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3D thin-film nanofluid flow with heat transfer on an inclined disc by using HWCM

  • Azzh Saad Alshehry , Humaira Yasmin EMAIL logo and Rasool Shah
Published/Copyright: October 25, 2023
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Open Physics
From the journal Open Physics Volume 21 Issue 1

Abstract

Momentum and heat transmission influence the coated physical characteristics of wire product. As a result, understanding the polymeric movement and heat mass distribution is crucial. An increase in thermal efficiency is necessary for the wire covering technology. So, the aim of this work is to investigate the influence of nanomaterials on the heat and mass transport processes in wire coating analyses. A thin film nanofluid is used to investigate heat and mass transfer in three dimensions over a rotating inclined disc. Both the suction and injection effects of nanofluids and the thermal radiation of their fluxes are taken into account. By employing similarity variables, the set of governing equations can be transformed into a differential equation system. The necessary differential equation system is solved using the Haar wavelet collocation method. Plots and observations of the velocity distribution, concentration, and thermal fields within the boundary layer across an inclining, steadily rotating plane are made. Flow characteristics change as a result of varying embedded factors such as S , Sc , N b , Pr , and thermophoretic parameters. Evidence suggests that as the number of rotation parameters grows, the thermal boundary layer weakens.

Keywords: Haar wavelet collocation method; thermal radiation; heat production/consumption; nanofluid; spinning disc; thin film flow

1 Introduction

In order to generate the nonlinear differential equation, a similarity transformation was applied to the original fluid flow model. A set of differential equation system is derived from these equations. The first-order differential equations have been converted into a set of fractional differential equations with the help of the Haar wavelet collocation method (HWCM) [1,2,3].

The removal of saturated vapor from a fluid condensate during cooling is an important step in the development of chemical and mechanical engineering. Many researchers have replicated this issue under varying situations. Nanoparticle deferrals in fluids exhibit substantial endowment enhancement at low nanoparticle concentrations. Nanofluids have been the subject of extensive study because of the critical importance of direct heat transfer enrichment in various manufacturing applications, transportation systems, and nuclear power plants [4,5]. It has also been detailed how nanofluid can be used as a “smart fluid” in which heat transfer can be controlled to increase or decrease as needed. This study’s overarching objective is to have a conversation about the myriad ways in which nanofluids are currently being and will be used in the future, with a focus on the enhanced heat transfer properties that can be regulated and the specific characteristics that these nanofluids keep [6,7]. Chemical and applied sciences rely heavily on solving the problem of liquid condensation from cool, saturated vapors. Several studies have examined this phenomenon under different frameworks.

Adding nanoparticles to regular fluids improves their thermal performance significantly. Bhatti et al. [8] studied the simultaneous effects of a changing magnetic field on Jeffrey nanofluid. Ellahi [9] considered the flow of a magnetohydrodynamic (MHD) non-Newtonian nanofluid whose viscosity changes with temperature. Hatami and Ganji [10] investigated nanofluid laminar flow between rotating discs with heat transfer, using the microchannel heat sink as a cooling medium and the least square method and porous media approach as cooling mechanisms, respectively. Throughout the extant literature, the common fluid is predominantly used as a low-thermal-conductivity base fluid. The outputs of these thermal systems are incredibly modest. Improving fluid thermal activity by dispersing microscopic particles called nanoparticles throughout the fluid. Sheikholeslami et al. [11] studied the fluid flow over an inclined plane.

There are numerous practical uses for studying time-dependent fluid flow in engineering and other scientific disciplines. Attia [12] studied flow on a disc using suction and injection. In their research, Bachok et al. [13] analyzed the fluid dynamics of a moving porous plate. With the use of nanofluids, they were able to improve heat transfer. Freidoonimehr et al. [14] analyzed numerically the streaming of a nanoliquid across porous expanding media. Makinde et al. [15] looked into the effect of varying viscosity on the streaming of a nanoliquid. Akbar et al. [16] analyzed the 2D stream of a nanoliquid by employing a magnetic field, and numerical results were derived by a shooting method. The MHD stream of a nanoliquid produced by a rotating disc was investigated by Ramzan et al. [17] under partial slip conditions. Recent studies [18,19,20,21,22,23] provide a comprehensive investigation of nanofluid streaming with various properties.

