Comparative study of copper nanoparticles over radially stretching sheet with water and silicone oil

Umer Hayat; Shakil Shaiq; Kottakkaran Sooppy Nisar; Azeem Shahzad; Aamir Farooq; Muhammad Kamran; Nehad Ali Shah

doi:10.1515/ntrev-2023-0200

Article Open Access

Comparative study of copper nanoparticles over radially stretching sheet with water and silicone oil

Umer Hayat , Shakil Shaiq , Kottakkaran Sooppy Nisar , Azeem Shahzad , Aamir Farooq , Muhammad Kamran and Nehad Ali Shah

Published/Copyright: March 15, 2024

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From the journal Nanotechnology Reviews Volume 13 Issue 1

Abstract

Copper nanoparticles are widely used in many sectors and research endeavors owing to their unique properties, including a large surface area, catalytic capabilities, and high thermal and electrical conductivity. The selection of the base fluid for copper nanoparticles should be contingent upon the anticipated application requirements since various fluids exhibit distinct characteristics that could potentially impact the mobility of the nanoparticles. The present investigation analyzes heat transfer phenomena occurring across a radially stretched surface. The research explores the effects of different states of Cu nanoparticles when combined with base fluids, such as water and silicone oil, on the heat transfer process. The momentum and energy equations are transformed into nonlinear ordinary differential equations by applying the similarity transformation. The boundary value problem-fourth-order (BVP4C) method numerically solves the governing ordinary differential equation for the modeled problem. In addition, the influence of various factors such as the slip parameter, solid volume fraction, Eckert number, Prandtl number, and unsteadiness parameter are examined. It has been discovered that blade-shaped nanoparticles transfer heat as quickly as possible via silicone oil and water. However, for platelet-shaped nanoparticles, a minimum heat transfer rate has been noted. The maximum skin friction coefficient is observed in platelet-shaped nanoparticles, while blade-shaped nanoparticles have the lowest skin friction coefficient.

Keywords: copper nanoparticles; silicone oil; stretching sheet; shape factor; slip parameter

Nomenclature

U: stretching velocity ( m / s )
q w: heat flux (surface) ( J / m 2 s )
ϕ: solid volume fraction ϕ ( nm )
C f: skin friction coefficient
Ψ: stream function ( m 2 / s )
k f: thermal conductivity of base fluid ( W / mK )
k nf: thermal conductivity of nanofluid ( W / mK )
B 0: magnetic field ( kg / s 2 A )
σ: electrical conductivity ( W / mK )
S: unsteadiness parameter
U: stretching velocity ( m / s )
μ: dynamic viscosity ( kg / m s )
τ w: shear stress ( kg / m s 2 )
ρ f: density of base fluid ( kg / m 3 )
R e: Reynolds number
c p: specific heat ( J / Kg K )
ρ s: density of nanoparticles ( kg / m 3 )
μ nf: viscosity of nanofluid ( kg / ms )
β: film thickness
M: magnetic parameter
Pr: Prandtl number
Nu: Nusselt number
ν: viscosity (kinematic) ( m 2 / s )

