A numerical analysis of the blood-based Casson hybrid nanofluid flow past a convectively heated surface embedded in a porous medium

Humaira Yasmin; Ali M. Mahnashi; Waleed Hamali; Showkat Ahmad Lone; Zehba Raizah; Anwar Saeed

doi:10.1515/phys-2023-0193

Article Open Access

A numerical analysis of the blood-based Casson hybrid nanofluid flow past a convectively heated surface embedded in a porous medium

Humaira Yasmin , Ali M. Mahnashi , Waleed Hamali , Showkat Ahmad Lone , Zehba Raizah and Anwar Saeed

Published/Copyright: February 16, 2024

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From the journal Open Physics Volume 22 Issue 1

Abstract

The analysis of the fluid flow with the energy transfer across a stretching sheet has several applications in manufacturing developments such as wire drawing, hot rolling, metal extrusion, continuous casting, paper production, and glass fiber fabrication. The current examination presents the hybrid nanofluid flow past a convectively heated permeable sheet. The ferrous oxide (Fe₃O₄) and Gold (Au) nanoparticles have been dispersed in the blood. The significances of thermal radiation, inclined magnetic field, and space-dependent heat source have been observed in this work. The modeled equations are presented in the form of partial differential equations and reformed into the set of ordinary differential equations (ODEs) by using the similarity substitution. The Matlab built-in package (bvp4c) is employed to resolve the transform nonlinear set of ODEs. The significance of flow constraints versus the velocity and temperature profiles is demonstrated in the form of Figures and Tables. The numerical outcomes for the physical interest quantities are presented in tables. It has been perceived from the results that raising the angle of inclination from 0° to 90° reduces both the velocity and energy profile. The escalating values of Eckert number, constant heat source, and space-dependent heat source factor accelerate the temperature profile. The velocity and temperature distributions are very effective in the cases of hybrid nanofluid (Au–Fe₃O₄/blood) when compared to nanofluid (Au/blood). The skin friction and rate of heat transfer are very effective in the cases of hybrid nanofluid (Au–Fe₃O₄/blood) when compared to nanofluid (Au/blood).

Keywords: Hybrid nanofluid; MHD; space-dependent heat source; thermal-dependent heat source; porous media; Joule heating

Nomenclature

B 0: magnetic field (A/m)
Pr: Prandtl number
k: thermal conductivity (W m⁻¹ K⁻¹)
Rd: Radiation factor (J m⁻² s⁻¹)
Fe₃O₄: ferrous oxide nanoparticle
Ec: Eckert number
T w: surface temperature (K)
C fx: skin friction
M: magnetic factor (A/m)
u w ( x ) = a x: stretching velocity (m/s)
C p: specific heat (J kg⁻¹ K⁻¹)
_Re: Reynolds number
Q E: space-dependent heat source factor
Au: gold nanoparticle
Q I: thermal-dependent heat source factor
B T: thermal Biot number
Nu x: Nusselt number

Greek symbols

σ: electrical conductivity (S/m)
ρ: density (kg/m³)
λ > 0: stretching case of the sheet
φ 1: volume fractions of iron oxide
φ 2: volume fractions of gold
λ < 0: shrinking case of the sheet

