J. Nig. Soc. Phys. Sci. 5 (2023) 1103 Journal of the Nigerian Society of Physical Sciences Thermal distribution of magneto-tangent hyperbolic flowing fluid over a porous moving sheet: A Lie group analysis A. B. Disua, S. O. Salawub,∗ aDepartment of Mathematics, National Open University, Abuja, Nigeria bDepartment of Mathematics, Bowen University, Iwo, Nigeria Abstract An investigation of magneto-hyperbolic tangent fluid motion through a porous sheet which stretches vertically upward with temperature-reliant thermal conductivity is scrutinized in this study. The current model characterizes thermal radiation and the impact of internal heat source in the heat equation plus velocity and thermal slipperation at the wall. The translation of the transport equations is carried out via the scaling Lie group technique and the resultant equations are numerically tackled via shooting scheme jointly with Fehlberg integration Runge-Kutta scheme. The results are publicized through various graphs to showcase the reactions of the fluid terms on the thermal and velocity fields. From the investigations, it is found that rising values of the material Weissenberg number, slip and suction terms damped the hydrodynamic boundary film whereas the heat field is prompted directly with thermal conductivity. DOI:10.46481/jnsps.2023.1103 Keywords: Thermal conductivity, Magneto-Tangent hyperbolic liquid, Porous sheet, Scaling Lie group, Thermal radiation Article History : Received: 03 October 2022 Received in revised form: 03 November 2022 Accepted for publication: 11 December 2022 Published: 24 February 2023 © 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: T. Latunde 1. Introduction Magnetohydrodynamic is the interaction of an electromag- netic field with conducting fluids. Rheostat flow kinematics is a realistic application of magnetohydrodynamic where a mag- netic field is used in conventional fluids since the heat exchange rate is unavailable for some sheet materials. Other uses include; thermal insulation, power storage, and so on. In the light of this, Abbas et al. [1] investigated a lateral stretching sheet of MHD power-law fluid with differing thermal conductivity. The au- thors reported a shrinking boundary layer structure with growth ∗Corresponding author tel. no: +234 8032056439 Email address: kunlesalawu2@gmail.com (S. O. Salawu) in the magnetic field term at all phase of fluid categories consid- ered in the report. More so, Salawu et al. [2] analyzed magne- tohydrodynamic viscous liquid flow across a nonlinear stretchy plate where the model ordinary derivative equations of the sys- tem was solved using the collocation-approximation. Sarkar and Makinde [3] explored the viscous fluids heat transport and magnetohydrodynamic flow across an exponentially stretching layer accounting for viscous dissipation and radiation effect. Nadeem et al. [4] explored the Magnetohydrodynamic rheolog- ical fluid flow on an angular boundary layer flow. The analysis showed that a boost in the magnetic field strength, the regular and tangential velocity profiles diminish while the skin friction coefficient increase. Meanwhile, the studies conducted on the flow along a stretch- 1 Disu & Salawu / J. Nig. Soc. Phys. Sci. 5 (2023) 1103 2 ing sheet has become a great deal of interest among researchers in respect of significant contributions in manufacturing and en- gineering activities. Sakiadis [5] analyzed the continuously mov- ing flat surface on a laminar boundary layer and obtained a computational solution for the boundary layer equations. Crane [6] extended this by reporting on the time-independent linearly stretchy flow and specified the solutions in a closed form. Such a study was also extended by Wang [7] by incorporating par- tial slip effect, whereas Tshivhi et al. [8] investigated such a concept over a flat stretchable sheet when the flow is ini- tiated spontaneously from rest. Okedoye et al. [9] analyzed