1 Introduction

A nanofluid is a suspension of nanoparticles in a base fluid, such as ethylene glycol or water. Nanotechnology has led to the development of improved thermal properties, with nanoparticles being more appealing due to their higher coverage area and heat transmission ability. This technology has applications in domestic refrigerator freezers, chilled circulation, semiconductor conditioning, photovoltaic water boiling, power plants, and other freezing temperature exchange equipment. Nanofluids [1] offer a variety of potential uses due to their enhanced heat transmission. Numerous applications of nanofluids [2] have been reported in the literature, covering areas like biomedical applications, lubrication, surface coating, and the crude oil sector. This article goes into great detail about the latest advancements in nanofluids [3]. This collection of academic articles on nanofluids describe the significance and influence of nanofluids and nanoparticles on current advancements. The method of suspending metal particles in fluids to enhance thermal conductivity is well-recognized in academic circles. According to Choi [4], the first proposition of the nanofluids notion may be attributed to him. The researcher observed that the incorporation of nanoparticles into the underlying fluid resulted in enhanced thermal properties. Wang et al. [5] examined the thermal conductivity of mixtures, including nanoparticles and fluids. In his seminal work, Sakiadis [6] invented the boundary layer investigation across a solid surface. Crane [7] developed a precise answer by modeling the flow pattern for the expansion of a problem that was similar. The effects on a stretching surface were studied by Erickson et al. [8], who also took into account the effects of suction or blowing. The effect of particle form on nanofluid flows has been studied experimentally and theoretically in a number of studies. Enhancing thermal performance and heat transmission capabilities is the primary objective of incorporating nanoparticles into such processes. A fractal model was proposed by Wang et al. [9] for predicting the effective thermal conductivity of liquids with nonmetallic nanoparticles, enhancing the effective medium theory, and accurately predicting the variation of effective thermal conductivity with nanoparticle suspension. The influences of nanoparticle shapes of A l 2 O 3 nanoparticles with silicon oil base fluid are investigated by Masood et al. [10] on a radially extended spinning disc while taking into account some important physical parameters. Copper nanoparticles [11–13] with specific shapes on a stretching sheet can enhance mechanical strength, flexibility, and conductivity, making it ideal for flexible electronics, wearable devices, sensors, electronic textiles, and displays. The controlled shape factor influences the sheet’s surface properties, affecting adhesion, friction, and catalytic activity, showcasing the versatility and adaptability of copper nanoparticles on stretchable substrates. In addition, copper nanoparticles have antibacterial qualities that make them useful in applications like food and medical packaging where hygienic conditions are critical. All things considered, the incorporation of copper nanoparticles [14–17] into stretched sheets creates opportunities for sophisticated materials with enhanced functional, electrical, and mechanical properties, spurring innovation across a range of industries. Leong et al. [18] looked at the thermal conductivity characteristics of Cu − Ti O 2 hybrid nanofluids and contrasted them with Cu and Ti O 2 nanofluids. In this research, Sadiq [19] examined the heat transmission in thin film flow over an unstable stretched plate of nanofluids. Turkyilmazoglu [20] has investigated in this work the flow and heat transfer characteristics of a rotating disc immersed in five distinct nanofluids. According to the findings, the nanofluids with Cu nanoparticles had the highest rate of heat transmission, while those with Ti O 2 nanoparticles had the lowest rate. Timofeeva et al. [21] described the effects of particle shapes on the thermophysical properties of alumina in this article in detail. This study [22] explores the effects of radiation and particle form on the Marangoni boundary layer flow and heat transfer of a copper–water nanofluid driven by an external temperature. They came to the conclusion that, in comparison to other nanoparticle shapes, sphere nanoparticles with low thermal conductivity improved heat transmission more successfully. Examining the effects of nanoparticle form on an A l 2 O 3 nanofluid with silicon oil-base fluid on a radially extended spinning disc was the aim of this paper. Jiang’s study [23] on nanoparticle formation in a rectangular cavity found that blade-shaped nanoparticles enhance heat transfer during boiling. He also examined the Marangoni boundary layer flow and heat transfer of a copper–water nanofluid, finding that sphere nanoparticles with low thermal conductivity improved heat transmission better than other types. Saranya and Al-Mdallal [24] found that blade-shaped nanoparticles have the strongest heat transfer capacity, while nanofluids, including platelet-shaped nanoparticles, exhibit the most skin friction. Shaiq et al. [25] studied the impact of an induced magnetic field on heat transfer and nanofluid stagnation point flow, focusing on copper and titanium dioxide particles and ethylene glycol as the base fluid. Hayat and Shahzad [26] explored heat and flow transfer on a multi-shaped Ag-Nps surface subjected to radial stretching, finding that platelet and sphere shapes have the highest and lowest transfer rates.

