Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 5s (2025) 145 https://internationalpubls.com Magnetohydrodynamic Effect on 3D Nanofluid Flow via Stretching/Shrinking Surface S. Rakmaiah1*, Uma M2, Sriram Yavagoni3*, Ch. Neeraja4, Y. Sugandhi Naidu5 1Assistant Professor, Department of Mathematics, Vignana Bharathi Institute of Technology, Aushpur, Hyderabad- 501301, Telangana, India 2Department of Mathematics, M S Ramaiah Institute of Technology, Bangaluru-560054, Karnataka, India 3Assistant Professor, Department of Mathematics, Vignana Bharati Institute of Technology, Telangana, Hyderabad, India 4Associate Professor, Department of Humanities and Sciences, CMR Technical Campus, Kandlakoya, Hyderabad- 501401, Telangana, India 5Assistant Professor, Department of ECE, Aditya University, Surampalem, Andrapradesh-533437, India *Corresponding Author Mail Id:- rakmajirao@gmail.com, Sriram.yavagoni@gmail.com, neeraja.maths@gmail.com, sugandhi.nusetti@gmail.com Article History: Received: 05-10-2024 Revised: 27-11-2024 Accepted: 04-12-2024 Abstract: This study presents a numerical technique to analyze the MHD effect on the motion of 3D NFs via shrinking or SS under convective BCs. The investigation incorporates convective and VSC. The gov. eqs. are transformed into a set of coupled NLODEs using appropriate ST. These transformed nonlinear equations are calculated using R-K-F method combined with the shooting technique. The influence of various physical parameters on , and distributions is illustrated graphically. Other that, the study evaluates SF coefficient and HTR for different NFs parameters. Keywords: nanofluid, MHD, Shrinking or SS, VS. Introduction: The study of Casson NFs is a type of NNF that combines the viscoplastic properties of a CF with the enhanced thermal and physical characteristics of NFs, which are fluids containing suspended NPs. The Casson model describes the yield stress behavior, where the fluid begins to motion only after a certain threshold shear stress is exceeded, while the inclusion of NPs improves heat transfer, viscosity, and thermal conductivity. CNFs are widely studied for applications in biomedical engineering, such as blood flow modeling, as well as in HT systems like cooling technologies, microfluidics, and energy- efficient industrial processes. Their unique combination of properties makes them suitable for optimizing thermal performance and understanding complex flow behaviors under various conditions. Yousuf Ali et al. [1] examined the CNFs MHD unstable BL characteristics in the simultaneous transmission of thermoelectric and radiation on a SS. Suresh Kumar et al. [2] finding the motion velocity is a diminishing function of the TR parameter has observed in Hall parameter. Khan et al. [3] examines the MHD on CNFs in a porous medium .Shek Akbar et al. [4] consider the 2D Casson HNFs motion inside the channel. Rehman et al. [5] reported that, the MHD CSC motion of a HNFs, considering the impact of VD via SS is presented. mailto:rakmajirao@gmail.com mailto:Sriram.yavagoni@gmail.com mailto:neeraja.maths@gmail.com mailto:sugandhi.nusetti@gmail.com https://www.sciencedirect.com/topics/earth-and-planetary-sciences/magnetohydrodynamics https://www.sciencedirect.com/topics/engineering/simultaneous-transmission https://www.sciencedirect.com/topics/engineering/simultaneous-transmission https://www.sciencedirect.com/topics/engineering/thermoelectrics https://www.sciencedirect.com/topics/physics-and-astronomy/magnetohydrodynamics https://www.sciencedirect.com/topics/physics-and-astronomy/hybrid-nanofluid Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 5s (2025) 146 https://internationalpubls.com The MHD is behavior of ECF, such as plasmas, liquid metals, in the presence of MF and EF. Governed by modified Navier-Stokes and Maxwell’s equations, MHD has wide-ranging applications, including the study of astrophysical phenomena (e.g., solar flares and planetary magnetospheres), fusion energy research, liquid metal cooling in reactors, MHD generators, and flow control in aerospace. It is crucial for understanding and engineering systems involving high-temperature plasmas or conductive fluids. Alamirew and Awgichew [6] examines the motion of MHD Casson NFs via vertically SS with effect of VD, MC in a spongy medium. Ismail et al. [7] developed the thermal instability of Tri-hybrid Casson NFs with TR PM. Khan et al. [8] present the constitution of MHD steady 3D CNFs motion containing gyrotactic microorganism via SS. Recently, some of authors developed numerical techniques applied in 3D NFs motion via SS was developed [9-11]. Mathematical Analysis: Here, we consider the 3D CNFs motion with MHD effect via SS. the physical model of the problem as predicted in Fig. 1. It is considered that liquid motion taken by 1 1x y − surface with VS. The 1z direction has taken by negligible. Under these basic gov. eq’s are shown below: 1 1 1 1 1 1 0, u v w x y z    + + =    (1) ( ) 22 1 01 1 1 1 1 1 1 1 12 1 1 1 1 , f Bu u u u u v w u x y z z        + + = −     (2) ( ) 22 1 01 1 1 1 1 1 1 1 12 1 1 1 1 , f Bv v v v u v w v x y z z        + + = −     (3) ( ) * 2 * * * 2 * ** * *1 1 1 1 1 1 1 1 2 * * 1 1 1 1 11 ,T m B DT T T T TC T u v w D x y z z z T zz                 + + = + +                    (4) ( ) ( ) * * * * 2 * 2 * 1 1 1 1 1 1 1 1 2 2* 1 1 1 1 1 ,T B DC C C C T u v w D x y z Tz z         + + = +               (5) Corresponding B.Cs are * * *1 1 1 1 1 0 1 1 1 2 0 1 1 1 * * * *1 1 0 1 1 1* * * * * 1 1 1 1 1 ( ) U ( ) N , ( ) ( ) N , , , (T T ), 0, 0, , , w w w w w f u v u u x x v v x V x z z at z o C w w T T k h z u v T T C C as z       = = + = = +     =    = = − = −    → → → → →  (6) https://www.sciencedirect.com/topics/engineering/mixed-convection https://www.sciencedirect.com/topics/earth-and-planetary-sciences/thermal-radiation Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 5s (2025) 147 https://internationalpubls.com The following dimensionless functions and the similarity variables are: ( ) 1 1 1 1 1 1 1 1 * * * * 1 1 1 1 * * * * , '( ), '( ), w ( ) ( ) , ( ) , ( ) . f f w w a z u a x f v a y g T T C C a f g T T C C                = = =    − −  = − + = = − −  (7) Utilizing the above dimensions, Eq. (1) is identically satisfied and translate Eqs. (2)-(5) ( ) 2 ''' ''( g) ' ' 1f f f f Mf= − + + + − (8) ( ) 2 ''' ''( g) ' ' 1g g f g Mg= − + + + − (9) ( )( )2 '' Pr ( ) ' ' ' 'b tf g N N    = − + + + (10) '' Pr( ) ' (N ) ''t bLe f g N  = − + − (11) With subject to the boundary conditions are: (0) S, g(0) 0, '(0) 1 A ''(0), g'(0) Bg''(0), (0) 1, '(0) (1 (0)) 0, '( ) 0, g'( ) 0, ( ) 0, ( ) 0 as f f f Bi f            = = = + = +   = + − =  → → → → → (12) Results and Discussion The physical effect of M (“Magnetic Parameter”) on fluid motion component ' ( )g  for the case ( )0  and ( )0  as predicts on Fig. 2. It is clear that the nanofluid flow velocity is slowly down along y -direction when the sheet is stretching ( )0  . Physically, a drag force like resistive type force is create disturbance by the fluid particles of the vertical MF to the electrically conducting liquid. This force has to reduce the motion of the fluid over a stretching surface. Fig. 3 presented the S on liquid motion ( ), ( )    for the cases of ( )0  , ( )0  . It is clear the liquid ( )  is slow reduction via SS with various enlarge values of S . Physically, the larger values of mass flux effect in fluid particles and the liquid resistance slow down then its liquid motion BL thickness is reducing. The impact of  on velocity component ' ( )g  along *y - direction is explored through in Fig. 4 for the cases of ( )0S  and ( )0S  . It is clear the liquid motion is monotonically enhances via SS with various enlarge values of . Because, the liquid motion convergent to surface area very fast then the surface is injection case. Fig. 5 depicts the physical parameter bN on ( ), ( )    . It is noticed that ( )  of the NF enhances via surface while opposite motion of fluid ( )  with higher values of bN . Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 5s (2025) 148 https://internationalpubls.com Conclusions: The main out comes of the present study are mentioned below: • The temperature of Brownian motion parameter is declined while opposite trend follows concentration with higher statistical values of bN . References [1] Md. Yousuf Ali, Sk. R. E. Rabbi, S. F. Ahmmed, Md. N. Nabi, A.K. Azad and S.M. Muyeen, Hydromagnetic flow of Casson nano-fluid across a stretched sheet in the presence of thermoelectric and radiation, International Journal of Thermofluids, 21 (2024) 100484. [2] Y. Suresh Kumar, S. Hussain, K. Raghunath, F. Ali, K. Guedri, S. M. Eldin and M. Ijaz Khan, Numerical analysis of magnetohydrodynamics Casson nanofluid flow with activation energy, Hall current and thermal radiation, Scientific Reports, 13 (2023) 4021. [3] D. Khan, P. Kumam, W. Watthayu and M.F. 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[12] Nadeem et al. [13] Gupta and Sharma [14] Ahmad and Nazar [15] 0.0 1.000000 1.00000 1.0004 1.0003181 1.0042 10 3.316624 3.31662 3.3165 3.3165824 3.3165 100 10.04987 10.04987 10.049 10.049864 10.049 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 5s (2025) 150 https://internationalpubls.com Nomenclature * *( , )x y Cartesian coordinate’s *T Temperature of the fluid 1 1 1, ,u v w velocity components along * * *, , zx y - axis *T fluid temperature far away from the surface A Velocity slip along x-axes 0 1a N * wT Constant fluid Temperature of the wall B Velocity slip along y-axes 0 2a N wU Stretching velocity *C Concentration U Free stream velocity * fC Skin friction coefficient * pc Specific heat Greek symbols *C Uniform ambient concentration  Density BD Brownian diffusion  Dimensionless concentration TD Thermophoresis diffusion 1 Boltzmann constant f Dimensionless stream function  Constant stretching/shrinking parameter 1 1 b a 'f Dimensionless velocity  Ratio of the nanoparticle to the fluid ( ) ( ) p f c c   S Constant mass flux parameter 0 1 0 w a  1 Kinematic viscosity of the fluid *k Thermal conductivity * Electrical conductivity Le Lewis number * m BD  =  Dimensionless temperature M Magnetic field parameter 2 1 0 1 f B a   = * m Thermal diffusivity ( )p fk c= tN Thermophoresis parameter * * * * D (T T ) T T w m     = − ( ) fc Heat capacity of the fluid bN Brownian motion coefficient * *D (C )B w f C  − = ( ) pc Heat capacity of the nanoparticle to the fluid Pr Prandtl number 1 * m     =     f Fluid density rq Radiative heat flux Subscripts Rex Reynolds number  condition at free stream dR Radiation parameter * *3 * * 16 3 m T kk   =