1 Introduction

The thermal transportation of various pure fluids can be enhanced by suspending tiny-sized particles like CNTs, metal ions, nano-powder, etc. Such fluids are nanofluids and are used as the best heat-conducting carrier fluids. Many of their applications comprise electric devices coolant, magnetic resonance imaging, and coolant of auto engines. The mixing of tiny particles in the pure fluid was first familiarized by Choi and Eastman [1] to expand the thermal performance of pure fluids. Acharya et al. [2] deliberated the effect of hydrothermal variations on radiative nanofluid flow subject to the impact of nanolayer as well as the diameter of nanoparticles and has established that with the use of nanoparticles, there is an enhancement of 84.61% in thermal performance of the fluid. Reddy et al. [3] numerical modeled the EMHD radiative transportation of Ti6Al4V-AA7075/H₂O nanofluids over a Riga surface. Shah et al. [4] used nanoparticles of gold in blood fluid flow between two surfaces with the influence of radiations and determined that with an upsurge in radiative factor and volume fraction, the heat flow progressed. Similarly, the thermal effectiveness of pure fluid can be heightened further by using two nanoparticles of different types and is termed a hybrid nanofluid. Chu et al. [5] described the hybrid nanofluid flow amid plates with the influence of particle shape and have established a good agreement with published results. Elsebaee et al. [6] modeled and simulated numerically the thermal transportation of silica-alumina hybrid nanoparticles past a slender surface. Khan et al. [7] studied the fluid flow with hybrid nanoparticles past a thermally heated needle subject to chemically reactive and bio-convective effects. Bharathi et al. [8] explored the thermally radiative flow of tangent hyperbolic trihybrid nanofluid flow across Darcy's porous surface. Alsaedi et al. [9] discussed hybrid MHD nanofluid flow amid two co-axial surfaces and deduced that there has been decay in temperature characteristics with higher values of copper volume fraction while Nusselt number has augmented in this process as depicted in their results during an investigation. Some recent studies may be found in the literature [10–15].

The fluids that are conducted electrically are counted as magneto-hydrodynamics (MHD) fluids. The flow of such fluid is fundamental in a wide range of industrial applications, like MHD generators, etc. Jayavel et al. [16] inspected the heat transference and irreversibility analysis through MHD Darcy–Forchheimer flow of hybrid nanoliquid over a wedge and cone. Kodi and Mopuri [17] examined MHD fluid flow over a vertical permeable and inclined surface using the impact of chemical reactions and heat absorption. Vishalakshi et al. [18] explored MHD fluid flow with mass transportation and slip conditions upon a stretching/shrinking surface and have verified that velocity and energy transition rise with the radiations and viscoelastic factors effects. Islam et al. [19] discussed the effects of Hall current on MHD fluid flow between two surfaces and deduced that augmentation in magnetic factor upsurge linear velocity and thermal characteristics while declining the rotational profiles of velocity. Sharma et al. [20] scrutinized MHD convective fluid flow past a rotating stretchable disk and highlighted that magnetic field strength was responsible for growth in Nusselt number. Raizah et al. [21] tested MHD fluid flow past a circular cylinder with the impact of dissipative energy and thermal radiations. Rehman et al. [22] conducted comparative work on thermal transportation for MHD fluid flow subject to thermal radiations past a plane as well as a cylindrical region and have proved that the thermal regime has more strength in the case of the cylindrical surface. Several researchers have recently reported on MHD fluid flow [23,24,25,26].

The fluid flow mechanisms through the porous surface in many problems are vibrant in science and industry, specifically in the petroleum industry. The thermal transportation phenomenon through a porous medium is practiced in many areas of science like geophysics and ceramic engineering, etc. Algehyne et al. [27] studied thermal transportation performance for fluid flow in a vertical porous sheet and determined that velocity characteristics have reduced both for nanoparticles and hybrid nanoparticles in the case of higher values of Darcy’s and Forchheimer numbers. Qureshi et al. [28] examined mathematically the influence of nanofluid flow through a porous sheet by injecting SWCNTs and deduced that the effects of the thickness of the nano-layer were quite significant in the case of Nusselt number growth. Duguma et al. [29] inspected the stability analysis and dual solutions of Cu–H₂O–Casson nanoliquid flow past a stretching and shrinking slippery porous surface. Kumar et al. [30] deliberated numerically the heat transfer and Williamson fluid flow across a porous enlarging cylinder with the influence of heat generation and Joule heating. Khan et al. [31] inspected radiative MHD fluid flow subject to a non-uniform source of heat, rotating frame, and impacts of the Darcy-Forchheimer model. Shamshuddin et al. [32] studied a mathematical model for chemically reactive MHD nanofluid flow on a porous surface with the influence of Buongiorno’s model and concluded that the Sherwood number has elevated with greater values of reaction factor. Alqahtani et al. [33] explored numerically the thermal transportation for radiative MHD Jeffery fluid flow in a permeable stretched surface. Duguma et al. [34] discussed the stagnation point flow of TiO₂–CoFe₂O₄/H₂O–Casson nanofluid over a slippery stretching cylindrical porous medium. Furthermore, some remarkable results may be found in previous studies [35–39].