slip fluid motion confined in a permeable stretchable material while the impact of partial slip due to vertical stretching sheet on stagnation-point flow with thermal transport was assessed by Zaimi and Ishak [10]. Salawu et al. [11] examined the cross- diffusion impact on magnetohydrodynamic fluid flow through a stretched sheet with velocity slip. A case of MHD dissipative fluid flow occasioned by a non-linearly stretched material with heat-mass transfer was numerically evaluated by Upreti et al. [12]. It was pointed out from the analysis that the thermal field is enlarged by the enhancement of the magnetic field. Such an investigation was also extended to the transport of Casson liq- uid configuration in a three-dimensional sheet with pores and Joule heating effect by Sreenivasulu et al. [13] while Fatun- mbi and Okoya [14] inspected hydromagnetic micropolar fluid thermal transport characteristics over a stretching material fea- turing the prescribed thermal flux and plate temperature heating conditions. The studies of non-Newtonian fluids have inspired scien- tists and engineers in recent times owing to its many uses in science and technology including food processing, drug and pharmaceutical productions, chemical engineering works and many more, Salawu et al. [15-16]. Examples of non-Newtonian fluids include gels, paints, blood, printers ink, lubricants with polymer ingredients, cosmetics and toiletries. Among various non-Newtonian fluid theories, there exists the tangent hyper- bolic fluid model commonly utilized in numerous laboratory and chemical engineering processes. This fluid model displays a shear thinning attributes such that there exists a decline in the viscosity as the shear rate rises, Hassan et al. [17]. This unique feature of tangent hyperbolic fluid makes it a sought af- ter in bio-engineering operations, for instance, the thinning at- tributes of blood flow in the body serves as a prevention to the obstruction of arteries and veins such that coagulation effect is minimized, Alsharif et al. [18]. In view of such striking char- acteristics, various researchers have applied this fluid model to analyze various flow problems under different configurations. Mamatha et al. [19] examined the motion of hydrodynamic tangent hyperbolic liquid mixed with dust particles in a porous stretching plate with convective heating. A numerical evalua- tion of such a phenomenon towards a stagnation-point in the occurrence of radiative heat, nonlinear convection, haphazard motion and thermo-migration of nanoparticles was scrutinized by Khan et al. [20]. Meanwhile, the distribution of such a liquid with nanomaterials mixture in a nonlinear stretchable material was scrutinized by Mahanthesh and Mackolil [21] for a stag- nation fluid. These researchers reported a rise in the viscous drag due to enhancement in the power-law index and magnetic field terms. Oyelakin and Sibanda [22] inspected the influence of exponentially based viscosity on the motion of hyperbolic tangent fluid. The report showed that a decrease in the viscos- ity triggered a spike in the velocity while lowering the heat and species intensity. Sophus and Ackerman [23] found point metamorphosis that mapped a given differential equation and introduced the Lie group analysis classical approach. This approach brings to- gether nearly every known technique of exact integration for all the associated ordinary and partial differential equations. Many researchers employed this technique to determine the similari- ties among given differential equations. Using this technique, the number of variables that control the partial differential sys- tems can be effectively reduced, Salawu and Dada [24]. The dilution of values transforms the partial differential system into ordinary systems. Using Lie group analysis approach, con- vective dynamics problems have been studied on different flow configurations in various science and