Spectral quasi-linearization and response optimization on magnetohydrodynamic flow via stenosed artery with hybrid and ternary solid nanoparticles: Support vector machine learning

Maddina Dinesh Kumar; Mamatha Sadananda Upadhya; Se-Jin Yook; Nehad Ali Shah; Chakravarthula Siva Krishnam Raju

doi:10.1515/ntrev-2025-0188

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Spectral quasi-linearization and response optimization on magnetohydrodynamic flow via stenosed artery with hybrid and ternary solid nanoparticles: Support vector machine learning

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Veröffentlicht/Copyright: 26. September 2025

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Aus der Zeitschrift Nanotechnology Reviews Band 14 Heft 1

Abstract

Advancements in nanotechnology have revolutionized the field of biomedical applications. Nanoparticles of molybdenum disulphide, copper, silver, aluminium oxide, and carbon nanotubes exhibit significant anticancer and antimicrobial properties. In this study, blood flow through cosine-shaped stenotic arteries was mathematically modelled and examined in an aligned magnetic field, radiation, and nanoparticles. Comparisons were made between the hybrid nanofluid flow in the cosine-shaped stenosis artery and other blood vessels for two cases: Case 1 – Blood + Cu + MOS 2 + Al 2 O 3 , and Case 2 – Blood + Ag + SWCNT + MWCNT . The flow governing equations are transformed into ODEs and solved numerically using spectral quasi-linearization with MATLAB (SQLM). Expressions for the temperature, velocity profile, and numerical and graphical representations were used to calculate and assess the Nusselt number and skin friction. The investigation was validated by comparing it with the published work. It was found that the Cu + MOS 2 + Al 2 O 3 combination effectively enhanced heat transmission in blood with improved radiation parameters and flow constraints, and the Ag + SWCNT + MWCNT combination effectively enhanced heat transmission in blood with improvement in magnetic parameters. This finding might be helpful in investigations on nano-haemodynamics and the therapy of haemodynamic disorders.

Keywords: spectral quasi-linearization; magnetohydrodynamic flow; stenosed artery; hybrid and ternary nanoparticles; response optimization; support vector machine learning