In the polymerization industry, wire coating is typically used as an insulating substance and protection against mechanical damage. This method involves pulling and submerging an exposed, warmed wire into the melting polymer [24,25]. During this procedure, the heated polymer is also extruded across a rolling wire. Because of its widespread application in science, technology, and engineering, thin-film flow research has recently come to the forefront. Non-viscous flow has real-world applications, such as in developing cables, wires, fibber coats, etc. Sandeep and Malvandi [26] investigated the thin-film fluid flow of non-Newtonian nanoliquids . According to Wang's [27] measurements, there is a variation in the thin-film fluid’s flow through the stretching sheet. Usha and Sridharan [28] analyzed unstable finite thin liquid as it moves past a straight surface. Liu and Andersson [29] discussed the film flow with heat on a surface. Aziz et al. [30] imagined fluid moving as a thin layer on a stretching sheet to generate internal heat. Tawade et al. [31] analyzed thin film thermal radiation and convection. A thorough investigation of thermal radiations can be further enhanced by referring to the comprehensive analysis provided in previous scholarly publications [32,33,34,35,36,37,38] and the diverse investigations encompassed within them. Anderssona et al. [39] pondered fluid film flow on a stretching sheet with heat transfer. Moreover, researchers [40,41,42,43] have considered the irregular motion of liquid films on a stretching surface to account for additional variations. Hatami et al. [44] used a steadily spinning disc to study the movement of nanofluids in three dimensions.

Considering the foregoing extensive discussion, this study seeks to examine the flow of nanofluid for cooling purposes in the wire surface coatings.

2 Mathematical formulation

Consider the flow of a nanofluid thin layer in three dimensions over a spinning disc that is deemed a wire. As can be seen in Figure 1, the rate at which the disc spins in its own plane, expressed as an angular velocity, is given by Ω . The slanted disc is at an inclination of degree β with respect to the horizontal plane. Here, h represents the nanofluid layer thickness and V w represents the spraying velocity. Since the liquid sheet is so thin in relation to the disc’s radius, the cumulative effect can be disregarded. As expected, the downward force of gravity (acceleration denoted by the symbol g ) is in effect. The film surface temperature is denoted by the letter T s , while the disc surface temperature is denoted by T d . A similar C s concentration is found on a film’s surface, while a C d concentration is found on the disc. By maintaining a fixed value for p ∘ at the film’s surface, the pressure is reduced to a simple function of the z-axis.

Figure 1 Schematic diagram.
Figure 1

Schematic diagram.

The basic flow equations are given as follows:

(1) ∇ . u ⇀ = 0 ,

(2) ρ u ∂ u ∂ x + v ∂ u ∂ y + w ∂ u ∂ z + ∂ u ∂ t = μ ∂ 2 u ∂ x 2 + ∂ 2 u ∂ y 2 + ∂ 2 u ∂ z 2 + ρ g sin β ,

(3) ρ u ∂ u ∂ x + v ∂ u ∂ y + w ∂ u ∂ z + ∂ v ∂ t = μ ∂ 2 v ∂ x 2 + ∂ 2 v ∂ y 2 + ∂ 2 v ∂ z 2 ,

(4) ρ u ∂ u ∂ x + v ∂ u ∂ y + w ∂ u ∂ z + ∂ w ∂ t = μ ∂ 2 w ∂ x 2 + ∂ 2 w ∂ y 2 + ∂ 2 w ∂ z 2 + ρ g cos β − p z ,

(5) [ uvw ] ∇ . T + ∂ T ∂ t = α ( ∇ 2 . T ) − ( ρ c p ) p ( ρ c p ) f × D b ∂ C ∂ x ∂ T ∂ x + ∂ C ∂ y ∂ T ∂ y + ∂ C ∂ z ∂ T ∂ z + Dt T ∂ T ∂ x 2 + ∂ T ∂ y 2 + ∂ T ∂ z 2 ,

(6) u ∂ C ∂ x + v ∂ C ∂ y + w ∂ C ∂ z + ∂ C ∂ t = D b ( ∇ 2 C ) + Dt T ∘ ∂ 2 T ∂ x 2 + ∂ 2 T ∂ y 2 + ∂ 2 T ∂ z 2 .

