Teknomekanik, Vol. 6, No. 2, pp. 103-121, December 2023 e-ISSN: 2621-8720 p-ISSN: 2621-9980 Numerical study on heat and flow transfer of biomagnetic fluid with copper nanoparticles over a linear extended sheet under the influence of magnetic dipole and thermal radiation Mohammad Ghulam Murtaza1, Maria Akter 1 and Mohammad Ferdows2* 1 2 Department of Mathematics, Comilla University, Cumilla, BANGLADESH Research Group of Fluid Flow Modeling and Simulation, Department of Applied Mathematics, University of Dhaka, Dhaka, BANGLADESH * Corresponding Author: ferdows@du.ac.bd Received Oct 28th 2023; Revised Dec 04th 2023; Accepted Dec 06th 2023 © The Author(s) Published by Universitas Negeri Padang. This is an open-access article under the: https://creativecommons.org/licenses/by/4.0/ Cite this https://doi.org/10.24036/teknomekanik.v6i2.26972 Abstract: A steady, two-dimensional flow of biomagnetic fluid namely blood flow with copper nanoparticles across a stretchable sheet that is affected by a strong magnetic field and thermal radiation is investigated in this study. Copper nanoparticles (Cu-NPs) were used for this study because of their important applicability in biomedical research. Thus, the properties of copper nanoparticles render it an antibacterial, antimicrobial, and anti-fungal material. Similarity substitutions were applied to reduce the nonlinear partial differential equations to ordinary differential equations. Utilizing the MATLAB R2018b software bvp4c function technique, the physical solution was established. This model's pertinent dimensions, such as the ferromagnetic parameter, the magnetic field parameter, the radiation parameter, the suction parameter, the ratio parameter, the slip parameter, and the Prandtl Number, were computationally and graphically inspected about the dimensionless velocity, temperature, skin friction, and heat transfer rate. One of the pivotal observations was that a rise in the ferromagnetic parameter and Prandtl number drops the temperature and velocity, correspondingly. A cross-case analysis with the outcome of other published research is also executed for divergent parameter values. Based on the investigations, copper nanoparticles may be advantageous for biomedical purposes and lessen the hemodynamics of stenosis. Owing to the research, copper nanoparticle-concentrated blood exhibits a reduced flow impedance and a larger temperature changeability compared to sheer blood. Keywords: Biomagnetic Fluid Dynamics; Copper Nanoparticles; Blood; Magnetic Dipole 1. Introduction Understanding, modulating, assessing, or mending human organs and tissues using internal or external magnetic fields is the purpose of the interdisciplinary field of biomagnetic fluid dynamics. Biomagnetic fluid uses have evolved to involve diagnosing neurological or cardiac disorders as well as comprehending the underlying mechanics of the human brain and heart throughout the last several decades. Researchers in the realms of biomedical engineering and its linked sciences have spurred the introduction of innovative biotechnologies. The fluid flow is analyzed using ferrohydrodynamic (FHD) and magneto-hydrodynamic (MHD) principles, and the magnetic field is adequately covered by a computational grid, in the current model. The broad modern utilizations of the investigation of limit layer conduct overextending sheet issues, for example, the streamlined expulsion of plastic sheets, the displacement of a polymer sheet from colour, the buildup cycle of metallic plates in cooling showers, and some designing applications, like the development of paper, metal turning, the production of food sources, the streamlined expulsion of plastic and elastic sheets, and so on, have drawn attention to the research. Mansur et al.