1 Introduction

The human body’s circulatory system is responsible for moving blood, which distributes nutrients and oxygen. It consists of blood, blood arteries, and the heart. Blood vessels include arteries, veins, and capillaries. The human cardiovascular system maintains the body’s equilibrium by controlling blood circulation, delivering nutrients and oxygen, removing waste products, and fighting infections and diseases. The obstruction of blood flow is the cause of health-related difficulties. One of the primary causes of cardiovascular disease (CVD) is the reduction in blood vessel size. According to data from the WHO, CVD accounted for 17.9 million fatalities in 2019, or 32% of all deaths worldwide, making it the leading cause of death. According to the WHO, an estimated 8.9 million deaths worldwide were attributed to ischemic heart disease (a type of heart disease caused by reduced blood flow to the heart) in 2019. In addition, an estimated 2.6 million deaths were attributed to stroke, a type of CVD that can be caused by a blockage of blood flow to the brain. In 2020, COVID-19 affected the myocardial systems. Haemodynamics is the study of the physical principles that govern blood circulation in an organism. Haemodynamics has been researched in the context of exercise physiology and sports medicine and is significant in diagnosing and treating cardiovascular disorders. Abnormal haemodynamics can contribute to various health problems, including hypertension, heart failure, and atherosclerosis. Atherosclerosis is the leading cause of death in many countries. Plaque builds up in arteries, a condition known as atherosclerosis. Here, the initial thickening of the stenotic arteries affects blood circulation. Severe stenosis results in critical blood flow conditions by reducing blood supply, which causes carotid artery occlusion and stroke. In a few instances, plaque that has built up in the arteries may separate, enter the bloodstream, and lodge in the blood vessels in the brain. An ischemia attack may result from this condition. As a result of all these serious problems, fluid mechanics has attracted the interest of numerous scholars over the years. The equations of motion proposed by Muskat [1], Brinkman [2], and Stefanoyska [3] could be altered because blood is a fluid, and this is useful for learning more about the vascular wall and the characteristics of blood flow. Bioengineers will benefit from this to build artificial organs and cure cardiovascular illnesses. By enabling targeted drug administration, advancing imaging methods, and strengthening the mechanical qualities of cardiovascular devices, nanotechnology can completely transform the treatment of CVD. Umadevi et al. [4] observed heat transport rate in various flow models by considering the combination of nanoparticles and stated that combining nanoparticles in the base fluid improved heat transfer. The application of nanotechnology could resolve the problem of stenotic arteries. Rathore and Sandeep [5] proved that blood-based nanoparticles are helpful in cancer therapy. Mishra et al. [6] reported that an increase in copper nanoparticles improved heat dissipation and thermal conductivity in stenosed arteries. Tang et al. [7] investigated Oldroyd B nanofluid flow in stenosis arteries and predicted the cause of atherosclerosis. Sajid et al. [8] considered tetra hybrid radiative nanofluid flow in stenosed arteries and stated that the temperature within arteries increases in strength with thermal radiation and viscous dissipation. Nadeem et al. [9] studied four triangular, overlapping (w shape), trapezoidal, and composite forms of stenotic regions. Shahzadi et al. [10] investigated a ternary nanofluid in an obliquely stenosed artery. In their study, Omamoke and Amos [11] showed improvement in a slip at the stenosis wall that decreased the blood velocity, volumetric flow rate, and acceleration. Following their assessment, Gandhi et al. [12] declared that the postponement of nanoparticles in the bloodstream influences the treatment of haemodynamic complaints. Patel and Patel [13] demonstrated that heat transfer improved with heat generation and thermal radiation. Waqas et al. [14] disclosed in their study that nanoparticles (silver and gold) improved the performance of blood flow. Minnam Reddy et al. [15] demonstrated that stopping lamina-shaped silver nanoparticles in blood aids in bacterial death. Kolin [16] was the first to introduce an electromagnetic field in medical research. Hence, Korchevskii and Marochnik [17] considered using a magnetic field to regulate blood flow. Suri and Pushpa [18], Vardanyan [19], and Sud et al. [20] theoretically examined the influence of magnetic fields by taking into account different blood flow models. They observed that the magnetic field reduced the blood flow rate. When there is reduced blood flow in the arteries due to arterial diseases such as arteriosclerosis or arterial stenosis, when performing cardiac surgeries, a magnetic field can be used as a blood pump. Zain et al. [21] discovered that the presence of a magnetic field altered blood flow. Zain and Ismail [22] observed that a magnetic field increased flow reversal by 26% for Newtonian fluids, 39% for shear-thinning fluids, and 27% for shear-thickening fluids. Dolui et al. [23] observed that magnetic force controls the flow velocity. Gandhi et al. [24] studied blood flow via the irregular stenotic artery and noticed an improvement in the magnetic number and increased heat transfer rate. Sharma et al. [25] examined the flow of MHD hybrid nanofluids via the inclined stenotic artery. Ahmed et al. [26] disclosed that porosity and an inclined magnetic field significantly influence biofluid flow. A mathematical study by Gangadhar and Dinarvand [27,28,29] showed that the cooling effects of radiation absorption as a rotating sphere interact with the stagnation point flow of a triple-nanoparticle nanofluid. This research evaluates the irreversible behaviour of electromagnetohydrodynamic (EMHD) flows using a magnesium oxide and silver hybrid nanofluid under nonlinear thermal radiation analysis. The evaluation of Maxwell fluid flow through spiralling discs with homogeneous–heterogeneous chemical processes employs both machine learning and numerical approaches. For increased processing power and accuracy, regularized machine learning methods must be used with nonlinear thermal radiation models to predict efficiency. Akbar et al. [30,31,32] investigated how thermophoretic diffusion affected the microbic flow analysis of a chemically reacting nanofluid in a microchannel with flexural walls, its use in drug delivery, the biological structural analysis of blood Casson fluid flow in catheterized diverging tapered stenosed arteries with emerging shaped nanoparticles, and the microbic flow analysis of a chemically reacting nanofluid in a microchannel with flexural walls with thermophoretic diffusion.

1.1 Novelty of the present study

Researchers have investigated nano or blood flow in different flow geometries in the abovementioned studies. A “ternary hybrid nanofluid” refers to a new, unique class of heat transfer fluid that has been developed scientifically. A ternary hybrid nanofluid is a joint fluid containing three types of colloidal nanoparticles. The effects of magnetic fields and radiation on blood moving through arteries with cosine-shaped stenosis were investigated. We then expanded on their study by considering a ternary hybrid nanofluid, which is a colloidal mixture of a conventional fluid containing three different types of nanoparticles. We then analysed the blood flow with cosine form stenosis arteries in two cases. Case 1: Blood + Cu + MOS 2 + Al 2 O 3 and Case 2: Blood + Ag + SWCNT + MWCNT . RSM with ANOVA was adopted to examine the influence of the input variables on heat transfer.

