Insights into the thermal characteristics and dynamics of stagnant blood conveying titanium oxide, alumina, and silver nanoparticles subject to Lorentz force and internal heating over a curved surface

Nomenclature

p

pressure field ( ML / T 2 )

k

dimensionless curvature parameter

ρ mnf

nanofluid density ( M / LT )

Pr

dimensionless Prandtl number

υ mnf

fluid kinematic viscosity ( L 2 / T )

ϕ 2

concentration of titanium ( mol / m 3 )

α mnf

nanofluid thermal diffusivity ( L 2 / T )

C f

skin friction

σ mnf

electrical conductivity ( T 3 A 2 / ML 3 )

μ mnf

nanofluid dynamic viscosity ( M / L 3 )

ϕ 1

concentration of silver ( mol / m 3 )

M

dimensionless magnetic parameter

k mnf

nanoparticles thermal conductivity ( ML / T 3 K )

ε

dimensionless suction parameter

( ρ C p ) mnf

heat capacity ( ML 2 / T 2 K )

ϕ 3

concentration of aluminum ( mol / m 3 )

Nu s

Nusselt number

1 Introduction

Ag is often used to represent silver, which is a chemical element. It is considered to possess high reflectivity and is a metal with a lustrous white color. It contains special thermal characteristics and has various industrial uses. The flows involving silver nanoparticles have been interpreted by many researchers. Waqas et al. [1] studied the stenotic artery flow incorporating silver and gold nanoparticles. The pertinence of usage of the silver–gold combination was described, which was based on the potential drug transport phenomenon. The flow phenomenon was studied under a magnetic field effect. The enhancement in heat transfer through a water-based nanofluid containing Ag–H₂O was observed by Suleman et al. [2]. The results of this study were mainly based on a numerical analysis that was performed using a classical shooting method. Zhang et al. [3] explored numerical Ag–MgO/water hybrid nanofluid flow via a rotating disc, and the magnesium oxide and silver particles, owing to their special thermal features, were taken into account. They noticed an increasing effect on the velocity and heat energy transfer due to the spinning disc. Mousavi et al. [4] used the bvp4c built-in function in MATLAB to simulate the zinc oxide–silver/water flow neglecting the velocity slip effect. Their observation showed that skin friction and velocity would enhance when the disc rotated faster. A spinning flow of the hybrid nanofluid, which was prepared by mixing gold and silver particles in a liquid solution of ethylene glycol was premeditated by Alqahtani et al. [5]. In this work, the energy transport mechanism is discussed, specifically, using the Ag–Au nanoparticles.

Aluminum and oxygen when combined yield aluminum oxide, which is a chemical compound in nature. Aluminum oxide can be naturally found in different minerals but it can also be prepared synthetically. Ahmad et al. [6,7,8] investigated the water-based hybrid nature flow of nanofluids comprising aluminum oxide (Al₂O₃) and copper (Cu) nanoparticles. In their work, they examined a significant enhancement in the heat transfer in the case of hybrid nanocomposition of Al₂O₃–Cu/H₂O rather than the usual nanofluid case. An experimental study was performed by Murtadha and Hussein [9] to examine solar panel cooling incorporating the aluminum oxide nanofluid. Different flow rates at different concentrations of nanoparticles were used to elaborate the experimental results. Rahman et al. [10] carried out a study and discussed the heat transfer increase due to the correlation of the Cu–Al₂O₃ hybrid nanofluid considering water as a base fluid. Hassan et al. [11] reported the thermal performance of polyvinyl alcohol solutions when they interacted with aluminum oxide. The physical aspects of the parameters were numerically analyzed.