[1]have been utilizing the Buongiorno model to investigate the progression of stagnation 103 Teknomekanik, Vol. 6, No. 2, pp. 103-121, December 2023 e-ISSN: 2621-8720 p-ISSN: 2621-9980 © The Author(s) Published by Universitas Negeri Padang. This is an open-access article under the: https://creativecommons.org/licenses/by/4.0/ highlighting a porous sheet of extending/contracting with the impact of pull in a nanofluid. They observed that the skin friction decreases as the sheet is extended, but rises as the impact of suction increases. Jat et al. [2], [3] studied the MHD flow of the boundary layer past the stagnation point of an extending linear surface. Wang [4] studied the stagnation point flow caused by a linearly shrinking surface. Aman et al. [5] observed the continuous flow of a two-dimensional stagnation point in a viscous fluid within the magnetic field through a linear sheet of stretching and shrinking. The findings indicate that there are two distinct options for shrink wrap and stretch wrap. The concept of fluid flow over a stretching boundary layer was also investigated by Crane [6]. The fluid behaviour of viscous flow through a shrinking sheet was examined by Miklavčič et al. [7]. The impact of thermal radiation on blood flow subject to magnetic dipole over an extended sheet was discussed by Alam et al. [8]. Mahapatra and Gupta [9], [10] addressed the thermal performance in the flow of a stagnation point across a sheet stretched over a viscoelastic fluid. Vajravelu, on the other hand, was the first researcher to examine the flow of viscous fluid past a nonlinear stretching sheet [11]. It was discovered that the fluid has consistently received heat flow from the stretching sheet. Similarity solutions for the stagnation point flow were examined by Bachok and Ishak [12] past a nonlinear sheet of stretching and shrinking. A well-known shooting method was used to solve the problem computationally. The behaviour of nanofluid flow caused by extendable non-linear sheets was further investigated by Rana et al. [13] Matin et al. [14] investigated what MHD meant for the drift of joined convective in a nanofluid past an extending sheet along entropy heat generation. A mathematical analysis of electrically conducting fluid over a stretchable boundary layer through a porous medium under chemical reaction and transverse magnetic field was discussed by Cortell [15]. Moreover, in [16] Cortell discussed the behaviour of viscous fluid flow under two surface temperature conditions. One is constant surface temperature and the other one is prescribed surface temperature. Raptis et al. [17] investigated thick drift over a nonlinear stretching sheet in the presence of a chemical response and a magnetic field. Tzirtzilakis first discussed the mathematical model of biomagnetic fluid which is based on Haik et al. in [18]. The extended mathematical model of BFD using MHD and FHD principles through a stretchable cylinder was discussed by Alam et al. [19] and shows that blood flow has a significant influence on the boundary layer compared to conventional blood flow. Recently a respectable study of boundary layer flow under several circumstances has been conducted by [20], [21], [22]. Due to the staggering number of potential applications in the fields of biomedicine, and bioengineering, nanoparticle evaluation has quickly