1.2 Applications of the current study

Magnetic targeting of drugs: The targeted drug delivery function of nanoparticles is improved through magnetic targeting owing to the tilting magnetic effect, which aids in the treatment of CVD and cancer therapy.
Treatment for hyperthermia: Radiative heat transfer facilitates nanoparticle-based hyperthermia therapy to ensure controlled heating for efficient tumor cell killing.
Tissue engineering and synthetic organs: The efficacy of vascular grafts and artificial organs for biomedical purposes is maximized through heat regulation via sources and heat sinks.
Use of cryotherapy and biomedical cooling: In cryotherapy, hybrid and ternary nanoparticles improve thermal regulation, which helps with tissue preservation and pain management.

2 Mathematical formulation

A continuous flow of incompressible hybrid blood embedded with Cu + Mos 4 + Al 2 o 3 (Case 1) and Ag + SWCNT + MWCNT (Case 2) nanoparticles. The fluid is assumed to flow through an artery constriction in a cosine form and be Newtonian, L 0 / 2 with a broad, unimpeded region R 0 , s positioned against the flow. Here, the liquid’s flow is parallel to the x -direction, and the r -axis is vertical to the flow. The increase in stenosis and artery radius is regarded as R ( x ) and λ , respectively. A powerful magnetic field B 0 is applied at an angle α , as shown in Figure 1, to the flow. Additionally, the effects of thermal radiation and shifting heat sinks and sources were considered. Given the thinness of blood, radiative heat transmission is viewed as ( q 1 ) r is taken as 4 α ν 2 ( T − T 0 ) , where α ν is the coefficient of radiation absorption. Figure 2 shows the stenosed portion of an artery.

Figure 1

An illustration of an artery with a cosine-shaped stenosisis shown, and the yellow area highlights fatty deposits in the artery Rathore and Sandeep [5].

Figure 2

Side view from the arteries Rathore and Sandeep [5].

The tensed section’s variable is [33]

(1) R ( x ) = R 0 − 1 + cos 4 π x L 0 λ 2 , − L 0 4 ≺ x ≺ L 0 4 R 0 , otherwise .

The governing equations and boundary layer constraints based on the aforementioned assumption are as follows Rathore and Sandeep [5,33]:

Continuity:

(2) ( r u ) x + ( r v ) r = 0 .

The momentum and energy equations are framed according to the study by Sarwar and Hussain [33]:

(3) ρ hnf ( u u x + v u r ) = μ hnf r − 1 ( r u r ) r − σ hnf B 0 2 u cos 2 α ,

(4) ( ρ C p ) hnf ( u T x + v T r ) = k hnf r − 1 ( r T r ) r − ( q 1 ) r + q ‴ ,

with the constrained conditions framed based on the study by Sarwar and Hussain [33]:

(5) v ( r ) = 0 , u ( r ) = u w , and T = T 1 at r = R T → T 0 and u → 0 as r → ∞ .

The stream function will complete equation (2) as

v = − ψ x / r , u = − ψ r / r .

Equations (3) and (4) are converted as follows:

(6) ρ hnf ( ψ r ( r − 1 ψ r ) x / r − ψ x ( r − 1 ψ x ) r / r ) = ( r ( r − 1 ψ r ) r ) r μ hnf r − 1 − ψ r cos 2 α σ hnf M 2 r − 1 ,

(7) ( ρ C p ) hnf ( r − 1 ψ r T x − r − 1 ψ x T r ) = k hnf r − 1 ( r T r ) r − ( q 1 ) r + q ‴ ,

where q ‴ = ρ K u w ( x ) v f ( A ⁎ ( T 1 − T 0 ) f ′ + B ⁎ ( T − T 0 ) ) is the source/sink parameter for energy, and ( q 1 ) r = 4 α ν 2 ( T − T 0 ) is a factor in thermic radiation.

The following hybrid nanoparameters are taken into consideration:

(8) u = u 0 L 0 − 1 x f ′ ( ξ ) , v = − R r − 1 f ( ξ ) ( u 0 ν f L 0 − 1 ) 0.5 , ξ = − ( R 2 − r 2 ) 2 R ( u 0 ν f − 1 L 0 − 1 ) 0.5 T = T 0 + ( T 1 − T 0 ) θ ( ξ ) , ψ = R f ( ξ ) ( u 0 x 2 ν f L 0 − 1 ) 0.5 .

The thermophysical characteristics of the ternary hybrid thermal conductivity and viscosity are defined as follows:

k Thnf k hnf = ( k 3 + 2 k hnf − 2 φ 3 ( k hnf − k 3 ) ) ( k 3 + 2 k hnf + φ 3 ( k hnf − k 3 ) ) k hnf k nf = ( k 2 + 2 k nf − 2 φ 2 ( k nf − k 2 ) ) ( k 2 + 2 k nf + φ 2 ( k nf − k 2 ) ) k nf k f = ( k 1 + 2 k f − 2 φ 1 ( k f − k 1 ) ) ( k 1 + 2 k f + φ 1 ( k f − k 1 ) ) ,

μ Thnf = μ f ( 1 − φ 1 ) 2.5 ( 1 − φ 2 ) 2.5 ( 1 − φ 3 ) 2.5 .