Various mathematical models are applied to examine the behavior of fluids and flow properties. The famous model of Buongiorno’s [40] has also been established to focus on seven different procedures and processes inside fluid flow. In these seven effects, thermophoresis and Brownian motion effects are more substantial. Mishra and Upreti [41] conducted a comparative analysis of the fluid flow across a curved surface by using the impact of Buongiorno’s model and realized that fluid motion has weakened and heat flow has expanded for progression in the Brownian factor. Elattar et al. [42] numerically analyzed the flow of nanofluid film subject to heat production with Buongiorno’s model. Upreti et al. [43] studied the effect of homogeneous–heterogeneous reactions on the MHD Sisko nanofluid flow across an elongating surface with the convective conditions and Buongiorno’s model. Safdar et al. [44] examined EMHD nanoparticle fluid flow on a stretched surface with the impact of activated energy, buoyancy force, microorganisms, and Buongiorno’s model and have deduced that with augmentation in mixed convection factor, there has been a growth in Nusselt number while velocity field degenerated with incrimination in viscosity factor. Puneeth et al. [45] implemented a modified Buongiorno’s model for the examination of chemically reactive trihybrid nanofluid flow using bioactive mixers. Rasool et al. [46] investigated MHD Maxwell nanoliquid flow past an extending surface and proved that the porosity factor supported the drag force/skin friction. Hussain et al. [47] debated Maxwell nanofluid flow in a solar collector by using modified Buongiorno’s model and explored that temperature field upsurges with growth in radiation factor and has declined with upsurge in Schmidt number and reactive factor.

In different fields, various chemical reactions are needed to analyze different flow systems. Some of these reactions are associated with catalysts. These reactions can be seen in many applications and processes at the industrial level. Shoaib et al. [48] discussed the production of entropy for fluid flow with quadratic order chemical reaction and highlighted that skin friction has augmented with growth in radiation factor. Bilal et al. [49] analyzed computationally a time-based MHD trihybrid nanofluid using the impacts of chemically reactive activation energy and deduced that the mass curve intensified with the upsurge in Lewis number, reaction, and activation factors. Ahmed et al. [50] explored numerically the hydrothermal fully developed hybrid nanoliquid flow across an annular sector. Goud et al. [51] discussed computationally the significance of chemically reactive effects on MHD fluid flow past a Riga plate and have proved that with growth in Hartman number, there is a growth in velocity characteristics at free stream. Sajid et al. [52] discussed the impacts of MHD on the fluid flow with the consequence of chemical reaction and heat source/sink. Zeeshan et al. [53] inspected the non-Newtonian fluid flow across a horizontal surface. Shah et al. [54] examined the bioconvective impact on nanofluid flow subject to chemical reactions and microorganisms past an extending surface. Wang et al. [55] discussed the production of entropy optimization with Darcy–Forchheimer model-based fluid flow with the effect of a chemical reaction and noticed that velocity profiles have weakened with slip and porosity factors.

The interaction of nanoparticles (NPs) with carrier red blood cells (RBCs) is profoundly altered by attachment to the surface of these cells, resulting in a deceleration of their clearance from the bloodstream, and facilitation of their transfer from the surface of these cells to the vascular cells. Many drug delivery purposes benefit from these changes in NP pharmacokinetics imposed by carrier RBCs. Using blood as a nanoparticle, many researchers have studied different types of mathematical problems for different types of physical surfaces [56,57,58,59,60]. The significance of blood-based Casson hybrid nanofluid flow comprising of gold and ferrous oxide nanoparticles contains many healthcare and biomedical engineering. Those fluids which contain metallic nanoparticles like Au and Fe₃O₄ nanoparticles can significantly enhance the thermal conductivity of the base fluid. The present investigation deals with the blood-based Casson hybrid nanofluid flow comprising gold and ferrous oxide nanoparticles over a stretching surface using porous media. According to the authors' knowledge, this topic has not yet been presented numerically. Additionally, some external forces like thermal radiation and Joule heating are taken into consideration. The sheet surface is kept with a hot working fluid (blood). Also, a non-Newtonian Casson model is adopted to hold the non-Newtonian fluid properties. Thus, the following research questions have to be answered:

In which case, i.e., blood-based nanofluid or hybrid nanofluid, do the embedding factors have dominant impacts on the flow profiles?
What is the role of angle of inclination in terms of velocity and temperature of the blood-based nanofluid or hybrid nanofluid?
How do the skin friction and local Nusselt number increase/decrease via differential entrenching factors?

In order to answer the above research questions, we have presented this article section-wise. Section 2 shows the problem formulation. Section 3 demonstrates the numerical analysis of the present model. Section 4 displays the code validation of the present analysis. Section 5 displays the results and discussion while Section 6 presents the concluding remarks.

2 Formulation of problem

Assume the two-dimensional flow of a Casson hybrid nanofluid past as a stretching surface using porous media. Gold and ferrous oxide are taken as two different nanoparticles and dispersed in blood which is used as base fluid. The sheet is stretched along x -axis with velocity u w ( x ) = a x with a > 0 . The strength of the magnetic field is B 0 that is employed perpendicular (along y -axis) to the nanofluid flow as shown in Figure 1. The surface temperature is denoted by T w . Also, the reference temperature is denoted by T f such that T f > T w . The ambient temperature is denoted by T ∞ . Additionally, the Joule heating, heat source, exponential heat source and thermal radiation effects are considered in this work.

Figure 1

Geometrical view of the flow problem.

The rheological equation of a Casson fluid flow is defined as:

(1) τ i j = 2 μ B + P y 2 π e i j , π > π c 2 μ B + P y 2 π c e i j , π < π c ,

where μ B shows the plastic dynamic viscosity, P y shows the yield stress, π = e i j e i j where e i j is the ( i , j ) th deformation rate component and π c is the critical value of this product.

The governing equations in vector form can be written as:

(2) ∇ ⋅ V = 0 ,

(3) ∂ V ∂ t + ( V ⋅ ∇ ) V = − 1 ρ h n f ∇ P + μ h n f ρ h n f 1 + 1 β ∇ 2 V − σ h n f ρ h n f ( J × B ) − μ h n f ρ h n f K p 1 + 1 β V ,

(4) ∂ T ∂ t + ( V ⋅ ∇ ) T = − ∇ ⋅ q ( ρ C p ) h n f + Q t ( T − T ∞ ) ( ρ C p ) h n f + J 2 ( ρ C p ) h n f σ h n f + Q e ( T f − T ∞ ) ( ρ C p ) h n f e x p − n a ν f y w h e r e q = q R + q F , q F = − k h n f ∇ T , q R = − 4 σ ⁎ 3 k ⁎ ∇ T 4 .

Thus, the leading equations in partial differential form can be described as [61,62,63,64]:

(5) ∂ u ∂ x + ∂ v ∂ y = 0 ,

(6) u ∂ u ∂ x + v ∂ u ∂ y = 1 + 1 β μ hnf ρ hnf ∂ 2 u ∂ y 2 − μ hnf ρ hnf 1 + 1 β u K p − σ hnf ρ hnf B 0 2 u sin 2 γ ,

(7) u ∂ T ∂ x + v ∂ T ∂ y = 1 ( ρ C p ) hnf k hnf + 16 σ ⁎ T ∞ 3 3 k ⁎ ∂ 2 T ∂ y 2 + σ hnf ( ρ C p ) hnf B 0 2 u 2 sin 2 γ + 1 ( ρ C p ) hnf Q t ( T − T ∞ ) + 1 ( ρ C p ) hnf Q e ( T f − T ∞ ) exp − n a ν f y ,

In the above equations, u , v are the velocity components, x , y are the coordinate axes, μ is the dynamic viscosity, ρ is the density, K p is the permeability of the porous medium, σ is the electrical conductivity, γ is the angle of inclination, C p is the specific heat, k is the thermal conductivity, σ ⁎ is the Stefan-Boltzmann constant, k ⁎ is the mean absorption coefficient, Q t is the constant heat source coefficient, and Q e is the exponential heat source coefficient.

