J. Nig. Soc. Phys. Sci. 5 (2023) 1054 Journal of the Nigerian Society of Physical Sciences Study of MHD SWCNT-Blood Nanofluid Flow in Presence of Viscous Dissipation and Radiation Effects through Porous Medium M. Ramanujaa,b,∗, J. Kavithac, A. Sudhakara, V. NagaRadhikab aDepartment of Mathematics, Marri Laxman Reddy Institute of Technology and Management, Dundigal, Hyderabad – 500 043, India bDepartment of Mathematics, GITAM Institute of Technology and Management, Bangalore, Karnataka – 561203, India cDepartment of Mathematics D K, Government College for Women, SPSR Nellore-524003, India Abstract In this analysis, a computational study is conducted to examine the two-dimensional flow of an incompressible MHD SWCNT-blood nanofluid, saturated mass and porous medium .In addition to viscous dissipation, thermal radiation is taken into consideration. We developed the mathemati- cal model and useful boundary intensity approximations to diminish the structure of partial differential equations based on the fluid for blood-based SWCNT underflow assumptions. Converted the partial differential equations by applying corresponding transformations to arrive at ODE’s. The above results are solved numerically by the Runge-Kutta 4th order technique. Noticed that there is desirable conformity when interpolated with the numerical one. The effects exhibited the velocity of SWCNT-blood nanofluid enhanced for defined standards of the viscosity parameter. Rise in temperature when various parameters like Prandtl number, Eckert number, and slip parameter are applied on SWCNT-blood. The impact of fluid flow on blood-based SWCNT is discussed graphically, and our results are tabulated along with illustrations. The design concepts, such as the Nusselt quantity and the local skin friction, conform to the analytical approach. Velocity reductions with an increase in CNT’s volume fraction, whereas enhancement in the blood temperature, is noted, which is directed to the rise in the heat mass transfer rates. DOI:10.46481/jnsps.2023.1054 Keywords: SWCNT, Blood, Viscous dissipation, Radiation, Nusselt number, Skin friction Article History : Received: 11 September 2022 Received in revised form: 20 November 2022 Accepted for publication: 28 November 2022 Published: 14 January 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: S. Fadugba 1. Introduction The study of non-Newtonian fluids like water, mineral oil, and ethylene glycol was reported in many papers for more than a decade, and their applications could be found in industrial sectors such as chemical manufacturing, microelectronics, air ∗Corresponding author tel. no: +91 9550754250 Email address: mramanuja09@gmail.com (M. Ramanuja) conditions, engineering, petroleum industry, paper production, aerodynamic heating, coating, and polymer processing, etc. A range of substances such as blood, mud, polymers, and paint depicts a non-Newtonian fluid description. However, no single model in literature deals with multiple non-Newtonian fluids. But the properties of these fluids are multiple in themselves be- cause of their low thermal conductivity, which hampers their functions during heat exchangers. As a consequence, there may be a demand to expand its thermal conductivity. In contempo- 1 Ramanuja et al. / J. Nig. Soc. Phys. Sci. 5 (2023) 1054 2 rary applications, due to commercial aspects, the flow involv- ing Casson nanofluid creates critical interest in present-day re- searchers. Many substances in the actual field, like mud, malt, condensed milk, glues, sugar solution, emulsions, soaps, paints, etc., exhibit Newtonian fluid properties. But the actual situa- tion is to assemble a single constitutive equation that follows the defined Casson nano -fluid’s properties. It also plays a vital function in nuclear physics within geographical flows. Many researchers have identified different effects resting on Casson nanofluid. Salman et al. [1] developed a combination of viscous dis- sipation with radiation parameters. The cone angle has a sig- nificant result on heat transfer and fluid flow conduct within the porous medium. The consequences of viscous dissipation, holmic dissipation, thermal radiation, and mass exchange out- comes on uneven hydromagnetic boundary thickness float of a stretching plane were developed by Anjali Devi et al. [2]. Abd El Aziz [3] is interested in the impact