engineering branches, Za- kir and Zaman [25]. Similarly, Ullah and Zaman [26] engaged this approach while studying the transport and thermal effects of a tangent hyperbolic flowing liquid through a stretched plate with Navier slip effect. Further, Ullah et al. [27] engaged this approach to extend the work of [26] by incorporating suc- tion/injection coupled with heat generation. The authors exam- ined the partial differential equations representing a natural con- vective unstable flow movement using the Lie symmetry trans- formation approach. The classical Lie group transformation is applied twice sequentially in this study to change the transport model into a set of ordinary derivative equation. The above studies however ignored the impact of variable heat conduction in the temperature field. Thermal conductivity describes the characteristic quantity of fluids that allows them to conduct heat. For accurate prediction of thermal propagation processes, the influence of temperature-based thermal conduc- tivity has to be considered. Shahzad et al. [28] investigated such an effect on a viscous fluid in the existence of a stretching layer by utilizing the shooting process and the perturbation pro- cedure in analyzing the numerical solution. Similarly, Alsherif et al. [29] took into account the case of a stretching cylinder, considered a viscous fluid flow alongside variable thermal con- ductivity. The investigation depicted that growing the curvature of the cylinder causes the fluid temperature to rise rapidly. An examination of temperature based thermal conductivity coupled with thermal radiation impact of a viscous fluid in a porous stretching material was evaluated by Hayat et al. [30]. Ullah et al. [31] reported on power-law convective MHD liqud flow across a linearly stretchy plate alongside thermal conductivity influence. Recently, such a concept has been widely investi- gated by various researchers, Aziz and Shams [32] on diverse flow configurations and conditions. In view of the discussion above and the consequential appli- cations of essential fluids parameters in manufacturing and en- gineering works, the present work aim to determine the motion and thermal transport of hydromagnetic tangent hyperbolic liq- uid over a permeable vertical stretchy surface using Lie group analysis approach. In particular, this study extends that of [26, 2 Disu & Salawu / J. Nig. Soc. Phys. Sci. 5 (2023) 1103 3 Figure 1. Configuration of the flow model 31] by considering porous media with the inclusion of vari- able heat conductivity, thermal radiation and a buoyancy effect which were ignored by previous authors. A unique similar- ity transformation approach is developed using the Lie group analysis which is adopted for transforming the nonlinear par- tial derivative transport model into a more simplified ordinary derivative form. The resultant set of outlining equations is nu- merically tackled using the shooting algorithm in conjunction with Fehlberg integration Runge-Kutta method. The physical characteristics of dimensionless terms obtained are clarified us- ing graphs with appropriate discussion. 2. Problem Formulation Analysis The underlisted assumptions have been identified as crucial for the formulation of the governing equations for the current investigation. Is is assumed that the fluid movement is time- independent, incompressible tangent hyperbolic fluid. The fluid movement is designed in a two-dimensional porous plate which stretches upwardly in a vertical route as displayed in Figure 1. The flow is routed in x axis while y axis runs perpendicular to x axis. A restriction is placed on the flow in the region y > 0. There is slippery in the momentum and energy boundary lay- ers. There is a surface mass flux on the sheet having a velocity of vw(x) as expressed in Eq. (7). With the imposition of an ex- ternal magnetic field normal to x direction but ignoring that of the induced magnetic filed