
==== Front
Heliyon
Heliyon
Heliyon
2405-8440
Elsevier

S2405-8440(24)12200-1
10.1016/j.heliyon.2024.e36169
e36169
Research Article
Influence of engine oil-infused multi-walled carbon nanotubes and titania nanoparticles on a vertically inclined porous surface
Sarfraz Mahnoor mahnoor@math.qau.edu.pk
⁎
Khan Masood
Department of Mathematics, Quaid-i-Azam University, Islamabad 44000, Pakistan
⁎ Corresponding author. mahnoor@math.qau.edu.pk
13 8 2024
30 8 2024
13 8 2024
10 16 e361697 5 2023
8 8 2024
12 8 2024
© 2024 The Authors. Published by Elsevier Ltd.
2024

https://creativecommons.org/licenses/by-nc/4.0/ This is an open access article under the CC BY-NC license (http://creativecommons.org/licenses/by-nc/4.0/).
This study analyses the flow of a hybrid nanofluid, combining engine oil with multi-walled carbon nanotubes and titania nanoparticles. The flow occurs over a vertically inclined, electrically conducting, heat-producing/absorbing surface that is both permeable and expanding/contracting. The analysis incorporates influential factors such as buoyancy forces, heat source/sink effects, and convective conditions with Cattaneo-Christov theory and Hamilton-Crosser model. The mathematical model is numerically solved using the bvp4c solver in MATLAB. The expansion/contraction of the surface significantly impacts the boundary layer thickness, leading to changes in velocity, temperature, and various physical parameters. This study is significant due to the nanoparticles’ enhanced optical and mechanical properties, offering potential applications in diverse fields. A notable finding is the reduced fluid velocity and temperature within a porous medium with permeability. These findings present opportunities for enhancing heat and fluid transmission in various systems, including those related to energy storage.

Highlights

• The flow of a hybrid nanofluid, combining engine oil with MWCNT and TiO2 is considered.

• The flow occurs over a vertically inclined, electrically conducting, heat-generating/absorbing surface that is both permeable and expanding/contracting.

• The analysis incorporates influential factors such as the Cattaneo-Christov theory, buoyancy forces, heat source/sink effects, and convective conditions.

• The mathematical model is numerically solved using the bvp4c solver in MATLAB.

• These findings present opportunities for enhancing heat and fluid transmission in various systems, including those related to energy storage.

Keywords

Buoyancy force
Carbon nanotubes
Convective boundary conditions
Engine oil
Hybrid nanofluid
Nanoparticles
Porous surface
==== Body
pmc Nomenclature

T∼∞	Ambient temperature	
τrz,τrθ	Axial and Azimuthal wall-shear stress	
β*	Biot number	
(x,y,z)	Cartesian coordinates	
ϱ	Density	
Θ,∼F∼',H∼,G,∼P∼	Dimensionless temperature, velocities, and pressure	
η	Dimensionless variable	
μ	Dynamic viscosity	
σ	Electric conductivity	
hw	Heat transfer coefficient	
α	Horizontal line angle	
ν	Kinematic viscosity	
B0	Magnetic field strength	
M	Magnetic parameter	
ϕ	Nanoparticle volume fraction	
εp	Permeability coefficient	
ϵ*	Permeability parameter	
Pr	Prandtl number	
p∼	Pressure	
Re	Reynolds number	
T∼	Temperature	
k	Thermal conductivity	
Bc	Thermal expansion	
βt	Thermal relaxation time	
u∼,v∼,w∼	Velocity components	
f∼w	Wall transmission constraint	
Subscript and Superscript	
f	Base fluid	
hf	Binary hybrid nanofluid	
x,z,'	Derivative w.r.t x,z,η, respectively	
∞	Free stream	
nf	Mono nanofluid	
nf2,nf1	MWCNT, TiO2 nanoparticles	