Titanium oxide (TiO₂) usually occurs naturally and is used widely in white pigment. It has various commercial applications. A mathematical approach toward finding the solution of nanofluid flow involving titanium oxide was performed by Arulprakasajothi et al. [12]. The friction factor was reduced but the Nusselt number increased by the nanoparticle fraction of TiO₂. An electromagneto-hydrodynamics flow was considered by Ramesh et al. [13] where they studied the thermal performance of iron oxide and titanium oxide. The Eckert number was considered in the energy equation. However, a numerical technique was used in the dimensionless form of the governing equations which was obtained after similarity transformation. The oxide forms of titanium and aluminum were amalgamated in the blood by Khan et al. [14] to prepare the hybrid mixture of blood-based nanofluids. A vertical oscillating plate showed the flow mechanism. The mixed convection phenomenon was also considered, and simulations were done by the Caputo fractional simulation technique. Ahmad et al. [15] interpreted the role of titanium oxide (TiO₂) in the augmentation of the heat transmission rate. A linear stretched surface was considered as the geometry of the problem. It was assumed that the hybrid mixture would increase thermal efficiency more rigorously as compared to the mono nanofluids. An algorithmic approach based on the fifth-order Runge–Kutta method was employed by Hafeez et al. [16] to compute the solutions of the ternary nanofluid flow consisting of TiO₂, CuO, and Al₂O₃.

Blood flows have potential use in biosciences. Recently, many studies have been reported on the blood-based nanofluid flows due to their prominent applications. Sharma et al. [17] incorporated the response surface methodology to develop the model describing the blood-based nanofluid flow under the prominent viscosity and Hall effects. How the physical parameters altered the Nusselt number and skin friction were examined. A peristaltic blood flow was considered by Noreen et al. [18]. Blood is considered to be flowing within a vertical channel (an artery). The axial velocity and the pressure gradient were computed by numerical solutions. Elelamy et al. [19] provided a mathematical approach to finding the theoretical aspects of the Casson micropolar blood flow which was non-Newtonian. The magnetic permeability caused variations in the temperature of the fluid. Abeer et al. [20] and Saeed et al. [21] used the homotopy analysis method for the exploration of blood-based tri-hybrid (CuO–TiO₂–Al₂O₃) and hybrid (SWCNT and MWCNT) flows, respectively. Makinde and Animasaun [22,23] investigated the magnetohydrodynamic flows past an upper surface of a paraboloid with allowance for a chemical reaction.

Ternary nanofluids refer to a type of nanofluid that contains three different types of nanoparticles suspended in a liquid. Nanofluids are engineered fluids that are created by suspending nanoparticles in a base fluid, such as water or oil. The addition of nanoparticles to the base fluid can significantly enhance its thermal, electrical, and mechanical properties, and Li et al. presented an analysis of using it [24]. Ternary nanofluids are of particular interest in research and development because they can offer even greater improvements in these properties compared to binary nanofluids, which contain only two types of nanoparticles [25,26]. By carefully selecting the types and concentrations of nanoparticles in a ternary nanofluid, it is possible to form a fluid with tailored properties that are optimized for a specific application. However, creating ternary nanofluids can be challenging, as the addition of a third nanoparticle type can lead to more complex interactions and behavior between the nanoparticles and the base fluid (see the study of Al Oweidi [27]). As a result, researchers must carefully study the thermodynamic, rheological, and transport properties of ternary nanofluids to fully understand their behavior and potential applications (see the study of Shahzad et al. [28]). Ternary nanofluids can be used as a catalyst in various chemical reactions, such as Fischer–Tropsch synthesis, where they can improve the conversion rate and selectivity of the reaction [29]. For further relevant work, see previous studies [30–35].

The purpose of this work is to investigate the novel flow and thermal characteristics of tri-hybrid nanofluids that are mainly based on silver, titanium dioxide, and aluminum dioxide. The proposed nano-composition (Ag–TiO₂–Al₂O₃/blood) has not been interpreted before being subjected to suction and Lorentz force. The analysis comprehends the features of ternary hybrid nanofluid and it also portrays the thermal characteristics of the Newtonian case of the nanofluid. The main effects of all physical parameters of the problem on the velocity, temperature, concentration, and surface drag are discussed in detail.