The density equation for a ternary hybrid nanofluid is as follows:

ρ Thnf ρ f = ( 1 − φ 1 ) ( 1 − φ 2 ) ( 1 − φ 3 ) + φ 3 ρ sp3 ρ f + φ 2 ρ sp2 ρ f + φ 1 ρ sp2 ρ f .

The ternary hybrid nanofluid’s heat capacity is as follows: ( ρ c p ) Thnf ( ρ c p ) f = φ 1 ( ρ c p ) sp1 ( ρ c p ) f + ( 1 − φ 1 ) ( 1 − φ 2 ) ( 1 − φ 3 ) + φ 3 ( ρ c p ) sp3 ( ρ c p ) f + φ 2 ( ρ c p ) sp2 ( ρ c p ) f .

The ternary hybrid electric conductivity is defined as follows:

σ Thnf σ hnf = ( σ 3 + 2 σ hnf − 2 φ 3 ( σ hnf − σ 3 ) ) ( σ 3 + 2 σ hnf + φ 3 ( σ hnf − σ 3 ) ) σ hnf σ nf = ( σ 2 + 2 σ nf − 2 φ 2 ( σ nf − σ 2 ) ) ( σ 2 + 2 σ nf + φ 2 ( σ nf − σ 2 ) ) σ nf σ f = ( σ 1 + 2 σ f − 2 φ 1 ( σ f − σ 1 ) ) ( σ 1 + 2 σ f + φ 1 ( σ f − σ 1 ) ) .

The following nomenclature was used for the transformation:

E 1 = μ hnf μ f , E 2 = ρ hnf ρ f , E 3 = ( ρ C p ) hnf ( ρ C p ) f , E 4 = k hnf k f , E 5 = σ hnf σ f .

Equations (6) and (7) were modified using equation (8) and hybrid nanoparameters as follows:

(9) E 1 ( ( 2 γ ξ + 1 ) f ‴ ( ξ ) + 2 γ f ″ ( ξ ) ) − cos 2 α f ′ ( ξ ) M 2 E 5 + E 2 ( f ( ξ ) f ″ ( ξ ) − ( f ′ ( ξ ) ) 2 ) = 0 ,

(10) E 4 ( ( 2 ξ γ + 1 ) θ ″ ( ξ ) + 2 γ θ ' ( ξ ) ) − Ra θ ( ξ ) + ( A ⁎ f ′ ( ξ ) + B ⁎ θ ( ξ ) ) + Pr E 3 f ( ξ ) θ ′ ( ξ ) = 0 .

The restrictions are non-dimensional:

(11) f ′ ( 0 ) = 1 , f ( 0 ) = 0 , θ ( 0 ) = 1 , at r = R , θ → 0 , f ′ → 0 , as r = 0 .

The Prandtl number is used for the nondimensional values in equations (9) and (10) Pr = μ C p k f , flow constraint γ = ν f L 0 u 0 R 2 0.5 , magnetic field parameter M = B 0 2 L 0 σ f u 0 ρ f , and thermal radiation parameter Ra = 4 α ν 2 L 0 ν f u 0 k f .

The coefficient of drag C f x includes the thermal transmission factor and Nu x is defined as follows:

(12) C f x = ρ f − 1 2 τ w u w 2 , Nu x = x q w ( T 1 − T 0 ) − 1 k f ,

where

(13) τ w = μ hnf ( u r ) | r = R is drag force and q w = − k hnf ( T r ) | r = R , heat flux .

After converting to nondimensional, equations (12) and (13) are

(14) Re x 0.5 C f x = E f ″ 1 ( 0 ) , Re x 0.5 Nu x = E 4 θ ′ ( 0 ) ,

where Re x 0 .5 is the Reynolds quantity .

3 Numerical procedure

The boundary conditions of equation (11) from the studies of Mallikarjuna et al. [34] and Trefethen [35] are applied to nonlinear ODEs (9) and (10) using the SQLM numerical approach. The approximations f r and θ r are used to linearize formulae (9) and (10).

We obtain

(15) b 1,1 3 f r + 1 ‴ + b 1,1 2 f r + 1 ″ + b 1,1 1 f r + 1 ′ + b 1,1 0 f r + 1 = R 1 ,

(16) b 2,2 2 θ r + 1 ″ + b 2,2 1 θ r + 1 ′ + b 2,2 0 θ r + 1 + b 2,1 0 f r + 1 = R 2 .