of thermal radiation along with mixed heat and mass transfer on hydromagnetic with the flow over a porous stretching level. MHD float was scientifi- cally clarified with radiation via a stretching sheet surrounded by a porous medium, as specified and examined by Anjali et al. [4]. Makinde et al. [5] have analyzed the chemical response results from the stretching surface in the existence of interior heat invention. The radiation and chemical response outcomes on the MHD boundary layer glide of a stretching surface were examined by Seini et al. [6]. Abdul et al. [7] have investigated, via a stretching flat surface, the effect of thermal radiation on MHD fluid flow. The pressure of thermal radiation on the MHD flow via the stretching surface was discussed by Chen et al. [8]. Further, Raju et al. [9] concentrated on resting the movement of Casson fluid through a slippery wedge and noticed a decrease in the increasing temperature area estimations of the Eckert num- ber. Electrically directing Casson liquid flow over an item that is neither a perfect level/vertical slanted/cone in the presence of a steady, attractive field is also a significant concern. Sa- betha et al. [10] determined the thermal radiation results on hy- dromagnetic free convection drift previously and impetuously began vertical plate. Hiteesh [11] studied the absence of trans- verse magnetic discipline, the border layer regular drift and heat transfer of a viscous incompressible fluid resulting from stretch- ing plate with viscous dissipation impact. The convection heat transfer side by way of a constantly shifting heated vertical plate, including suction or injection an- alyzed by Al-Sanea [12]. Unsteady free convection and mass change glide over a limitless vertical permeable plate in the commentary of suction/injection are to come upon with learn- ing about by Takhar et al. [14]. The impact of suction /in- jection on unsteady free convection Couette float and warmth switch of an active viscous fluid with vertical permeable plate is explained by Jha et al. [16]. Shamshuddin et al. [17] developed a particular cover of flow that isn’t dissipative, and in-depth graphical illustrations are offered for the fine of the magnetic subject parameter. Fur- ther, Shah et al. employed the dissimilar case of nanoparti- cles. Additionally, entropy optimization with activation elec- tricity and chemical response is also studied. The 2nd regulation of Thermodynamics is utilized to discover the entropy technol- ogy in velocity. Heat and mass switch of Williamson nanofluid causes a magnetohydrodynamic perimeter layer to move with the flow across a stretched sheet explored by Reddy et al. [19]. The magnetohydrodynamic (MHD) flow of Casson nanofluid impact over an extended surface was developed by Hayat et al. [20]. Dawar et al.[22] investigated the MHD Casson-nanofluid, carbon nanotube, and radioactive heat transfer revolving chan- nels. Ali et al. [21] studied the blood moves with the Casson fluid below the influence of MHD magnetohydrodynamics in axis-symmetric cylindrical tubes. The MHD Williamson fluid via a bent sheet and below the influence of non-thermal heat source or sink CNTs is analyzed by Kumar et al. [23]. C. Sus [24] for the first time, nanoflu- ids were proposed by elevating nanometer and sized particles interested in the bottom fluid. The (SWCNTs) single-walled carbon annotates own a better heat transfer assessment, and sur- face drag compels (MWCNTs) multiwall carbon annotates de- scribed by Haq et al. [25]. Further, Liu et al. [26] examined glycol, ethylene, and engine oil with the existence of MWC- NTs, and they concluded that CNTs with ethylene glycol have an advanced thermal conductivity. Recently, The approach of nanofluid precedent above a stretching sheet was obtained by Needed and Lee [27]. The boundary deposit flow of nanoparti- cles concluded a stretching/shrinking exterior had been exam- ined by Nadeem Bejan [28]. Further, Hayat et al.