2 Mathematical formulation

This work examines a triple hybrid nanofluid formed by suspending Ag, TiO₂, and Al₂O₃ in blood as the base liquid in a predictable two-dimensional, inflexible MHD stagnant point movement. Figure 1 shows how a flow designer functions. In this case, mass sucking is equivalent to vw, so the radius R defines the permeable arched stretching/shrinking surface, and the temperature source/sink can be found by the curvy directs (r, s), where r is the normal to tangent vector at every point on the surface and s is the arc length coordinate alongside the path of movement route. The configuration ensures that a modest distortion corresponds to a significant amount of the circumference R. It is additionally presumed that the mass transfer speed is vw, and when vw > 0, it denotes sucking and B ₀ is the strength of the magnetic field.

Figure 1

The flow configuration model of a curved surface.

Pressure does not continue as a constant during a border sheet, as given by Waini et al. [36]. Subsequently, it is incredible in the direction of the pressure gradient arranged by a curved surface. Now, it is theoretical that u _e(s) = as with a > 0 and T(s) = T _∞ + T ₀(s⁄L), where T _∞ is the constant ambient heat, T ₀ characterizes the source heat, and L is distance. It is conceivable to simplify the equations and reduce their similarity by choosing a heat that has already been settled upon for the surface. Numerous conceptual considerations are put forward in studying ternary nanofluids. The ternary nanofluid is anticipated to be stable. The base fluid and the nanoparticles are thought to be in thermal balance, meaning they are traveling at a comparable pace. Consequently, the effects of nanoparticle aggregating and settling are not considered. The nanoparticles are assumed to be of the same size and cylindrical in shape. In ternary hybrid nanofluids, these presumptions and boundary layer approximations result in controlling equations [36,37].

3 Numerical approach: quasi-linearization method (QLM)

QLM is a generalized form of the Newton–Raphson method which basically provides an algebraic sequence of functions. It transforms the nonlinear equations into linear ones from which the numerical solutions are obtained. The usual numerical techniques may not provide the solutions of equations (9) and (10) due to the involvement of coupled and non-linear terms. A constant of integration will also appear upon integrating the differential equations. There is no such mechanism that could directly find the value of this integration constant, and the only possibility is to find its value by hit and trial rule which may require a lot of time. The linearization of the system (9) and (10) will not face such type of inadequacy. However, quasi-linearization is the best approach for determining the approximate solutions of the problem. A detailed procedure of this technique is elaborated in our earlier work [38]. This technique is mainly based on the steps given in Figure 2a. Figure 2b represents the comparison of the present numerical results with those presented by Ali et al. [39].

Figure 2

(a) A flow chart of QLM. (b) Comparison of the present numerical results with those of Ali et al. [39].

4 Results and discussions

The QLM is implemented to provide the numerical solutions of equations (9) and (10), together with the boundary requirement (11) and physical parameters (14), and (15) which have been described in the previous section. In this section, information and positions are utilized to demonstrate the numerical results.

Fixed values of the parameters are: k = 5 , M = 2 , ε = 0.04 , Pr = 6.2 , Rn = 0.5 .

For Newtonian case: ϕ 1 = 0.00 ; ϕ 2   =   0.00 ; ϕ 3   =   0.00 .

For ternary hybrid nanofluids: ϕ 1 =   0.03 ; ϕ 2   =   0.06 ; ϕ 3   =   0.1 .

The data are the profiles of axial velocity F ( η ) , radial velocity F ′ ( η ) , and temperature θ ( η ) . These profiles contain the Newtonian and ternary (Ag–TiO₂–Al₂O₃/blood) hybrid nanofluid curves, which meet the far-field boundary requirements (11). All data present the greater velocity with the Newtonian fluid, whereas the higher fluid temperature with the ternary hybrid nanofluid. All the controlling parameters, such as k = 0.1 , 0.2 , 0.3 , 0.4 (the curvature parameter), M = 3 , 5 , 7 , 9 (the magnetic field parameter), Pr = 1 , 3 , 5 , 7 (the Prandtl number), Rn = 0 , 1 , 2 , 3 (the heat source/sink parameter), and Pr = 1 , 3 , 5 , 7 (the suction parameter) are selected to produce these profiles as depicted in Figure 3 and S5.