influence and electric field as well. Likewise, it is supposed that the radiative heat flux is negligi- ble towards the x axis whereas it is applicable along y direction. Furthermore, the assumption of varying thermal conductivity is held valid with the inclusion of heat source. Other fluid at- tributes are are constant apart from the non-uniformity of the density in the momentum body force and the thermal conduc- tivity. Boussinesq approximation coupled with boundary layer approximation are applied in this study for the derivation of the main equations. For this study, the tangent hyperbolic fluid ten- sor is described as [26,31] τ = [µ∞ + (µ0 + µ∞) tanh(Γγ) m]γ, (1) In Eq. (1), τ describes the tensor stress while µ∞ depicts vis- cous shear rate at infinity whereas µ0 signifies the zero vis- cous shear rate and Γ describes the material constant of time- dependent whereas m connotes the power-law exponent while γ is expressed as: γ = ( 1 2 ΣiΣ jγi jγ ji ) 1 2 = ( 1 2 Π ) 1 2 . (2) In Eq. (2), Π = 12 tr((∇V ) T + ∇V )2. The case µ∞ = 0 is ac- counted for owing to low influence of viscosity at infinity. Also taking into account the tangent hyperbolic fluid detailing shear thinning characteristics, with the assumption that Γγ < 1, Eq. (1) then reduces to: τ = µo[Γγ m]γ = µo[(1 + Γγ− 1) m]γ ≈ µo[(1 + m(Γγ− 1))]γ (3) 2.1. The Governing Equations Combining the above-mentioned assumptions for the devel- opment of the transport model, Eqs. (4-6) describes the trans- port equations for the present investigation (see [20,26,31]). ∂u ∂x + ∂v ∂y = 0, (4) u ∂u ∂x + v ∂u ∂y = (1 − m) ν ∂2u ∂y2 + √ 2νmΓ ( ∂u ∂y ) ∂2u ∂y2 − σB2 ρ u + gβT (T − T∞) − ν kp u, (5) u ∂T ∂x + v ∂T ∂y = 1 ρC p ∂ ∂y ( k(T ) ∂T ∂y ) − 1 ρC p ∂qr ∂y + Qo ρC p (T − T∞) + ν C pkp u2 + σB2 ρC p u2, (6) The respective flow boundary constraints are stated below u = cx + β ∂u ∂y , v = vw(x), T = Tw + G ∂T ∂y at y = 0, (7) u −→ 0, T −→ T∞ as y −→∞. (8) The thermal flux radiation qr in Eq. (6) is indicated in Eq. (9) as (see Sumalatha and Bandari [33]) qr = − ( 4σ∗ 3k∗ ) ∂T 4 ∂y (9) From the above Eqs.(4-9), u and v describe flow rtae modules in respect to x and y axes. The symbols kp,β and σ represent porous medium permeability, velocity slip factor and electrical conductivity whereas the density, volumetric thermal expansion coefficient, magnetic flux density and the thermal slip factor are sequentially denoted by ρ,βT , B and G. Also, T signals the fluid temperature, g denotes gravitational acceleration, ν is the kinematic viscosity, vw describes surface mass flux, c defines 3 Disu & Salawu / J. Nig. Soc. Phys. Sci. 5 (2023) 1103 4 stretching rate and Qo describes coefficient of heat source/sink, σ∗ connotes Stefan-Boltzmann constant while the coefficient absorption mean is taken as k∗. By the application of the Rosse- land approximation and assuming that the heat variation is low in the flow field, so that Taylor’s series can utilized to expand T 4 to get T 4 ≈ 4T 3∞ − 3T 4 ∞, The temperature-based thermal conductivity is also specified as (see Animasaun [34]): k(T ) = k∞[1 + ζ(T − T∞)], (10) in which k∞ denotes the upstream heat conduction, ζ typifies the thermal conductivity parameter. To transmute the outlin- ing flow equations into dimensionless system, the underlisted quantities adopted: x x = ( a ν ) 1 2 , y y = ( a ν ) 1 2 , u u = 1 (aν) 1 2 , v v = 1 (aν) 1 2 , T = (Tw − T∞)θ + T∞. (11) Dropping the bar and substituting u = ∂ψ ∂y and v = − ∂ψ ∂x into Eqs. (5-6) taking cognizance Eqs (9) and (10), the underlisted are obtained( ∂ψ ∂y ∂2ψ ∂x∂y − ∂ψ ∂x ∂2ψ ∂y2 ) = (1 − m) ∂3ψ ∂y3 + √ 2maΓ ( ∂2ψ ∂y2 ) ∂3ψ ∂y3 − ( σB2 aρ + ν akp ) ∂ψ ∂y + gBT (Tw − T∞) a 3 2 ν 1 2 θ, (12) ( ∂ψ ∂y ∂θ ∂x − ∂ψ ∂x ∂θ ∂y ) = ( k∞ µcp (1 + ζθ) + 16σ∗ 3µcpk∗ T 3∞ ) ∂2θ ∂y2 + k∞ µcp ζ ( ∂θ ∂y )2 + Qo aρC p θ+ u2wσB 2 aρC p(Tw − T∞) ( ∂ψ ∂y )2 + u2wν akpρC p(Tw − T∞) ( ∂ψ ∂y )2 , (13) Also, the boundary conditions (7-8) transform to: ∂ψ ∂y = c a