become a hotbed of intense research interest. The terminology "nanofluid" implies a composition of a substrate liquid and nanoparticles having distinctive chemical and bodily attributes. Non-metals like nitrides and metals like carbon nanotubes and graphite contain the majority of the nanoparticles. Manufacturing, computers, microelectronics, transportation, biomedicine, food processing, fuel cells, solid-state lighting, and the heat transfer rates of microchips are just a few examples of their applications. Choi was the first person to use the word "nanofluid"[23]. Recently, the heat and flow characteristics of blood flow containing magnetic particles past a stretchable cylinder under the magnetic dipole effect were described by Ferdows et al. [24]. Ferdows et al. [25] discussed the blood behaviour in three cases namely FHD, MHD and extension of MHD and FHD i.e. BFD. They observed that blood temperature was the highest found in the case of BFD compared to MHD and FHD, where ferromagnetic numbers play a key role over the flow boundary layer. A comprehensive comparison of magnetic and non-magnetic particles when injected into blood flow was found [26]. Using Newtonian heating, Hayat et al. [27] examined homogeneous-heterogeneous reactions in the stagnation point flow of carbon nanotubes. Utilizing Ti and Ti-alloy nanoparticles on blood (as the base fluid), Reddy et al. [28] found the importance of nanofluid across a non-linear stretching sheet with the velocity slip. 104 Teknomekanik, Vol. 6, No. 2, pp. 103-121, December 2023 e-ISSN: 2621-8720 p-ISSN: 2621-9980 However, the aforementioned, studies reveal that the study of biomagnetic fluid containing nanoparticles over an extendable sheet using the ferrohydrodynamics concept has not yet been studied to date with the author's best knowledge. Blood is considered a base fluid which is electrically non-conducting and has properties of magnetization. Additionally, the impacts of velocity slip and thermal radiation are also considered. The addressed govern flow problem is expressed in PDEs form which is converted into ODEs with the help of suitable similarity transformations. A bvp4c technique is then used to show the physical insight of governing dimensionless parameters. Material dan methods 2. © The Author(s) Published by Universitas Negeri Padang. This is an open-access article under the: https://creativecommons.org/licenses/by/4.0/ 2.1 Mathematical formulation Assuming a laminar flow of blood-Cu in two dimensions through a stretching/shrinking sheet controlled by velocity slip. The non-linear velocity for the stretching/shrinking sheet is 𝑢𝑤 (𝑥) = 𝑐𝑥 𝑚 , and the power index is represented by "m" and indicates stretching when c>0 and shrinking sheetc<0, respectively. The surface temperature is assumed Tw and the ambient temperature far away from the sheet is T∞ such that Tw0 precedes the suction and S<0 precedes the injection respectively), 𝜆1 >0 represents first order velocity slip, A is the ratio of free stream velocity c∞ to stretching velocity c. Also, Prandtl number 𝑃𝑟 = (𝜇𝑐𝑝 ) 𝑓 𝑘𝑓 , viscous dissipation parameter 𝜆 = 𝑇 𝑐𝜇𝑓2 𝜌𝑓 𝑘𝑓 (𝑇𝑐−𝑇𝑤 ) , dimensionless 𝑐 ∞ Curie temperature 𝜀 = 𝑇 −𝑇 , dimensionless distance 𝛼 = √𝜐 𝑑, radiation conduction parameter 𝑤 ∞ 𝑓 © The Author(s) Published by Universitas Negeri Padang. This is an open-access article under the: https://creativecommons.org/licenses/by/4.0/ 16𝜎 ∗ 𝑇∞ 3 𝛾 𝜇° 𝑘(𝑇𝑐 −𝑇𝑤 )𝜌𝑓 𝑁𝑟 = , ferromagnetic parameter 𝛽 = . ∗ 3𝑘 𝑘𝑓 2𝜋 𝜇𝑓 2 In this research work, skin friction co-efficient 𝐶𝑓 and local Nusselt number