The modified boundary conditions include

(17) f r + 1 ′ ( 0 ) = 1 , f r + 1 ( 0 ) = 0 , θ r + 1 ( 0 ) = 1 , at r = R , θ r + 1 ( ξ ) → 0 , f r + 1 ′ ( ξ ) → 0 , as ξ → ∞ .

The coefficients of (15) and (16) are as follows:

b 1,1 3 = A 1 ( 2 γ ξ + 1 ) , b 1,1 2 = 2 A 1 γ + A 2 f r , b 1,1 1 = − A 5 cos 2 α M 2 − 2 f r ′ , b 1,1 0 = A 2 f ″ , R 1 = A 2 f r ″ f r − ( f r ′ ) 2

b 2,2 2 = A 3 ( 2 γ ξ + 1 ) , b 2,2 1 = 2 A 3 γ + Pr A 4 f r , b 2,2 0 = − R a , b 2,1 0 = A 4 Pr θ r ′ , R 2 = Pr A 4 f r θ r ′ .

Then, Chebyshev Gauss Lobatto points can be selected. z j = Cos π j n , j = 1 to n , where n is the collation number, using the spectral approach on the domain [ 0 , 1 ] after applying the transformations ξ = ( z + 1 ) 2 to z ∈ [ − 1 , 1 ] .

The resulting matrix system of equations B X = R is as follows:

B = B 11 B 12 B 21 B 22 , X = f r + 1 θ r + 1 , R = R 1 R 2 ,

where

B 11 = dia ( b 1,1 3 ) D 3 + dia ( b 1,1 2 ) D 2 + dia ( b 1,1 1 ) D 1 + dia ( b 1,1 0 ) I B 12 = 0 B 22 = dia ( b 2,2 0 ) I B 21 = dia ( b 2,1 2 ) D 2 + dia ( b 2,1 1 ) D + dia ( b 2,1 0 ) I .

In this case, I represents a matrix of identities with order D , N + 1 and o for the zero matrix, respectively, for the Chebyshev differentiation matrix.

In the present study, the spectral method converges when the spectral points are N = 100, and the error tolerance for the spectral method is 10−7, which is typically very small and ensures high accuracy.

4 Results and discussion

The nonlinear ODEs in equations (9) and (10) with well-defined BCs (11) are determined using the numerical technique of the spectral method. The influence of magnetic parameter ( M ) , radiation factor ( Ra ) , and flow constraint ( γ ) on velocity f ′ ( ξ ) and temperature fields θ ( ξ ) of blood-based hybrid nanofluid flow in cosine-shaped stenosis arteries is compared for two cases: Case 1 ( Blood + Cu + MoS 4 + Al 2 O 3 ) and Case 2 ( Blood + Ag + SWCNT + MWCNT ) (Figures 11–14). In the figures, black represents case 1, and blue represents case 2.

4.1 Analysis of results by the response surface method (RSM)

The heat transfer rate for Case 1 and Case 2 is demonstrated by the preparation of a 3D surface plot with quadratic regression in Figures 3–8. RSM is one of the statistical and mathematical hybrid processes capable of constructing trials with the correct parameters. Here, M , R a , γ are essential variables, and the Nusselt number is an indicator of the reaction. The essential variables and responses have a strong positive association, which is statistically significant. The quadratic models for Nus1 and Nus2 to the coded coefficients 2 ≤ X 1 ≤ 4 , 0.6 ≤ X 2 ≤ 1.2 , 1 ≤ X 3 ≤ 2 are as follows:

Nus 1 = 3.232 + 1.67 M + 1.65 Ra − 5.72 γ − 0.160 M ⁎ M − 1.68 Ra ⁎ Ra + 1.29 γ ⁎ γ − 0.227 M ⁎ Ra + 0.114 M ⁎ γ + 1.21 Ra ⁎ γ . Nus 2 = 3.355 + 1.71 M + 1.72 Ra − 5.85 γ − 0.165 M ⁎ M − 1.73 Ra ⁎ Ra + 1.32 γ ⁎ γ − 0.237 M ⁎ Ra + 0.115 M ⁎ γ + 1.24 Ra ⁎ γ .

$Figure 3 Surface plot of Nus-1 for variations in Ra \text{Ra} and M M .$

Figure 3

Surface plot of Nus-1 for variations in Ra and M .

$Figure 4 Surface plot of Nus-2 for variations in Ra \text{Ra} and M M .$

Figure 4

Surface plot of Nus-2 for variations in Ra and M .