The boundary conditions are [61,62,63,64]:

(8) u = u w ( x ) λ , v = 0 , − k hnf ∂ T ∂ y = h f ( T f − T ) , at y = 0 , u → 0 , T → T ∞ as y → ∞ ,

where λ > 0 shows the stretching case of the sheet, and λ < 0 shows shrinking case of the sheet.

The effective characteristics of the nanofluid and hybrid nanofluid are defined as [65]:

(9) μ nf = μ f ( 1 − φ 1 ) 2.5 , μ hnf = μ f ( 1 − φ 1 ) 2.5 ( 1 − φ 2 ) 2.5 ,

(10) ρ nf = ρ f ( 1 − φ 1 ) + φ 1 ρ N 1 , ρ hnf = ( 1 − φ 2 ) { ρ f ( 1 − φ 1 ) + φ 1 ρ N 1 } + φ 2 ρ N 2 ,

(11) ( ρ C p ) nf = ( ρ C p ) f ( 1 − φ 1 ) + ( ρ C p ) N 1 φ 1 , ( ρ C p ) hnf = { ( ρ C p ) f ( 1 − φ 1 ) + φ 1 ( ρ C p ) N 1 } ( 1 − φ 2 ) + ( ρ C p ) N 2 φ 2 .

(12) k nf k f = k N 1 + 2 k f − 2 φ 1 ( k f − k N 1 ) k N 1 + 2 k f + φ 1 ( k f − k N 1 ) , k hnf k nf = k N 2 + 2 k f − 2 φ 2 ( k f − k N 2 ) k N 2 + 2 k f + φ 2 ( k f − k N 2 ) .

(13) σ nf σ f = σ N 1 + 2 σ f − 2 ( σ f − σ N 1 ) φ 1 σ N 1 + 2 σ f + ( σ f − σ N 1 ) φ 1 , σ hnf σ nf = σ N 2 + 2 σ f − 2 ( σ f − σ N 2 ) φ 2 σ N 2 + 2 σ f + ( σ f − σ N 2 ) φ 2 .

Here, N 1 and N 2 represent the first and second nanoparticles with φ 1 and φ 2 as their volume fractions, respectively. Thermophysical features of the nanoparticles and base fluid are defined in Table 1.

Table 1

Thermophysical features of pure fluid and nanoparticles [65]

Physical property	Blood	Fe 3 O 4	Au
ρ	1,063	5,180	19,300
C p	3,750	670	129
k	0.429	9.7	318
σ	4.3 × 10⁻⁵	0.74 × 10⁶	4.1 × 10⁶

To reduce the above system of Eqs. (5)–(7), the following similarity variables are described as [64]:

(14) u = a x f ′ ( η ) , v = − ν f a f ( η ) , T = T ∞ + ( T f − T ∞ ) θ ( η ) , η = y a ν f .

So, Eq. (5) is satisfied, and the remaining equations are as follows:

(15) ξ 1 ξ 2 1 + 1 β f ‴ ( η ) − f ′ 2 ( η ) + f ″ ( η ) f ( η ) − ξ 3 ξ 2 M sin 2 γ f ′ ( η ) + ξ 1 ξ 2 K 1 + 1 β f ′ ( η ) = 0 ,

(16) 1 ξ 5 ( ξ 4 + Rd ) θ ″ ( η ) + Pr f ( η ) θ ′ ( η ) + Pr ξ 5 ( ξ 3 M Ec sin 2 γ f ′ 2 ( η ) + Q E exp ( − n η ) + Q I θ ( η ) ) = 0 .