Nu are the quantities of practical interest and are defined as: 𝜏 𝐶𝑓 = 𝜌 𝑢𝑤 2 (14) 𝑓 𝑤 𝑥𝑞 𝑤 𝑁𝑢 = 𝑘 (𝑇 −𝑇 ) 𝑓 𝑤 (15) ∞ The surface shearing stress 𝜏𝑤 and the convective heat 𝑞𝑤 are given by: 𝜕𝑢 (16) 𝜏𝑤 = 𝜇𝑛𝑓 (𝜕𝑦)𝑦=0 𝜕𝑇 𝜕𝑦 (17) 𝑞𝑤 = −𝑘𝑛𝑓 ( )𝑦=0 Therefore, finally, we have the following form: 1 𝐶𝑓 𝑅𝑒 2 = 1 (1−𝜙)2.5 1 𝑘 (18) 𝑓ʹʹʹ(0) & 𝑁𝑢𝑅𝑒 2 = − 𝑛𝑓 𝜃ʹ(0) 𝑘𝑓 2.3 Numerical method for solution For the solver (bvp4c) function in MATLAB software, the non-linear boundary value problem represented by Eq. (11) and (12) with the boundary condition Eq. (13) is solved consistently. To set the equations in MATLAB, new variables are added to Eq. (13) and (11) to make them firstorder differential equations. The following are the brand-new initial variables: 𝑓 = 𝑦1, 𝑓ʹ = 𝑦2 , 𝑓ʹʹ = 𝑦3 , 𝜃 = 𝑦4 , 𝜃ʹ = 𝑦5 Equations and boundary conditions undergo the following transformations into a system of firstorder ordinary differential equations: 𝑓ʹ = 𝑦2 , 𝑓ʹʹ = 𝑦2 ʹ = 𝑦3, 𝑓ʹʹʹ = 𝑦3 ʹ = −(1 − 𝜙)2.5 (1 − 𝜙 + 𝜙 107 𝜌𝑠 𝜌𝑓 ) (𝑦1 𝑦3 − (𝑦2 2 − 𝐴2 )) + (1 − 𝜙)2.5 2𝐵𝑦4 (𝑥+𝛼)4 (19) Teknomekanik, Vol. 6, No. 2, pp. 103-121, December 2023 e-ISSN: 2621-8720 p-ISSN: 2621-9980 𝜃ʹʹ = 𝑦5 ʹ = 1 ( 𝑘𝑛𝑓 𝑘𝑓 +𝑁𝑟) (𝜌𝐶𝑝) 2𝐵𝜆(𝜀+𝜃) [− (1 − 𝜙 + 𝜙 (𝜌𝐶𝑝) 𝑠 ) Pr(𝑦1 𝑦5 − 𝑛𝑦2 𝑦4 ) + (𝑥+𝛼)5 𝑦1 ] 𝑓 (20) Under the following delimitation conditions: 𝑦1 (0) = 𝑆, 𝑦2 (0) = 1 + 𝜆1 𝑦3 (0), 𝑦4 (0) = 1, 𝑦2 (∞) = 𝐴, 𝑦4 (∞) = 0 (21) By a given proximate point, Eq. (19), (20) and (21) are compacted numerically as an initial value problem. The aforementioned simplifications are accomplished using the MATLAB software's bvp4c function. © The Author(s) Published by Universitas Negeri Padang. This is an open-access article under the: https://creativecommons.org/licenses/by/4.0/ 3. Results and discussion For numerical accuracy and validation of the present results, a comparison has been made with 1 𝑘 earlier studies of Mukhopadhyay [30] for several values of Prandtl number for 𝑁𝑢𝑅𝑒 2 = − 𝑘𝑛𝑓 𝜃ʹ(0) 𝑓 and this analysis is stated in Table 1. 1 𝑘 Table 1: Value of 𝑁𝑢𝑅𝑒 2 = − 𝑘𝑛𝑓 𝜃ʹ(0) for several values of Prandtl number 𝑓 Pr 1 2 3 Mukhopadhyay 0.9547 1.4714 1.8961 Present Study 0.9542 1.4748 1.8937 Since, in this model, blood is considered as base fluid and Cu as magnetic particles. So in the computational process, we utilize the following values that are captured in Table 2. Table 2: Values of blood and Cu [31], [32] Properties C  jkg − 1K − 1  p     kgm− 3    − 1   Wm K − 1    Base Fluid nano particles Blood Cu 3 3.9 x 10 385 1050 8933 0.5 400 However, the values of dimensionless parameters are that used in this model are for the numerical solution, specific values for the dimensionless parameters must be determined. In the case of fluid, like human blood, the body temperature is set at Tw = 370C = 3100K and the ambient temperature is T∞ = 410C = 3170K [7]. For the given values, we have the following dimensionless parameters: the dimensionless temperature e = 78.5 [33], Prandtl Number Pr = 21, 23, 25 [19], radiation conduction parameter Nr = 0.2, 0.8, 1.0, viscous dissipation parameter λ = 0.01, ferromagnetic number B = 1, 5, 6 [34], dimensionless distance α = 0.5, 0.75, 1 [34], volume friction ϕ = 0.01, 0.05, 0.15 [34], suction parameter S = 0.2, 0.5, 1.0, first order velocity slip parameter λ1 = 0.1, 0.5, 1.0, ratio parameter A = 0.8, 0.9, 1.0, dimensionless Curie number ε = 78.5 and linear stretching sheet n = 1. Unless otherwise noted in the relevant graphs, the values indicated above are conserved as common. 