$Figure 5 Surface plot of Nus-1 for variations in Ra \text{Ra} and γ \gamma .$

Figure 5

Surface plot of Nus-1 for variations in Ra and γ .

$Figure 6 Surface plot of Nus-2 for variations in Ra \text{Ra} and γ \gamma .$

Figure 6

Surface plot of Nus-2 for variations in Ra and γ .

$Figure 7 Surface plot of Nus-1 for variations in M M and γ \gamma .$

Figure 7

Surface plot of Nus-1 for variations in M and γ .

$Figure 8 Surface plot of Nus-2 for variations in M M and γ \gamma .$

Figure 8

Surface plot of Nus-2 for variations in M and γ .

Figures 3–8 show the combined influence of Ra & M , Ra & γ , M & γ on the Nusselt number for Case 1 ( Blood + Cu + MOS 2 + Al 2 O 3 ) and Case 2 ( Blood + Ag + SWCNT + MWCNT ). It has been noticed that heat transfer improves in all cases with increasing values of both parameters. Ra & M , Ra & γ , M & γ and the highest heat transfer occurs at the magnetic parameter. ( M ) is at a high level, the radiation parameter ( Ra ) is at a low level, and the flow constraint ( γ ) is at a low level in both cases. Figure 3 illustrates the surface plot of Nus-1 for variations in Ra and M , Figure 4 shows a surface plot of Nus-2 for variations in Ra and M , Figure 5 shows a surface plot of Nus-1 for variations in Ra and γ , Figure 6 shows the surface plot of Nus-2 for variations in Ra and γ , Figure 7 shows the surface plot of Nus-1 for variations in M and γ , and Figure 8 shows the surface plot of Nus-2 for variations in M and γ .

4.2 Support vector machine (SVM) learning and various factors influence velocity and temperature profiles

Figures 9 and 10 compare truth and predicted values for Nus1 (Case 1) and Nus2 (Case 2). SVMs are linear models that solve regression and classification issues. It can efficiently resolve both linear and nonlinear problems, as well as real-world difficulties. The fundamental concept of SVMs is a line or hyperplane that divides the data into classes is the fundamental concept of SVMs.

Figure 9

Comparison of truth and predicted results for Nus-1.

Figure 10

Comparison of truth and predicted results for Nus-2.

Figures 11 and 12 characterize the influence of the flow parameter ( γ = 0.5 , 1 , 1.5 , 2 ) on f ′ ( ξ ) and θ ( ξ ) profiles. It is observed that the fluid velocity f ′ ( ξ ) and temperature θ ( ξ ) increase with an increase in γ . In both Case 1 and Case 2, an increase in γ increases the viscosity of the fluid and a decrease in the radius of the artery, hence f ′ ( ξ ) decreases and θ ( ξ ) increases. Case 1 Blood + Cu + MoS 4 + Al 2 O 3 has higher f ′ ( ξ ) and temperature θ ( ξ ) compared with Case 2 Blood + Ag + SWCNT + MWCNT . Figures 13 and 14 describe the results the radiation factor ( Ra = 0.3 , 0.6 , 0.9 , 1.2 ) on velocity f ′ ( ξ ) and temperature θ ( ξ ) ; improving radiation augmented f ′ ( ξ ) and θ ( ξ ) increases the hybrid nanofluid in both cases. The improvement in radiation resulted in more heat in the fluid, and the combination of nanoparticles also helped; thus, the combination of Cases 1 and 2 is more suited to medical treatment. Table 1 lists the thermophysical properties of the nanoparticles and base fluid. Table 2 displays the skin friction variation and Nusselt number for case 2; it also shows that increased magnetic and radiation parameters increased the skin friction and flow constraints in scenarios 1 and 2. Conversely, increasing the magnetic, radiation, flow constraint, and Prandtl numbers increases the heat transfer rate. The levels and codes of the primary input parameters are shown in Table 3. The RSM model applies a sequential plan, presents unique design considerations, and calculates the best correlations. In this case, a central composite design (CCD) is employed, and to provide an accurate model, the CCD considers extreme combinations of crucial components. The experimental design includes 20 experiments, as shown in Table 4. The ANOVA table in Table 5 shows that the model is statistically significant at a 95% significance level. The high determination coefficient (Table 5) shows that the model is statistically significant at the 95% significance level. A high determination coefficient (R2 = 97.76 %) assures the accuracy of the model.

$Figure 11 Influence of γ \gamma on f ′ ( ξ ) f^{\prime} (\xi ) .$

Figure 11

Influence of γ on f ′ ( ξ ) .