With boundary conditions,

(7) u = − Ω x , v = − Ω y , w = 0 , T = T s , C = C s , at z = 0 , u z = 0 , v z = 0 , w = − W , T = T ∘ , C = C ∘ , p = p ∘ , at z = h .

Let us consider the following non-dimensionalized parameters:

(8) u = − Ω x 1 − b t g + Ω x 1 − b t f ′ + g ( − bt + 1 ) − 1 2 k sin β Ω , v = − Ω y ( − bt + 1 ) − 1 2 g + Ω y ( − bt + 1 ) − 1 2 f ′ + ( − bt + 1 ) − 1 2 gh sin β Ω ,

(9) w = − 2 Ω υ f 1 − b t f , η θ = T − T d T ∘ − T d , η ϕ = C − C d C ∘ − C d , η = z Ω υ f ( 1 − b t ) .

With the help of the above defined parameters, one can obtain the governing non-dimensionalized equations:

(10) D 3 f − ( Df ) 2 + g 2 − 2 f D 2 f − S Df + η D 2 f 2 = 0 ,

(11) D 2 K − KD ( f ) − h ( η ) g ( η ) + 2 fD ( K ) + 1 − S ( K + η D ( K ) ) 2 = 0 ,

(12) D 2 g − 2 gD ( f ) + 2 D ( g ) f − S D ( g ) + η D ( g ) 2 = 0 ,

(13) D 2 h − K ( η ) g ( η ) − hD ( f ) + 2 fD ( h ) − S ( h − η D ( h ) ) 2 = 0 ,

D 2 θ + 2 Pr fD ( θ ) + N b θ ′ ϕ ′ + N t ( θ ′ ) 2 + S ( η θ ′ + η 2 θ '' ) 2 = 0 , ( 14 )

(15) D 2 ( ϕ ) + 2 ScfD ( ϕ ) + N t N b θ '' + S ( η ϕ ′ + η 2 ϕ '' ) 2 = 0 .

With boundary conditions,

(16) f ( 0 ) = Df ( 0 ) = 0 , D 2 f δ = g ( 0 ) = 0 , D g δ = K ( 0 ) = D ( K ) δ = 0 , h ( 0 ) = θ ( 0 ) = D ( h ) δ = ϕ ( 0 ) = 0 , θ ( δ ) = ϕ ( δ ) = 1 ,

where the parameters Pr , Sc , N b , N t , and S are defined as follows:

(17) Pr = μ ρ f , Sc = μ ρ f D , N b = ( ρ c ) p D b C h ( ρ c ) f , N t = ( ρ c ) p D t T h ( ρ c ) f T s , S = α Ω ,

(18) δ = ε Ω υ f ( 1 − bt ) , f ( δ ) = w 2 Ω υ , θ ′ = 1 δ .

The equation δ = ε Ω υ f ( 1 − b t ) yields the constant normalized thickness. The phenomenon under consideration is understood by means of the condensation or spraying velocity, denoted as f ( δ ) = w 2 υ Ω = α . The pressure can be determined using the process of integrating Eq. (4). For Pr = 0 and θ ( δ ) = 1 , the exact solution is represented by

(19) θ ' ( 0 ) = 1 δ .

The equation δ = ε Ω υ f ( 1 − b t ) represents the asymptotic limit for a small ( δ ) . The non-monotonic nature of the decline of θ ' ( 0 ) with increasing ( δ ) can be observed from the oscillatory behavior of the curves for large ( Pr ) :

(20) N u = ∂ T ∂ z T o − T d = δ θ ′ ( 0 ) .