108 Teknomekanik, Vol. 6, No. 2, pp. 103-121, December 2023 e-ISSN: 2621-8720 p-ISSN: 2621-9980 © The Author(s) Published by Universitas Negeri Padang. This is an open-access article under the: https://creativecommons.org/licenses/by/4.0/ The temperature and velocity profiles for various ratio parameter values (A) are shown in Figures 2 and 3. The figures show that both the velocity profile and temperature increase with increasing proportion thresholds in both the unadulterated blood and copper-adulterated blood cases. As the ratio parameter (A) increases, the velocity and temperature profiles change as can be seen. The sheet and liquid both are moving at the same speed when A = 1.0 is used. The non-dimensional flow function of the nano-fluid becomes increasingly positive between A= 0.8 & and A= 0.9. . Figure 2: Effect of A on velocity profile ƒʹ(η) Figure 3: Effect of A on temperature profile θ(η) The two figures 4 and 5 illustrate the velocity profile and heat flow for varying values of the nondimensional range factor (α). The stats indicate that for two very different sheer blood and copper sheer blood, the velocity profile declines and the heat flow increases with increasing values of the non-dimensional range component. 109 © The Author(s) Published by Universitas Negeri Padang. This is an open-access article under the: https://creativecommons.org/licenses/by/4.0/ Teknomekanik, Vol. 6, No. 2, pp. 103-121, December 2023 e-ISSN: 2621-8720 p-ISSN: 2621-9980 Figure 4: Effect of α on velocity profile ƒʹ(η) Figure 5: Effect of α on temperature profile θ(η) The influence of the speed profile and temperature profile for various upsides of ferromagnetic number (β) is depicted in Figures 6 and 7. The graph demonstrates that the acceleration and temperature curve in copper sheer blood as well as sheer blood diminish as the ferrous quantity factors rise. 110 © The Author(s) Published by Universitas Negeri Padang. This is an open-access article under the: https://creativecommons.org/licenses/by/4.0/ Teknomekanik, Vol. 6, No. 2, pp. 103-121, December 2023 e-ISSN: 2621-8720 p-ISSN: 2621-9980 Figure 6: Effect of β on velocity profile ƒʹ(η) Figure 7: Effect of β on temperature profile θ(η) Figures 8 and 9 illustrate the speed profile and heat flux considering multiple values of the firstorder speed drop coefficient (𝜆1 ). As can be demonstrated from the figures, both pure blood and copper blood undergo temperature rises and a drop in the velocity profile with rising values for the slip effects at the fundamental level. 111 © The Author(s) Published by Universitas Negeri Padang. This is an open-access article under the: https://creativecommons.org/licenses/by/4.0/ Teknomekanik, Vol. 6, No. 2, pp. 103-121, December 2023 e-ISSN: 2621-8720 p-ISSN: 2621-9980 Figure 8: Effect of λ1 on velocity profile ƒʹ(η) Figure 9: Effect of λ1 on temperature profile θ(η) Demonstration of speed curve and heat flux profile for assorted values of radiation conduction constant (Nr) are represented by Figures 10 and 11. As the radiation conductance factor is elevated in sheer blood and copper sheer blood, the speed curve and heat flux diminish. 112 © The Author(s) Published by Universitas Negeri Padang. This is an open-access article under the: https://creativecommons.org/licenses/by/4.0/ Teknomekanik, Vol. 6, No. 2, pp. 103-121, December 2023 e-ISSN: 2621-8720 p-ISSN: 2621-9980 Figure 10: Effect of Nr on velocity profile ƒʹ(η) Figure 11: Effect of Nr on temperature profile θ(η) Figures 12 and 13 illustrate the speed curve and heat flux for assorted values of the volume friction factor (ϕ). Analyzing the statistics, it tends to be apparent that the speed profile reduces and a surge in heat flux concomitant growing upsides of volume contact boundary in the two contexts for undiluted blood and copper undiluted blood. 