J. Nig. Soc. Phys. Sci. 5 (2023) 1435 Journal of the Nigerian Society of Physical Sciences Numerical investigation of nonlinear radiative flux of non-Newtonian MHD fluid induced by nonlinear driven multi-physical curved mechanism with variable magnetic field K. M. Sannia,∗, A. D. Adesholab, T. O. Aliub aDepartment of Mathematics, COMSATS University Islamabad, Chak Shahzad Road, Islamabad, 44000 bDepartment of Mathematics and Statistics, Kwara State University Malete Abstract This paper discusses two-dimensional heat flow of an incompressible non-Newtonian hydromagnetic fluid over a power-law stretching curved sheet. The energy equation of the flow problem considers a radiative flux influenced by viscous dissipation and surface frictional heating. Lorentz force and Joule heating are taken in the consequence of applied variable magnetic field satisfying solenoidal nature of magnetism. The governing equations are reduced to boundary-layer regime using dimensionless quantities and the resulting PDEs are converted into ODEs by suitable similarity variables. The flow fields; velocity and temperature are computed numerically by implementing Keller-Box shooting method with Jacobi iterative technique. Error analysis is calculated to ensure solutions’ convergence. Interesting flow parameters are examined and plotted graphically. Results show that velocity is increased for large number of fluid rheology and opposite effects are recorded for increasing curvature, Lorentz force, and stretching power. Flow past a flat and curved surfaces are substantial in validation of this present work. DOI:10.46481/jnsps.2023.1435 Keywords: Power-law, cross fluid, radiation, dissipation, MHD, curved surface, Joule heating. Article History : Received: 01 March 2023 Received in revised form: 09 May 2023 Accepted for publication: 10 May 2023 Published: 11 June 2023 c© 2023 The Author(s). Published by the Nigerian Society of Physical Sciences under the terms of the Creative Commons Attribution 4.0 International license (https://creativecommons.org/licenses/by/4.0). Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Communicated by: O. Adeyeye 1. Introduction Several studies that addressed heat transfer problems in thermal engineering have been documented over the years. At- tentions have been given towards heat flow and thermal en- ergy exchange through a physical system that involve heat mechanisms like convection, conduction as well as radiation. This is due to undisputed applications of heat generation, heat conversions, and heat transfer of energy by phase changes in ∗Corresponding author tel. no: Email address: annikennie13@gmail.com (K. M. Sanni ) various devices including condensers, boilers, and evapora- tors. However, fluids flow accompanied by thermal energy over stretchable materials have important applications in many in- dustrial processes which are not limited to- cooling of metallic sheets, liquified and plastic film, glass blowing, paper produc- tion, drawing of plastic film, glass fiber, food processing, poly- mer sheet extrusion from dye among several others. Neverthe- less, evolution of boundary-layer study of heat transfer for non- Newtonian fluids tremendously plays a vital role in processing of material and manufacturing products. Non-Newtonian fluids have been extensively investigated in account for the dynamic behaviours of most natural, biological, and industrial fluids. 1 Sanni et al. / J. Nig. Soc. Phys. Sci. 5 (2023) 1435 2 So far there is no unique model that exhibits all the charac- teristics of non-Newtonian fluids. In response to this observa- tion, various rheology models have been established with cer- tain precisions and limitations. Nonlinear stretching terms such as power-law, exponential, and quadratic have extensively ex- amined to have greater impacts on flow problems physically in view of real-life scenarios. These forms of stretching exhibi- tion are more challenging in exploring the physical significance of flow situations. Literature on this subject is so enormous to be discussed here hence, the present investigation concen- trates only on recent published articles. Hayat et al. [1] ex- amined the heat transport influenced by Cattaneo-Christov heat flux of a stagnation point flow induced past a nonlinear stretch- ing surface. They concluded that non-Fourier heat flux re- duces the heat flow as compared with Fourier expression. Rout and Mishra [2] discussed the heat transfer of a hydromagnetic nanofluid over an unsteady stretching surface. Turkyilmazoglu [3] investigated heat transfer of a microp- olar fluid flow past a stretching porous sheet. He presented a unique close form solution which satisfied the entire physical parameters of the problem. Zeeshan et al. [4] documented the flow of viscous ferro-fluid induced by a stretching surface with magnetic dipole