This article [27] analyzes 2D magnetohydrodynamics (MHD) Jeffrey fluid flow over a permeable stretching surface with viscous dissipation and a heat source or sink. It reveals that the Eckert number increases with temperature and thermal boundary layer thickness, but oppositely with Pr. The impact of partial slip and heat radiation on the MHD flow of a Jeffrey nanofluid containing gyrotactic microorganisms that move along a vertical stretching surface is examined in this article [28]. It was discovered that whereas the Richardson number falls with heat radiation, the Nusselt number rises. The study [29] examined the flow through a porous wedge while taking magneto-hydrodynamic effects into account. The nanofluid contained gyrotactic microorganisms. Adnan et al. [30] conducted a study on A l 2 O 3 , Ti O 2 , and Cu nanomaterials over a magnetized disk and measured their impact on flow properties. Platelet-shaped nanomaterial decreased nanofluid velocity, while a stronger magnetic field counteracted the fluid’s velocity. Ramesh et al. [31] investigated the flow of stagnation-point hybrid carbon nanotubes (CNTs) over a rotating sphere under thermal radiation and thermophoretic particle deposition. Results show that primary velocity increases with acceleration but decreases with secondary velocity. The study also reveals that concentration decreases with decreasing Schmidt number and thermophoretic parameters, and heat dispersion rate increases with solid volume fraction. The behavior of hybrid CNTs flowing through a slipping surface with an induced magnetic field is investigated by Ramesh and Madhukesh [32]. The aggregation of single-walled carbon nanotubes and multi-walled carbon nanotubes nanomaterials with water-base liquid is taken into consideration in this study, and the effects on thermal performance, chemical reaction, and activation energy were examined. According to the study, higher magnetic Prandtl numbers increase the magnetic field and response rate, while activation energy increases the concentration field. In this study [33], a hybrid nano-liquid containing magnetic ferrite F e 3 O 4 nanoparticles and Cu nanoparticles was used to explore the heat transfer in a plate-cone viscometer. The investigation revealed that augmenting the volume percentage of both nanoparticles led to a notable enhancement in the thermal transmission rate and base fluid velocity. The unsteady MHD nanofluid flow over a vertical plate was studied by Aleem et al. [34]. The results show that the temperature of Ag − water nanofluid is higher because of its better thermal conductivity, while the velocity of A l 2 O 3 − water is higher because of its smaller particle count. The study [35] examined nanofluid flow in porous medium, finding that increasing solid volume percentage improves thermal delivery but reduces microorganism density, concentration, and velocity. Suction and porous characteristics also reduce velocity. The motion of Reiner–Rivlin fluid on a stretching surface under the influence of different physical parameters is investigated in this work by Hiremath et al. [36]. The effect of inclination angle on velocity and temperature profiles is the main focus of the analysis. The findings indicate that at an inclined angle of 90°, there is a maximum thermal distribution, a minimum velocity, and a minimum and maximum surface drag force.

An overview of heat transfers and fluid flow over an unsteady stretched sheet will be provided in this article. Because of their wide range of applications in several fields, copper nanoparticles were considered for this study. This work aims to fill the gap in the literature by taking into account heat transfer and copper nanoparticle flow over a stretching sheet in the presence of a magnetic field. The behavior of different physical parameters on temperature, skin friction, velocity, and the Nusselt number for various shape factors are elucidated via the presentation of graphical representations and tabulated data. Finally, an analysis of the effects of better particle shape will be presented.

The format of the manuscript is as follows: the creation of physical models and problem formulation are covered in Section 2; the mathematical concept and technique of the simulated problem are covered in Section 3; the effects of various physical properties on temperature, skin friction coefficient, heat transfer coefficient, thin film thickness, and velocity are examined in Section 4; we compared our results for limiting the case value of the parameter to recent literature in order to validate our findings; and Section 5 contains the concluding remarks.