113 © The Author(s) Published by Universitas Negeri Padang. This is an open-access article under the: https://creativecommons.org/licenses/by/4.0/ Teknomekanik, Vol. 6, No. 2, pp. 103-121, December 2023 e-ISSN: 2621-8720 p-ISSN: 2621-9980 Figure 12: Effect of ϕ on velocity profile ƒʹ(η) Figure 13: Effect of ϕ on temperature profile θ(η) Illustrations of the temperature profile with varying rates of Prandtl number (Pr) are delineated in Figure 14. It is seen that the temperature profile increased with the enhancement of the Prandtl number. The Prandtl number transmits the fraction of the energy distribution to warm dispersion. As a consequence, the Prandtl number is the proportion of the kinematic consistency of the liquid to its conductance of radiation. This indicates that a greater Prandtl number ends in reduced warm conductivity and this is the explanation why growing Prandtl number attributes has the consequence of lessened heat flux stinginess. 114 © The Author(s) Published by Universitas Negeri Padang. This is an open-access article under the: https://creativecommons.org/licenses/by/4.0/ Teknomekanik, Vol. 6, No. 2, pp. 103-121, December 2023 e-ISSN: 2621-8720 p-ISSN: 2621-9980 Figure 14: Effect of Pr on temperature profile θ(η) For varied upsides of the adhesion boundaries (S), Figures 15 and 16 illustrate the speed profile and temperature profile. From the figures, it very well may be noticed that the speed profile and the temperature both abatement with growing up sides of the attraction boundary in the two scenarios for undiluted blood and copper undiluted blood. Figure 15: Effect of S on velocity profile ƒʹ(η) 115 © The Author(s) Published by Universitas Negeri Padang. This is an open-access article under the: https://creativecommons.org/licenses/by/4.0/ Teknomekanik, Vol. 6, No. 2, pp. 103-121, December 2023 e-ISSN: 2621-8720 p-ISSN: 2621-9980 Figure 16: Effect of S on temperature profile θ(η) Finally, the variations of physical quantities such as coefficient of skin friction ƒʹʹ(0) and local Nusselt number are graphically shown through Figures 17 and Figure 18 under several values of Prandtl number, radiation parameter, volume fraction and ratio parameter, respectively about the ferromagnetic number. 1.2 1.2 Nr=0.2, l=0.01, a=.5 f=0.01, A=0.2, e=78.5 n=1 1.0 Pr=21, l=0.01, a=.5 f=0.01, A=0.2, e=78.5 n=1 1.0 0.8 0.8 f²(0) 0.6 0.6 f²(0) 0.4 0.4 Nr=0.2, 0.5, 0.8 Pr=21, 23, 25 0.2 0.2 0.0 0.0 a 1 2 3 b 4 5 6 b 1 2 3 b 4 5 6 5 6 1.2 Pr=21, Nr=0.2, l=0.01 a=.5, A=0.2, e=78.5 n=1 1.0 1.4 Pr=21, Nr=0.2, l=0.01, a=.5, f=0.01, e=78.5 n=1 1.2 0.8 1.0 f²(0) 0.6 f²(0) 0.8 f=0.01, 0.05, 0.15 0.4 A=0.8, 0.9, 1.0 0.6 0.2 0.4 0.0 c 1 2 3 b 4 5 6 d 1 2 3 b 4 Figure 17: Skin friction co-efficient ƒʹʹ(0) with β for different values of (a) Pr (b) Nr (c) ϕ (d) A 116 Teknomekanik, Vol. 6, No. 2, pp. 103-121, December 2023 e-ISSN: 2621-8720 p-ISSN: 2621-9980 -10 Nr=0.2, l=0.01, a=.5, f=0.01, A=0.2, e=78.5 n=1 -12 -12 -13 q¢(0) Pr=21, l=0.01, a=.5 -10 f=0.01, A=0.2, e=78.5 n=1 -11 -14 q¢(0) -14 -16 -15 Pr=21, 23, 25 -18 -16 -17 © The Author(s) Published by Universitas Negeri Padang. This is an open-access article under the: https://creativecommons.org/licenses/by/4.0/ -20 1 a Nr=0.2, 0.5, 0.8 2 3 b 4 5 6 b 2 3 b 4 5 6 5 6 -10 Pr=21, Nr=0.2, l=0.01, a=.5, A=0.2, e=78.5 n=1 -6 1 Pr=21, Nr=0.2, l=0.01, a=.5, f=0.01, e=78.5 