effect and thermal radiation. Their re- sults showed that increased magnetic parameter diminished the velocity field and enhanced the temperature distribution. The numerical solution of Carreau nanofluid flow induced by a non- linear stretching porous surface is carried out by Mohamed et al. [5] in which the impact of rheology parameter of the fluid showed kinetic differences between pseudo-plastic and dilatant nanofluid. Prasnnakumara et al. [6] examined the flow of MHD Sisko nanofluid with nonlinear radiation over nonlinear stretch- ing sheet whereas collocation method is applied by Feroz et a. [7] to analyze the melting heat of Sisko fluid past a moving sheet. Lund et al. [8] investigated dual solution of hydromag- netic Williamson fluid flow with slip condition. Khalil et al. [9] presented a numerical treatment of MHD Casson fluid flow with thermal stratification and dual convection. The heat trans- port of MHD power-law fluid by virtue of Lie-group analysis is provided by Mohamed et al. [10]. Hayat et al. [11] carried out numerical simulation of MHD Cross fluid past a stagna- tion point stretching flat sheet with energy equation under cer- tain flow conditions. Khan et al. [12] examined heat flow of a Cross fluid along axisymmetric channel in a radially stretched plate. Effects of mixed convection and thermal radiative heat flux of a Cross fluid over a stretching plane surface are inves- tigated by Manzur et al. [13]. For more relevant studies for non-Newtonian fluids in the light of Ellis fluid, Eyring-Powell fluid, Maxwell fluid, Sutterby fluid, second and third grade flu- ids, and etc., interested readers are referred [14-26]. The fluid flow induced by curved stretching sheet has at- tracted the interest of researchers in the last decade because of its applications in engineering of stretchable curved materials. Initiated by the fundamental paper of Sajid et al. [27], Abbas et al. [28] included heat transfer analysis in an electrically con- ducting Newtonian fluid flow over stretching curved surface. The results provided in these articles have been recently re- vised by Sanni et al. [29, 30] with the accurate magnetic field included in the momentum equations. The results are modi- fied having satisfied solenoidal magnetic property, ∇.B = 0, as well as the geometry. The consequences of internal heat generation in a flow of nanofluid induced by stretched curved surface is studied by Saba et al. [31]. Hayat et al. [32] of- fered the impacts of homogeneous-heterogeneous reaction on electrically conducting Micropolar fluid past a curved stretch- ing sheet whereas in an unsteady permeable curved sheet, Saleh et al. [33] analyzed the shrinking and stretching flow scenario. The flow of MHD viscous fluid is documented by Naveed et al. [34] under dual solution effects with shrinking curved sheet likewise Sanni et al. [35] documented the viscous fluid flow caused by a nonlinear power-law driven curved surface. Nadeem et al. [36] worked on magneto nanofluid over a stretch- ing curved surface in the presence of variable viscosity and carbon nanotubes. Studies from the aforementioned literature revealed that absolutely, no investigation of heat transport of non-Newtonian Cross fluid over a power-law stretching curved surface has been addressed. Therefore, the prime objective of this study is to investigate heat transfer of a Cross fluid un- der certain flow conditions. It is worth mentioning that non- Newtonian rheological model of some fluids like Sisko fluid, Bingham fluid, and power-law fluid can be recovered with def- inite constraints being met from Cross fluid viscosity equation. For instance, it turns Sisko model if ξ � ξ0, Bingham model taking n = 1, and power-law model for ξ � ξ0 and ξ � ξ∞ [37]. Characterizing parameters, n and Γ, of the fluid curve fit- ting are more important due to strong flow prediction at low and high shear rates. In addition, applied variable magnetic field is considered due to solenoidal nature of magnetism and to elucidates its consequences on the Lorentz force and surface frictional heating. This work is prepared in sections that is the basic equations for the flow momentum, problem description accompanied by the heat transport, numerical computations, and results and discussion. Special cases of Newtonian fluid for flat and curved surfaces remain vital to authenticate the present findings. The results obtained are significantly useful in ther- mal engineering and manufacturing processing of stretchable sheets. 