2 Formulation

In the context of the MHD phenomenon, we examine the behavior of a two-dimensional boundary layer flow in water and silicone oil, both serving as base fluids. This flow is subjected to the presence of Cu nanoparticles with different shapes as they travel through a radially extending sheet with a stationary center. The nanoparticles now under discussion include three distinct forms: blade, cylinder, and platelets. The variables U and T represent the surface velocity and temperature, respectively, of a stretched sheet. Figure 1 illustrates a schematic representation of the nanofluid flow through a porous stretched sheet and the corresponding coordinate system.

Figure 1

Representation of the modeled problem.

The cylindrical polar coordinate system ( r , θ , z ) is employed for mathematical modeling. Due to the flow’s rotational symmetry, all physical parameters are independent of θ , and the velocity field has the form v . = . [ u ( r , z ) , 0 , w ( r , z ) ] , where u and w are the velocity components along the radial r and axial z directions, respectively. Stretching a sheet produces the nanofluid flow; the fluid flow field is unaffected by pressure gradients. Table 1 shows the thermo-physical characteristics of nanofluids and base fluids. In addition, it is considered that nanoparticles are uniform in size and shape. Nanofluid and nanoparticle phases are also taken into account in thermal equilibrium with stretching velocity U = . br 1 − α t , where b and α are dimensional constants. U ( r , t ) reflects that fixed elastic sheet at origin when radially stretched with force in the direction of positive r and effective stretching rate. b 1 − α t increase with time as 0 ≤ α ≤ 1 . The temperature profile on the radially stretching sheet is formulated as T s = T 0 − T r b r 2 2 ν f ( 1 − α t ) − 3 2 , where T 0 , T r , and ν f are ambient, reference temperature, and kinematic viscosity of the base fluid, respectively. Magnetic field B ( t ) = B 0 ( 1 − α t ) 1 2 is applied in z direction (Table 2).

Table 1

Thermo-physical characteristics of copper (Cu) nanoparticles and base fluids [37]

Base fluid/Nanoparticles	Specific heat C p ( J / kg K )	Thermal conductivity K ( W / mK )	Density ρ ( kg / m 3 )
Water ( H 2 O )	4,179	0.613	997.1
Silicone oil	1,966	0.1	818
Copper (Cu)	385	401	8,933

Table 2

Viscosity and structure factor parameters for nanoparticles [38,39]

Nanoparticles\Parameters	A 1	A 2	mm
Blade	14.6	123.3	8.26
Cylinder	13.5	904.4	4.82
Platelets	37.1	612.6	5.72

Under the aforementioned assumptions, governing equations can be written as [40] follows:

(1) ∂ u ∂ r + . u r + ∂ w ∂ z = 0 ,

(2) ∂ u ∂ t + u ∂ u ∂ r + w ∂ u ∂ z = μ nf ρ nf ∂ 2 u ∂ z 2 − σ nf ρ nf B 2 ( t ) u ,

(3) ∂ T ∂ t + u ∂ T ∂ r + w ∂ T ∂ z = α nf ∂ 2 T ∂ z 2 + μ nf ( ρ C p ) nf ∂ u ∂ z 2 .

where u and w are the velocity components along the radial r and axial z directions, respectively, and T denotes the nanofluid temperature.

The physical properties [39,41] are as follows:

density: ρ nf = ( 1 − ϕ ) ρ f + ϕ ρ s ,

thermal diffusivity: α nf = K nf ( ρ C p ) nf ,

electrical conductivity: σ nf = σ f ( 1 − ϕ ) σ f + ϕ σ s ,

dynamic viscosity: μ nf = μ f ( 1 + A 1 ϕ + A 2 ϕ 2 ) , and

thermal conductivity: k nf k f = K s + ( m − 1 ) k f + ( m – 1 ) ( k s − k f ) ϕ k s + ( m – 1 ) k f − ( k s − k f ) ϕ , where φ , K nf , ( ρ C p ) nf , A ₁ , and A ₂ are the volume fraction, thermal conductivity, heat capacitance, and viscosity enhancement coefficients of the nanofluid, whereas K and nm are thermal conductivity and shape factor of nanoparticles, respectively.

