Comparative Numerical Study of Thermal Features Analysis between Oldroyd-B Copper and Molybdenum Disulfide Nanoparticles in Engine-Oil-Based Nanofluids Flow

Abstract

Apart from the Buongiorno model, no effort was ably accomplished in the literature to investigate the effect of nanomaterials on the Oldroyd-B fluid model caused by an extendable sheet. This article introduces an innovative idea regarding the enforcement of the Tiwari and Das fluid model on the Oldroyd-B fluid (OBF) model by considering engine oil as a conventional base fluid. Tiwari and Das’s model takes into account the volume fraction of nanoparticles for heat transport enhancement compared to the Buongiorno model that depends significantly on thermophoresis and Brownian diffusion impacts for heat transport analysis. In this paper, the thermal characteristics of an Oldroyd-B nanofluid are reported. Firstly, the transformation technique is applied on partial differential equations from boundary-layer formulas to produce nonlinear ordinary differential equations. Subsequently, the Keller-box numerical system is utilized to obtain final numerical solutions. Copper engine oil (Cu–EO) and molybdenum disulfide engine oil (MoS₂–EO) nanofluids are considered. From the whole numerical findings and under the same condition, the thermodynamic performance of MoS₂–EO nanofluid is higher than that of Cu–EO nanofluid. The thermal efficiency of Cu–EO over MoS₂–EO is observed between 1.9% and 43%. In addition, the role of the porous media parameter is to reduce the heat transport rate and to enhance the velocity variation. Finally, the impact of the numbers of Reynolds and Brinkman is to increase the entropy.

1. Introduction

In 1995, Choi [1] first presented the idea of nanofluid. Their main observation was that the thermal conductivity increased when a low rate of copper nanoparticles was immersed in water. Consequently, numerous mixtures of nanoparticles and liquids have been investigated for improving transport phenomena and thermophysical properties, such as thermal conductivity. Characteristic materials utilized for synthesizing nanofluids include metals, inorganic oxides, carbon-based resources, and other porcelains. Meanwhile, a popular selection of cleaners contains aquatic, ionic liquids, glycols, biological liquids, and refrigerants. Other fluids discovered in research works contain oils, biofluids, mixtures, fuels, alcohols, melted salt eutectics, etc. The preferred fluid properties are important for the selected process, to facilitate the optimal combination procedure for a specific nanofluid. Consequently, it is critical to develop a collection of mixture procedures for the nanofluids’ choice group, when subjected to objective properties and processes [2,3,4,5,6].

Another significant issue in nanofluids preparation is long-term stability. Therefore, the nanomanufacturing field was established, producing nanoparticles with various scopes, forms, and construction. Nonetheless, protecting nanoparticle suspension homogeneously in the base fluid for a sufficient long time is a challenging task. Therefore, many methods have been developed to grow the suspensions’ stability, and they have fundamentally produced short-range stability. However, no specific investigations have been conducted to confirm long-range stability. The strategic reason for keeping nanoparticle suspensions stable for extended periods of time is to avoid nanoparticle agglomeration. For instance, high thermal conductivity is desirable in heat transmission processes. At the same time, non-Newtonian rheological performance and load-carrying capacity are carefully selected in lubrication processes to vanish heat rapidly.

One significant consideration in a nanofluid composition is the cost and practicability of scaling up to a combination of outsized capacities. Unfortunately, such engineering considerations are usually ignored in critical investigations. Still, they are fundamental to the commercial achievement of nanofluids, if this technology were to enter conservative and novel manufacturing processes [7,8,9,10].