1 Introduction

Hybrid nanofluids (HNFs) comprise a combination of diverse nanoparticles immersed in a base fluid. They can be classified into four categories: carbon, metal, metal oxide, and mixed/hybrid metal-based nanofluids (NFs). Due to their enhanced thermos-electrical conductivity properties, HNFs have the potential for numerous applications in various fields. In his work, Iijima [1] presented the discovery of carbon nanotubes, their structure, properties, and potential applications. Choi and Eastman [2] demonstrated the capacity to significantly increase thermal conduction by suspending nanoparticles in them. Hamilton and Crosser [3] developed a model to predict the thermal conductivity of two-component systems based on their thermal conductivities and volume fractions. The potential applications of HNFs were presented in the works of Sarkar et al. [4] and Muneeshwaran et al. [5]. Esfe et al. [6] investigated the rheological behavior of a HNF consisting of MWCNT-TiO2 and developed a neural network model. Li et al. [7] synthesized and characterized a water-based HNF of titania and developed a model to predict its thermal performance. Other interesting discoveries related to HNFs are addressed in Refs. [[8], [9], [10], [11]].

The investigation of thermal energy transfer over a stretching/shrinking surface has drawn substantial attraction due to its applicability in diverse fields. Initially, Wang [12] investigated free convection flow with its heat transfer, which was later expanded upon by Golra and Sidawi [13] by considering the consequences of suction/blowing. Further developments in this area were made by Chamkha [14], who evaluated the impact of magnetic fields and radiation on stretching surfaces. In his work, he explored how changes in controlled parameters influence profiles of physical quantities. Raju et al. [15] scrutinized the model for inclined magnetic radiative effects. Ramudu et al. [16] researched and extended the model with the influences of porous boundaries. At the same time, Jha and Samaila [17] assessed the development of Roseland radiation. Pattnaik et al. [18] incorporated dissipation effects into their model to analyze mixed convective flow. Their research uncovered practical uses for thermo-magnetic coating processes that incorporate nanomaterials with specific microstructural features. Additional studies on the subject can be found in Refs. [[19], [20], [21], [22], [23]].

This study examines the flow and thermal characteristics of HNF (heat-producing nanofluids) on a vertically inclined porous plate. Unlike previous studies focusing on single nanofluids, our research is unique because it uses an HNF comprised of engine oil, multi-walled carbon nanotubes (MWCNT), and titania. MWCNTs consist of multiple layers of graphene sheets arranged in a tubular fashion due to the hexagonal lattice structure of graphene. We also analyze the effects of free convection and convective conditions. The problem is solved using the bvp4c function in MATLAB. The advantages of these nanocomposites include improved electro-thermal conductivity (using Cattaneo-Christov theory and Hamilton-Crosser model), chemical stability, and antibacterial properties.

2 Mathematical formulation

Consider a laminar flow of electrically conducting heat-producing HNF past a semi-infinite vertically inclined porous plate with Cattaneo-Christov theory, as shown in Fig. 1. The base fluid is engine oil mixed with a combination of Titania and MWCNT (see Table 1, Table 2 for their thermophysical properties). The assumptions made are.♦ The model is formulated in (x,y,z) with V represented as [u˜(x,z),v˜(x,z),w˜(x,z)] and uniform B=[0,B0,0], which is applied normal to the y-axis. It is induced along the x and z-axes, handling the plate's stretching and suppressing convective flow.

♦ The dissipation effects are neglected. As there is no surface tension, all variables are only dependent on the x and z coordinates, and not on the y -coordinate.

♦ The buoyancy force is observed along the x and y-axes, and the plate is heat up by convection from T˜w and hw. The ambient temperature distribution is set to be T˜∞.

♦ The plate stretches along the x-axis with u˜(x,0)=cx. Wall suction/blowing constraint w0 is provided along the z-component, and the plate is vertically inclined at α along the y-axis w.r.t the horizontal line.

Fig. 1 Flow mechanism.

Fig. 1

Table 1 Thermophysical characteristics - for mono NF and binary HNF (see Refs. [7,8]).