For a given system, the associated boundary conditions [40] of differential equations are

u = U + K ∂ u ∂ z , v = 0 , T = T s , at y = 0 , u → 0 , T → T ∞ , as y → ∞ .

To reduce the system dimensionless, the following conversion [40] was implemented:

(4) ψ = r 2 U Re − 1 2 f ( η ) , θ ( η ) = T 0 − T T r b r 2 2 ν f ( 1 − α t ) 3 2 , η = z r Re 1 2 .

Independent variable is η , ψ is the Stokes theorem function, and Re = rU ν f is the local Reynold number, and u = − 1 r ∂ ψ ∂ z , w = 1 r ∂ ψ ∂ r ; therefore, we can determine the velocity components as follows:

(5) u = U f ′ ( η ) and w = − 2 U Re − 1 2 f ( η ) .

Using a set of similarities found earlier, Eqs. (2) and (3) are turned into a set of ordinary differential equations with boundary conditions.

(6) f ‴ − ϵ 3 ϵ 1 M f ′ − 1 ϵ 1 S f ′ + η 2 f ″ − 1 ϵ 1 ( f ′ ) 2 + 2 ϵ 1 ff ″ = 0 ,

(7) θ ″ + P r ϵ 2 E c ϵ 1 ( f ″ ) 2 − P r ϵ 2 S 2 ( 3 θ + η θ ′ ) − 2 f ′ θ + 2 f θ ′ = 0 .

Subject to the boundary conditions

(8) f ( 0 ) = 0 , f ′ ( 0 ) = 1 + Kf ″ ( 0 ) , f ′ ( ∞ ) = 0 , θ ( 0 ) = 1 , θ ( ∞ ) = 0 ,

where ϵ 1 , ϵ 2 , and ϵ 3 are three constants that are defined as follows:

ϵ 1 = 1 + A 1 ϕ + A 2 ϕ 2 1 – ϕ + ϕ ρ s ρ f , ϵ 2 = k nf k f 1 – ϕ + ϕ ( ρ C p ) s ( ρ C p ) f , ϵ 3 = 1 − ϕ + ϕ σ s σ f 1 – ϕ + ϕ ρ s ρ f .

M = σ f B 0 2 r ρ f . U , Pr = ν f . ( ρ . C p ) f k f , Ec = U 2 C p . ( T w . − . T 0 ) , S = α c , K = A ν f U w x .

Since shear stress and heat transfer rate can be defined as follows:

C fr = τ w ρ . ( u w ) 2 , Nu r = r q w ( r ) k f . [ T f − T ∞ ] ,

where τ w = μ nf . ∂ u ∂ z z = 0 and q w ( r ) = − k nf . ∂ T ∂ z z = 0 are shear at wall and the wall heat flux. After using the transformations defined above, we have the following final form:

(9) C f Re 1 2 = ( 1 + A 1 φ + A 2 φ 2 ) f ″ ( 0 ) ,

(10) Nu Re − 1 2 = − K nf K f θ ′ ( 0 ) .

M is the magnetic parameter, S is the unsteadiness parameter, Pr is the Prandtl number, K is the slip parameter and Ec is the Eckert number. Eqs. (9) and (10) represent the skin friction coefficient and the heat transfer coefficient, respectively, also known as the Nusselt number.

3 Numerical solution

The solution is found using the boundary value problem-fourth-order (BVP4C) method [42] approach in this article. The proposed BVP4C method is widely recognized for its important characteristics, such as handling single boundary value problems (BVPs) and a faster convergence with lower error. The basic strategy of BVP4C is a popular Simpson’s method that can be found in several scripts.

The systems of nonlinear differential Eqs. (6) and (7) will be solved numerically using the proper method BVP4C, based on the boundary conditions (8). To use this method, Eqs. (6) and (7) have been changed into first-order differential equations.