2. Model Description Consider rheological Cross fluid model and Navier-Stokes equation of an incompressible fluid with body force as [37]: S = η∞ + (η0 −η∞) [ 1 1 + (Γγ̇)n ] A1, (1) ρ DV Dt = −p + ∇.S + F, (2) ∇.V = 0, (3) where S is the extra stress tensor with η0 and η∞ being low and infinite shear viscosity shear rate, n, Γ, and γ̇ = √ trace(A1 )2 2 2 Sanni et al. / J. Nig. Soc. Phys. Sci. 5 (2023) 1435 3 Figure 1: Problem Geometry represent fluid behavioural index, material constant, and strain rate tensor respectively. Fluid density denoted by ρ, velocity field V = (v, u, 0), p symbolizes the pressure, F the body force, and ∇ = êr ∂ ∂r + ês R R+r ∂ ∂s , in which êr and ês are unit vector in radial, r− and axial, s− directions. Presenting 2D curvilin- ear coordinates system (r, s, 0), the first Rivlin-Erickson tensor A1 = ∇V + (∇V)T is given by A1 = [ 2 ∂v ∂r ∂u ∂r + R R+r ∂v ∂s − u R+r ∂u ∂r + R R+r ∂v ∂s − u R+r 2R R+r ∂u ∂s + 2v R+r ] . (4) 3. Problem Formulation The steady 2D flow of an incompressible electrically con- ducting fluid driven by nonlinear curved stretching sheet of ra- dius R is considered. The magnetic field B(r) = RB0R+r êr is vari- ably imposed satisfying the solenoidal magnetic property and perpendicular to flow direction. A resistive force under the ef- fect of magnetic Reynolds number yields F = J × B. Electrical field varnishes as E ≈ 0 induced a negligible magnetic field implying that the current density J = σ(V × B). Then we get F̄ = (−σ RB0 r + R u, 0, 0), (5) where B0 is the strength of magnetic field and σ gives conduc- tivity of the fluid. Physical problem geometry is represented in Fig. 1. Utilizing Eqs. (1) - (5), governing flow equations can be expressed as following ∂v ∂r + R r + R ∂u ∂s + v r + R = 0 (6) v ∂v ∂r + Ru R + r ∂v ∂s − u2 r + R = − 1 ρ ∂p ∂r −N1 [ S rs ∂ ∂r ( 1 1 + (Γγ̇)n ) + RS ss R + r ∂ ∂s ( 1 1 + (Γγ̇)n )] +N2 [ 1 R + r ∂ ∂r ( RS rr R + r ) + R R + r ∂S rs ∂s − S ss R + r ] (7) v ∂u ∂r + Ru R + r ∂u ∂s + uv r + R = − R ρ(R + r) ∂p ∂s −N1 [ S rr ∂ ∂r ( 1 1 + (Γγ̇)n ) + RS sr R + r ∂ ∂s ( 1 1 + (Γγ̇)n )] +N2 [ 1 R + r ∂ ∂r ( RS sr R + r ) + R R + r ∂S ss ∂s + S sr R + r ] − σR2 B2o ρ(R + r)2 u, (8) such that N1 = Γn η0 n 2ρ (γ̇) n−2 2 and N2 = η0 ρ (1 − Γn(γ̇) n 2 ). The corresponding stress components S rr , S rs, and S ss are given as S rr = 2 ∂v ∂r (9) S rs = ∂u ∂r + R R + r ∂v ∂s − u R + r (10) S ss = 2R R + r ∂u ∂s + 2v R + r . (11) Relevant boundary conditions associate with this problem are u(0)|r=0 = as m, v(0)|r=0 = 0, (12) u(∞)|r→∞ = 0, ∂u(∞) ∂r |r→∞ = 0, (13) in which a is a constant (l1−mt−1) with l being the surface length, and m the stretching power. Employing suitable scale together with dimensionless characteristics of the form u∗ = u U∞ , v∗ = vl U∞δ , s∗ = s l , r∗ = r δ , R∗ = R δ , p∗ = P ρu2w , b = η0nU 2n∞ δ2n . (14) After using Eq. (14), Eqs (6)-(8) give the boundary-layer equa- tions as u2 r + R = ∂P ∂r , (15) v ∂u ∂r + Ru R + r ∂u ∂s + uv R + r = − 1 ρ − σR2 B20 ρ(R + r)2 u, −N3 ( (n + 1) ∂2u ∂r2 − n − 1 R + r ∂u ∂r + n − 1 (R + r)2 u ) + η0 ρ ( ∂2u ∂r2 + 1 R + r ∂u ∂r − u (r + R)2 ) , (16) in which N3 = nbΓn ( ∂u ∂r − u R+r )n whereas Eq. (6) is identically satisfied. Existing models of Newtonian fluid for both flat and curved surface are recovered in the limiting state as R −→ ∞ at ze- roth pressure, P = 0 and material constant Γ = 0 (see refs. 20,21,27). Relevant similarity variables are calculated as follow u = asmh′(ξ),ξ2 = r2 aρs(m−1) η0 , P = a2 s2 N(ξ), (17) γ2 = R aρs(m−1) η0 , M2 = σB20a 2 η0 , Res = aρs(m+1) η0 , (18) 3 Sanni et al. / J. Nig. Soc. Phys. Sci. 5 (2023) 1435 4 v = −a R R + r √ η0 s(m−1) ρa [( m + 1 2 ) h(ξ) + ξ ( m − 1 2 ) h′(ξ) ] , (19) where h(η) denotes the flow stream, h′(η) represents the flow speed, Res the Reynolds number, and U∞ the ambient velocity. By virtue of Eqs. (17)-(19), Eqs. (15) and (16) become (h′)2 ξ + γ = N′(ξ) (20) 2mγ ξ + γ N(ξ) = h′′′ + h′′ ξ + γ − h′ (ξ + γ)2 + γ ξ + γ [( m + 1 2 ) hh′′ − m(h′)2 ] −Nn4 [ (n + 1)h′′′ + (1 − n)h′′ ξ + γ + (n − 1)h′ (ξ + γ)2 ] − M2γ2h′ (ξ + γ)2 + γ (ξ + γ)2 ( m + 1 2 ) hh′, (21) subject to h(0)|ξ=0 = 0, h ′(0)|ξ=0 = 1, h′(∞)|ξ=∞ = 0, h ′′(0)|ξ=∞ = 0. (22) In these equations, M2 = σB 2 o a 2 η0 represents the Hartmann num- ber and We = a2ΓRes the Weissenberg number whereas N4 = We ( h′′ − h ′ ξ+γ ) . In the limit of γ approaches infinity at N = 0, equations (∀m ≥ 0) of fluid flow past a flat surface is recovered and it substantiates the accuracy of present model. h′′′+ ( m + 1 2 ) hh′′−m(h′)2 = (Weh′′)n(n+1)h′′′−M2h′,(23) Utilizing Eq. (20) in Eq. (21), the pressure term varnishes and we get h′′′′ + 2h′′′ ξ + γ − h′′ (ξ + γ)2 + h′ (ξ + γ)3 + γ ξ + γ [( m + 1 2 ) hh′′′ − ( 3m − 1 2 ) h′h′′ ] − γ (ξ + γ)3 ( m + 1 2 ) hh′ + γ (ξ + γ)2 [( m + 1 2 ) hh′′ − ( 3m − 1 2 ) (h′)2 ] = Nn4 [ (n + 1)h′′′′ + h′′′ ξ + γ + (n − 1)h′′ (ξ + γ)2 − (n − 1)h′ (ξ + γ)3 ] + nNn4( h′′ − h ′ ξ+γ ) [(n + 1)(h′′′)2 − 2nh′′h′′′ ξ + γ + 2nh′h′′′ (ξ + γ)2 + (n − 1)(h′′)2 (ξ + γ)2 ] − nNn4( h′′ − h ′ ξ+γ ) [ (2n − 2)h′h′′ (ξ + γ)3 + (n − 1)(h′)2 (ξ + γ)4 ] + M2γ2 (ξ + γ)2 ( h′′ − h′ ξ + γ ) . (24) 4. Heat Transfer Analysis Under the present flow conditions, the heat transport by