Table 1Viscosity	μhf=μf(1−ϕnf1)−2.5(1−ϕnf2)−2.5,μnf=μf(1−ϕnf1)−2.5,	
Density	ϱhf=ϕnf2ϱnf2+(1−ϕnf2)[(1−ϕnf1)ϱf+ϕnf1ϱnf1],ϱnf=ϕnf1(ϱnf1)+(1−ϕnf1)(ϱf),	
Specific Heat Capacity	(ϱcp)hf=ϕnf2(ϱcp)nf2+(1−ϕnf2)[(1−ϕnf1)(ϱcp)f+ϕnf1(ϱcp)nf1],(ϱcp)nf=ϕnf1(ϱcp)nf1+(1−ϕnf1)(ϱcp)f,	
Thermal Expansion	(ϱBc)hf=ϕnf2(ϱBc)nf2+(1−ϕnf2)[(1−ϕnf1)(ϱBc)f+ϕnf1(ϱBc)nf1],(ϱBc)nf=ϕnf1(ϱBc)nf1+(1−ϕnf1)(ϱBc)f,	
Thermal Conductivity (Hamilton-Crosser Model)	khf=knf2+(l−1)knf−(l−1)ϕnf2(knf−knf2)knf2+(l−1)knf+ϕnf2(knf−knf2)×knf,knf=knf1+(l−1)kf−(l−1)ϕnf1(kf−knf1)knf1+(l−1)kf+ϕnf1(kf−knf1)×kf,	
Electrical Conductivity	σhfσnf=σnf2+(l−1)σnf−(l−1)ϕnf2(σnf−σnf2)σnf2+(l−1)σnf+ϕnf2(σnf−σnf2),σnfσf=σnf1+(l−1)σf−(l−1)ϕnf1(σf−σnf1)σnf1+(l−1)σf+ϕnf1(σf−σnf1).	

Table 2 Experimentally verified values for Table 1 (see Refs. [7,8]).

Table 2Physical	Engine Oil	TiO2	MWCNT	
Properties	
ϱ	884	4250	1600	
cp	1910	686.2	796	
Bc ( × 10−5)	70	0.90	2.6	
k	0.144	8.9528	3000	
σ	10−10-10−12	>10−10	106-107	
Pr	58	-	-	

The governing equations are (see Ref. [14])(1) u˜x+w˜z=0,

(2) u∼u∼x+w∼u∼z=μhfϱhfu∼xx+u∼zz+gϱBchfϱhfcosαT∼w−T∼∞−σhfB02ϱhfu∼−μhfεpϱhfu∼,

(3) u∼v∼x+w∼v∼z=μhfϱhfv∼xx+v∼zz+gϱBchfϱhfsinαT∼w−T∼∞−μhfεpϱhfv∼,

(4) u∼w∼x+w∼w∼z=−1ϱnfp∼z+μhfϱhfw∼xx+w∼zz−σhfB02ϱhfw∼−μhfεpϱhfw∼,

(5) u˜T˜x+w˜T˜z+λt(u˜2T˜xx+w˜2T˜zz+2u˜w˜T˜xz+u˜u˜xT˜z+u˜w˜xT˜z+w˜u˜zT˜x+w˜w˜xT˜z)=khf(ϱcp)hf(T˜xx+T˜zz)+Q*(ϱcp)hf{(T˜−T˜∞)+λt(u˜T˜x+w˜T˜z)},

BCs are (see Ref. [12])(6) u˜=cx,v˜=0,w˜=w0,−khf∂T˜∂z=hw(T˜w−T˜),atz=0,u˜=v˜=w˜z=0,T˜=T˜∞,asz→∞,

with(7) u∼=cxF∼'η+ΓcosαH∼η,v∼=ΓsinαG∼η,w∼=−νfc12F∼η,T∼=T∼∞+T∼w−T∼∞Θ∼η,p∼=cϱνfP∼η.

Where in Eq. (7) η=zcνf12andΓ=gBcT∼w−T∼∞c. Using Eq. (7) in Eqs. (1), (2), (3), (4), (5), (6), we have(8) F˜‴+(μhfμf){(ϱhfϱf)[F˜F˜″−(F˜′)2]−(σhfσnf)M2F˜′}−ϵ*F˜′=0,F˜(0)=f˜w,F˜′(0)=1,F˜′(∞)=0,

(9) H˜″+(μhfμf){(ϱhfϱf)(F˜H˜′−H˜F˜′)+((ϱBc)hf(ϱBc)f)Θ˜−(σhfσnf)M2H˜}−ϵ*H˜=0,H˜(0)=0,H˜(∞)=0,