[29] obtained the nonmaterial fluid flow in a circulating method. The nanofluid flow throughout the en- tropy generation considers the circular heat source, which is studied by Nouri et al. [30]. Das et al. [31] have analyzed the MHD flow of nanofluid through a porous medium. Fur- ther, Sheremet et al. [32] examined the identical fluid in the crimped cavity. The flow of nanofluid in an enlarged porous sheet is observed by Alharbi et al. [33]. Zueco[34] exploited a community simulation technique (NSM) to check the conse- quences of viscous dissipation and radiation on unsteady free convection MHD on a vertical porous plate. Hamzeh et al. [45] aims to investigate the properties of heat transfer and magneto- hydrodynamics Casson nanofluid in the presence of a free con- vection boundary layer fluid flow on a stretching sheet using CNTs in human water/blood as the base fluid. The unsteady separated stagnation-point flow of hybrid nanofluid with viscous dissipation and Joule heating is inves- tigated numerically in this examined by Amira Zainal et al. [46]. Zainal et al. [47] have investigated viscous dissipation and MHD hybrid nanofluid flow towards an exponentially stretch- ing/shrinking surface. Taza et al. [48] have studied heat and mass transmission more conveniently, such as in hybrid-powered engines, pharmaceutical processes, microelectronics, domes- tic refrigerators, and engine cooling. Aditya et al. [49] have been examined by considering Buongiorno’s two-component non-homogeneous model with the inclusion of electrification of nanoparticles and viscous dissipation. Adedire et al. [50] examined the concentration profiles in the single and the inter- connected multiple-compartment systems with sievepartitions to transport chemical species with second-order chemical re- 2 Ramanuja et al. / J. Nig. Soc. Phys. Sci. 5 (2023) 1054 3 action kinetics. Ramanuja et al. [51] have examined Casson nanofluid flow over a growing or contracting porous medium with different permeability and thermal radiation.[54] have ob- tained numerical experiments show that the methods compete favourably with existing processes and efficiently solve stiff and oscillatory problems. Nanofluids are obtained by reacting oxides, metals, car- bides, or carbon nanotubes (CNTs) with nanoparticles. Gen- erally, in base fluids, nanoparticles are regularly floating, con- sisting of water, kerosene, ethylene glycol, and engine oil in some areas. CNTs used inner nanofluids, which are available in two types in carbon nanotubes (SWCNTs), and colloidal de- ferment of nanoparticles in a base fluid is used to create these fluids. This model investigates the stagnation flow on SWCNT and blood nanofluids of MHD fluid in the existence of a porous medium, viscous dissipation, and investigation under the im- pact of injection/suction in addition to thermal radiation param- eters are studied and analyzed through this examination. Nu- merical techniques solved the resulting equations. Symbolic computational software such as MATLAB bvp4c solver is used. The effects are presented graphically. 2. Formation of the Problem The physical description of the problem is illustrated in fully- developed steady-mixed convection flow of human blood uti- lized as base fluid, and SWCNT as the nanoparticles over a state, incompressible, laminar flow of SWCNT-based nanoflu- ids saturated with human blood is embedded in the medium porous surface that allows the liquid to enter or exit during pro- gressive developments or constrictions. The porous plates are separated by distancea. One part of the cross-segment converse to separation by 2a (t)between the walls, which is to a great extent less than the channel’s width and length. A consistent segment of the flow field is shown in the Cartesian coordinate system, which is chosen within such a manner as exposed in fig.1. One and the other channel partitions are perceived to have distinct permeability issues and expansion or convention are systematically at a dependents-time velocity v0which repre- sents the uniform suction v0 > 0 and injection v0 < 0channel which is supposed to be infinite in the distance. Because of their magnetic characteristics, these nanoparticles are consid- ered. In the proposed issue, the Casson liquid model is exposed to blood and SWCNTs nanoparticles which are scattered into it for upgraded heat move. The above assumptions portrays leading equations in sup- port of the nanofluid flow by 2-dimensional boundary cover equations are assumed as a continuity equation, momentum equation, and the energy equations as mentioned by Vijayalak- shmi et al. [41]; Bestman [42]; Srinivas et al. [43] and Radha krinshnama charya et al.