In the same way, the Sherwood number is defined as follows:

(21) S h = ∂ C ∂ z C o − C d = δ θ ′ ( 0 ) .

3 Methodology

Assume that A and B are constants, then for x ∈ [ A , B ] , the definition of the i th Haar wavelet family is

(22) H i ( x ) = 1 for x ∈ [ a 1 , b 1 ) − 1 for x ∈ [ b 1 , c 1 ) 0 elsewhere ,

(23) a 1 = k m 1 , b 1 = k + 1 2 m 1 , c 1 = k + 1 m 1 ,

where m 1 = 2 j is the highest attainable resolution, we will refer to this quantity as M = 2 j , and each of the 2 M subintervals of the interval [ A , B ] has length x = ( B , A ) / 2 M , where M is the number of subintervals ( 2 M ) . A translation parameter k = 0,1 , … , m 1 − 1 and a dilatation parameter j = 0,1 , … , J . The formula for the wavelet number i is i = m 1 + k + 1 .

For the Haar function and its integrals

(24) p i 1 ( x ) = ∫ 0 x H i ( x ′ ) ,

(25) p i , l + 1 ( x ) = ∫ 0 x p i , l ( x ′ ) d x ′ , l = 1 , 2 …

In order to calculate these integrals, we use Eq. (22)

(26) p i 1 ( x ) = x − a 1 c 1 − x 0 for x ∈ [ a 1 , b 1 ) for x ∈ [ b 1 , c 1 ) elsewhere ,

(27) p i 2 ( x ) = 1 2 ( x − a 1 ) 2 for x ∈ [ a 1 , b 1 ) 1 4 m 1 2 − 1 2 ( c 1 − x ) 2 for x ∈ [ b 1 , c 1 ) 1 4 m 1 2 for x ∈ [ c 1 , 1 ) 0 elsewhere ,

(28) p i 3 ( x ) = 1 6 ( x − a 1 ) 3 for x ∈ [ a 1 , b 1 ) 1 4 m 1 2 ( x − b 1 ) − 1 6 ( c 1 − x ) 3 for x ∈ [ b 1 , c 1 ) 1 4 m 1 2 ( x − β ) for x ∈ [ c 1 , 1 ) 0 elsewhere .

We also present the notation shown below:

(29) C i 1 = ∫ 0 L p i 1 ( x ′ ) d x ′ ,

(30) C i 2 = ∫ 0 L H i ( x ′ ) d x ′ .

Summation of the Haar wavelet function can be written as follows:

(31) f ( x ) = ∑ i = 1 ∞ a i H i ( x ) .

Using wavelets, we may approximate the highest order derivatives of f , θ , g , h , K , and ϕ given by problems (10)–(14) and (15) to build a straightforward and precise HWCM

(32) f ''' ( η ) = ∑ i = 1 2 M a i H i ( η ) ,

(33) g '' ( η ) = ∑ i = 1 2 M b i H i ( η ) ,

(34) θ '' ( η ) = ∑ i = 1 2 M c i H i ( η ) ,

(35) ϕ '' ( η ) = ∑ i = 1 2 M d i H i ( η ) ,

(36) h '' ( η ) = ∑ i = 1 2 M e i H i ( η ) ,

(37) K '' ( η ) = ∑ i = 1 2 M f i H i ( η ) .

Integrating Eq. (32), we obtain the values f ''' ( η ) , f '' ( η ) , f ′ ( η ) , and f ( η ) , respectively,

(38) f ''' ( η ) = ∑ i = 1 2 M a i p i , 1 ( η ) − 1 L C i , 1 ,

(39) f '' ( η ) = ∑ i = 1 2 M a i p i , 2 ( η ) − 1 L η C i , 1 ,

(40) f ′ ( η ) = ∑ i = 1 2 M a i p i , 3 ( η ) − 1 L η 2 2 C i , 1 ,

(41) f ( η ) = ∑ i = 1 2 M a i p i , 4 ( η ) − 1 L η 3 3 C i , 1 .

Assume L is a large number. The numerical solution for the given system is obtained by substituting Eqs (32)–(41) into Eqs (10)–(15). In the next section, we see a visual representation of the results of the proposed solution.

