakshmi Narayana (2024): Viscous dissipation effect on steady free convective hydromagnetic heat transfer flow of a reactive viscous fluid in a bounded domain, African Journal of Biological Sciences, Vol. 6(Si4), pp. 4287-4295 https://internationalpubls.com 774 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 -0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 y Figure (3): Veloicty profiles for different values of M U Gr = 2 Gr = -2 Pr=0.71,K=0.5,R=2.0,=0.4,t=0.6,a=0.6 M = 0.5, 1, 1.5, 2 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 -0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 y Figure (4): Veloicty profiles for different values of  U  = 0.2, 0.6, 1, 1.4 Pr=0.71, M=1,K=0.5,R=2.0,=0.4,t=0.6;a=0.6 Gr = 2 Gr = - 2 Communications on Applied Nonlinear Analysis ISSN:1074-133x Vol31No.7s (2024) https://internationalpubls.com 775 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 y Figure (5): Veloicty profiles for different values of K U Pr=0.71,M=1,R=2.0.=0.4,Gr=2,a=0.6 K = 0.5, 1, 1.5, 2 t = 0.9 t = 0.5 t = 0.2 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 -0.5 0 0.5 1 1.5 2 2.5 3 y Figure (6): Veloicty profiles for different values of R U Pr=0.71,M=1,K=0.5, =0.4,Gr=2,t=0.8 a = 0.2 a = 0.6 a = 1.2 R = 1, 2, 3, 4 Communications on Applied Nonlinear Analysis ISSN:1074-133x Vol31No.7s (2024) https://internationalpubls.com 776 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 y Figure (7): Veloicty profiles for different values of Pr U M=1,K=0.5,R=2.0, =0.4,Gr=2,t=0.8,a = 0.6 Pr = 0.71, 0.9, 1, 7 0 1 2 3 4 5 6 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 y Figure (8): Temperature profiles for different values of R T Pr=0.71,=0.4,t=0.8 R = 0.5, 1, 1.5, 2 Communications on Applied Nonlinear Analysis ISSN:1074-133x Vol31No.7s (2024) https://internationalpubls.com 777 0 1 2 3 4 5 6 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 y Figure (9): Temperature profiles for different values of  T Pr=0.71,t=0.8,R=1.0  = 0.1, 0.3, 0.5, 0.7 0 1 2 3 4 5 6 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 y Figure (10): Temperature profiles for different values of Pr T =0.4,t=0.8,R=1.0 Pr = 0.71, 0.9, 1, 7 Communications on Applied Nonlinear Analysis ISSN:1074-133x Vol31No.7s (2024) https://internationalpubls.com 778 0 1 2 3 4 5 6 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 y Figure (11): Temperature profiles for different values of t T t = 0.2, 0.4, 0.6, 0.8 Pr=0.71,=0.4,R=1.0 0 1 2 3 4 5 6 -12 -10 -8 -6 -4 -2 0 t Figure (12): Nusselt number for different values of R N u Pr=0.71,R=1.0  = 0.5, 1, 1.5, 2 Communications on Applied Nonlinear Analysis ISSN:1074-133x Vol31No.7s (2024) https://internationalpubls.com 779 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 -5 0 5 10 t Figure (13): Skin friction for different values of Gr U Pr=0.71,M=1,K=0.5,R=2.0,=0.4, a=0.3 Gr = 5, 10, 15, 20 Communications on Applied Nonlinear Analysis ISSN:1074-133x Vol31No.7s (2024) https://internationalpubls.com 780