A numerical study on thin film flow and heat transfer enhancement for copper nanoparticles dispersed in ethylene glycol

Umer Hayat; Ramzan Ali; Shakil Shaiq; Azeem Shahzad

doi:10.1515/rams-2022-0320

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A numerical study on thin film flow and heat transfer enhancement for copper nanoparticles dispersed in ethylene glycol

Umer Hayat , Ramzan Ali , Shakil Shaiq and Azeem Shahzad

Published/Copyright: June 9, 2023

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From the journal REVIEWS ON ADVANCED MATERIALS SCIENCE Volume 62 Issue 1

Abstract

The current study examines thin film flow and heat transfer phenomena with some additional effects such as magnetohydrodynamic, viscous dissipation, and slip condition over unsteady radially stretching surfaces for various shapes of copper ( Cu ) nanoparticles dispersed in ethylene glycol ( EG ) . The effective thermal conductivity of a nanofluid made of Cu nanometer-sized particles distributed in EG is significantly higher than that of pure EG. Partial differential equations are transformed into ordinary differential equations using the proper transformations. An effective convergent technique (i.e., BVP4C) is used to compute the solutions of nonlinear systems. MATLAB software is used to perform the calculations. The effect of numerous emerging physical characteristics on temperature and velocity, such as unsteadiness parameter ( S ) , slip parameter ( K ) , Hartmann number ( M ) , solid volume fraction ( ϕ ) , and Eckert number ( EC ) is investigated and illustrated graphically. The physical quantities, such as the skin friction coefficient and the Nusselt number, are calculated, described, and displayed in tabular form. It is observed that blade-shaped Cu nanoparticles had the lowest surface drag, highest heat transfer rate, and minimum film thickness compared to the brick and cylinder-shaped nanoparticles. According to our detailed investigation blade-shaped Cu nanoparticle is the most suited solution for manufacturing unsteady radially stretching modules.

Keywords: thin film; copper nanoparticles; unsteady radial stretching surface; partial slip; shape factors; viscous dissipation

1 Introduction

A significant amount of thermal energy is required for many industrial activities, including the manufacturing of food and paper. As a result, industrialists, engineers, scientists, and researchers have had such a significant problem with the high quantity of heat required to produce these products. However, base fluids are incapable of providing the quantity of heat required for many manufacturing operations. Furthermore, some investigators have suggested replacing traditional base fluids with a new fluid type with better heat transmission capabilities. As an outcome, a new category of fluids described as nanofluids has arisen. Nanofluids are preferred in various manufacturing engineering and medical appliances due to their improved thermophysical properties. Microparticles of various metals, oxides, and carbon nanomaterials are combined with base fluids to form nanofluids. These particles’ thermal equilibrium is stable and saturable in common fluids. The molding and extrusion processes both require heating and cooling. The integration of nanoparticles is one aspect that has elevated heat transfer dynamics to new heights. Because big particles can settle down owing to gravity’s influence, the size of metal particles is critical. Thin films are used in coating procedures for aerospace, chemical production equipment, maritime vessel structures, and other heat exchangers, so understanding flow and heat transfer is vital. Sakiadis [1] pioneered the study of boundary layers across a continuous solid surface by assuming constant speed in his seminal paper from 1961. Many researchers have theoretically, numerically, and experimentally investigated the challenges of a stretching sheet in many physical situations. Because of its widespread application, research into heat and mass transfer via a thin film’s stretched surface has gotten a lot of interest. Crane [2] developed a flow pattern and found a precise solution to an expansion of a similar problem in 1970. In 2002, Erickson et al. [3] expanded on Sakiadis’ study by examining the effects of suction or blowing on the stretched surface. The study of heat transfer caused by stretching surfaces of thin film has recently attracted much interest due to its widespread use. Some nonlinear issues in mechanics lend themselves to a thorough numerical analysis. In the electrical sector, thin film fluids are used for resistors and capacitors. It is also employed in polymer suspension, plastic sheets, food waste disposal, cable and fiber layers, exchangers, and many more applications. Thin liquid film applications comprise liquid sensor technology, diffusion barriers, and lens antireflection coating.