4 Results and discussion

This study investigates fluid flow characteristics in thin-film nanofluids within a three-dimensional framework. Specifically, we examine the behavior of these nanofluids across an inclined and rotating surface, with a particular emphasis on the associated heat and mass transfer processes. The focus of our work has been to analyze the impact of different embedded characteristics. The findings were achieved using the HWCM. Figure 1 depicts the schematic state of the flow. Variations in the value of the parameter ( S ) and their effects on the axial velocity, radial velocity, drainage flow, and induced flow are depicted in Figures 2–5. Higher values of the parameter ( S ) signify more significant fluctuations in the fluid velocity. This effect is considerably obvious and it holds true for all of the aforementioned different types of fluid motion that emerge with greater values of ( S ) , as the thickness of the momentum barrier layer increases. This result is modified in Figure 6, where the temperature profiles continue to fall as ( S ) increases. In reality, increasing the amount of S cools the system and slows down the rate at which heat is transferred from the sheet to the fluid. Molecular collisions within the fluid are slowed down physically but only slightly.

Figure 2 Variation of “S” on f(η) for N b = N t = Sc = 0.5, Pr = 6.7.
Figure 2

Variation of “S” on f(η) for N b = N t = Sc = 0.5, Pr = 6.7.

Figure 3 Variation of “S” on f'(η) for N b = N t = Sc = 0.5, Pr = 6.7.
Figure 3

Variation of “S” on f'(η) for N b = N t = Sc = 0.5, Pr = 6.7.

Figure 4 Variation of “S” on k(η) for N b = N t = Sc = 0.5, Pr = 6.7.
Figure 4

Variation of “S” on k(η) for N b = N t = Sc = 0.5, Pr = 6.7.

Figure 5 Variation of “S” on h(η) for N b = N t = Sc = 0.5, Pr = 6.7.
Figure 5

Variation of “S” on h(η) for N b = N t = Sc = 0.5, Pr = 6.7.

Figure 6 Variation of “S” on θ' for N b = N t = Sc = 0.5, Pr = 6.7.
Figure 6

Variation of “S” on θ' for N b = N t = Sc = 0.5, Pr = 6.7.

As illustrated in Figure 7, the concentration profile rises as the unsteadiness parameter ( S ) is increased. This is because a thicker momentum boundary layer results in a higher concentration profile. The point-wise rate of change inherent in this situation makes this effect more efficient and transparent.

Figure 7 Variation of “S” on ϕ'(η) for N b = N t = Sc = 0.5, Pr = 6.7.
Figure 7

Variation of “S” on ϕ'(η) for N b = N t = Sc = 0.5, Pr = 6.7.

As can be seen in Figure 8, the heat transfer rate improves with increasing values of ( N t ) and ( N b ) . As can be seen in Figure 9, the concentration rate rises as ( Sc ) (Schmidt number) is changed. Actually, the kinematic viscosity increases as ( Sc ) grows larger, and the Sherwood number decreases with the increase in concentration of the chemical species.

Figure 8 Variation of “N t, N b” on θ'(η) for S = Sc = 0.5, Pr = 6.7.
Figure 8

Variation of “N t, N b” on θ'(η) for S = Sc = 0.5, Pr = 6.7.

Figure 9 Variation of “Sc” on ϕ'(η) for S = N t = Np = 0.5, Pr = 6.7.
Figure 9

Variation of “Sc” on ϕ'(η) for S = N t = Np = 0.5, Pr = 6.7.

Figure 10 depicts the relationship between the Prandtl number ( Pr ) and the heat transfer rate. When the Prandtl number increases, the thermal boundary layer thins out, slowing the rate at which the system cools. In the initial stages, this impact takes on a quite distinct form. This is because there is a ceiling on the effect caused by the point-wise rate of change.

Figure 10 Variation of “P r” on θ'(η) for Sc = N t = N b = 0.5.
Figure 10

Variation of “P r” on θ'(η) for Sc = N t = N b = 0.5.