(10) G˜″+(μhfμf){(ϱhfϱf)F˜G˜′+((ϱBc)hf(ϱBc)f)Θ˜}−ϵ*G˜=0,G˜(0)=0,G˜(∞)=0,

(11) P˜′+(μfμhf)F˜″+(ϱhfϱf)F˜′F˜−M2(μhfμf)(σhfσnf)F˜−ϵ*F˜=0,P˜(0)=0,

(12) (ϱcp)hf(ϱcp)fΘ˜″+(khfknf)Pr{F˜Θ˜′−βt(F˜2Θ˜″−F˜′Θ˜′)}+Prδ{Θ˜−βt(F˜Θ˜′)}=0,(khfknf)Θ˜′(0)=β*{Θ˜(0)−1},Θ˜(∞)=0,

where M=σfB02cϱf,f∼w=−w0cνf,ϵ*=νfεpc,Pr=νfαf,βt=λtc,δ=Q*cϱcpf,β*=hwkfcνf.

The horizontal wall stresses are given as(13) τxz=μhf[u˜z|z=0,τyz=μhf[v˜z|z=0.

Eq. (13) gives(14) τxz=ϱfνfc12μfμhfcxF∼''0+Γ cosαH∼'0,τyz=ϱfνfc12μfμhfΓ sinαG∼'0.

2.1 Solution Approach

The bvp4c function in MATLAB is a numerical method that specifically tackles boundary value problems (BVPs) associated with ODEs. The method transforms the BVP into a set of algebraic equations using the collocation method, which is then solved using a nonlinear solver. For a broad range of industrial and engineering applications, bvp4c efficiently solves ODEs. The user inputs the ODE, BCs, and initial guess to obtain a numerical solution.

3 Discussion and results

The numerical results of controlled parameters on the physical quantities are analyzed here. These quantities include F˜′(η),H˜(η),G˜(η), and Θ˜(η) and solutions are established through the numerical and asymptotic analysis. The ranges of some parameters are in-between ϕnf1,ϕnf2ϵ[0,0.05],Mϵ[0,3],δϵ[−1,2],βtϵ[0,2], and β*ϵ[−1,1]. The solid lines illustrate the behavior of the traditional NF (TiO2 +engine oil), whereas the dashed lines depict the behavior of the binary HNF (TiO2 +MWCNT + engine oil). The brick-shaped nanoparticles, i.e., l=3.7 are represented in red, while the blade-shaped nanoparticles, i.e., l=8.9 are represented in blue.

Fig. 2(a–d) illustrate how ϵ* influences F˜′(η),H˜(η),G˜(η), and Θ˜(η) for l=3.7,8.9 on binary HNF and traditional NF. The increase in ϵ* leads to a rise in both F˜′(η) and H˜(η) because higher ϵ* indicates there are more void spaces within the porous medium, thus the fluid encounters less resistance along these directions, these interconnected pores create pathways that facilitate free movement of fluid. Consequently, F˜′(η) and H˜(η) increase as the fluid can navigate through the medium more easily, see Fig. 2(a and b). On the other hand, the reduction in G˜(η) is due to the presence of more voids disrupts the continuous flow. The fluid tends to spread out into the available pores and divert some of the flow from the main axial stream, which causes dispersion, and it reduces G˜(η) as the fluid is no longer confined to a narrow path but instead flows through a more distributed network of pores, as shown in Fig. 2(c). Physically, increment in ϵ* enhances the convective heat transfer within the porous medium and leads to a more rapid dissipation of heat. As surface area increased due to the void spaces, which allows for more effective heat exchange. Additionally, the increased F˜′(η) and H˜(η) further promotes energy distribution and dissipation of thermal energy, which decrease Θ˜(η). Additionally, leading behavior is noted for l=8.9 - traditional NF over l=3.7 - traditional NF and binary HNF. Fig. 3(a–c) demonstrate the effect of f˜w on the flow field for l=3.7,8.9 on binary HNF and traditional NF. f˜w is affected by suction (f˜w>0) and injection (f˜w<0). When f˜w is applied, F˜′(η) decreases because the wall restricts outward or inward fluid movement, adhering to the no-slip condition where the fluid's velocity relative to the wall surface is zero, as shown in Fig. 3(a). Conversely, H˜(η) increases (see Fig. 3(b)) due to the conservation of angular momentum, as the restricted radial flow enhances the fluid's rotational motion near the walls. Fig. 3(c) shows that G˜(η) decreases due to viscous drag and flow redistribution, where frictional forces at the wall oppose fluid motion, and reduced radial flow diminishes overall axial momentum. Θ˜(η) decreases due to enhanced convective heat transfer near the walls. The reduced F˜′(η) and G˜(η) result in a thicker thermal boundary layer where conduction dominates, and the increased H˜(η) induces secondary flows, further promoting heat loss to the walls, as shown in Fig. 3(d). Fig. 4(a–d) illustrate how M affects physical quantities such as F˜′(η),H˜(η),G˜(η), and Θ˜(η) for l=3.7,8.9 on binary HNF and traditional NF. F˜′(η) and H˜(η) typically decrease due to the induced Lorentz force, which acts perpendicular to both the magnetic field and the direction of fluid motion. It creates a damping effect that opposes and reduces F˜′(η) and H˜(η), as shown in Fig. 4(a and b). In the radial direction, the magnetic field restricts the outward or inward movement of fluid particles and decreases F˜′(η), while in the azimuthal direction, the force resists tangential motion around the axis and reduces H˜(η). On the other hand, G˜(η) increases because the damping of F˜′(η) and H˜(η) redistributes momentum towards the axial direction, reducing flow resistance and allowing for faster axial flow, see Fig. 4(c). Θ˜(η) tends to increase as the reduced F˜′(η) and H˜(η) lead to less convective heat transfer, retaining more thermal energy within the fluid. Additionally, the increased G˜(η) improves axial heat transport, and the Joule heating effect, where the interaction between the magnetic field and electric currents generates additional thermal energy and raises Θ˜(η), as shown in Fig. 4(d).Fig. 2 (a)-2(d): Impact of porosity parameter.