S. Maity [4] evaluated viscous dissipation and uniform film thickness of hydromagnetic flow of a thin nano-liquid film over an unsteady stretched sheet. Fakour et al. [5] explored nanofluid’s heat transfer and unsteady flow on a stretching surface. In 2019, Dinarvand and Nademi Rostami [6] published a comprehensive study on zinc oxide – gold hybrid nanofluid for swirling flow. Sadiq [7] examined heat transfer in thin film flow over an unstable stretch plate. In this study, the effect of altering viscosity and thermal conductivity on laminar flow and heat transfer in a liquid layer on a horizontal stretched sheet is demonstrated by Prasad et al. [8]. Khader [9] considered a fractional-order mathematical model and investigated the effects of various parameters. In 2020 Iqbal et al. [10] explored thin film flow over-stretching sheets. Thermophoresis characteristics and Brownian motion were used to define the properties of heat and mass transfer. Ananta Kumar et al. [11–13] investigated the physical characteristics of steady and unsteady flow in heat transfer. Stretching, porous media, coagulated sheet, flow past cone, and non-stretching are among the geometrical models under consideration. Shahzad et al. [14] analytically investigated unsteady magnetohydrodynamic (MHD) boundary layer flow in a radial sheet. Thin film flow is discussed by Gul et al. [15,16] for various geometries and shape factors. The impact of particle shape on nanofluid has been studied in several experimental and theoretical studies. The primary purpose of utilizing nanoparticles is to improve heat-transfer capability and thermal performance. Thermal conductivity and viscosity of several types of alumina ( Al ) nanoparticles were investigated [17] in a fluid comprising equal volumes of EG − H 2 O . The examination of experimental data was aided by theoretical modeling. Ganesh Kumar et al. [18] examined the effect of nanoparticles on the moving plate for different thermophysical parameters of a copper (Cu)–water nanofluid. Due to an exponential temperature, Chaudhary and Kanika [19] addressed numerical solutions for a continuous MHD boundary layer flow for various shapes of nanoparticles. Due to the flat surface, Hafeez et al. [20] conducted a comparative assessment of Cu nanoparticles on the nano-liquid flow for different shape factors. Turkyilmazoglu considered the flow and heat transmission characteristics of a revolving disc immersed in five distinct nanofluids in this study [12]. In 2016, Lin et al. [22] investigated the impact of radiation on heat transport and Marangoni flow for exponential temperature in a Cu–water nanofluid. A numerical analysis of nanofluid thermal capillary convection around a gas bubble was reported by Jiang et al. [23] in 2020.

The nanofluid flow was simulated using a mixture model and five different shapes of nanoparticles. According to this study [24], ethylene glycol ( EG ) has a lower effective thermal conductivity than a nanofluid made of Cu nanometer-size particles dispersed in base fluid with the same volume percent of dissimilated oxide nanoparticles. Shaiq et al. [25] examined how an induced magnetic field affected heat transfer and nanofluid stagnation point flow in this article. The ability to transfer heat in the presence of Cu and titanium dioxide particles was numerically investigated using EG as the base fluid. Shaiq et al. [25] focused on nanofluid movement in the presence of temperature-dependent viscosity. For comparison investigation, Si O 2 and Mo S 2 nanoparticles were suspended in propylene glycol. Heat transport in nanofluids and mixed nanofluids was also investigated. In this study, Hayat et al. [26] investigated the flow and heat transfer through a multi-shaped Ag-Nps surface subject to radial stretching. The results of this study indicate that the heat and flow transfer rates are highest and lowest for the platelet shape and sphere shapes, respectively.