virtue of energy equation follows ρC p DT Dt = ∇(K∇T ) + Φ + ∇.q + Qo, (25) where Φ = 2S (S 2rr + S 2 rs + S 2 ss) is viscous dissipation term, Qo = 1 σ (J̄ × J̄) the ohmic heating term and C p, T , K, and q rep- resent fluid- heat capacity, temperature, thermal conductivity, and ohmic heating respectively. By Rosseland expression, radiative flux q̄ gives q̄ = − 4σ∗ 3γ∗ ∂T 4 ∂r (26) such that γ∗, σ∗ are mean spectral absorption coefficient and Stefan-Boltzmann constant respectively. It worth nothing that for small heat kinetics, Eq. (26) can be linearized. However, this investigation is subjected to substantial high nonlinear ra- diation. The radiative heat flux is now expanded as q̄ = − 16σ∗ 3γ∗ T 3 ∂T ∂r (27) Incorporating Eq. (27) into Eq. (25), the temperature boundary layer region becomes v ∂T ∂r + Ru R + r ∂T ∂s = π(1 + RnT 3) ∂2T ∂r2 + 3πRnT 2 ( ∂T ∂r )2 + π R + r ∂T ∂r + σR2 B20u 2 ρC p(R + r)2 + η0 ρC p ( ∂u ∂r − u R+r )2 1 + [ Γ ( ∂u ∂r − u R+r )]n . (28) T → T∞ and T = Tw referred as the ambient and surface temperature respectively whereas π = K ρC p denotes fluid thermal diffusivity and Rn = 16σ ∗ 3γ∗K being the radiation parameter. Require boundary conditions T (0)|r=0 = Tw(0) = 0, T (0)|r→∞ → T∞. (29) Using an existing similarity quantity in non-dimensional form θ(ξ) = T − T∞ Tw − T∞ , (30) Invoking Eqs. (17) - (19) and (30) in Eq. (28) we get (1 + Rn)[1 + (θw − 1)θ]3 Pr θ′′ = 3Rn(1 − θw)[1 + (θw − 1)θ]2 Pr (θ′)2 − θ′ Pr(ξ + γ) − γ(m + 1)hθ′ 2(ξ + γ) − Ec ( h′′ − h ′ ξ+γ )2 1 + [ We ( h′′ − h ′ ξ+γ )]n = γ2w(h′)2 Pr(γ + ξ)3 (31) 4 Sanni et al. / J. Nig. Soc. Phys. Sci. 5 (2023) 1435 5 Corresponding boundary conditions become θ(0)|ξ=0 = 1,θ(∞)|η=∞ = 0, (32) in which Pr = η0C p K , θw = Tw T∞ , Ec = U 2 ∞ C p (Tw−T∞) , Rd = 16σ ∗T 3∞ 3k∗K0 , and w = σB20 U 2 ∞a 2 K(Tw−T∞) are known as Prandtl number, temperature parameter (θw ≥ 1), Eckert number, radiation parameter, and Ohmic heating parameter respectively. and Br = PrEc the Brinkman number. It worth mentioning that radiative flux arises when θw > 1. In response to industrial significance of this work from engineering point of view, physical quantities such as surface drag force and rate of heat transfer are calculated as Csk = τrs|r=0 1 2 ρu 2 w , Nu = sq̄ K(T − Tw) , (33) where uw = asm, τrs|r=0 = η0( ∂u∂r − uR+r ) 1+[Γ( ∂u∂r − uR+r )] n |r=0 , and q̄ = −K ∂T ∂r |r=0. Utilizing this expression together with Eqs. (15)-(19) into Eq. (33), we obtain − 1 2 R 1 2 esCsk = h′′(0) − 1 γ 1 + [ We ( h′′(0) − 1 γ )]n (34) and R − 1 2 es Nu = −θ ′(0). (35) These equations are indispensable in thermal engineering and flow control. 5. Computations Approach Analytic approach to solve the boundary value problems, BVPs obtained in Eqs. (24) and (31) subjected to Eqs. (22) and (32) remains a task and open for future investigation. This section presents a numerical method. Firstly, BVPs initializes into initial value problem, IVPs using h = s1, h ′ = s2, h ′′ = s3, h ′′′ = s4,θ = s5,θ ′ = s6, (36) After effecting Eq. (36), Eqs. (24) and (31) are linearised into first order of the form  s′1 s′2 s′3 s′4 s′5 s′6  =  s2 s3 s4 −2βs4 + β2 s3 −β3 s2 −βγ [( m+1 2 ) s1 s4 + βγ ( 3m−1 2 ) s2 s3 ] −(We)n(s3 −βs2)n[(n + 1)s′4 + βs4 + (n − 1)β 2 s3 − (n − 1)β3 s2] −n(We)n(s3 −βs2)n−1[(n + 1)s24 − 2nβs3 s4 + 2nβ 2 s2 s4 +(n − 1)β2 s23 − (2n − 2)β 3 s2 s3 + (n − 1)β4 s22] +M2γ2β2(s3 −βs2)s6 − 1 1+RnD31 [ γβPr ( m+1 2 ) s1 s6 + 3Rn(θw − 1)D21 s 2 6 + βs6 ] − 1 1+RnD31 ( PrEc(s3−s2 )2 1+[We(s3−s2 )]n + wγ2β2 s22 )  , (37) subject to s1(0) s2(0) s3(0) s4(0) s5(0) s6(0)  =  0 1 z1 z2 1 z3  (38) where D1 = 1 + (θw − 1)θ and β = 1 ξ+γ whereas z1, z2, and z3 are initial missing conditions. Employing Keller-Box shooting technique together with Jacobi iterations [38]. Let w′ = w1, w ′ 1 = w2, w ′ 2 = w3, ..., w ′ k−1 = H(ξ, w1, w2, w3, ..., wk−1), (39) wk+1i − w k i−1 δh = (w1) k i−1/2, (w1)k+1i − (w1) k i−1 δh = (w2) k i−1/2, ... (wn−1)k+1i − (wn−1) k i−1 δh = H2(ξ k i−1/2, w k i−1/2, (w1) k i−1/2, ..., (wn−1) k i−1/2) (40) Implementing Eq. (40) explicitly, we can write (δh)−1(sk+1i − s k i−1) = −0.5((s1) k i + (s1) k i−1), sk+10 = 0, (δh)−1((s1) k+1 i − (s1) k i−1) = −0.5((s2) k i + (s2) k i−1), (s1) k+1 0 = 1, (δh)−1((s2) k+1 i − (s2) k i−1) = −0.5((s3) k i + (s3) k i−1), (s2) k+1 0 = z1, (s4) k+1 i − (s4) k i−1 = −δh[2β((s4) k i + (s4) k i−1) 5 Sanni et al. / J. Nig. Soc. Phys. Sci. 5 (2023) 1435 6 −β2((s3) k i + (s3) k i−1)] −δh[((s3) k i + (s3) k i−1) −β((s2) k i + (s2) k i−1)] n A1 −δh[((s3) k i + (s3) k i−1) −β((s2) k i + (s2) k i−1)] n−1 A2 −δhβ3((s2) k i + (s2) k i−1) + δhM 2γ2β2 A5 −δhγβ ( m + 1 2 ) A3 + δhγβ ( 3m + 1 2 ) A4 (s3) k+1 0 = z2, (s5) k+1 i − (s5) k i−1 = −0.5δh((s6) k i − (s6) k i−1), (s5) k+1 0 = 1, (s6) k+1 i − (s6) k i−1 = − δh 1 + RnD21 A6, (s6) k+1 0 = z3, where zi for i = 1(1)3 are missing initial conditions. These constants are randomly selected using MATLAB solver. Their choices of values are changed and the accuracy is determined with corresponding end point. The iterations loop with the step size (δh = 0.01) until the solutions are found satisfying the criterion |‖w(n+1)‖2 −‖wn‖2| < �. 