Fig. 2

Fig. 3 (a)-3(c): Impact of wall transmission constraint.

Fig. 3

Fig. 4 (a)-4(d): Impact of magnetic parameter.

Fig. 4

The behavior of δ,β*, and βt is observed through Fig. 5(a–d) on Θ˜(η) for l=3.7,8.9 on binary HNF and traditional NF. δ indicates the amount of energy absorbed/generated in any mechanical system. δ>0 corresponds to heat generation/source, which converts the fluid hotter and thickens the boundary layer, while absorption/sink (δ<0) has the opposite effect. However, due to δ>0, the energy transport is augmented, which raises the energy and elevates Θ˜(η), as presented in Fig. 5(a). A higher value of β* indicates that surface's internal resistance is higher than its external resistance. A lower β* means that surface's resistance (due to conduction) is much less than the external resistance of its environment. Fig. 5(b) shows that increasing β* causes the energy transport to elevate as well. The system's behavior is affected by βt as shown in Fig. 5(c). When βt is small, the system can quickly adjust to changes in temperature. Conversely, a large βt means that the system will take longer to adjust to temperature changes. Therefore, boost in βt leads to a reduce in Θ˜(η). Moreover, dominant behavior is noted for l=8.9 - traditional NF.Fig. 5 (a)-5(d): Impact of heat source/sink, Biot number, and thermal relaxation.

Fig. 5

Table 3, Table 4, Table 5 demonstrate a comparison of the values of F˜″(η),H˜′(η),G′˜(η), and Θ′˜(η) for Pr and M, with Golra and Sidawi [13] in the absence of M,ϕnf1, and ϕnf2. The numerical values obtained exhibit exact agreement with those reported in Golra and Sidawi [13]. The tables reveal that, as Pr increases, Θ′˜(η) and F˜″(η),H˜′(η), and G′˜(η) decrease. However, as M increases, the values of G′˜(η) and Θ′˜(η) are enhanced, while H˜′(η) decreases. This is because M suppresses the velocity components perpendicular to its direction, thereby reducing Θ′˜(η) and F˜″(η),H˜′(η), and G′˜(η). Nonetheless, as M continues to increase, the velocity components parallel to the magnetic field direction are intensified, increasing Θ′˜(η) and F˜″(η),H˜′(η), and G′˜(η).:Table 3 Θ˜′(0) for Pr at M=δ=0 and F˜″(0) for M at δ=−0.1,Pr=6.2 with Golra and Sidawi [13].