In this article, Shaiq et al. [27] investigated the slip flow of H 2 O with tungsten ( W ) , Ti and tin ( Sn ) , and nanoparticles on a nonlinear stretching sheet. In this study, Khan and Mao [28] investigated the effects of thermophoresis, Brownian motion, and heat radiation on an incompressible steady flow of Oldroyd-B nanofluid near a plan porous stretched surface. In this study [29], the laminar flow of Ree-Eyring nanofluid is used to examine the heat and mass transport processes in the setting of motile bacteria. Mass transfer is examined when a chemical reaction is present, and convective conditions are applied to both the heat and the mass transfer. When these circumstances exist, differential transform method is used to solve the mathematical model. This study [30] investigates the hydro-magnetic flow of a hybrid nanofluid in two dimensions through a permeable stretching/shrinking sheet and also, the combined impact of heat generation, velocity slip, thermal radiations, and convective circumstances. In this article, Mehdi [32] provided flow and heat transfer study for laminar MHD nanofluid on two paralleled discs of indefinite length. In this article [31], primary focus was on the analysis of heat transfer and thin film flow over an asymmetric radial stretching surface. A mathematical model was proposed for the two-dimensional motion of the pair-stress gold nanoparticle nanofluid. Blade and cylinder-shaped nanoparticles have been found to have the highest heat transfer rates and lowest skin friction coefficients compared to brick-shaped particles.

The main objective of this study is to investigate MHD thin film flow and heat transfer enhancement over an unsteady radial stretching sheet of Cu − EG -based nanofluid, since metal nanoparticles play a vital role in many applications in industry. The integration of nanoparticles is one aspect that has elevated heat transfer dynamics to new heights. Thin films are used in coating procedures and heat exchanger equipment. Radial stretching sheets for various Cu nanoparticle shape parameters dispersed in EG have not yet been studied. So we consider here thin film flow and heat transfer effects of Cu nanoparticles of different shapes on radial stretching sheet using EG as base fluid.The hit-and-trial method calculates film thickness for fixed values of the unsteadiness parameter. Moreover, viscous dissipation and partial slip effects are also considered. The nonlinear governing system of partial differential equations converted into nonlinear ordinary differential equations (ODEs) by incorporating suitable transformation. Then, by using BVP 4 C system of ODEs is solved. Furthermore, we investigated the impact of physical parameters on velocity, temperature, skin friction, and Nusselt number (Tables 1 and 2).

Table 1

Characteristics of EG and Cu nanoparticles in view of refs [19,32,33]

Nanoparticles/base fluid	Specific heat C p ( J · kg − 1 · K − 1 )	Thermal conductivity K ( W· m − 1 · K − 1 )	Density ρ ( kg· m − 3 )	Electric conductivity σ ( S · m − 1 )
EG	2,430	0.253	1,115	3 . 14 × 10 6
Cu	385	401	8,933	5 . 96 × 10 7

Table 2

Values of shape factors ( m ) and viscosity of nanoparticles in view of refs [33,34]

Parameters/nanoparticles	Blade	Brick	Cylinder
A 1	14.6	1.9	13.5
A 2	123.3	471.4	904.4
m	8.26	3.72	4.82

2 Problem formulation

Consider a two-dimensional boundary layer thin film flow of EG -based nanofluid with different shapes of Cu nanoparticles (blade, brick, and cylinder) passing across a radially stretching sheet with a fixed center. Figure 1 shows a graphical depiction of the physical model under consideration. For mathematical modeling, cylindrical polar coordinates ( r , θ , z ) are taken. All the physical parameters are free from θ due to the flow’s rotational symmetry, and the velocity profile has the form V ̅ ( r , z ) = [ u ( r , z ) , 0 , w ( r , z ) ] . The nanofluid flow is caused by sheet stretching, and the pressure gradient does not affect the fluid flow field. Since we are considering a stretching sheet whose velocity is defined as U ( r , t ) = br 1 − α t , b and α are constants, and the temperature of the surface can be defined as T s = T 0 − T r b r 2 2 ν ( 1 − α t ) − 3 2 . h ( t ) is the film thickness parameter, and the magnetic field is chosen such that B o ( 1 − α t ) − 1 2 .

Figure 1

Graphical representation of the nanofluid flow on stretching sheet.

Under the above assumptions, the governing equations for momentum, mass, and energy, according to Ref. [14], can be written as

(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 .

The corresponding boundary conditions [14] of the differential equations for the given system are

(4) u = U + A ∂ u ∂ z , w = 0 , T = T s at z = 0 , ∂ u ∂ z = ∂ T ∂ z = 0 , w = d h d t as z → h .

Thermophysical parameters as given in ref. [34].