Related conditions are as follows:

(17) f ( 0 ) = 0 , f ′ ( 0 ) = λ , f ′ ( ∞ ) → 0 k hnf k f θ ′ ( 0 ) = − B T ( 1 − θ ( 0 ) ) , θ ( ∞ ) → 0 .

In the above equations, Pr is the Prandtl number, M is the magnetic factor, Rd is the radiative factor, Ec is the Eckert number, Q E is the space-dependent heat source factor, Q I shows the space-dependent heat source, and B T indicates the thermal Biot number. All these parameters are defined as [61,62,63,64]:

(18) ξ 1 = μ hnf μ f , ξ 2 = ρ hnf ρ f , ξ 3 = σ hnf σ f , ξ 4 = k hnf k f , ξ 5 = ( ρ C p ) hnf ( ρ C p ) f , M = σ f B 0 2 a ρ f , Ec = u w 2 ( C p ) f ( T f − T ∞ ) , B T = h f k f ν f a , Pr = ν f ( ρ C p ) f k f , K = μ f a ρ f K p , Rd = 16 σ ⁎ T ∞ 3 3 k ⁎ k f , Q E = Q t a ( ρ C p ) f , Q I = Q e a ( ρ C p ) f .

The quantities of importance like skin friction ( C fx ) and local Nusselt number ( Nu x ) are described as:

(19) C fx = τ w ρ f ( u w ( x ) ) 2 , Nu x = x q w k f ( T f − T ∞ ) ,

where

(20) τ w = μ hnf ∂ u ∂ y y = 0 , q w = − k hnf ∂ T ∂ y y = 0 − 16 σ ⁎ 3 k ⁎ ∂ T 4 ∂ y y = 0 .

Thus, Eq. (20) reduces as follows:

(21) Re x C fx = ξ 1 f ″ ( 0 ) , Nu x Re x = − ( ξ 4 + Rd ) θ ′ ( 0 ) ,

where Re x = a x 2 ν f is the local Reynolds number.

3 Numerical solution

To investigate the numerical solution of the proposed model presented in Eqs (15) and (16) with conditions at boundaries given in Eq. (17), the Matlab built-in package (bvp4c) is adopted [66,67,68,69,70]. To perform the bvp4c technique, first, we have to reduce the model equations into first-order differential equations. Thus, we have assumed the following relations:

(22) f ( ξ ) = ℵ ( 1 ) , f ′ ( ξ ) = ℵ ( 2 ) , f ″ ( ξ ) = ℵ ( 3 ) , f ‴ ( ξ ) = ℵ ′ ( 3 ) , θ ( ξ ) = ℵ ( 4 ) , θ ′ ( ξ ) = ℵ ( 5 ) , θ ″ ( ξ ) = ℵ ′ ( 5 ) .

Then the leading equation can be written as:

(23) ℵ ′ ( 3 ) = − ℵ ( 1 ) ℵ ( 3 ) − ( ℵ ( 2 ) ) 2 − ξ 3 ξ 2 M sin 2 γ ℵ ( 2 ) + ξ 1 ξ 2 1 + 1 β K ℵ ( 2 ) ξ 1 ξ 2 1 + 1 β ,

(24) ℵ ′ ( 5 ) = − Pr ℵ ( 1 ) ℵ ( 5 ) + Pr ξ 5 ( ξ 3 M Ec sin 2 γ ( ℵ ( 2 ) ) 2 + Q E exp ( − n η ) + Q I ℵ ( 4 ) ) 1 ξ 5 ( ξ 4 +Rd ) ,

with boundary conditions:

(25) ℵ a ( 1 ) − 0 , ℵ a ( 2 ) − λ , ℵ b ( 2 ) − 0 , k hnf k f ℵ a ( 5 ) + B T ( 1 − ℵ a ( 4 ) ) , ℵ b ( 4 ) − 0 .

The quantities of interest can be written as:

(26) Re x C fx = μ hnf μ f ℵ a ( 3 ) , Nu x Re x = − ( ξ 4 + Rd ) ℵ a ( 5 ) .