The numerical results of the HWCM are presented in Table 1. The study incorporates a comprehensive analysis utilizing numerical and analytical HAM calculations. The findings are shown in Tables 2–4, providing a complete comparison between the two methods. The data indicate a significant level of concurrence between the HAM and numerical results, suggesting a strong level of precision in our conclusions. The investigation demonstrates a minimal absolute error between our numerical results and the calculations based on the HAM. Therefore, it is apparent that our numerical computations are in complete agreement with the analytical HAM outcomes.

Table 1

Numerical solution of the HWC approach Sc = S = 0.5 , Pr = 6.7 , N t = N b = 0.5

η f ( η ) f ' ( η ) θ ' ( η ) k ( η ) ϕ ' ( η ) h ( η )
0 0.012222 0.002136 1.11918 0.001021 0.013215 0.092328
0.1 0.016141 0.0594279 1.30172 0.164447 0.059058 0.096225
0.2 0.219794 0.136644 1.41946 0.309963 0.109512 0.010135
0.3 0.249326 0.228574 1.56612 0.437877 1.152476 0.010774
0.4 0.280665 0.331357 1.93674 0.550058 1.188817 0.011539
0.5 0.290019 0.441102 2.27418 0.648589 1.219154 0.124272
0.6 0.323463 0.55441 2.35061 0.735506 1.243934 0.139431
0.7 0.357075 0.6686 3.57273 0.812652 1.263408 0.149434
0.8 0.370967 0.781725 3.92233 0.851608 1.277633 0.154355
0.9 0.395266 0.992468 4.01623 0.843696 1.286467 0.155058
1 0.402134 1.387695 4.05592 0.897643 1.589538 0.158441
Table 2

Analytical (HAM) and numerical approach (HWCM) to evaluating the function f ( η ) and f ' ( η ) are compared Sc = S = 0.5 , Pr = 6.7 , N t = N b = 0.5

( η ) HWCM calculations Ham calculations Absolute error
0.0 0.012222 0.002136 0.012223 0.002136 0.000001 4.873421 × 10−8
0.1 0.016141 0.059427 0.016141 0.059427 4.573421 × 10−8 5.743610 × 10−8
0.2 0.219794 0.136644 0.219794 0.136645 3.973421 × 10−7 −0.000001
0.3 0.249326 0.228574 0.249327 0.228574 0.000002 3.457235 × 10−8
0.4 0.280665 0.331357 0.280665 0.331289 4.172121 × 10−9 0.000068
0.5 0.290019 0.441102 0.290017 0.441102 0.000002 4.842367 × 10−6
0.6 0.323463 0.554410 0.323463 0.554395 6.873421 × 10−7 0.000015
0.7 0.357075 0.668656 0.357076 0.668653 0.000030 0.000026
0.8 0.370967 0.781725 0.370967 0.781725 2.871121 × 10−6 6.845295 × 10−8
0.9 0.395266 0.992468 0.395267 0.992466 0.000005 0.000002
1 0.402134 1.387695 0.402133 1.387694 0.000001 0.000135
Table 3

Analytical (HAM) and numerical approach (HWCM) to evaluating the function θ ' ( η ) and k ( η ) are compared Sc = S = 0.5 , Pr = 6.7 , N t = N b = 0.5

( η ) HWCM calculations Ham calculations Absolute error
0.0 1.11918 0.001021 1.11910 0.001021 0.000080 0.000000
0.1 1.30172 0.164447 1.30145 0.164442 0.000270 0.000005
0.2 1.41946 0.309963 1.41923 0.309961 0.000230 0.000002
0.3 1.56612 0.437877 1.56612 0.437871 2.172121 × 10−9 0.000006
0.4 1.93674 0.550058 1.93670 0.550058 0.000040 4.198120 × 10−5
0.5 2.27418 0.648589 2.27418 0.648586 4.172021 × 10−7 0.000003
0.6 2.35061 0.735506 2.35056 0.735500 0.000050 0.000006
0.7 3.57273 0.812652 3.57273 0.812651 6.170121 × 10−8 0.000001
0.8 3.92233 0.851608 3.92223 0.851603 0.000100 0.000005
0.9 4.01623 0.843696 4.01589 0.843689 0.000340 0.000007
1 4.05592 0.897643 4.05591 0.897638 0.000010 0.000005
Table 4