[44]. ∂u ∂x + ∂v ∂y = 0 (1) Figure 1. Schematic of problem ∂u ∂t + u ∂u ∂x + v ∂u ∂y = µn f ρn f ( 1 + 1 β ) ( ∂2u ∂x2 + ∂2u ∂y2 ) − µn f φ ρn f k ( 1 + 1 β ) u − B20σn f µn f u ρn f − 1 ρn f ∂p ∂x (2) ∂v ∂t + u ∂v ∂x + v ∂v ∂y = µn f ρn f ( 1 + 1 β ) ( ∂2v ∂x2 + ∂2v ∂y2 ) − µn f φ ρn f k ( 1 + 1 β ) v − B20σn f µn f u ρn f − 1 ρn f ∂p ∂y (3) ∂T ∂t + u ∂T ∂x + v ∂T ∂y = kn f (ρCP )n f ( ∂2T ∂y2 ) − 1 (ρCP )n f ∂qr ∂y + µn f (ρCP )n f ( 1 + 1 β ) ( ∂u ∂y )2 + q′′′ (ρCP )n f + Q0 (ρCP )n f (T − T0) (4) where uand v denote the velocity fundamental quantities, with the directions of x-axis andy-axis, pdenote the dimensional pres- sure t be the time,φ&k are the permeability and porosity of the permeable medium, φ (η) is the dimensionless concentration of the fluid, temperature T ,kn f denote the thermal conductivity of the nanofluid, β be the blood Casson fluid parameter,ρn f be the efficient density of the nanofluid, the efficient dynamic viscos- ity of the nanofluid be µn f , (ρCP )n f denote the heat electrical condenser of the nanofluid, and vdenote the kinematic viscos- ity. The non-uniform heat absorbed by generation per unit vol- ume q′′′is defined as: q′′′ = B(x0)m+1k vx0 [ f 1(η)A∗(T1 − T0) + B ∗(T − T0)] Here A∗represents the velocity of heat transfer for the space- dependent and B∗ represents an exponentially decaying param- eter of space and internal temperature-dependent heat absorp- tion. where A0 = v0a−1 and A1 = v1a−1 are the wall permeability quantities;T0, T1are the temperature of the upper and lower 3 Ramanuja et al. / J. Nig. Soc. Phys. Sci. 5 (2023) 1054 4 walls;Twis the temperature taking place at the wall;T∞is the temperature of free stream fluid flow. The substantial effects of the such as nanofluid ρn f ,µn f , (ρCP)n f , and kn f are involved in the outcomes of the distance distribution on CNTs are com- pensated for using spinning oblique nanotubes with a very large axial ratio and given as, which might be outlined. 2.1. Mathematical Model for the Thermal Physcial Property of a Nanofluid Table 1 Mathematical model for the thermal physical prop- erty of a nanofluid the viscosity, density, heat capacitance and the effective thermal conductivity of the nanofluid are defined as given by H.C. Brikman [52] and R. I. Hamilton et al. [53] respectively: Where, nthe nanoparticle is shape factor , vn f = µn f ρn f ,φ- the volume of the utility of nanoparticles,ρ f - Concentration of the base fluid, ρs-be the density of the nanoparticle, µ f -viscosity of the base fluid, (ρC p) f , (ρC p)s- The capacitance heat of the base fluid along with nanoparticle is a combination with solid nanoparticles, and k f ; ksthermal conductivities of the base fluid and nanoparticle correspondingly. The thermo-physical prop- erties of changed base fluids and nano-particles are revealed in Table 1. By introducing the complimentary flow utility, the same represents flow velocity components u and v can be writ- ten through conditions of the free flow function in flow. u = ∂ψ ∂y and v = − ∂ψ ∂x (5) Partial differential equations that are non-linear and condensed addicted to non-linear ODE’s deliberated for that purpose the stream function whereψ = ψ(x, y)routinely satisfy continuity equation, indicates stream function appropriate to mass conser- vation and f (η) is dimensionless flow function. u = xva−2 Fn(η, t), v = −va −1 F(η, t) ; ψ = xvF(η, t)/a (6) Here η = ya , Fn = ∂F ∂η The governing equations (2) are based on these assumptions are given by Vijayalakshmi et al. [41]: ∂u ∂t = µn f ρn f ( 1 + 1 β ) ∂2u ∂y2 − µn f φ ρn f k ( 1 + 1 β ) u − B20σn f µn f u ρn f − 1 ρn f ∂p ∂x (7) Usage of the irradiative heat flux is basic in Rosseland’s esti- mate for radiation Brewster [39], and the thermal flux is defined as: qr = − 4σ∗ 3k∗ ∂T 4 ∂y (8) Here and k∗ be the “absorption Specific” coefficient, σ∗- be the Stefan-Boltzmann constant. We had been constrained that the temperature variations contained by using the glide are satisfac- torily slighter such that the term T 4strength is stated as a direct function of temperature. This is consummated by increasing T 4 in Taylor’s sequence about T∞ and ignoring higher-order ex- pressions, thus assuming a small temperature difference in flow given below: T 4 � 4T 3∞T − 3T 4 ∞ (9) Using Eqs(9) and Eqs(8) becomes ∂qr ∂y = −16σ∗T 3∞ 3k∗ ∂2T ∂y2 (10) Under these assumptions, the leading equations are given by Vijayalakshmi et