6. Error Analysis Theorem: Let W1 = W1(h, h1, h2, h3), and W2 = W2(θ,θ1, h, h1, h2) be differentiable functions, the maximum error reached by the Keller-Box shooting technique together with Jacobi iterations for Eq. (24) is bounded. Proof: Discretising this equation following the Keller-Box method with Jacobi iterative scheme, one can write hn+1i − h n i−1 δh + (h1) n i−1/2 = 0, (h1)n+1i − (h1) n i−1 δh + (h2) n i−1/2 = 0, (h2)n+1i − (h2) n i−1 δh + (h3) n i−1/2 = 0, (h3)n+1i − (h3) n i−1 δh + (W1) n i−1/2 = 0 (41) in which the exact scheme can be written as hEi − h E i−1 δh + (h1) E i−1/2 = 0, (h1)Ei − (h1) E i−1 δh + (h2) E i−1/2 = 0, (h2)Ei − (h2) E i−1 δh + (h3) E i−1/2 = 0, (h3)Ei − (h3) E i−1 δh + (W1) n i−1/2 = 0, (42) subject to solution error at any grid point of the form (e1) n i = h n 1 − h E 1 , Figure 2: (a) to (e). Effects of γ, M, m, n, and We on f (ξ) (e2) n i = (h1) n i − (h1) E i , (e3) n i = (h2) n i − (h2) E i , 6 Sanni et al. / J. Nig. Soc. Phys. Sci. 5 (2023) 1435 7 Figure 3: (a) to (e). Effects of γ, M, m, n, and We on f ′(ξ) (e4) n i = (h3) n i − (h3) E i . (43) Figure 4: (a) to (e). Effects of γ, M, m, n, and We on θ(ξ) By virtue of Mean Value Theorem, we get W1(h n i , (h1) n i , (h2) n i , (h3) n i ) − W1(h E i , (h1) E i , (h2) E i , (h3) E i ) 7 Sanni et al. / J. Nig. Soc. Phys. Sci. 5 (2023) 1435 8 Figure 5: (a) to (e). Effects of Pr, Ec, Rn, w, and θw on θ(ξ) = (ē1) n i .∇W1(z1, z2, z3), (44) in which z1 = h n i + ε1(h2) n 1, z2 = (h1) n i + ε2(e2) n i , z3 = (h2) n i + ε3(e3) n i , (45) zi�[0, 1] for i = 1(1)3, and (ē1)ni = [(e1) n i , (e2) n i , (e3) n i ], whereas convergence error equations are define as (e1) n+1 = (e1) n i−1 + δh(e1) n i−1/2, (e2) n+1 i = (e2) n i−1 + δh(e2) n i−1/2, (e3) n+1 i = (e3) n i−1 + δh(ê1) n i−1/2∇W1, (46) from Eq. (46), once can obtain the inequalities |(e1) n+1 i | ≤ |(e1) n i−1| + δh|(e2) n i−1/2|, |(e2) n+1 i | ≤ |(e2) n i−1| + δh|(e3) n i−1/2|, |(e3) n+1 i | ≤ |(e3) n i−1| + δh|(ê1) n i−1/2.∇W1|. (47) Setting ∇W1 = [W̄ 11 , W̄ 2 1 , W̄ 3 1 ], Eq. (47) can be written as |(e3) n+1 i | ≤ |(e3) n i−1| + δh|Σ 3 j=1(e j) n i−1/2Ŵ j 1|, ≤ |(e3) n i−1| + δhΣ 3 j=1|(e j) n i−1/2W̄ j 1|, (48) such that |(e3) n+1 i | ≤ |(e3) n i−1| + δh|Σ 3 j=1(e j) n i−1/2Ŵ j 1|, ≤ |(e3) n i−1| + δhΣ 3 j=1|(e j) n i−1/2W̄ j 1|. (49) The maximum error is estimated in the form (e1) n i = maxi=1(1)N|(e1) n i |, (e2) n i = maxi=1(1)N|(e2) n i |, (e3) n i = maxi=1(1)N|(e3) n i |, (ē)n = max[maxi=1(1)N (e1 = 1(1)N) n i ], (50) where N denotes the number of nodes and Eqs. (47) and (48) can be expressed as en+11 ≤ e n 1 + δhe n 2 + M̄1O(δh) 2, en+12 ≤ e n 2 + δhe n 3 + M̄2O(δh) 2, ēn+1 ≤ (1 + 4δhΣ4j=1|W̄ j 1|)ē n + M̄3O(δh) 2. (51) Evaluating n = 0, 1, and n in the above expression, one can get ē1 ≤ (1 + 4δhΣ4j=1|W̄ j 1|)ē 0 + M̄3O(δh) 2, ē2 ≤ (1 + 4δhΣ4j=1|W̄ j 1|) 2ē0 +[1 + (1 + 4δhΣ4j=1|W̄ j 1|)]M̄3O(δh) 2, ēn ≤ (1 + 4δhΣ4j=1|W̄ j 1|) nē0 +[1 + (1 + 4δhΣ4j=1|W̄ j 1|) + ..., +(1 + 4δhΣ4j=1|W̄ j 1|) n−1]M̄3O(δh) 2. (52) Taking nth term series sum, we obtain ēn ≤ (1 + 4δhΣ4j=1|W̄ j 1|) nē0 8 Sanni et al. / J. Nig. Soc. Phys. Sci. 5 (2023) 1435 9 +  [1 + 4δhΣ4j=1|W̄ j 1|] n 4δhΣ4j=1|W̄ j 1|  M̄3O(δh)2, ≤ (1 + 4δhΣ4j=1|W̄ j 1|) nē0 +EXP(4δhΣ4j=1|W̄ j 1|)M̄3O(δh) 2. (53) Utilizing Eq. (53) into Eq. (52), one can get en1 ≤ (1 + δh)[(1 + 4δhΣ 4 j=1|W̄ j 1|) nē0 +EXP(4(n − 1)δhΣ4j=1|W̄ j 1|)M̄3O(δh) 2] +M̄1O(δh) 2, (54) en2 ≤ (1 + δh)[(1 + 4δhΣ 4 j=1|W̄ j 1|) nē0 +EXP(4(n − 1)δhΣ4j=1|W̄ j 1|)M̄3O(δh) 2] +M̄2O(δh) 2. (55) Equations (54) and (55) express the maximum error bounds of Eqs. (22) and (24). In similar ways, maximum error can be obtained in view of Eqs. (31) and (32) (see ref. [40]). 7. Results and Graphical Analysis This section presents the graphical illustrations of the flow quantities that is; the stream function h(ξ), velocity h′(ξ), and temperature θ(ξ) against characterizing parameters. Figures 2(a, b, and c) show the behaviour of stream function in re- sponse to curvature parameter γ, magnetic field M, and non- linear stretching power m. In these graphs, the stream function decreases for increas- ing these parameters which means that the flow trajectory can be minimized either by reducing the curvature or enhancing the Lorentz force as well as stretching power. On the other hand, in Figs. 2(d and e) the flow trajectory is enhanced by increasing the rheology parameters (n and We) of the fluid. This means that fluid resistance to flow reduces and the fluid behaves more likely as shear thinning. Figure 3(a) shows that the velocity h′(ξ) decreases steadily by increasing