Analytical (HAM) and numerical approach (HWCM) to evaluating the function ϕ ( η ) and h ( η ) are compared Sc = S = 0.5 , Pr = 6.7 , N t = N b = 0.5

( η ) HWCM calculations Ham calculations Absolute error
0.0 0.013215 0.092328 0.013215 0.092328 3.173464 × 10−8 5.174321 × 10−8
0.1 0.059058 0.096225 0.059036 0.096219 0.000022 0.000007
0.2 0.109512 0.010135 0.109506 0.010130 0.000006 0.000005
0.3 1.152476 0.010774 1.152465 0.010773 0.000011 0.000002
0.4 1.188817 0.011539 1.188815 0.011537 0.000002 0.000003
0.5 1.219154 0.124272 1.219145 0.124270 0.000009 0.000002
0.6 1.243934 0.139431 1.243933 0.139430 0.000001 0.000001
0.7 1.263408 0.149434 1.263407 0.149428 0.000001 0.000006
0.8 1.277633 0.154355 1.277628 0.154346 0.000005 0.000010
0.9 1.286467 0.155058 1.286458 0.155045 0.000009 0.000013
1 1.589538 0.158441 1.589529 0.158389 0.000009 0.000053

5 Conclusion

This study focuses on examining the complexities of the dynamics of three-dimensional nanofluid spraying over a spinning inclined disc, which is similar to the wire utilized in practical surface coating applications. The following discussion summarizes the novel findings discovered in the current investigation.

As the unsteadiness parameter ( S ) increases, the temperature field decreases due to the increase of the momentum barrier layer's thickness.

An increase in the Schmidt number ( Sc ) leads to a decrease in the Sherwood number as a result of an increase in the kinematic viscosity and concentration of the chemical species.

With larger values of the Prandtl number ( Pr ) , the Nusselt number decreases. Although the initial effect is very different, as the value decreases, we notice an increase in boundary layer flow turbulence.

Increasing the proportion of nanofluids in the system, the temperature profile decreases, and the lowest temperature is reached.

The velocity profile declines for parameter S due to the internal collision of the small fluid particles.

Acknowledgments

Princess Nourah bint Abdulrahman University Researchers Supporting Project number (PNURSP2023R183), Princess Nourah bint Abdulrahman University, Riyadh, Saudi Arabia. Thiswork was supported by the Deanship of Scientific Research, the Vice Presidency for Graduate Studiesand Scientific Research, King Faisal University, Saudi Arabia (Grant No. 4383).

  1. Funding information: This work was supported by Princess Nourah bint Abdulrahman University Researchers SupportingProject number (PNURSP2023R183), Princess Nourah bint Abdulrahman University, Riyadh, SaudiArabia. This work was supported by the Deanship of Scientific Research, the Vice Presidency forGraduate Studies and Scientific Research, King Faisal University, Saudi Arabia (Grant No. 4383).

  2. Author contributions: All authors have accepted responsibility for the entire content of this manuscript and approved its submission.

  3. Conflict of interest: The authors state no conflict of interest.

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Received: 2023-08-02
Revised: 2023-09-23
Accepted: 2023-10-02
Published Online: 2023-10-25

© 2023 the author(s), published by De Gruyter

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

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https://doi.org/10.1515/phys-2023-0122
Keywords for this article
Haar wavelet collocation method; thermal radiation; heat production/consumption; nanofluid; spinning disc; thin film flow
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Articles in the same Issue

  1. Regular Articles
  2. Dynamic properties of the attachment oscillator arising in the nanophysics
  3. Parametric simulation of stagnation point flow of motile microorganism hybrid nanofluid across a circular cylinder with sinusoidal radius
  4. Fractal-fractional advection–diffusion–reaction equations by Ritz approximation approach
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