Let

f = y ( 1 ) , y ( 1 ) ′ = y ( 2 ) , y ( 2 ) ′ = y ( 3 ) , θ = y ( 4 ) , y ′ ( 4 ) = y ( 5 ) .

Then, the differential equation and corresponding boundary equations can be written as follows:

(11) y ( 3 ) ′ = 1 ϵ 1 ( ϵ 3 My ( 2 ) ) + 1 ϵ 1 S y ( 2 ) + η 2 y ( 3 ) + ( y ( 2 ) ) 2 − 2 y ( 1 ) y ( 3 ) , y ( 5 ) ′ = P r ϵ 2 S 2 ( 3 y ( 4 ) + η y ( 5 ) ) + 2 y ( 2 ) y ( 4 ) − 2 y ( 1 ) y ( 5 ) − E c ϵ 1 ( y ( 3 ) ) 2 ,

(12) y ( 1 ) ( 0 ) = 0 , y ( 2 ) ( 0 ) = 1 + K y ( 3 ) ( 0 ) , y ( 2 ) ( ∞ ) = 0 , y ( 4 ) ( 0 ) = 1 , y ( 4 ) ( ∞ ) = 0 .

4 Results and discussion

On an unsteady, radially stretched sheet, we examined the axisymmetric flow of diverse forms of nanoparticles, such as blades, cylinders, and platelets of copper (Cu) nanoparticles. The dimensionless mathematical model is numerically solved using the BVP4C approach. A special focus is placed on influential traits, including the solid volume percentage, Prandtl number, partial slip condition, unsteadiness parameter, and Eckert number. In this section, we will look at how these characteristics affect temperature profiles using plotted figures and tables of various parameters, including K , S , Pr , ϕ , and Ec . The effects of the unsteadiness parameter S on the temperature profile are illustrated in Figure 2a–c. It describes that the temperature profile is a decreasing function of the unsteadiness parameter S in the boundary layer for both base fluids (water and silicone). As the unsteadiness parameter increases, the velocity of the stretching sheet also decreases, so it transmits a lesser amount of heat and mass from the sheet to the nanofluid in the boundary layer region. Figure 3a–c explains the influence of Pr on temperature profile for different shapes of nanoparticles. It can be observed from the figure that the temperature profile decreases with escalating values of Pr for both base fluids, water and silicone oil. The viscosity of nanofluids increases due to the Prandtl numbers, which reduces the temperature distribution. So, we can observe that Pr numbers are capable of enhancing the rate of cooling. Figure 4a–c indicates the influence of ϕ on the temperature profile for different shapes of nanoparticles. As can be seen in the graph, the temperature field grows as the value of ϕ . Escalating values of volume fraction cause a rise in the thermal conductivity, which increases the boundary layer thickness and causes an escalation in the temperature profile.

$Figure 2 (a) Blade shape: influence of S S on θ ( η ) . \theta (\eta ). (b) Cylinder shape: influence of S S on θ ( η ) . \theta (\eta ). (c) Platelets shape: influence of S S on θ ( η ) . \theta (\eta ).$

Figure 2

(a) Blade shape: influence of S on θ ( η ) . (b) Cylinder shape: influence of S on θ ( η ) . (c) Platelets shape: influence of S on θ ( η ) .

$Figure 3 (a) Blade shape: influence of Pr \Pr on θ ( η ) . \theta (\eta ). (b) Cylinder shape: influence of Pr \Pr on θ ( η ) . \theta (\eta ). (c) Platelets shape: influence of Pr \Pr on θ ( η ) . \theta (\eta ).$

Figure 3

(a) Blade shape: influence of Pr on θ ( η ) . (b) Cylinder shape: influence of Pr on θ ( η ) . (c) Platelets shape: influence of Pr on θ ( η ) .

$Figure 4 (a) Blade shape: influence of ϕ \phi on θ ( η ) . \theta (\eta ). (b) Cylinder shape: influence of ϕ \phi on θ ( η ) . \theta (\eta ). (c) Platelets shape: influence of ϕ \phi on θ ( η ) . \theta (\eta ).$

Figure 4

(a) Blade shape: influence of ϕ on θ ( η ) . (b) Cylinder shape: influence of ϕ on θ ( η ) . (c) Platelets shape: influence of ϕ on θ ( η ) .