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

Thermal diffusivity : α nf = K nf ( ρ C p ) nf

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

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

(5) Dynamic viscosity : μ nf μ f = ( 1 + A 1 ϕ + A 2 ϕ 2 )

A 1 and A ₂ are the viscosity enhancement coefficients while k s , ( ρ C p ) nf , and m are the thermal conductivity, heat capacitance, and shape factor of nanoparticles, respectively.

To make the system dimensionless, we introduced the following dimensionless variables as defined in ref. [14]:

(6) Ψ = r 2 U R e − 1 2 f ( η ) , η = z r R e 1 2 , θ ( η ) = ( T 0 − T ) T ref b · r 2 2 v ( 1 − t ) 3 2 ,

where η is an independent variable, Re = rU v f is a local Reynolds number, and Ψ is the stream function.

(7) u = − 1 r ∂ ψ ∂ z , w = 1 r ∂ ψ ∂ r .

As a result, the velocity components are calculated as:

(8) u = U f ′ ( η ) and w = − 2 U R e − 1 2 f ( η ) .

By utilizing the relationship defined in Eqs. (6)–(8), Eq. (1) is identically satisfied, and Eqs. (2) and (3) , along with boundary conditions defined in Eq. (4) take the following form:

(9) f ‴ − 1 ξ 1 ξ 3 M f ′ − S · f ′ + η 2 f ″ − ( f ′ ) 2 + 2 f f ″ = 0 ,

(10) θ ″ + Pr ξ 2 · Ec · ξ 1 ( f ″ ) 2 − Pr ξ 2 S 2 ( 3 θ + η θ ′ ) + 2 f ′ θ − 2 f θ ′ = 0 .

Subject to the boundary conditions

(11) f ( 0 ) = 0 , f ′ ( 0 ) = 1 + K f ″ ( 0 ) , θ ( 0 ) = 1 f ( β ) = S β 2 , f ′ ( β ) = 0 , θ ′ ( β ) = 0 .

Also β = b ν ( 1 − α t ) 1 2 h is the dimensionless film thickness constant and can be obtained from the rate at which film thickness varies by differentiating h w . r . t time in the form of d h d t . In the above governing system of equations ξ 1 , ξ 2 , ξ 3 are three constants that are defined as

(12) ξ 1 = 1 + A 1 ϕ + A 2 ϕ 2 1 – ϕ + ϕ · ρ s ρ f , ξ 2 = k nf k f 1 − ϕ + ϕ · ( ρ C P ) s ( ρ C P ) f , ξ 3 = 1 − ϕ + ϕ · σ s σ f 1 – ϕ + ϕ · ρ s ρ f .

Furthermore, we have the following significant non-dimensional physical parameters, which are defined as

(13) 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 s − T 0 ) , S = α b , K = A ν f U w r .

Shear stress and heat transfer rate can be defined as:

(14) C f r = τ w ρ ( u w ) 2 , N u r = r q w ( r ) k f [ T f − T s ] ,

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:

(15) C f Re 1 2 = ( 1 + A 1 ϕ + A 2 ϕ 2 ) f ″ ( 0 ) , Nu Re − 1 2 = − k nf k f θ ′ ( 0 ) .

3 Methodology

In this article, BVP 4 C [35,37] approach is used to find the solution. The suggested approach is well-known due to its significant properties, including faster convergence with reduced error. The fundamental approach of BVP4C is a widely used Simpson’s method that occurs in different codes [38–40]. The systems of nonlinear differential Eqs. (9) and (10) with boundary conditions are solved using the suggested approach, i.e., BVP4C. In order to use the technique, third-order ordinary differential Eq. (9) and second-order ordinary differential Eq. (10) have been reduced to first-order differential equations.

(16) Let f = y ( 1 ) , f ′ = y ( 1 ) ′ = y ( 2 ) , f ″ = y ( 2 ) ′ = y ( 3 ) , f ‴ = y ( 3 ) ′ , θ = y ( 4 ) , θ ′ = y ( 4 ) ′ = y ( 5 ) .