Accordingly, it is imperative to recognize, model, and maximize the combination environments to allow superior control over the transport instruments, such as similar or dissimilar nucleation of the nanoparticles compared to the originators or the development and monitoring of the accumulation of the nanoparticles generated. Recently, an increased number of studies on non-Newtonian fluids has been introduced by numerous researchers, because of their excellent reputation in many industrial and chemical activities, nuclear production, bioengineering, geophysics, and material processing. Therefore, the importance of non-Newtonian fluids is still puzzling, and thus, many non-Newtonian designs are being investigated to demonstrate their composite rheological perspective. Among these non-Newtonian fluids, the Oldroyd-B fluid (OBF) releases with synchronized relaxation and retardation times. Furthermore, OBF allows clarifying the creep and standard stress changes, which are related to simple shear flow. Nevertheless, the OBF does not define the shear condensing and shear withdrawing structures, unlike other polymeric materials. The OBF contains viscoelastic applications for Maxwell and viscous fluids for example. Based on such remarkable properties, the flow of OBF was thoroughly investigated in the literature [11,12,13,14,15,16,17,18]. Moreover, researchers have explained the effect of slip boundary conditions in nanofluids past the extending slip [19,20].

Entropy computes the amount of chaos in a system and its environment, which describes the impact of heat transfer on the alterations in kinetic energy and potential energy. The analysis of entropy generation can describe the heat losing progression of a fluid wave. Researchers have utilized the entropy generation analysis in fluid flow, aiming at two rotating disks moving with different rotation rates. It indicated that because of every thermal progression, the entropy measures the quantity of irreversibility. Therefore, conserving and warming are the most critical issues in numerous industrial fields of engineering investigations, which are utilized primarily on energy and electronic devices [21,22,23,24,25,26,27,28].

Based on the abovementioned literature, none of the published research works independently considered a Tiwari─Das nanofluid scheme [29] over the stretching sheet, together with the impacts of controlling parameters such as penetrable materials, variant thermal conductance, radiation heat flux, and slip conditions. Hence, the current study emphasizes the mentioned controlling parameters and entropy on an OBF using a specific numerical method, namely, the Keller-box method.

2. Mathematical Formulation

This analysis considers a steady, laminar two-dimensional heat transfer flow of the incompressible Oldroyd-B nanofluid through an extending sheet with first-order slip, convective endpoint conditions, and passive control of different nanoparticles. Equation (1) characterizes the moving flat horizontal surface with the nonuniform expanding velocity $U_w$, where $b$ is a positive constant. (Figure 1).

$$U_w\left(x,0\right)=bx$$

(1)

2.1. Model Equations

The basic time-independent governing equations of mass, momentum, and thermal energy of a viscous Oldroyd-B nanofluid flow employing customary boundary-layer approximations, subjected to a porous media, and variable thermal conductivity with radiation heat flux are [30]:

$$\frac{\partial v_1}{\partial x}+\frac{\partial v_2}{\partial y}=0,$$

(2)

$$\begin{array}{ll}{v}_1\frac{\partial v_1}{\partial x}+v_2\frac{\partial v_1}{\partial y}+& {\lambda }_1\left[v_1^2\frac{{\partial }^2{v}_1}{\partial x^2}+v_2^2\frac{{\partial }^2{v}_1}{\partial y^2}+2uv\frac{{\partial }^2{v}_1}{\partial x\partial y}\right]\\ & =\frac{{\mu }_{nf}}{{\rho }_{nf}}\left[\left(\frac{{\partial }^2{v}_1}{\partial y^2}\right)-\frac{v_1}{k}\right]+\frac{{\mu }_{nf}}{{\rho }_{nf}}\left[{\lambda }_2\left(v_1\left(\frac{{\partial }^3{v}_1}{\partial x\partial y^2}\right)-\frac{\partial v_1}{\partial x}\left(\frac{{\partial }^2{v}_1}{\partial y^2}\right)+\frac{\partial v_1}{\partial y}\left(\frac{{\partial }^2{v}_2}{\partial y^2}\right)+v_2\left(\frac{{\partial }^3{v}_1}{\partial y^3}\right)\right)\right],\end{array}$$

(3)

$$v_1\frac{\partial \yen }{\partial x}+v_2\frac{\partial \yen }{\partial y}=\frac{1}{{(\rho C_p)}_{nf}}\left[\frac{\partial }{\partial y}\left({\kappa }_{nf}^{*}(\yen )\frac{\partial \yen }{\partial y}\right)\right]-\frac{1}{{(\rho C_p)}_{nf}}\frac{\partial q_r}{\partial y}.$$