al. [41]; Bestman [42]; Srinivas et al. [43] ∂T ∂t = kn f (ρCP )n f ( ∂2T ∂y2 ) + 1 (ρCP )n f  16σ∗T 313k∗ ∂ 2T ∂y2  + µn f (ρCP )n f ( 1 + 1 β ) ( ∂u ∂y )2 + q′′′ (ρCP )n f + Q0 (ρCP )n f (T − T0) (11) The temperature of the nanofluid in the channel can be calcu- lated as follows: Tw = T∞ + B ( x a )m1 θ(η) (12) The dimensionless form of temperature from Eq. (12) where θ- dimensionless temperature function, η- Similarity variable, m1 denote the index power-law of the temperature and B is the constant of the fluid. The pressure gradient of the kind by Vijayalakshmi et al. [41]; Bestman [42]; Srinivas et al. [43] and Radhakrinshnama charya et al. [44] is thought to generate the pulsatile flow. A(1 + ceiwt) = − 1 ρn f ∂p ∂x (13) By inserting the dimensionless variables and parameters listed below: x = X h , y = Y a , t = ωt ′ , P = P Aρn f h u = ωu ′ A ,θ(η) = T − T0 T1 − T0 (14) At this moment, we eliminate pressure commencing from equa- tions(12) using (13) and (14) with reference from Vijayalak- shmi et al. [41] the following is obtained; (1 + ceiωt) = − ∂p ∂x (15) A2 A1R (1 + 1 β ) ∂2U ∂y2 − 1 A1 ∂p ∂x − 1 A1 ( A5 M 2 + A2 Da ) U − ∂U ∂t = 0 (16) Eqs(12) using (14) (15) Eqs becomes:( A4 A3 + 4 3A3 Rd ) 1 R Pr θ ′′ + A2 A3 EcA∗ R ( ∂U ∂y )2 + B∗QH A3R θ− ( ∂θ ∂t ) = 0 (17) 4 Ramanuja et al. / J. Nig. Soc. Phys. Sci. 5 (2023) 1054 5 Table 1. Mathematical model for the thermal physical property of a nanofluid Physical Quantity Mathematical model Influential Dynamic viscosity of the nanofluid µn f = µ f (1 −φ) −2.5 The influential Density of the nanofluids ρn f = φps + (1 −φ)p f The Heat capacitance of nanofluid (ρCP)n f = φ (ρCP)s + (1 −φ)(ρCP) f Thermal conductivity of sphericalnanoparticles approximated kn f = k f [ 2k f +ks−2φ(k f −ks ) 2k f +ks +φ(k f −ks ) ] The electrical conductivity σn f = σ f [ 1 + 3(σ−1)(σ+2)−(σ−1)ϕ ] Using the following dimensionless similarity variables, where Darcy parameter, the frequency parameter, Eckert number, prandtl number, Heat source parameter, Radiation parameter. Da = k a2 , R = ωh2 v f , Ec = A2 (C p) f ω2(T1 − T0) Pr = (pC p) f vn f k f , QH = Q0a2 (ρC f ) f v f , A1 = φ ρs ρ f + (1 −φ) , A2 = (1 −φ) 2.5 , A3 = φ (ρcp)s (ρcp) f (1 −φ), Rd = 4T 31 σ ∗ k f k∗ A4 = [ 2k f + ks − 2φ(k f − ks) 2k f + ks + φ(k f − ks) ] A5 = 1 + 3 ( σs σ f − 1 ) φ( 2 + σs σ f ) + ( − σs σ f + 1 ) ϕ  (18) The corresponding boundary conditions are: f (1) = 1, f (−1) = 1, f ′ (1) = 0, and f ′ (−1) = 0. (19) θ(−1) = 1,θ(1) = 0, i f θ(0) = 1 + δθ′(0) (20) It was once initiated that there is an appropriate settlement be- tween analytical and numerical solutions. Dimensionless shear stress at the partitions is described as heat transfer. The pace of the partitions is a prerequisite for Nusselt quantity non-dimensional, which is characterized by Hatami et al. [40] τ = x(1 −φ)−2.5 R ( f ′′(η) ) η=−1,1 (21) Nux = − kn f k f ∂T ∂η /(T1 − T0) = −φ2θ(η)η = −1, 1 C f = 2µn f ρ f (uw (x)) ( ∂u ∂y ) y=0 = −φ2θ(η)η = −1, 1 qw = −kn f ∂T ∂y /y = 0 3. Results and Discussion We investigated in this study the combined property of ther- mal radiation, heat generation, and viscous dissipation resting on the SWCNT and blood nanofluid flow modal that incor- porates nanoparticle volume fraction. Within the numerical computation, the properties of the blood and SWCNT are uti- lized (reference Table 2). The consistency and accuracy of our accurate solutions and numerical trials of the significant pa- rameters are highlighted through graphs in this section. The governing equation (15) and (16) through the boundary condi- tions (18) and (19) were worked out by employing the Runge- Kutta strategy through the shooting method (MATLAB solver, bvp4c package software). To achieve these results, mathemat- ical computations are exposed by making an allowance for a distinct norm of non-dimensional governing parameters. The impacts of governing substantial parameters are explored in de- tail. Specifically; Eckert number(Ec), Magnetic parameter (M), Nanoparticle volume parameter(φ), Heat generation parameter (QH ), Prandtl number (Pr), Darcy number (Da), Casson param- eter (β), and A∗, B∗ are velocities of heat transfer for the space- dependent; with the following assigned values to the respective parameters: \ M = 2, A = 0.5, B = 0.1, m1 = 0.2, R = 2, Da = 0.5, QH = 0.1, A ∗ = 0.2, B∗ = 0.2, Nr = 0.5, Ec = 0.2 Through