Table 3Θ˜′(0)	F˜″(0)	
Pr	Golra and Sidawi [13]	Present	M	Golra and Sidawi [13]	Present	
0.07	−0.0656	−0.06568	0	−1.00180	−1.00183	
0.2	−0.1691	−0.16918	1	−1.41602	−1.41423	
0.7	−0.4539	−0.45429	2	−2.23731	−2.23762	
2.0	−0.9114	−0.91129	3	−3.16351	−3.16117	
7.0	−1.8954	−1.89522				
20.0	−3.3539	−3.35352				
70.0	−6.4622	−6.46186				

Table 4 H˜′(0) for Pr at M=δ=0 and Matδ=−0.1,Pr=6.2 with Golra and Sidawi [13].

Table 4H˜′(0)	
Pr	Golra and Sidawi [13]	Present	M	Golra and Sidawi [13]	Present	
0.07	5.746	5.7439	0	0.27781	0.27729	
0.2	2.309	2.3098	1	0.26234	0.26266	
0.7	0.948	0.94800	2	0.23241	0.23228	
2.0	0.5265	0.52613	3	0.20175	0.20166	
7.0	0.2829	0.28266				
20.0	0.1796	0.17045				
70.0	0.0956	0.09273				

Table 5 G˜′(0)forPr at M=δ=0 and M at δ=−0.1,Pr=6.2 with Golra and Sidawi [13].

Table 5G˜′(0)	
Pr	Golra and Sidawi [13]	Present	M	Golra and Sidawi [13]	Present	
0.07	8.690	8.6778	0	0.30358	0.30375	
0.2	3.287	3.2881	1	0.32932	0.32957	
0.7	1.208	1.2081	2	0.38625	0.38619	
2.0	0.6160	0.61604	3	0.45423	0.45341	
7.0	0.3094	0.30925				
20.0	0.1796	0.17995				
70.0	0.0956	0.09558				

4 Concluding remarks

In this investigation, we explored the attributes of a stable, laminar flow involving a HNF over a vertical plate, incorporating the Cattaneo-Christov theory with shape factor, magnetic field, buoyancy forces, and heat generation and absorption. Notably, our study utilized a blend of Titania and MWCNT nanoparticles suspended in engine oil as the base fluid. It is worth noting that dissipation effects were omitted. To obtain the solution for this complex problem, we employed the bvp4c method. The outcomes of this research carry significant implications with broad practical relevance, extending into various domains, including engineering, materials science, and nanotechnology. The summary of results obtained is given below.♦ The magnetic field functioned as an impediment to particle motion, restraining velocities in the radial and axial directions while augmenting it azimuthally and bolstering energy transfer.

♦ Permeability led to a decline in both fluid flow velocity and temperature.

♦ Stretching/shrinking influenced velocity and temperature by modifying their boundary layer thicknesses and induced forces associated.

♦ The increase in Prandtl number resulted in a significant growth in energy transport rate.

♦ Thermal transport took precedence in governing the heat transfer process, primarily due to the heat source/sink.

♦ Brick-shaped nanoparticles were dominant in behavior as compared to blade-shaped.

♦ Prandtl number caused a reduction in the numerical values of heat transfer rate and coefficients of skin friction concerning the magnetic number.

Limitation and future scope

While providing focused insights, the choice of engine oil with Titania and MWCNT limits the study's generalizability to different nanoparticle combinations. Excluding dissipation effects simplifies the study but overlooks their potential impact on the system's overall energy balance and temperature distribution. The future scope of research in this area is extensive and promising, as given below.♦ Extending the model to investigate non-Newtonian viscoelastic flows would provide valuable insights into the behavior of complex fluids like polymer solutions in similar configurations.

♦ Exploring the possibility of dual solutions and conducting bifurcation analysis can help us understand the system's multiple stable states and transitions.

♦ Statistical analysis can be employed to assess the uncertainty and variability in results, enhancing the robustness of the findings.

♦ Experimental validation should be pursued to corroborate computational results.

Data availability statement

Data will be made available on request.

CRediT authorship contribution statement

Mahnoor Sarfraz: Writing – review & editing, Writing – original draft, Visualization, Validation, Software, Methodology, Investigation, Formal analysis. Masood Khan: Writing – review & editing, Supervision, Resources, Project administration, Formal analysis, Data curation, Conceptualization.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
==== Refs
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