Figure 3

Variation in F ( η ) for different values of k .

4.1 Discussion

The profiles against the curvature parameter are shown in Figures 3–5 for the axial velocity, radial velocity, and temperature. In detail, Figure 1 shows the impact of k on the velocity. Figures 3 and 4 show that higher k indicates an inciting nature in the axial and radial velocities. Meanwhile, increasing k indicates a weak temperature distribution as displayed in Figure 5. Figures 6 and 7 show the magnetic field influence on the velocity in the axial and radial components, together with the temperature curves of the Newtonian and ternary hybrid nanofluid. The figures contradict the theory of the effect of the magnetic field on the velocity distribution, where the magnetic field containing the Lorentz force should act against the direction of the velocity itself. Figures 6 and 7 prove that the Lorentz force still plays a role by reducing the interaction between molecules. However, this force produces different results by increasing the velocity values. When the axial and radial velocities of the flow are increased, more thermal energy may be absorbed by fluid particles, and temperatures will become higher due to the effect of M (Figure 8). The temperature profile against the Prandtl number is shown in Figure S1, which indicates that the higher Pr suppresses the temperature profile because Pr increases the fluid viscosity. Consequently, the wall friction is augmented together with thermal conductivity. Therefore, finally, the temperature profile is decreased.

Figure 4

Variation in F ′ ( η ) for different values of k .

Figure 5

Variation in θ ( η ) for different values of k .

Figure 6

Variation in F ( η ) for different values of M .

Figure 7

Variation in F ′ ( η ) for different values of M .

Figure 8

Variation in θ ( η ) for different values of M .

Figure S2 gives the variation of temperature distribution within the boundary layer for diverse values of heat source/sink parameters for Newtonian and ternary hybrid nanofluid cases. The increase in the velocities distribution in the axial (Figure S3) and radial components (Figure S4) can be observed in conjunction with increasing suction ε. The association between heat and ε is shown in Figure S5. These parameters remain constant unless they are declared in certain tables: k = 5 , M = 2 , ε = 0.04 , Pr = 6.2 , Rn = 0.5 . Meanwhile, the fixed values for the Newtonian and ternary hybrid nanofluids are ϕ ₁ = 0.00; ϕ ₂ = 0.00; ϕ ₃ = 0.00 and ϕ ₁ = 0.03; ϕ ₂ = 0.06; ϕ ₃ = 0.1, respectively. The influence of k and M on f ″ ( 0 ) and θ ′ ( 0 ) are shown in Table 5. An additional conclusion that can be inferred from Table 2 is that when k increases, the shear stress also increases. This observation can be explained by the fact that the velocity gradient at the wall is higher for the curved surface compared to the curved surfaces which tend to be flattening at the same position. Table 5 shows that the convective heat transmission amount of a curved surface with a small curvature is more significant than the same surface with a higher curvature. This proves that heat transfer reduces when k increases. Moreover, it can be noted from Table 5 that f ″ ( 0 ) is an augmenting function of the magnetic field parameter and θ ′ ( 0 ) decreases with the same parameter. Table 6 lists the values of the shear stress and heat transfer for the suction parameter ε; the results indicate that Newtonian fluid has a lower heat transfer rate compared with the ternary hybrid nanofluid. The behavior of the heat transfer acted by Pr and Rn is shown in Table 7. There is a decrease in the heat transfer subjected to the increasing heat source/sink. With a stronger heat source, a larger region of the thermal boundary layer decelerates the heat transfer rate.