Figure 2, the result ofA∗on the temperature distribu- tion θ(ς) is illustrated and from this, we conclude that A∗ starts declining against the temperature profile which is being en- larged after a certain range. It is observed from Figure 3 that the impact of B∗on temperature profile θ(ς) decreases. After these factors, the speed profiles are accelerated. It can be seen from Figure 4 the difference of Eckert number (Ec) through the temperature profile. The incidence of Eckert number in Casson nanofluids enhances the development of thermal vitality, which results in the advance with temperature distributions and in con- sequence of thermal deposit thickness. The result of the unde- niable viscosity of nanofluids supplies vitality from the waft because of the rise in heat electricity through frictional heating and transforms it into interior electricity. Variations of various heat-generating parameter values are dependent on the temperature; the profile is revealed in Fig- ure.5. When QH grows positively, heat production takes up res- idence in the thermal limit layer. The Casson nanofluid thermal electricity improves due to a large amount of heat. This process raises the thickness of the thermal boundary layer, implying that 5 Ramanuja et al. / J. Nig. Soc. Phys. Sci. 5 (2023) 1054 6 Figure 2. Impact ofA∗onθ(ς) Figure 3. Impact ofB∗on θ(ς) warmness strength is activated and, as a result, the temperature of the fluid rises. The impression of the thermal slip parameter δon the tem- perature profile and the velocity profile is depicted through Fig- ure 6. We observed that the thermal slip parameter leads to in- creases in the temperature distribution and the thermal bound- ary layer thickness; besides, this outmost impact is noticed at the outside of the channel. Figure 7 shows the impact of the slip parameter on the rate of Casson nanofluid. We noticed that the slip parameter δ at a certain point the velocity of the Casson fluid enhanced. Figure 8 depicts the effect of Da on velocity profiles of Cas- son nanofluid, and it is observed that velocity accelerates with rising values of Darcy number. The impact of attractive bound- ary M on velocity and temperature profiles is shown through Figure 9 and Figure 10. From this, we noticed that the veloc- Figure 4. Impact ofEcon θ(ς) Figure 5. Impact of QH on θ(ς) Figure 6. Impact of δ on θ(ς) 6 Ramanuja et al. / J. Nig. Soc. Phys. Sci. 5 (2023) 1054 7 Figure 7. Impact ofδ on f ′(ζ) Figure 8. Impact ofDaon f ′(ζ) ity profiles decline for SWCNT with rising values of M result- ing thickness of the boundary layer is reduced at a faster rate. Physically, the Lorentz energy, which opposes movement, oc- curs due to the used transverse magnetic flow and is responsible for reducing fluid velocity. Besides, as the temperature profiles are improved, the temperate limit layer thickness expands. The impact of blood parameters β on the velocity and tem- perature distributions is shown in Figure 11 and Figure 12; it is noticed that for the increasing value ofβ, the velocity profile de- creases for SWCNT. It is because of blood with β will increase the plasticity of blood fluid expands with motive the deceler- ation in velocity. It’s due to the blood’s malleability, when β decreases, the flexibility of the fluid increases, causing the pace to slow down. In addition, the temperature profile of the flow escalates for increasing values of M. In Figure 13, the impact of volume fraction φon the temper- ature profile for human blood-based nanofluid with SWCNT is displayed; it is noticed that for both human blood flow and SWCNT, as the φ rises, the temperature of the nano-fluid also increases. It is additionally referred to as those changes in φ in- Figure 9. Impact of M on f ′(ζ) Figure 10. Impact of M on θ(ς) dicating the adjustments in temperature after which shows the significance of nanofluid. The impact of A on velocity and temperature profiles are portrayed through Figure 14 and Figure 15; it is noticed that the temperature profile θ(ς)escalates and the velocity profile f ′(ζ) decelerates with the impact ofA. In Figure 16, the outcomes of the m1 on the temperature pro- file are displayed; the temperature of the nanofluid decreases to SWCNT with increasing value of m1, which leads to a decline in velocity boundary deposit thickness. 