radius of curvature (as the surface becomes flat). The velocity graphs plotted in Figs. 3(b and c) elucidate reductions in flow velocity as a consequence of opposing Lorentz force due to applied high magnetic field. This observation substantiates the impact of the Lorentz force and stretching power m and confirms the above conclusion made for the flow trajectory. In other words, the flow velocity can be regulated with the aid of geometry parameters that is radius of curvature, magnetic field, and stretching power. To investi- gate the rheology effects n- fluid power-law behavior and We- Weissenberg number, figures 3(d and e) are plotted. In these figures, the velocity increases for increasing n and We. Physically, shear thinning (low viscosity) of the fluid shows less resistance to flow and invariably enlarges the mo- mentum boundary-layer. Figure 4 is given to illustrates the in- fluence of nonlinear radiation θw > 1 on temperature profile as compared to linear radiation θw = 1. In Fig. 4(a), we observe that increasing radius of curvature γ decreases the temperature profile θ(ξ) as well as thermal boundary layer region. This ob- servation on the other hand shows that, heat flows more quickly over curved surface which can be controlled by value of γ. A striking difference is obvious when θw = 1.5 as compared with θw = 1. Effect of increasing magnetic field on temperature is displayed in Fig. 4(b). The thermal characteristics is increased due to opposing Lorentz force. A usual contribution in thermal engineering study with reason being that the force induced heat from the curved surface to the fluid. The effect of stretching power on temperature profile is plotted in Fig. 4(c). This graph shows that increasing m shrinks the thermal region as well as associated thermal boundary layer. This observation infers that either any side of the thermal spectrum can also be controlled by virtue of stretching power of the fluid velocity. Figures 4(d and e) elucidate the impacts of fluid parameters (n and We) on temperature profile. In both graphs, the ther- mal kinetic energy decreases for large fluid power-index and Weissenberg number. This observation implies that shear thin- ning fluid absorbed more heat due to weak or free viscous bond. Figure 5 maintains the usual interpretations of the various phys- ical parameters involved in this problem. Figure 5(a) shows a decrease in temperature profile as a consequence of lowering thermal conductivity of the fluid for large Pr. Figure 5(b) in- dicates a positive response of viscous heating in enhancing the thermal energy of the fluid by increasing Ec. This observation is more significant when θw = 1.5. The effect of radiation on temperature profile is given in Fig. 5(c). As habitual source of heat, increasing Rn enlarges the thermal boundary layer thick- ness and temperature profile. The effect of Ohmic heating from surface to the temperature field is given in Fig. 5(d). In this graph, the thermal characteristics increase slightly for increasing w due to additional heat generated at the surface to the fluid and invariably enhances fluid temperature. Finally, the impact of temperature difference on temperature as well as thermal boundary layer thickness is substantiated in Fig. 5(e) showing a positive influence of nonlinear radiation. In other words, increasing temperature difference θw parameter expands the temperature and associated boundary thickness. Table 1 compares the present results of heat transfer with published work in the limiting case when n = 0, Rn = 0, M = 0, w = 0, Ec = 0, and We = 0. It can be seen that the results in the lit- erature are special cases of the present study. Table 2 presents the impact of various parameters on the skin friction coefficient and rate of heat transfer for the present problem. 8. Concluding Remark Modelling of nonlinear radiative heat transfer of an elec- trically conducting Cross-fluid under variable applied magnetic field has been studied. Governing equation is conducted in the presence of viscous dissipation and Ohmic heating. The fluid motion is induced by power-law stretching velocity over curved surface mechanism.The behaviours of flow field- velocity as well as temperature against interesting parameters are summa- rized. 