Table 5

Change in the shear stress and heat transfer rate for various values of k and M

		f ″ ( 0 )		θ ′ ( 0 )
k	M	Newtonian case	Ternary case of hybrid nanofluids	Newtonian case	Ternary case of hybrid nanofluids
0.1		−1.203238616629410	−1.259578166385444	−0.555094490410257	−0.443910424529403
0.2		0.591823760046073	0.514802306034179	−1.030097125481055	−0.828961490026320
0.3		1.067970150592235	0.981804387289453	−1.394922781265906	−1.132797556986670
0.4		1.239132602295912	1.149268009599802	−1.668437712930249	−1.366794042494590
	3	1.686277239651979	1.261786968376755	−3.260276600035172	−2.809430460133622
	5	2.620266613722611	1.871398759283882	−3.352713655141870	−2.879428653940952
	7	3.586223889675000	2.525473608085416	−3.420647625707729	−2.936120577235041
	9	4.559797513595254	3.193281891289979	−3.471778128921688	−2.980952856077146

Table 6

Change in the shear stress and heat transfer rate for various values of ε

ε	f ″ ( 0 )		θ ′ ( 0 )
	Newtonian case	Ternary case of hybrid nanofluids	Newtonian case	Ternary case of hybrid nanofluids
1	1.058650792336024	0.973895167470928	−8.046345793990495	−6.488935867727203
3	1.828905137060133	1.622767436195716	−21.205294385699691	−16.704695557817573
5	2.723427076691820	2.389116363907817	−34.456276171561939	−27.176208353688637
7	3.662563539492680	3.199403894669217	−47.263019055568591	−37.446009795628839

Table 7

Change in the heat transfer rate for various values of Pr and Rn

Pr	Rn	θ ′ ( 0 )
		Newtonian case	Ternary case of hybrid nanofluids
1		−1.172136251081740	−1.046581613409547
3		−2.004968974201304	−1.762915866537146
5		−2.613896722269793	−2.289312058933390
7		−3.129109076232328	−2.732516485490855
	0	−3.011935135301480	−2.629217711295500
	1	−4.043186641655977	−3.539006876378919
	2	−4.866417487532693	−4.266047375379184
	3	−5.568631606197178	−4.886564013028472

5 Conclusions

This study considered the flow and thermal properties of a ternary nanofluid, where the base fluid is engine oil, and the nanoparticles submerged in the engine oil are copper, titanium dioxide, and silicon dioxide. The ternary hybrid nanofluid flows around a stagnation region of a stretching/shrinking curved surface, where this fluid system is subjected to additional factors such as magnetic field, suction, and heat source/sink. The appropriate similarity conversions and non-dimensional quantities were implemented to reduce the governing equations system and transform this system into a simplified version. A numerical calculation, such as QLM, was chosen to obtain the final solutions. The impact of diverse factors on the velocity in the axial and radial components, temperature, shear stress, and heat transfer are illustrated in figures and tables. Furthermore, a comparison thermal efficiency of the ternary hybrid nanofluid and conventional Newtonian fluid is also being discussed.

Fixed values of the parameters are:

k = 5 , M = 2 , ε = 0.04 , Pr = 6.2 , Rn = 0.5

For Newtonian case:

ϕ ₁= 0.00; ϕ ₂ = 0.00; ϕ ₃ = 0.00;

For ternary hybrid nanofluids:

ϕ ₁ = 0.03; ϕ ₂ = 0.06; ϕ ₃ = 0.1.

From the findings as presented previously, the following results can be deduced:

1. The axial and radial velocities for both fluids (Newtonian and ternary hybrid nanofluids) are increased due to the increasing function of the curvature parameter, magnetic field, and suction parameter. These parameters increase the thickness of the momentum boundary layer and cause an increase in the axial and radial velocities.

2. The thermal boundary layer is suppressed by all the involved parameters such as the curvature parameter, magnetic field, suction parameter, Prandtl number, and heat source/sink. Therefore, there is always a decrease in the temperature for the ternary hybrid nanofluid and Newtonian fluid under these restricting parameters in this fluid flow model.

3. Since velocity and shear stress are related to each other, the increase of the velocity profile due to the related parameters also causes the shear stress to increase with the same governing parameters (curvature parameter, magnetic field, suction parameter).

4. The direct relationship between the temperature and heat transfer decreases the heat transfer by the curvature parameter, magnetic field, suction parameter, Prandtl number, and heat source/sink.

In the future, the present method might be used in several physical and technical problems [48,49,50,51,52,53,54,55,56,57,58,59].