3.1. Physical Properties of Base Fluid and Nano-Particles Thermo substantial properties regarding the base fluid and nanoparticles of carrier fluid human blood and SWCNT nanopar- ticles are given below Table 2 3.2. Skin Friction and Nusselt Number for the Cases of Un- steady Contraction/Expansion From Table 3 it is seen that the coefficients of space with temperature-dependent A∗and B∗with relatively high-temperature 7 Ramanuja et al. / J. Nig. Soc. Phys. Sci. 5 (2023) 1054 8 Table 2. Physical properties of Base fluid & Nanoparticles (Blood and SWCNT) Physical properties Solid Nanoparticles SWCNT Base fluid blood cp(j/kg k) 425 3617 κ(w/m k) 6600 0.52 ρ (kg/m3) 2600 1050 Figure 11. Impact of β on f ′(ζ) Figure 12. Impact ofβon θ(ς) source/sink, the skin friction coefficient stable in nature whereas the Nuselt number will be enhanced. With the improved values of Da, both skin friction and Nusullt number decrease. The en- hanced values βreduce skin friction on velocity, in addition, to enhancing the heat transfer moderately. The pores of skin fric- tion remain stable, convenient and incomplete gradual reduc- tion inside heat transfer velocity used with increasing values ofNr &QH . In this case of M, friction (C f )and Nusselts num- Figure 13. Impact of φon θ(ς) Figure 14. Impact of A on f ′(ζ) ber decrease. For the effect of the parameters Ec, φ andA, both the values of skin friction, Nusselts number declines, whereas form1, both skin friction and Nusselt number increases. 3.3. Skin Friction and Nusselt Number for Steady Contraction Situation In the above Table 4, we can see that the enhanced values of A∗as well as B∗have no impact resting on skin friction, al- though these values concentrated the heat transfer velocity. The skin friction values for the variations in Daremained constant whereas local Nusselt number values will be increasing. The enhanced values of β covers the oscillatory nature for pores 8 Ramanuja et al. / J. Nig. Soc. Phys. Sci. 5 (2023) 1054 9 Table 3. The impact of various parameters on skin friction ( C f ) and Nusselt number (Nux)for the cases of unsteady contraction/expansion A∗ B∗ Da β Nr QH M Ec δ φ A m1 C f Nux 0.0 19.471388 3.193296 0.2 19.471388 3.290621 0.4 19.471388 3.387946 0.0 19.471388 3.015712 0.2 19.471388 3.290621 0.4 19.471388 3.646890 0.001 66.175569 3.259443 0.05 24.559828 3.307261 0.1 21.627181 3.298364 0.1 19.471388 3.290621 0.3 8.683747 3.562513 0.5 6.435490 3.624260 0.5 19.471388 3290621 1.0 19.471388 5.542735 1.5 19.471388 7.858197 -0.1 19.471388 3.015712 0.0 19.471388 3.144902 0.1 19.471388 3.290621 1 19.246748 3.292558 10 21.302554 3.274551 20 23.677717 3.252965 0.1 19.471388 3.516654 0.2 19.471388 3.290621 0.3 19.471388 3.064588 -0.2 8.072545 3.021844 0.0 11423853 3.118250 0.2 19.471388 3.290621 0.1 19.471388 3.290621 0.3 35.873231 8.780344 0.5 19.180772 27.79143 0.1 34.402685 4.064004 0.3 27.006243 3.661321 0.5 18.710125 3.254065 0.0 19.471388 3.267488 1.0 19.471388 3.401005 2.0 19.471388 3.579480 and skin friction although there is an insignificant variation for heat transfer rate. Influence of mis not affecting skin friction whereas heat transfer velocity is decreased for the same m val- ues. There is an oscillatory characteristic in pores and skin fric- tion and a fast increased within the heat transfer velocity which is meant for rising values ofA. Raising values of the slip param- eter φescalate the both skin friction and Nusselt number. For the influence ofm1, the skin friction values remain constant and the local Nusselt number decrease. 4. Validation Similarly, Table 4 is organized to explain Nu at the employ- ing the Runge-Kutta strategy through the shooting method esti- mated for different values of relevant model variables for both SWCNTs and MWCNTs Nano fluids. From Table 3 and Table 4, it can be observed that the weight of rate of heat transport accelerates for a high magnitude of both φ and Pr and declines for a higher value of the Eckert number(Ec). Eckert number (Ec) is related to the dissipation term, and the more consider- able importance of Eckert number (Ec) enhances the thermal field. Therefore, the opposite result for the higher significance of the Eckert number (Ec) verses Nu is perceived Table 5 5. Conclusion In this article, the stagnation flow on SWCNT and blood nanofluids of MHD fluid in the existence of a porous medium, viscous dissipation, and injection/suction