1. The flow stream patterns, velocity, and associated boundary-layer thickness are increased for large number 9 Sanni et al. / J. Nig. Soc. Phys. Sci. 5 (2023) 1435 10 Table 1: Comparison of Nusselt number, −θ′(0) with published results Pr Hammad Wang Khan et al. G and S Ijaz et al. Present [17] [18] [12] [21] [14] Results 0.2 0.139100 0.169100 0.164037 0.139100 0.196550 0.196502 0.7 0.453900 0.453900 0.418299 0.453900 0.454446 0.454369 20 3.353900 3.353900 3.256030 3.353900 3.359500 3.353902 70 6.462200 6.462200 6.366620 6.462200 6.462290 6.462200 Table 2: Numerical values of the skin-friction coefficient and rate of heat transfer for n = 2. Parameters θw = 1 θw = 1.5 γ M m Ec Pr Rn We w Re0.5s Csk −θ(0) −θ(0) 5 0.3 0 0.2 2 0.9 2 0.5 0.24141 0.48065 0.26879 - 0.4 - - - - - - 0.24029 0.47929 0.26750 - 0.5 - - - - - - 0.23888 0.47758 0.26590 - - 1 - - - - - 0.22868 0.66205 0.25535 - - 2 - - - - - 0.21036 0.82918 0.22628 10 - 0 0.1 - - - - 0.24128 0.50082 0.28532 - - - 0.2 - - - - - 0.48188 0.27561 - - - 0.3 - - - - - 0.46295 0.26590 - - - 0.2 - - - - - 0.48188 0.33336 - - - - 3 - - - - 0.46474 0.38631 5 - - - 2 0.5 - - 0.23888 0.55891 0.35957 - - - - - 0.7 - - - 0.51446 0.30477 - - - - - - 1 - 0.49760 0.41772 0.22763 - - - - - - 1.5 - 0.32994 0.45326 0.24932 10 - - - - - - 0.5 0.24128 0.48188 0.27561 - - - - - - - 1.5 - 0.20253 0.13530 - - - - - - - 2 - 0.06285 0.06495 of fluid rheology n and Weissenberg number We. Oppo- site effects are observed for increasing geometry curva- ture γ, magnetic field M, and stretching power m. That is, the flow stream together with the velocity and associ- ated boundary-layer thickness are decreased. 2. The temperature profile together with thermal boundary- layer are reduced for large curvature and stretching power. 3. Increase in magnetic field enhances the thermal region and consequently enlarges its thermal boundary-layer. 4. Existing interpretation of thermal profiles for Eckert number Ec, Prandtl number Pr, radiation Rn parameter, and Ohmic heating parameter w, are well established. 5. The present work is comparatively substantiated by the impact of nonlinear radiation through temperature differ- ence θw parameter. Therefore, profound results are well recorded in both linear and nonlinear radiations. 6. The practical significance of the present results is that, the flow as well as heat transfer phenomena can be con- trolled, maintained, and optimized through various stud- ied parameters. Thus, theses results obtained are useful in polymer dynamics of stretchable materials and curved mechanism. References [1] T. Hayat, M. Zubair, M. Ayub, M. Waqas & A. Alsaedi, “ Stagnation point flow towards nonlinear stretching surface with Cattaneo-Christov heat flux”, Eur Phys J Plus 131 (2016) 355. [2] B.C. Rout & S. R. Mishra, “ Thermal energy transport on MHD nanofluid flow over a stretching surface: A comparative study”, Eng Sci Tech Int J 21(1) (2018) 69. [3] M. Turkyilmazoglu ”Flow of a micropolar fluid due to a porous stretching sheet and heat transfer”, Int J Nonlinear Mech 83 (2016) 64. [4] A. Zeeshan, A. Majeed & R Ellahi, “Effect of magnetic dipole on viscous ferro-fluid past a stretching surface with thermal radiation”, J Mol Liq 215 (2016) 554. [5] R. Mohamed, M. L. Eid Kasseb, T. Mohammed & M. Sheikholeslami, “Numerical treatment for Carreau nanofluid flow over a porous nonlinear stretching surface”, Results Phys 8 (2018) 1193. 10 Sanni et al. / J. Nig. Soc. Phys. Sci. 5 (2023) 1435 11 [6] B.C. Prasannakumara, B. J. Gireesha, M. R. Krishnamurthy & K. G. Ku- mar, “MHD flow and nonlinear radiative heat transfer of Sisko nanofluid over a nonlinear stretching sheet”, Inf Med Unlocked 9 (2017) 132. [7] A. S. Feroz, M. Usman, R. Ul Haq & W. Wang, “Melting heat transfer analysis of Sisko fluid over a moving surface with nonlinear thermal ra- diation via collocation method”, Heat Mass Transfer 126 (2018) 1042. [8] L. A. Lund, Z. Omar & K llyas, “Analysis of dual solution for MHD flow of Williamson fluid with Slippage”, Heliyon 5(3) (2019) 20. [9] R. Khalil, A. Qaiser, M. Y. Malik & U. Ali, “Numerical communication for MHD thermally stratified dual convection flow of Casson fluid yields by stretching cylinder”, Chinese J Phys 55(4) (2017) 1614. [10] A. Mohamed & A. A. Afify ”Lie group analysis of hydromagnetic flow and heat transfer of a power-law fluid over stretching surface with temperature-dependent viscosity and thermal conductivity”, Int J Mod Phys 27(11) (2016) 20. [11] T. Hayat, M. K. Ijaz, M. Tamoor, M. Waqas & A. Alsaedi ”Numerical Simulation of heat transfer in MHD Stagnation point flow of Cross fluid model towards a stretched surface”, Results Phys 7 (2017) 1827. [12] M. Khan, M. Manzur & M. Rahman, “On axisymmetric flow and heat transfer of Cross fluid over a radially stretching sheet”, Results Phys 7 (2017) 3772. [13] M. Manzur, M. Khan & M. Rahman, “Mixed convection heat transfer to Cross fluid with thermal radiation: Effects of buoyancy assisting and opposing flows”, Int J Mechanical Sci 139 (2018) 523. [14] M. K. Ijaz, W. Waqas, T. Hayat & A. Alsaedi, “Magneto-hydrodynamical numerical simulation of heat transfer in MHD stagnation point flow of Cross fluid model towards a stretched surface”, Phys Chem Liq 56(5) (2017) 595. [15] I-C. Liu, “Flow and heat transfer of an electrically conducting fluid of second grade in a porous medium over a stretching sheet subject to a transverse magnetic