Then equations can be written as

(17) f ‴ = y ( 3 ) ′ = 1 ξ 1 ( ξ 3 My ( 2 ) ) + 1 ξ 1 S y ( 2 ) + η 2 y ( 3 ) + ( y ( 2 ) ) 2 − 2 y ( 1 ) y ( 2 ) θ ″ = y ( 5 ) ′ = Pr ξ 2 S 2 ( 3 y ( 4 ) + η y ( 5 ) ) + 2 y ( 2 ) y ( 4 ) − 2 y ( 1 ) y ( 5 ) − Ec ξ 1 ( y ( 3 ) ) 2 .

Boundary conditions are

(18) y ( 1 ) ( 0 ) = 0 , y ( 2 ) ( 0 ) = 1 + K * y ( 3 ) ( 0 ) , y ( 1 ) ( β ) = S β 2 , y ( 3 ) ( β ) = 0 , y ( 4 ) ( 0 ) = 1 , y ( 5 ) ( β ) = 0 .

4 Results and discussion

This section will discuss the impact of significant physical quantities on the velocity field and temperature profile. Furthermore, the effects of different parameters on skin friction and Nusselt number are tabulated in Tables 3 and 4, respectively.

Table 3

Numerical values of C f R e 1 2 for different shapes of nanoparticles

Physical parameters				C f R e 1 2
S	M	K	ϕ	Blade	Brick	Cylinder
0.8	1	0.5	0.04	−1.3806372	−1.4084744	−2.0028919
	2			−1.5250238	−1.5568293	−2.2410259
	3			−1.6270705	−1.6618743	−2.4146735
	1	0.5	0.04	−1.3806372	−1.4084744	−2.0028919
		1		−0.92976068	−0.95070175	−1.4098995
		1.5		−0.69853509	−0.71513246	−1.0850528
	1	0.5	0.03	−1.1466836	−0.83213919	−0.97327714
			0.05	−1.5433607	−1.7870293	−2.8305737
			0.07	−1.9259177	−2.5183941	−4.0025894
1.2	1	1	0.04	−1.2979466	−1.3243533	−1.8887102
	2			−1.4064652	−1.4361517	−2.076237
	3			−1.4830092	−1.5151536	−2.2125964
	1	0.5	0.04	−1.2979466	−1.3243533	−1.8887102
		1		−0.85696339	−0.87636182	−1.3025482
		1.5		−0.6378889	−0.65305838	−0.9916653
	1	0.5	0.03	−1.1466836	−0.83213919	−0.97327714
			0.05	−1.5433607	−1.7870293	−2.8305737
			0.07	−1.9259177	−2.5183941	−4.0025894

Table 4

Numerical values of heat transfer coefficient for different shapes of nanoparticles

Physical parameters						NuR e 1 2
S	M	K	ϕ	Pr	Ec	Blade	Brick	Cylinder
0.8	1	0.5	0.04	203.63	1	19.701875	19.628003	16.801244
	2					17.527383	16.04193	14.238168
	3					16.022229	14.577435	12.049867
	1	0.5	0.04		1	19.701875	19.628003	16.801244
		1		203.63		22.048962	20.386566	20.118935
		1.5				22.633993	20.961003	21.078421
	1	0.5	0.03		1	19.332128	22.276709	18.89272
			0.05	203.63		19.028968	17.666435	16.696626
			0.07			19.702386	17.23969	15.896321
	1	0.5	0.04	10	1	5.196157	4.9735616	4.2350976
				100		15.477932	13.499045	12.378161
				300		26.097064	22.739536	20.749328
	1	0.5	0.04		0.5	25.016914	24.27786	21.787183
				203.63	1	21.687454	18.900592	17.271322
					1.5	18.357994	15.311054	12.322179
1.2	1	1	0.04		1	24.424599	24.366068	21.476829
	2			203.63		23.187061	21.335286	20.109157
	3					22.382342	20.540808	18.807287
	1	0.5	0.04		1	24.424599	24.366068	21.476829
		1		203.63		26.325622	24.366468	24.376789
		1.5				26.747948	24.78526	25.107634
	1	0.5	0.03		1	23.795362	28.5327	25.153507
			0.05	203.63		23.354382	21.560441	20.602761
			0.07			24.526786	21.496232	20.413397
	1	0.5	0.04	10	1	5.6281952	5.4235312	4.6624117
				100		16.815745	14.866192	13.875806
				300		28.502749	25.199329	23.441638
	1	0.5	0.04	203.63	0.5	25.504757	26.36869	23.804501
					1	23.644045	20.901922	19.463202
					1.5	20.297327	17.370245	14.650392

4.1 Velocity profile

We will look at how various parameters affect the velocity. For various shapes of Cu nanoparticles dispersed in the base fluid, i.e., EG range of different physical parameters are 0 ≤ k ≤ 1.5 , 1 ≤ M ≤ 3 , 0.5 ≤ Ec ≤ 1.5 , ϕ = 0.03 , 0.05 , 0.07 , and Pr = 10 , 100 , 300 .