x + β √ a ν ∂2ψ ∂y2 , ∂2ψ ∂x2 = vw √ aν ,θ = 1 + G √ a ν ∂θ ∂y at y = 0, ∂ψ ∂y → 0, θ → 0 as y →∞. (14) 3. Lie Group Scaling Transformations The Lie scaling technique depends on theory formulated to find all symmetry transformations that keep the system of equations unchanged. It helps in reducing the number of in- dependent variables and in consequence transforms the PDEs to an ODEs. Using this method to generate similarity vari- ables involves finding the invariant solution which does not alter the structure of the given equation under study. In this section, the simplified format of the Lie group transformation approach is employed to derive the new similarity transforma- tions for the transport equations. As such, the outlining flow equations can be changed to ordinary derivative equations. Fol- lowing [27,31,35] the transformation variables are defined Υ : x∗ = xeεγ1, y∗ = yeεγ2, ψ∗ = ψeεγ3, θ∗ = θeεγ4, Γ∗ = Γeεγ5 (15) In Eq. (15), ε depicts the parameter of the group whereas the transformation variables are represented by γ1,γ2,γ3,γ4,γ5. Also, Eq. (15) is called point transformation for the set of coordinates system (x, y,ψ,θ, Γ) transforms into (x∗, y∗,ψ∗,θ∗, Γ∗). The sub- stitution of the transformation Eq. (15) into Eq. (12) and (13) results to the form: eε(γ1 +2γ2−2γ3 ) ( ∂ψ∗ ∂y∗ ∂2ψ∗ ∂x∗∂y∗ − ∂ψ∗ ∂x∗ ∂2ψ∗ ∂y∗2 ) = eε(3γ2−γ3 )(1 − m) ∂3ψ∗ ∂y∗3 + eε(5γ2−2γ3−γ5 ) ( √ 2mΓ ( ∂2ψ∗ ∂y∗2 ) ∂3ψ∗ ∂y∗3 ) − eε(γ2−γ3 ) ( σB2 aρ + ν akp ) ∂ψ∗ ∂y∗ + gBT (Tw − T∞) a 3 2 ν 1 2 θ∗e−εγ4, (16) eε(γ1 +γ2−γ3−γ4 ) ( ∂ψ∗ ∂y∗ ∂θ∗ ∂x∗ − ∂ψ∗ ∂x∗ ∂θ∗ ∂y∗ ) = eε(2γ2−γ4 ) ( k∞ µcp (1 + ζθ∗) + 16σ∗ 3µcpk∗ T 3∞ ) ∂2θ∗ ∂y2 + eε(2γ2−2γ4 ) k∞ µcp ζ ( ∂θ∗ ∂y∗ )2 + Qo aρC p θ∗e−εγ4 + eε(γ2−γ3 ) ( u2wσB 2 aρC p(Tw − T∞) + u2wν akpρC p(Tw − T∞) ) ( ∂ψ∗ ∂y∗ )2 , (17) Similarly, the boundary conditions transform to: eε(γ2−γ3 ) ∂ψ∗ ∂y∗ = c a e−εγ1 x∗ + β √ a ν ∂2ψ∗ ∂y∗2 eε(2γ2−γ3 ), ∂2ψ∗ ∂x∗2 eε(γ2−γ3 ) = vw √ aν , e−εγ4θ∗ = 1 + G √ a ν ∂θ∗ ∂y∗ eε(γ2−γ4 ) at e−εγ1 y∗ = 0, eε(γ2−γ3 ) ∂ψ∗ ∂y∗ → 0, e−εγ4θ∗ → 0 as y∗ →∞. (18) The preceding system of equations is invariant under the group transformation if the underlisted relationship exist among the exponents: γ1 + 2γ2 − 2γ3 = 3γ2 −γ3 = 5γ2 − 2γ3 −γ5 = γ2 −γ3 = −γ4 (19) γ1 + γ2 −γ3 −γ4 = 2γ2 −γ4 = 2γ2 − 2γ4 = −γ4 (20) 4 Disu & Salawu / J. Nig. Soc. Phys. Sci. 5 (2023) 1103 5 solving Eq. (19) and (20) to obtain the following relations: γ1 = γ3,γ2 = 0,γ4 = γ1,γ5 = −γ1 (21) Eq. (21) can then be introduced into Eq. (15) to obtain the criterion for the transformation as: Υ : x∗ = xeεγ1, y∗ = y,ψ∗ = ψeεγ1,θ∗ = θ, Γ∗ = Γe−εγ1 (22) Applying Taylor’s series to expand Eq. (22) in the power of ε to the first order to obtain: x∗ − x = xεγ1, y ∗ − y = 0, ψ∗ −ψεγ1, θ∗ − θ = 0, Γ∗ − Γ = −xεγ1, (23) Taking Eq. (23), the following characteristic equation were ob- tained: d x xγ1 = dy 0 = dψ xγ1 = dθ 0 = dΓ −xγ1 , (24) the following similarity transformations are derived by solving Eq. (24) (see Ulla and Zaman, 2017): η = y, ψ = x f (η), θ = θ(η), Γ = x−1Γo (25) The non-dimensional ODEs obtained with corresponding bound- ary condition via the similarity transformations (25) into Eqs. (16-18) are as follows: (1 − m) f ′′′ + mWe f ′′′( f ′′) + f f ′′ − (M2 + Da) f ′ − ( f ′)2 + Grθ′ = 0 (26) (1 + ζθ + Nr)θ′′ + ζθ′2 + Pr(Qθ + f θ′) + PrEcM2 f ′2 + PrEcDa f ′2 = 0 (27) f ′(0) = λ + α f ′′(0), f (0) = S, θ(0) = 1 + θ′(0), (28) f ′ → 0, θ → 0 as η →∞. (29) In Eqs. (26-29), We = √ 2aΓ symbolizes the Weissenberg num- ber, Nr = 16σ ∗ 3k∗k T 3 ∞ defines radiation parameter, M 2 = σB 2 aρ de- notes Hartmann number, Q = Q0aρcp typifies the heat source/sink factor and b = √ a ν G is the thermal slip parameters whereas α = √ a ν β represents the velocity slip, Da = νakp characterizes the Darcy number and ζ implies thermal conductivity parame- ter. The primes signifies differential with respect to η, λ = ca is the stretching parameter, Ec = u 2 w C p (Tw−T∞) is Eckert number, Gr = gBT (Tw−T∞) a 