field”, Int. J. Nonlinear Mech 40 (2005) 474. [16] T. Hayat, A. Shafiq & A. Alsaedi, “Effect of Joule Heating and Thermal Radiation in flow of third grade Fluid over Radiative surface”, PLoS ONE 9(1) (2014) 153. [17] A. A. Hamad, “Analytical solution of natural convection flow of a nanofluid over a linearly stretching sheet in the presence of magnetic field”, Int Commun Heat Mass 38 (4)(2011) 492. [18] A. B. Disu & S. O. Salawu, “Thermal distribution of magneto-tangent hy- perbolic flowing fluid over a porous moving sheet: A lie group analysis”, J. Nig. Soc. Phys. Sci. 5 (2023) 1103. [19] P. B. Sharma, M. Kapalta, A. Kumar, D. Bains, S. Gupta & P. Thakur, “Electrodynamic Convection Dieletric Rotating Oldroydian Nanofluid in Porous Medium”, J. Nig. Soc Phys Sci 5 (2023) 1137. [20] C. Y. Wang, “Free convection on a vertical stretching surface”, J Appl Math Mech 9 (1989) 420. [21] R. S. R. Gorla & R. I. Sidawi, “Free convection on a vertical stretching surface with suction and blowing”, Appl Sci Res 52 (1994) 257. [22] M. Jalil & S. Asghar, “Flow of power-law fluid over a stretching surface: A Lie group analysis”, Int J Nonlinear Mech 48 (2018) 71. [23] M. Mustafa, J. A. Khan, T. Hayat & A. Alsaedi, “Simulation for Maxwell fluid flow past a convectively heated exponentially stretching sheet with nanoparticles”, AIP Adv 5(3) (2015) 133. [24] M. Khan, R. Malik, A. Munir & W.A. Khan, “Flow and heat transfer in Sisko fluid with convective boundary conditions”, PLoS ONE 9(10) (2014) 989. [25] T. Hayat, M. Farooq, A. Alsaedi & Z. Iqbal, “Melting heat transfer in the stagnation point flow of Powell-Eyring fluid”, J Thermophys Heat 27 (2013) 766. [26] E. Azhar, Z. Iqbal, S. Ijaz & E. N. Maraj, “Numerical approach for stag- nation point flow of Sutterby fluid impinging to Cattaneo-Christov heat flux model”, Pramana – J Phys 91 (2018) 61. [27] M. Sajid, N. Ali, T. Javed & Z. Abbas, “Stretching a curved surface in a viscous fluid”, Chinese Phys Lett 27 (2010) 703. [28] Z. Abbas, M. Naveed & M. Sajid, “Heat transfer analysis for stretching flow over a curved surface with magnetic field”, J Eng Thermophys 22 (2013) 345. [29] K. M. Sanni, Q. Hussain & S. Asghar, “Heat transfer analysis for non- linear boundary driven flow over a curved stretching sheet with a variable magnetic field”, Front Phys 8 (2020) 113. [30] K. M. Sanni, Q. Hussain & S. Asghar, “Thermal Analysis of a Hydromag- netic Viscoelastic Fluid Over a Continuous Curved Stretching Surface in the Presence of Radiative Heat Flux”, Arab J Sci Eng 2020 2020 8. [31] F. Saba, N. Ahmed, S. Hussain, U. Khan & S. T. Mohyd-Din, “Thermal Analysis of Nanofluid Flow over a Curved Stretching Surface Suspended by Carbon Nanotubes with Internal Heat Generation”, Appl Sci 8 (2018) 395. [32] T. Hayat, R. Sajjad, R. Ellahi, A. Alsaedi & T. Muhammad, “Homogeneous-heterogeneous reaction in MHD flow of Micropolar fluid by a Curved Stretching Surface”, J Mol Liq 240 (2017) 220. [33] S. H. Saleh, M. Arfin, R. Nazar & I. Pop, “Unsteady Micropolar Fluid over a Permeable Curved Stretching Shrinking Surface”, Math Probl Eng 2017 (2017) 13. [34] M. Naveed, Z. Abbas & M. Sajid, “Dual Solutions in Hydromagnetic Viscous Fluid Flow Past a Shrinking Curved Surface”, Arab J Sci Eng 43 (2018) 1194. [35] K.M. Sanni, S. Asghar, M. Jalil & N. F. Okechi, “Flow of viscous fluid along a nonlinearly stretching curved surface”, Results Phys 7 (2017) 7. [36] S. Nadeem, Z. Ahmed & S. Saleem, “Carbon nanotubes effect in magneto nanofluid flow over a curved stretching surface with variable viscosity, Microsyst Technol”, 25 (2018) 2888. [37] M. M. Cross, “Rheology of non-Newtonian fluids: A new flow equation for pseudoplastic systems”, J Colloid Sci 20(5) (1965) 437. [38] Yasir N & Shoaib Arif M, “Keller-Box shooting method and its applica- tion to nanofluid flow over convectively heated sheet with stability and convergence. Num Heat Transfer, B: Fundamentals”, 76 (2019) 180. 9. APPENDIX A1 = (We) n (n + 1) (s4)k+1i − (s4)ki−1 δh + β((s4) k i + (s4) k i−1)  + (We)n(n − 1)β2((s3) k i + (s3) k i−1) − (We)n(n − 1)β3((s2) k i + (s2) k i−1) A2 = n(We) n(n + 1)((s4) k i + (s4) k i−1) 2 − 2n2β(We)n((s3) k i + (s3) k i−1)((s4) k i + (s4) k i−1) + 2n2β2(We)n((s2) k i + (s2) k i−1)((s4) k i + (s4) k i−1) + n(n − 1)(We)nβ2((s3) k i + (s3) k i−1) 2 − n(2n − 2)(We)nβ3((s2) k i + (s2) k i−1)((s3) k i + (s3) k i−1) + (n − 1)β4((s2) k i + (s2) k i−1) A3 = ((s1) k i + (s1) k i−1)((s4) k i + (s4) k i−1) A4 = ((s2) k i + (s2) k i−1)((s3) k i + (s3) k i−1) A5 = (((s3) k i + (s3) k i−1) −β((s2) k i + (s2) k i−1)) A6 = γβPr ( m + 1 2 ) ((s1) k i + (s1) k i−1)((s6) k i + (s6) k i−1) + 3Rn(θw − 1)D 2 1((s6) k i + (s6) k i−1) 2 + β((s6) k i + (s6) k i−1) + PrEc(((s3)ki + (s3) k i−1) −β((s2) k i + (s2) k i−1)) 2 1 + [We(((s3)ki + (s3) k i−1) −β((s2) k i + (s2) k i−1))] n + wγ2β2((s2) k i + (s2) k i−1) 2 11