$Figure 12 Influence of γ \gamma on θ ( ξ ) \theta (\xi ) .$

Figure 12

Influence of γ on θ ( ξ ) .

$Figure 13 Influence of Ra \text{Ra} on f ′ ( ξ ) f^{\prime} (\xi ) .$

Figure 13

Influence of Ra on f ′ ( ξ ) .

$Figure 14 Influence of Ra \text{Ra} on θ ( ξ ) \theta (\xi ) .$

Figure 14

Influence of Ra on θ ( ξ ) .

Table 1

Thermophysical properties of the hybrid nanofluid and nanoparticles with a uniform shape

	Nomenclature of nanoparticles and base fluid	ρ (kg/m³)	Cp (J/kgK)	K (W/mK)
Base fluid	Blood	1,050	3,617	0.52
Solid particles 1	Molybdenum disulphide ( MOS 2 )	5,060	397.21	904.4
	Copper (Cu)	8,933	385	400
	Aluminium oxide ( Al 2 O 3 )	3,970	765	40
Solid particles 2	Silver (Ag)	10,500	235	429
	Single-wall carbon nanotubes ( SWCNT )	2,600	425	6,600
	Multi-walled carbon nanotubes ( MWCNT )	1,600	796	3,000

Table 2

Skin friction coefficients and local Nusselt numbers as physical parameters

			Skin friction		Nusselt number
γ	M	Ra	Case 1	Case 2	Case 1	Case 2
0.5			0.28646681	0.28646681	0.79449888	0.79655704
1			0.26525694	0.26525694	0.86611274	0.86964282
1.5			0.24542172	0.24542172	0.94334611	0.94872326
2			0.22688500	0.22688500	1.02659447	1.03426681
	1		0.2864661	0.28646681	0.79449888	0.7965574
	2		0.57293362	0.57293362	1.59311408	1.58899775
	3		0.85940043	0.85940043	2.38349663	2.38967112
	4		1.14586724	1.14586724	3.17799551	3.18622816
		0.3	0.08594004	0.08594004	0.23834966	0.23896711
		0.6	0.17188009	0.17188009	0.47669933	0.47793422
		0.9	0.25782013	0.2378	0.55782013	0.71504899
		1.2	0.34376017	0.34376017	0.95339865	0.95586845

Table 3

Key RSM parameters, their levels, and symbols

Key factors	Symbols	Levels
Key factors	Symbols	−1 (Low)	0 (Medium)	1 (High)
M	X 1	2	3	4
Ra	X 2	0.6	0.9	1.2
γ	X 3	1	1.5	2

Table 4

Design and results of heat transfer rate experiments true values (Nus) and predicted values using the SVM regression model

Runs	Coded values			Real values			Response (true values)		Predicted values
Runs	X 1	X 2	X 3	M	R a	γ	Nus-1	Nus-2	Nus-1	Nus-2
1	−1	−1	−1	2	0.6	1	2.23255734	2.25355943	2.286	2.322
2	1	−1	−1	4	0.6	1	4.46511467	4.50711887	4.419	4.455
3	−1	1	−1	2	1.2	1	2.23255734	2.25355943	2.232	2.257
4	1	1	−1	4	1.2	1	4.46511467	4.50711887	3.965	3.992
5	−1	−1	1	2	0.6	2	2.23255734	2.25355943	2.188	2.208
6	1	−1	1	4	0.6	2	3.46445096	3.47857129	3.921	3.943
7	−1	1	1	2	1.2	2	1.73222548	1.73928564	1.735	1.744
8	1	1	1	4	1.2	2	3.46445096	3.47857129	3.468	3.479
9	−1	0	0	2	0.9	1.5	1.73222548	1.73928564	2.21	2.232
10	1	0	0	4	0.9	1.5	3.46445096	3.47857129	3.943	3.967
11	0	−1	0	3	0.6	1.5	2.5983322	2.60892946	3.303	3.332
12	0	1	0	3	1.2	1.5	2.59833822	2.60892846	2.85	2.868
13	0	0	−1	3	0.9	1	2.38349663	2.38967112	3.325	3.356
14	0	0	1	3	0.9	2	2.83003833	2.84616977	2.828	2.844
15	0	0	0	3	0.9	1.5	3.07978341	3.10280042	3.077	3.1
16	0	0	0	3	0.9	1.5	3.07978341	3.10280042	3.077	3.1
17	0	0	0	3	0.9	1.5	3.07978341	3.10280042	3.077	3.1
18	0	0	0	3	0.9	1.5	3.07978341	3.10280042	3.077	3.1
19	0	0	0	3	0.9	1.5	3.07978341	3.10280042	3.077	3.1
20	0	0	0	3	0.9	1.5	3.07978341	3.10280042	3.077	3.1