Figure 5a–c shows that as the partial slip parameter K is increased, the temperature distribution decreases. It is easy to see how the slip parameter influences the profile as it moves away from the stretched sheet and toward the center. The temperature profile variation for several types of copper nanoparticles is shown in Figure 6. Temperature for both base fluid water and silicone oil exhibits an ascending order for all configurations (blade, cylinder, and platelet). The values of the numerous parameters that have been provided lead to the conclusion that platelets composed of copper nanoparticles have a notably increased heat transfer rate. However, the form of the blade indicates a comparatively lower capacity to transfer heat.

$Figure 5 (a) Blade shape: impact of K K on θ ( η ) . \theta (\eta ). (b) Cylinder shape: impact of K K on θ ( η ) . \theta (\eta ). (c) Platelets shape: impact of K K on θ ( η ) . \theta (\eta ).$

Figure 5

(a) Blade shape: impact of K on θ ( η ) . (b) Cylinder shape: impact of K on θ ( η ) . (c) Platelets shape: impact of K on θ ( η ) .

Figure 6

Temperature profile fluctuation for platelets, cylinder, and blade shape.

Figure 7a and b describes the effect of the unsteadiness parameter on skin friction and the Nusselt number. We observe that in skin friction, for increasing values of S , both base fluids effects are close enough for all shapes. But for low values of S , there is a difference between silicone oil and water. On the other hand, for Nusselt number blade shape, nanoparticle show a quick increase for increasing values of S . Figure 8a and b describes the effect of ϕ on skin friction and the Nusselt number for different shapes of Cu nanoparticles. We can observe that for increasing values of ϕ , both skin friction and Nusselt number behave totally differently. In skin friction, silicone oil has much higher values than the base fluid, water, for all shapes of nanoparticles. For the Nusselt number, it is totally different, i.e., base fluid values are much higher than the silicone oil values for increasing values of ϕ .

Figure 7

(a) Effects of S on skin friction coefficient. (b) Effects of S on Nusselt number.

$Figure 8 (a) Effects of ϕ \phi on skin friction coefficient. (b) Effects of ϕ \phi on skin friction coefficient.$

Figure 8

(a) Effects of ϕ on skin friction coefficient. (b) Effects of ϕ on skin friction coefficient.

Table 3 shows the dimensionless shear stress at the sheet computed for multishape Cu nanoparticles. It can be shown that as the unsteadiness parameter, magnetic parameter ( M ) grows, the skin friction coefficient also increases, whereas the skin friction coefficient decreases with the increasing values of slip parameter.

Table 3

Skin friction coefficient values for different shapes of nanoparticles

Physical parameters			Silicone oil			Water
S	K	M	Blade	Cylinder	Platelets	Blade	Cylinder	Platelets
0.8		1	3.02889	4.32004	5.17398	1.27201	1.64396	1.8681
1			3.02889	4.32005	5.17399	1.29324	1.67272	1.9015
1.2			3.02889	4.32005	5.17399	1.31363	1.70042	1.93373
	0.5	1	3.02889	4.32004	5.17398	1.27201	1.64396	1.8681
	1		1.53151	2.18909	2.6251	0.88447	1.17509	1.35422
	1.5		1.02486	1.46597	1.7587	0.68176	0.91957	1.06815
		1	3.02889	4.32005	5.17399	1.29324	1.67272	1.9015
		3	3.05769	4.36909	5.23825	1.51376	1.97697	2.2581
		5	3.06667	4.38442	5.25837	1.65602	2.17803	2.49652

In Table 4, heat transfer rate (Nusselt number) is calculated at the stretching sheet for different shapes of copper nanoparticles. It can be observed that the rise in Prandtl ( Pr ) and unsteadiness parameter S results in the increase of Nusselt number. Nusselt number decreases by escalating values of Eckert number ( Ec ) and volume friction ϕ (Table 5).