4 Code validation

Validation of the obtained results of − θ ′ ( 0 ) from the previous and present analysis is shown in Table 2. This table shows the comparison of present and previously published results which can confirm and validate the code of this problem. Additionally, one can say that the designed mathematical model is valid and can be extended for further analysis.

Table 2

Validation of − θ ′ ( 0 ) for different values of Pr while all other factors are zero

Pr	− θ ′ ( 0 )
Pr	Hamad [71]	Present results
0.07	0.06556	0.0655797
0.2	0.16909	0.1690994
0.7	0.45391	0.4539206
2.0	0.91136	0.9113705
7.0	1.89540	1.8954015

5 Results and discussion

We have analyzed the blood-based Casson hybrid nanofluid flow composed of Au and Fe₃O₄ NPs over a convectively heated permeable surface. The default values of embedded factors are taken as Pr = 21 , M = 1.0 , K = 0.5 , Rd = 0.5 , Ec = 0.1 , Q E = Q I = 0.3 and B T = 0.5 . The significances of flow constraints against the velocity and energy profiles are displayed in the form of figures as well as tables. Figure 1 exhibits the physical representation of the hybrid nanoliquid flow. Figures 2 and 3 elaborate the effect of magnetic factor (M) on f ′ ( ξ ) and θ ( ξ ) for both cases hybrid (Au–Fe₃O₃/blood) and nanofluid (Au/blood). Physically, the magnetic effect generates the resistive force which contests to the motion of fluid, and results in lowering the velocity curve as shown in Figure 2. That resistive effect is known as Lorentz force, stated as the combination of the electric and magnetic force due to electromagnetic flux on a point charge. On the other side, the energy profile rises for flourishing impact of (M). Actually, the free particles move over the sheet surface and some of them crash due to friction force, these collisions amongst particles of fluid produce heat and “resistance” to the fluid motion. Figures 4 and 5 show the impression of inclination angle ( γ ) on f ′ ( ξ ) and θ ( ξ ) for both cases hybrid (Au–Fe₃O₄/blood) and nanofluid (Au/blood). It can be detected that both the temperature and velocity curve drop for upsurge in angle of inclination. Physically, the increasing angle of inclination boosts the Lorentz force which reduces the velocity profile. On the other hand, the greater angle of inclination reduces the thermal boundary layer thickness which results reduction in the temperature profile. Figures 6 and 7 show the impression of Eckert number (Ec) and heat source factor (space-dependent) Q E on θ ( ξ ) , respectively. Physically, the Eckert number is the relation between advection transpport and heat dissipation and used to describe the energy transport dissipation. So, the inverse relation of heat dissipation potentional with the Ec enhances the fluid temoperature as displayed in Figure 6. Figure 7 demonstrates that the influence of heat source term Q E on the energy curve θ ( ξ ) . The higher Q E augments θ ( ξ ) . Physically, the specific heat capacity of base fluid deteriorates with the effect of Q E which results in the advancement of the energy curve. Figure 8 demonstrates that the influence of heat source term Q I on the energy curve θ ( ξ ) . The higher Q I augments θ ( ξ ) . Physically, the specific heat capacity of base fluid deteriorates with the effect of Q I which results in the advancement of the energy curve. Table 1 particularizes the experimental values used in the approximation of the model equations of nanoparticles Fe 3 O 4 and Au , and blood. Tables 3 and 4 reveal numerically the impression on Re x C fx and Nu x Re x . From Table 3, it can be noticed that the variability of the magnetic term gradually enhances the skin friction. Because due to resistive effect produced by magnetic field enhances the drag force Re x C fx . Table 4 shows the statistical outcomes for both the cases nanofluid ( Au / blood ) and hybrid nanoliquid ( Fe 3 O 4 − Au / blood ) versus the variation of distinct physical constraints. From Table 4, it can be noticed that the energy transmission rate, in case of ( Fe 3 O 4 − Au / blood ) hybrid nanoliquid is much hgher as compared to the nanofluid case.