in addition to ther- mal radiation parameters are studied and analyzed through this examination. The resulting equations were solved by numeri- cal techniques. The tables and graph values for the tempera- 9 Ramanuja et al. / J. Nig. Soc. Phys. Sci. 5 (2023) 1054 10 Table 4. The impact of various parameters on skin friction ( C f ) and Nusselt number (Nux)for steady contraction situation A∗ B∗ Da β Nr QH M Ec δ φ A m1 C f Nux 0.0 19.997073 2.555168 0.2 19.997073 2.647360 0.4 19.997073 2.739552 0.0 19.997073 2.462266 0.2 19.997073 2.647360 0.4 19.997073 2.880635 0.001 68.577628 2.542118 0.05 25.222523 2.654748 0.1 22.208947 2.651121 0.1 19.997073 2.647360 0.3 9.274754 2.907525 0.5 7.085115 2.970431 0.5 19.997073 2.647360 1.0 19.997073 4.751474 1.5 19.997073 6.972890 -0.1 19.997073 2.462266 0.0 19.997073 2.549819 0.1 19.997073 2.647360 1 19.769175 2.649027 10 21.854765 2.633497 20 24.264288 2.614779 0.1 19.997073 2.860118 0.2 19.997073 2.647360 0.3 19.997073 2.434603 0.2 8.119202 2.475313 0.0 11.560063 2.542268 0.2 19.997073 2.647360 0.1 19.997073 2.647360 0.3 36.522610 6.779180 0.5 91.966641 20.72406 0.1 35.32447 3.026218 0.3 27.732563 2.846351 0.5 19.215453 2.625739 0.0 19.997073 2.765707 1.0 19.997073 2.328076 2.0 19.997073 2.100096 Table 5. The numerical values of the Nusselt number −φ ′ (1) Pr Ec φ −φ ′ (1) SWCNTs MWCNTs 20 1.5 0.01 0.067257 0.201779 21 0.107184 0.254538 22 0.127169 0.307473 20 1.6 0.067655 0.202565 1.6 0.068047 0.203369 1.5 0.02 0.246742 0.517264 0.03 0.427643 0.835802 ture field, skin-friction coefficient, velocity profiles, local Nus- selt number with the effects of parameters magnetic enclosure thermal radiation, thermophoresis, Prandtl number, permeabil- ity parameter, and Eckert number are exposed and obtained. The numerous observations of the current examined about the following conclusions. 1. Human blood as well as the base fluid is drastically utilised first time to attain the solution for Casson nano-fluids. It’s extremely noticed that rate decreases with increasing CNT quantity portion, and advances in CNT quantity di- vision will increase the blood temperature, which affects in and gives an improvement to the heat transfer velocity. 2. The velocity expressed increases through increasing the velocity fraction parameter about the temperature and con- centration profiles are decreased and increased byA. 3. An increase of the SWCNT solid φquantity section and Eckert quantity yields an addition with nano-fluids tem- 10 Ramanuja et al. / J. Nig. Soc. Phys. Sci. 5 (2023) 1054 11 Figure 15. Impact of A on θ(ς) Figure 16. IImpact of m1 on θ(ς) perature, leading to the direction of sudden radiation in the heat transfer rates. 4. Increasing values of the slip parameterδreduces the pace subject. The concept of base fluid provides parameter enhancement in the temperature. 5. The viscous dissipation affects the flow within the tem- perature profile and decreases with the insignificant value of the (Pr)Prandtl quantity. 6. An Eckert number (Ec)shows a small effect C f decreased but Nux has increased with the enlargement of the (Ec) Eckert number. The temperature distribution θ(η) is in- creased as the Eckert number (Ec) and Radiation param- eter increase. 7. The thermal radiation, heat generation/absorption, and permeability parameters throughout decreases with ad- vancement in prenatal number, the unsteadiness, the suc- tion, and the magnetic parameter. 8. As an essential position in dissipating heat the temper- ature of the fluid decreases through enhancement with nanoparticle quantity section for SWCNT as a result of elevated thermal conductivity. An expansion in the SWCNT’s quantity suction increases the Casson-nanofluid temper- ature, which affects inflation in the heat transfer velocity. 9. SWCNTs hold an identical consequence resting on the velocity with temperature profiles for Casson nanofluid flow. The resistance impedance in the direction of flow confirms higher results used for the SWCNT case than for the pureblood case. 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Sci. 5 (2023) 1054 13 ρn f Efficient density of the nanofluid (ρCP )n f Heat electrical condenser of the nanofluid v Kinematic viscosity k f ; ks Thermal conductivities T fluid temperature K Porous permeability R Radiation Da Darcy Number λ1 Jeffrey parameter θ Dimensionless temperature 13