Table 5

ANOVA table for Nus

Source	Degrees of freedom	Adjusted sum of squares		Adjusted mean square		F-value		P-value
		Case-1	Case-2	Case-1	Case-2	Case-1	Case-2	Case-1	Case-2
Model	9	6.1026	6.2199	0.67807	0.6911	1.52	1.52	0.0261	0.0262
Linear	3	1.3948	1.4302	0.46493	0.47672	1.04	1.05	0.0415	0.0413
M	1	0.2166	0.227	0.2166	0.227	0.49	0.5	0.0502	0.0496
Ra	1	0.0191	0.0208	0.01912	0.02079	0.04	0.05	0.064	0.0635
γ	1	0.6357	0.6646	0.63575	0.6646	1.43	1.46	0.026	0.0255
Square	3	0.3042	0.3197	0.10141	0.10658	0.23	0.23	0.0675	0.067
M ⁎ M	1	0.0722	0.0767	0.07222	0.07669	0.16	0.17	0.0696	0.069
Ra ⁎ Ra	1	0.0648	0.0686	0.06481	0.06855	0.15	0.15	0.0711	0.0706
γ ⁎ γ	1	0.2914	0.3052	0.29138	0.30522	0.65	0.67	0.0438	0.0432
2-way interaction	3	0.4194	0.4363	0.1398	0.14544	0.31	0.32	0.0615	0.0611
M ⁎ Ra	1	0.045	0.0491	0.04503	0.04912	0.1	0.11	0.0757	0.0749
Ra ⁎ γ	1	0.0318	0.0323	0.03178	0.0323	0.07	0.07	0.0795	0.0795
γ ⁎ M	1	0.3235	0.3376	0.32345	0.33763	0.73	0.74	0.0414	0.0409
Error	10	4.4568	4.5489	0.44568	0.45489
Lack of fit	5	1.9646	2.0096	0.32744	0.33494	0.53	0.53	0.0771	0.0769
Pure error	5	0	0	0	0
Total	19	10.5594	0.003483
R 2 =	97.76%

5 Conclusion

This study considers the numerical simulation of hybrid blood flow in stenotic arteries that are radiative and magnetohydrodynamic. The flow and heat transformation are compared for two cases: Case 1 Blood + Cu + MOS 2 + Al 2 O 3 and Case 2 Blood + Ag + SWCNT + MWCNT . This investigation is mainly helpful when plaque is deposited in the arterial lumen of the blood vessel. The outcome of this work helps in understanding the role of physical parameters. M , Ra , γ in haemodynamic variables affect blood flow. This investigation helps design medical devices that utilize radiation, magnetic fields, and nanoparticles.

The main findings are as follows:

The SVM learning regression model was able to predict the truth values accurately.
A residual value approaching 1 represents the accuracy of the current study model’s accuracy.
Cu + MOS 2 + Al 2 O 3 hybrid blood flow enhances the velocity when compared with Ag + SWCNT + MWCNT blood flow.
The highest heat transfer occurs at the magnetic parameter ( M ) is at a high level, radiation parameter ( Ra ) is at a low level, and flow constraint ( γ ) is at a low level in both Cases 1 and 2.
Improvement in magnetic and radiation parameters has increased skin friction and flow constraint.

# These authors contributed equally to this work and should be considered first co-authors.

Funding information: This study was supported by the National Research Foundation of Korea (NRF) grant funded by the Korean government (MSIT) (RS-2024-00346834).
Author contributions: M.D.K.: data curation; software; and writing – original draft. S.M.U.: formal analysis; resources; and software. S-J.Y.: supervision; validation; and writing – review and editing. N.A.S.: methodology; resources; and software. C.S.K.R.: formal analysis; validation; writing – review and editing. Maddina Dinesh Kumar and Nehad Ali Shah contributed equally to this work and are co-first authors. All authors have accepted responsibility for the entire content of this manuscript and approved its submission.
Conflict of interest: The authors state no conflict of interest.
Data availability statement: The datasets generated and/or analysed during the current study are available from the corresponding author on reasonable request.

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Received: 2025-03-26

Revised: 2025-03-26

Accepted: 2025-06-12

Published Online: 2025-09-26

This work is licensed under the Creative Commons Attribution 4.0 International License.

Artikel in diesem Heft

https://doi.org/10.1515/ntrev-2025-0188

Schlagwörter für diesen Artikel

spectral quasi-linearization; magnetohydrodynamic flow; stenosed artery; hybrid and ternary nanoparticles; response optimization; support vector machine learning

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