n=1 -11 -8 -12 -13 q¢(0) -10 q¢(0) -14 -12 -15 -14 f=0.01, 0.05, 0.15 A=0.8, 0.9, 1.0 -16 -16 -17 1 2 3 b 4 5 6 d 1 2 3 b 4 Figure 18: Local Nusselt Number θʹ(0) with β for different values of (a) Pr (b) Nr (c) ϕ (d) A It is seen that both ƒʹʹ(0) and θʹ(0) are reduced for enlarging values of radiation parameter and Prandtl number. For boosting values of particle volume fraction it is found that the rate of heat transfer accelerates but skin friction is reduced. In the case of the ratio parameter, ƒʹʹ(0) enhanced but reverse phenomena are found for θʹ(0). 4. Conclusion A numerical investigation of blood-Cu has been conducted in this study in the presence of a magnetic dipole that passed through a stretching sheet. The governing PDEs are transformed into ODEs via suitable similarity variables and a well-known bvp4c technique is therefore applied to solve this problem computationally. Therefore, from the present numerical outcomes, we can summarize our findings as: (i) With enlarging values of the suction parameter, and radiation parameter both fluid velocity and temperature decline; whereas reverse phenomena are observed for the ratio parameter. (ii) Fluid velocity reduces for advancement values of dimensionless distance, first-order slip parameter, and volume fraction, while reverse results are found in temperature distributions case. (iii) For increment values of ferromagnetic number, fluid temperature decreases but velocity increases. 117 Teknomekanik, Vol. 6, No. 2, pp. 103-121, December 2023 e-ISSN: 2621-8720 p-ISSN: 2621-9980 (iv) Both skin friction and Nusselt number reduce for Prandtl number and radiation parameter. (v) For rising values of volume fraction, skin friction reduces but the rate of heat transfer increases. (vi) The rate of heat transfer of blood-Cu reduces for accelerating values of ratio parameter but increases in skin friction profile. Author contribution M.G. Murtaza and Maria Akter: Carried out the formal analysis, investigation, conceptualization and drafted the manuscript. M. Ferdows: Carried out methodology, review, editing and supervision. All authors read and approved the final manuscript. © The Author(s) Published by Universitas Negeri Padang. This is an open-access article under the: https://creativecommons.org/licenses/by/4.0/ Funding statement This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors. Acknowledgements We are grateful to Simulation Laboratory, Department of Mathematics, Comilla University for high configuration computer used in this work. 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This is an open-access article under the: https://creativecommons.org/licenses/by/4.0/ Teknomekanik, Vol. 6, No. 2, pp. 103-121, December 2023 e-ISSN: 2621-8720 p-ISSN: 2621-9980 NOMENCLATURE U, V Components of velocity Ue Free stream velocity viscosity under dynamic conditions 𝜇 𝜇𝑛𝑓 Effective friction coefficient 𝜌𝑛𝑓 Effective density of the nanofluid T Nanofluid temperature (𝜌𝐶𝑝)𝑛𝑓 Thermal capacity of the base fluid Volume friction of nanoparticles ϕ 𝑘𝑛𝑓 Nanofluid's thermodynamic resistance 𝜇𝑓 Viscosity under dynamic condition 𝜌𝑓, 𝜌𝑠 Densities of fluid and solid 𝑘𝑓, 𝑘𝑠 Thermal conductivities of fluid and solid H Slope amplitude M Magnetization K Pyro-magnetic co-efficient S Suction/injection parameter Pr Prandtl number Viscous dissipation parameter λ Dimensionless Curie temperature ε Dimensionless distance α Nr Radiation Conduction Parameter Ferromagnetic parameter β Cf Skin friction coefficient Nu Local Nusselt Number Surface shearing stress 𝜏𝑤 = Convective heat 𝑞𝑤 121