Table 4

Values of heat transfer coefficient for different shapes of nanoparticles

Physical parameters				Silicone oil			Water
S	ϕ	Pr	Ec	Blade	Cylinder	Platelets	Blade	Cylinder	Platelets
0.8				2.74095	2.5658	2.20483	3.86721	3.72805	2.60181
1.0				3.07347	2.88405	2.52778	4.07597	3.92202	2.82254
1.2				3.37408	3.17179	2.81965	4.27599	4.10843	3.031
	0.01			2.49677	2.44703	2.64957	4.13098	3.88522	3.57374
	0.03			2.49677	2.44703	2.43191	3.59584	3.54533	3.57374
	0.05			2.74095	2.5658	2.55784	3.86721	3.72805	3.77704
		4		2.24917	2.11406	3.04224	2.11351	3.04224	3.08605
		6		2.74095	2.5658	3.72805	2.55784	3.72805	3.77704
		8		3.15169	2.94007	4.30219	2.92384	4.30219	4.35533
			0.5	2.74095	2.5658	2.55784	3.86721	3.72805	3.77704
			1	2.66127	2.43079	2.38133	3.43522	3.20214	3.18943
			1.5	2.58156	2.29577	2.20483	3.00323	2.67623	2.60181

Table 5

Comparison of skin friction values for different values of slip parameter with existing literature [43] and [44]

K	Homotopy analysis method [43]	Homotopy pertubation method [44]	Present result
0	1.173721	1.178511	1.174964
0.05	1.079949	1.080820	1.081199
0.1	1.001834	1.00308	1.003089
0.5	0.650528	0.645304	0.651768
1	0.462510	0.458333	0.463691

5 Conclusion

Because of its remarkable conductivity, copper is a perfect material for enhancing stretched sheets’ electrical performance, which is especially important for flexible electronics and sensor applications. In addition to increasing surface area, nanoparticles’ small size may also improve the material’s reactivity and catalytic qualities. Here, we employ a mixture of water and silicone oil impregnated with copper nanoparticles. We look at the heat transmission and axisymmetric flow of multi-shaped copper (Cu) nanoparticles across a fluctuating, radially expanding surface. The BVP4C is used for the numerical solution of the dimensional-free mathematical model.

The following are the results of the present articles:

Blade-shaped nanoparticles are found to transmit heat at maximum rates in both base fluid, silicone oil and water. Its skin friction coefficient is the lowest when compared to those of cylinders and platelets.
In both base fluid scenarios, a minimal heat transfer rate is reported for platelet-shaped nanoparticles. It has the highest skin friction coefficient.
Growing volume fraction values result in increased thermal conductivity, which thickens the boundary layer and raises the temperature profile.
In the boundary layer area, less heat, and mass are transferred from the stretching sheet to the nanofluid due to its velocity. The temperature drops as a result when the unsteadiness parameter increases.
Temperature decreases with the increasing values of S , Pr , Ec , and K .
As the unsteadiness parameter ( S ) and magnetic parameters ( M ) grow, the magnitude of skin friction increases, while for the slip parameter K , it decreases.

The incorporation of thin film flow for various nanoparticle forms and the Cattaneo–Christove theory to address the problem of unsteady heat transfer in viscous fluids will be the subjects of our latest study. Tri-hybrid nanoparticles will also be included in our upcoming task.

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

Funding information: This study is supported via funding from the Prince Sattam bin Abdulaziz University project number (PSAU/2023/R/1444).
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: All data generated or analysed during this study are included in this published article.

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Received: 2023-09-06

Revised: 2023-11-28

Accepted: 2024-01-12

Published Online: 2024-03-15

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

Articles in the same Issue

https://doi.org/10.1515/ntrev-2023-0200

Keywords for this article

copper nanoparticles; silicone oil; stretching sheet; shape factor; slip parameter

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