Figure 2(a) and (b) shows the influence of the slip parameter K on the velocity field for S = 0.8 and S = 1.2 , respectively. As we can observe that for different values of K , there will be different values of film thickness β . Fluid near the sheet no longer moves at the same speed as the stretching sheet when slip effects occur. As K grows, slip velocity increases but fluid velocity drops because the stretched sheet’s dragging can only be partially coupled to the fluid under the slip parameter. It should be noticed that the slip parameter significantly affects the velocity. Figure 3(a) and (b) shows the influence of the Hartmann number M on the velocity field for S = 0.8 and S = 1.2 , respectively. As we can detect for increasing values of M , there will be a decrease in film thickness ( β ) and velocity as well. The Lorentz force has a retarding impact, according to our observations. Thus, when M increases, the velocity profile decreases.

$Figure 2 (a) Influence of K K on ( f ′ ( η ) ) ({f}^{^{\prime} }(\eta )) for S = 0.8 S=0.8 and (b) influence of K K on ( f ′ ( η ) ) ({f}^{^{\prime} }(\eta )) for S = 1.2 . S=1.2.$

Figure 2

(a) Influence of K on ( f ′ ( η ) ) for S = 0.8 and (b) influence of K on ( f ′ ( η ) ) for S = 1.2 .

$Figure 3 (a) Impact of M M on ( f ′ ( η ) ) ({f}^{^{\prime} }(\eta )) for S = 0.8 S=0.8 and (b) effect of M M on ( f ′ ( η ) ) ({f}^{^{\prime} }(\eta )) for S = 1.2 . S=1.2.$

Figure 3

(a) Impact of M on ( f ′ ( η ) ) for S = 0.8 and (b) effect of M on ( f ′ ( η ) ) for S = 1.2 .

4.2 Temperature

Figures 4–7 are drawn to show the effects of different parameters such as slip parameter, Eckert number, volumetric friction, and Prandtl number on the temperature profile. The effect of slip parameter K on temperature for different shapes of Cu nanoparticles is shown in Figure 4(a) and (b). From these figures, it is perceived that the film thickness and temperature decrease with increasing values of the slip parameter.

$Figure 4 (a) Impact of ( K ) \left(K) on temperature for S = 0.8 S=0.8 and (b) influence of ( K ) \left(K) on temperature for S = 1.2 . S=1.2.$

Figure 4

(a) Impact of ( K ) on temperature for S = 0.8 and (b) influence of ( K ) on temperature for S = 1.2 .

$Figure 5 (a) Influence of Ec {\rm{Ec}} on temperature for S = 0.8 S=0.8 and (b) influence of Ec {\rm{Ec}} on temperature for S = 1.2 . S=1.2.$

Figure 5

(a) Influence of Ec on temperature for S = 0.8 and (b) influence of Ec on temperature for S = 1.2 .

$Figure 6 (a) Influence of ϕ \phi on temperature and (b) influence of ϕ \phi temperature.$

Figure 6

(a) Influence of ϕ on temperature and (b) influence of ϕ temperature.

$Figure 7 (a) Influence of Pr {\rm{\Pr }} on temperature for S = 0.8 S=0.8 and (b) influence of Pr {\rm{\Pr }} on temperature for S = 1.2 . S=1.2.$

Figure 7

(a) Influence of Pr on temperature for S = 0.8 and (b) influence of Pr on temperature for S = 1.2 .