3 2 ν 1 2 x symbolizes the Grashof number, S = vw√ aν is the mass suction and Pr = µcp k∞ represents the Prandtl number. The incorporated engineering quantities in the current investi- gation include the wall friction C fx and the thermal gradient Nux which are orderly specified in Eq. (30) as: ρ(ax)2C fx = τw, Nux = xqw k(Tw − T∞) (30) Table 1. Skin friction coefficient as compared with previous studies when We = m = 0 M Akbar [36] Fathizadeh et al. [37] Present values 0 1.00000 1.00000 1.00000 1 −1.41421 −1.41421 −1.41421 5 −2.44948 −2.44948 −2.44949 10 −3.31662 −3.31662 −3.31663 50 −7.14142 −7.14142 −7.14143 100 10.0499 10.0499 10.0499 500 −22.38300 −22.38300 −22.38300 where τw = (1 − m) ∂u ∂y + mΓ √ 2 ( ∂u ∂y )2∣∣∣∣∣∣∣ y=0 , qw = −k∞ ( 1 + 16σ∗ 3k∗k∞ T 3∞ ) ∂T ∂y ∣∣∣∣∣∣ y=0 (31) The dimensionless form of Eq. (30) are specified in Eq. (31) as: Re 1 2 C f = [(1 − m) f ′′(0) + m 2 We( f ′′(0))2], Re− 1 2 Nux = − (1 + Nr) θ ′(0), (32) where Rex = ax2 ν signifies the local Reynolds number. 4. Numerical Solution Eqs. (26-27) comprises of a set nonlinear coupled differ- ential equations with it’s associated wall conditions. Owing to the non-linearity nature of the governing equations, Eqs (26- 27) subject to (28-29) are tackled numerically using shooting techniques alongside Runge-Kutta Fehlberg scheme by utiliz- ing a computer algebra symbolic code of Maple software. This algorithm relies on the adopted method. Except otherwise, the subsequent default values as been adopted for the study based on related previous analysis as n = 0.4, We = 0.3, � = 0.2, Nr = 0.3, M = 0.2, Da = 0.3, Pr = 3.0, Q = 0.3, Gr = 2.0, Ec = 0.01, λ = 0.7, S = 0.3, α = 0.2, b = 0.5. The numeri- cal code’s accuracy is validated by assessing the computational outcomes of the wall drag coefficient C f x offered in this study as compared with previously published works of Akbar [36] and Fathizadeh et al. [37] in respect to variations in the Hartmann number (M). As recorded in Table 1, the comparison showed a perfect harmony with the existing data in the literature un- der limiting circumstances and thus confirming the accuracy of our numerical code. Table 2 depicts the influences of some en- trenched parameters on the wall friction and heat gradient. As seen, an enhance or decline in the engineering quantities are ob- served due to the boundary layer viscosity. When the boundary film viscidness is stimulated the wall friction and Nusselt effect are raised, but when thinner boundary film viscidness noticed the diffuse more to the ambient leading to a decrease in the wall effects. 5 Disu & Salawu / J. Nig. Soc. Phys. Sci. 5 (2023) 1103 6 Table 2. Numerical values for the skin friction (C f ), heat gradient (Nux) M � λ Da We Ec Q C f Nux 0.2 0.2 0.7 0.3 0.3 0.01 0.3 0.8722973797 -0.5937107035 0.5 0.7677915589 -0.56754883934 1.0 0.4376848220 -0.48222768379 0.4 0.9077648755 -0.53183984435 0.7 0.9077648765 -0.53183984435 1.0 0.4765432364 -0.67408773863 1.5 0.2732784056 -0.80313254770 0.7 0.6779856282 -0.54475633672 1.0 0.6779856282 -0.54475633672 0.5 0.8498608064 -0.59162423311 0.7 0.8300639857 -0.58974089749 0.03 0.8735104313 -0.59064728168 0.07 0.8735104313 -0.59064728168 1.0 1.1540832041 0.0316335380 2.0 1.8442011437 1.9856306670 Figure 2. Plot of Da&λ on velocity f ′(η) 5. Discussion of Outcomes This aspect displays and discusses the reactions of the di- mensionless flow rate and energy profiles due to variations in the physical flow parameters. These physical parameters in- clude the stretchy term (λ), Grashof (Gr), Prandtl (Pr), Weis- senberg (We) and Darcy (Da) numbers, heat source term (Q), power-law exponent term (m), velocity slip term (α), radiation parameter (Nr), Hartmann number (M), mass suction param- eter (S ), thermal conductivity parameter (ζ), and temperature slip term (b). Figures 2-6 describe the influences of various physical flow parameter on the velocity field. Fig. 2 illustrates the effects of