$Figure 2 Variation in f ′ ( ξ ) f^{\prime} (\xi ) via M M .$

Figure 2

Variation in f ′ ( ξ ) via M .

$Figure 3 Variation in θ ( ξ ) \theta (\xi ) via M M .$

Figure 3

Variation in θ ( ξ ) via M .

$Figure 4 Variation in f ′ ( ξ ) f^{\prime} (\xi ) via γ \gamma .$

Figure 4

Variation in f ′ ( ξ ) via γ .

$Figure 5 Variation in θ ( ξ ) \theta (\xi ) via γ \gamma .$

Figure 5

Variation in θ ( ξ ) via γ .

$Figure 6 Variation in θ ( ξ ) \theta (\xi ) via Ec \text{Ec} .$

Figure 6

Variation in θ ( ξ ) via Ec .

$Figure 7 Variation in θ ( ξ ) \theta (\xi ) via Q E {Q}_{\text{E}} .$

Figure 7

Variation in θ ( ξ ) via Q E .

$Figure 8 Variation in θ ( ξ ) \theta (\xi ) via Q I {Q}_{\text{I}} .$

Figure 8

Variation in θ ( ξ ) via Q I .

Table 3

Variation in skin friction via M

M	Re x C fx
M	Fe 3 O 4 ‒ Au / blood	Au ‒ blood
0.1	1.1157	1.1065
0.2	1.1165	1.1069
0.3	1.1173	1.1074

Table 4

Variation in Nusselt number via M , Rd , Q E , and Ec

M	Rd	Q E	Ec	Nu x Re x
M	Rd	Q E	Ec	Fe 3 O 4 ‒ Au / blood	Au ‒ blood
0.1	0.5	0.1	0.1	0.3052	0.2969
0.2				0.3038	0.2939
0.3				0.3024	0.2905
	1.0			0.4094	0.4157
	1.1			0.4303	0.4274
	1.2			0.4511	0.4373
		0.1		0.5112	0.4964
		0.2		0.4467	0.4075
		0.3		0.3783	0.2989
			1.0	0.3035	0.2896
			2.0	0.3007	0.2846
			3.0	0.2979	0.2789

6 Conclusion

In this work, hybrid nanofluid flow consisting of Fe₃O₄ and Au NPs over a convectively heated permeable sheet has been investigated. The flow of fluid is formulated using the system of partial differential equations (PDEs). The modeled equations of PDEs are transformed into the set of ordinary differential equations (ODEs) by using the similarity substitution. The Matlab built-in package (bvp4c) is employed to solve the transform nonlinear set of ODEs. The main conclusions are as follows:

By raising the angle of inclination, both the velocity and temperature profiles are reduced. Furthermore, the magnetic factor increases the temperature distribution while reduces the velocity distribution.
Growth in Eckert number, thermal-dependent heat source and the space-dependent heat source factor accelerates the temperature profile.
The energy transfer rate boosts with the consequence of the radiation factor, whereas drops with the impact of the Eckert number.
The velocity and temperature distributions are very effective in the cases of hybrid nanofluid (Au–F_e3O₄/blood) when compared to nanofluid (Au/blood).
The skin friction and rate of heat transfer are very effective in the cases of hybrid nanofluid (Au–Fe₃O₄/blood) when compared to nanofluid (Au/blood).

Funding information: The authors extend their appreciation to the Deanship of Scientific Research at King Khalid University, Abha, Saudi Arabia, for funding this work through the Research Group Project under Grant Number (RGP.2/505/44). This work was supported by the Deanship of Scientific Research, the Vice Presidency for Graduate Studies and Scientific Research, King Faisal University, Saudi Arabia (Grant No. 5580).
Author contributions: 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 analyzed during the current study are available from the corresponding author on reasonable request.

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Received: 2023-11-15

Revised: 2024-01-10

Accepted: 2024-01-13

Published Online: 2024-02-16

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

Articles in the same Issue

https://doi.org/10.1515/phys-2023-0193

Keywords for this article

Hybrid nanofluid; MHD; space-dependent heat source; thermal-dependent heat source; porous media; Joule heating

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