Figure 5(a) and (b) demonstrates that temperature upsurges for increasing values of Ec . It is because the frictional heating mechanism causes the nanofluid particles to retain energy for rising values of Ec . The variation of volumetric friction ( ϕ ) on temperature profile and film thickness is illustrated in Figure 6(a) and (b). The temperature rises for increasing values of ϕ . A rise in volume fraction values causes thermal conductivity to increase, which thickens boundary layers and elevates temperatures. Figure 7(a) and (b) demonstrates how the temperature profile is decreased by rising the Prandtl number ( Pr ) . Physically, thermal diffusivity decreases for escalating values of the Prandtl number. Consequently, the fluid temperature declines due to this alteration in thermal diffusivity.

4.3 Film thickness ( β ) effects on velocity and temperature

We used an appropriate method to determine various values of β for all shape factors of nanoparticles. Nanoparticles with cylinder shapes alter their values more quickly than those with brick and blade shapes as can be observed in Figure 8(a) and (b). Figure 8(a) shows the change in velocity for different shapes of Cu nanoparticle. Figure 8(b) shows the temperature change for different Cu nanoparticle shapes. Blade, brick, and cylinder are the order in which the temperature profile rises. For fixed values of the various parameters, we may conclude that maximum and minimum temperature is noted for the cylinder shape and a blade shape, respectively. Moreover, for both velocity and temperature, the maximum and minimum values of film thickness β are observed for cylinder shape and blade shape nanoparticles of Cu.

$Figure 8 (a) Effect of β \beta on velocity and (b) effect of β \beta on temperature.$

Figure 8

(a) Effect of β on velocity and (b) effect of β on temperature.

4.4 Impact on skin friction and heat transfer coefficient

From Table 3, we can observe that the skin friction coefficient is decreasing for slip parameter K; on the other hand, it increases for ϕ and Hartmann number, M . From the tabulated representation of the Nusselt number, we can notice that the heat transfer coefficient (Nusselt number) increases with increasing values of K , ϕ , and Pr , and it decreases for increasing values of M and Ec .

4.5 Comparison

In this part, we compare our results with prior research in a limiting sense. Table 5 demonstrates how accurate and efficient the results are.

Table 5

Result in comparison of skin friction coefficient values when M = 0 , K = 0 , ϕ = 0 , Ec = 0

S	β	f ″ ( 0 )
S	β	[10]	[36]	[34]	Present
0.4	4.981455	−1.134098	−1.134	−1.134096	−1.134096
0.6	3.131710	−1.195128	−1.195	−1.195125	−1.195125
0.8	2.151990	−1.245805	−1.246	−1.245806	−1.245805
1.0	1.543617	−1.277769	−1.278	−1.277769	−1.277769
1.2	1.1227783	−1.279171	−1.279	−1.279172	−1.277901

5 Conclusion

In this article, we consider an unsteady radial stretching sheet, multi-shape Cu nanoparticles dispersed in EG base fluid. On the stretching sheet, we investigate the thin film flow of these nanoparticles. We used an appropriate method to determine various values of β for all shape factors of nanoparticles. Using a mathematical method BVP4C, a numerical solution is produced. The following are the key findings of this article:

Maximum and minimum film thickness were noted for cylinder and blade-shaped nanoparticles.
Blade-shaped Cu nanoparticles had the lowest surface drag and the highest heat transfer rate.
Cylinder shape nanoparticles alter their values more quickly than those with brick and blade shapes for different values of β , i.e., film thickness.
In comparison to brick and cylinder, a blade-shaped Cu nanoparticle is the most effective for the unstable radial sheet.
The magnitude value of the skin friction coefficient decreases for the slip parameter K ; on the other hand, it increases for ϕ and Hartmann number, M .
The heat transfer coefficient (Nusselt number) increases with increasing values of K , ϕ , and Pr , and it decreases for increasing values of M , Ec , and ϕ .

Acknowledgments

The authors acknowledge the anonymous reviewers for valuable comments.

Funding information: Open Access funding was provided by the Qatar National Library.
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.

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Received: 2023-03-04

Revised: 2023-04-11

Accepted: 2023-04-21

Published Online: 2023-06-09

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

Articles in the same Issue

https://doi.org/10.1515/rams-2022-0320

Keywords for this article

thin film; copper nanoparticles; unsteady radial stretching surface; partial slip; shape factors; viscous dissipation

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