Figure 3. Behaviour Gr&We on velocity f ′(η) (Da) Darcy term on the dimensionless velocity in the existence of stretching parameter (λ). Evidently, there is a decrease in the velocity as (Da) increases. The flow behaviour in respect to a spike in Darcy number (Da) stimulates an opposition to the flow distribution that leads to a shrink boundary layer and thereby decelerates the fluid motion. In a related sense, an enhancement in the magnitude of the stretching term (λ) lowers the momen- tum boundary layer structure and consequently decelerates the locomotion. The impacts of Grashof number (Gr) and Weis- senberg term (We) on the dimensionless flow rate profile are presented in Fig. 3. It is evident from the graph displayed that the velocity drop significantly by a rise in (We) owing to an in- crease in the viscosity whereas there is an acceleration in the 6 Disu & Salawu / J. Nig. Soc. Phys. Sci. 5 (2023) 1103 7 Figure 4. Effect of S &m on velocity f ′(η) Figure 5. Reaction Q&M on velocity f ′(η) fluid motion as Grashof number increases due to enhancement in the buoyancy force. As (Gr) is raised, the buoyancy force dominates the viscous force and thus encourages the velocity distribution. Fig. 4 portrays the impact of mass suction term (S ) coupled with that of the power-law exponent (m) on the ve- locity distribution. It is evident that enhancing the magnitude of the power-law exponent (m) raised the viscosity and as a re- sult, there is a significant drop in fluid velocity. Also, this plot Figure 6. Effect of α&M on velocity f ′(η) shows raising the magnitude of S and (m), the hydrodynamic boundary structure thickness declines and the velocity deceler- ates. The evaluation of the heat generation (Q) term and the Hartmann number (M) is plotted in Fig. 5. Evidently, an elec- tromagnetic force is produced from the magnetic field interac- tion with the tangent hyperbolic electrically conducting liquid that create a drag in the flow movement as noted in the plot. The electro-conducting fluid’s interaction with the transverse magnetic field induces a retarding force on the liquid motion. Similarly, a hike in (Q) induces higher flow velocity rate owing to a decrease in the viscosity. Fig. 6 offers the behaviour of (α) on the liquid motion. In this plot, a declining trend is observed in the velocity field as the slip term (α) rises. Figures 7-12 offer the variations of some physical terms on the thermal field. Firstly, the temperature profile showing the impact of (λ) in the occurrence of radiative heat (Nr) parameter is plotted in Fig 7. The graph elucidates that advancement in (λ) causes the temperature to fall whereas growing (Nr) enhances the thermal profile. An advancement in the radiative heat flux corresponding to a rise in Nr while the Rosseland mean ab- sorption coefficient declines and as such, the thermal field is enhanced as found in this figure. The results of the Prandtl number (Pr) and power-law index (m) on the thermal distribu- tion are depicted in Fig. 8. The graph demonstrates that a boost in the (Pr) number lowers the thermal field by shrinking the energy boundary viscosity structure whereas the thermal prop- agation improves as (m) rises. The Prandtl number connotes the diffusivity of the momentum ratio to the diffusivity of the heat, and also influence the relative momentum shear stress and ther- mal boundary layer. Thus, a boost in the Pr implies a reduction in the energy boundary film and consequently leads to a decline in the heat transfer. The reactions of thermal generation term 7 Disu & Salawu / J. Nig. Soc. Phys. Sci. 5 (2023) 1103 8 Figure 7. Influence of λ&Nr on temperature θ(η) (Q) and Hartmann term (M) on the energy filed are displayed in Fig. 9. The graph portrays the fluid temperature exhibit- ing identical growing patterns on (Q) and (M). Typically, both parameters cause a rising trend in the thermal boundary layer. An enhancement in (M) induces a higher electromagnetic force which inspires an obstruction to the liquid motion and thus in- crease frictional heating effect which boosts the surface tem- perature. Similarly, a hike in Q is an indication of extra energy being generated and thus, a rise in the temperature as found in this figure. Fig. 10 elucidates the reactions of the thermal slip term (b) and Weissenberg term (We) on the fluid heat propagation. The temperature boundary structure shrinks and the tempera- ture falls with growth in b whereas the converse occurs with enhancement in We as noticed in this figure. A rise in (b) draws away the fluid from the heated region thereby lowers the tem- perature whereas as We rises in magnitude a frictional heat is generated due to rising viscosity. The reactions of mass suc- tion term (S ) and power-law exponent (m) are plotted in Fig. 11. This plot reveals that with advancement in S , the temper- ature distribution subsides whereas as (m) increases, the tem- perature distribution shoots up. Likewise, the plot showing the variation in Darcy term (Da) and thermal conductivity term ζ in respect to temperature is sketched in Fig. 12. It is noticeable that an increment in the (Da) and ζ boost the temperature dis- tribution due to extra heat generated by the resistance imposed on the fluid flow as Da increases. In Figure 13, the impact of Ecket number (Ec) on the heat propagation with variation in Hartmann number (M) is established. As seen, temperature distribution is raised due to an induce magnetic Joule heating that inspired the tangent hyperbolic fluid flow particles interac- tion. Also, the magnetic Joule heating effect is complemented Figure 8. Reactions of m&Pr on temperature θ(η) Figure 9. Impact of Q&M on temperature θ(η) by the porous Joule heating that creates fluid friction and re- sistant to free flow, thus, particles collision and random motion is encouraged to increase heat transfer. Therefore, rising heat distribution magnitude is observed all over the flow region. 6. Conclusion A computation solution has been performed on the motion and thermal propagation of hydromagnetic tangent hyperbolic 8 Disu & Salawu / J. Nig. Soc. Phys. Sci. 5 (2023) 1103 9 Figure 10. Impact of b&We on temperature θ(η) Figure 11. Effect of S &m on θ(η) liquid passing a vertically stretched surface with varying ther- mal conductivity. The flow model is in steady 2-dimensional and incompressible stretchable plate enclosed in permeable me- dia with the impact of radiative heat and internal thermal energy source. Lie group analysis generates the similarity transforma- tion which transformed the coupled differential equations with boundary conditions from partial to ordinary derivative equa- tions, The solution to the equations are the offered computa- Figure 12. Influence of Da&ζ on temperature θ(η) Figure 13. Effect of Ec&M on heat field θ(η) tionally via shooting approach alongside Fehlberg Runge-Kutta method. The solutions are given graphically and deliberated while comparison with published studies show good agreement. The study also reveals that: • The fluid velocity accelerates with enhancement in the heat source term Q and Grashof number Gr magnitude. However, augmenting the Darcy term Da, suction S term, velocity slip α, stretching term λ, Hartmann number M, 9 Disu & Salawu / J. Nig. Soc. Phys. Sci. 5 (2023) 1103 10 power-law exponent m as well as Weissenberg value We decelerates the velocity profiles. • A damped in the thermal boundary structure and the cor- responding surface temperature distribution falls with a rise in the mass suction term S , Prandtl number Pr and thermal slip term b. • A rising thermal boundary film viscosity structure is formed with increasing the fluid heat conduction term ζ, radiative heat term Nr, Darcy number as well as the heat source parameter Q. References [1] T. Abbas, S. Rehman, A. S. Muhammad & I. 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