(4)

The relevant boundary conditions are [31]:

$$\begin{array}{c}{v}_1\left(x,0\right)=U_w+N\nu \left(\frac{\partial v_1}{\partial y}\right), v_2\left(x,0\right)=V_w,-k_0\left(\frac{\partial \yen }{\partial y}\right)=h_f\left({\yen }_w-\yen \right),at y=0\\ v_1\to 0, \frac{\partial v_1}{\partial y}\to 0, \yen \to {\yen }_{\infty } as y\to \infty .\end{array}\}$$

(5)

where $V_w$ signifies suction (injection) parameter.

2.2. Thermophysical Features of the Oldroyd-B Nanofluid

Thermophysical characteristics are increased by the dispersion of nanoparticles into an aqueous fluid. The material parameters for the Oldroyd-B nanofluid are offered in

Table 1. Thermophysical features of the Oldroyd-B nanofluid.

Features	Nanofluid
$\mathrm{Viscosity} \left(\mu \right)$	${\mu }_{nf}={\mu }_f{(1-\varphi )}^{-2.5}$
$\mathrm{Density} \left(\rho \right)$	${\rho }_{nf}=\left(1-\varphi \right){\rho }_f+\varphi {\rho }_s$
$\mathrm{Heat} \mathrm{capacity} \left(\rho C_p\right)$	${(\rho C_p)}_{nf}=\left(1-\varphi \right){(\rho C_p)}_f+\varphi {(\rho C_p)}_s$
$\mathrm{Thermal} \mathrm{conductivity} \left(\kappa \right)$	$\frac{{\kappa }_{nf}}{{\kappa }_f}=\left[\frac{\left({\kappa }_s+2{\kappa }_f\right)-2\varphi \left({\kappa }_f-{\kappa }_s\right)}{\left({\kappa }_s+2{\kappa }_f\right)+\varphi \left({\kappa }_f-{\kappa }_s\right)}\right]$

[32,33]:

In Table 1$,$ ${\mu }_f$, ${\rho }_f$, ${(C_p)}_f$, and ${\kappa }_f$ are defined as the nanoparticle volumetric fraction coefficient, dynamic viscosity, intensity, effective heat capacitance, and thermal conductance of the base fluid, respectively. The other properties ${\rho }_s$, ${(C_p)}_s$, and ${\kappa }_s$ are denoted as the density, effective heat capacitance, and thermal conductance of the nanoparticles, respectively. The variable thermal conductance is defined as [33,34]:

$${\kappa }_{nf}^{*}(\yen )=k_{nf}\left[1+ϵ\frac{\yen -{\yen }_{\infty }}{{\yen }_w-{\yen }_{\infty }}\right]$$

(6)

2.3. Nanosolid-Particles and Base Fluid Attributes

The physical characteristics of the base fluid engine oil and various nanosolid particles are tabulated in

Table 2. Material characteristics of base fluid and nanosolid particles at 293 K.

Thermophysical	$\mathit{\rho } \left(\mathbf{k}\mathbf{g}/{\mathbf{m}}^3\right)$	${\mathit{c}}_{\mathit{p}} \left(\mathbf{J}/\mathbf{k}\mathbf{g}·\mathbf{K}\right)$	$\mathit{k} \left(\mathbf{W}/\mathbf{m}\mathbf{K}\right)$
Copper (Cu)	8933	385.0	401.00
Engine Oil (EO)	884	1910	0.144
Molybdenum disulfide (MoS2)	5060	397.21	904.4

below [35,36].

2.4. Rosseland Approximations

Using the Roseland approximation [37], one can write:

$$q_r=-\frac{4{\sigma }^{*}}{3k^{*}}\frac{\partial {\yen }^4}{\partial y},$$

(7)

Here, ${\sigma }^{*}$ is the Stefan─Boltzmann number and $k^{*}$ is the absorption coefficient.

3. The Solution to the Problem

The boundary value problem in Equations (2)–(5) is in the form of PDEs, which are converted into ODEs with the assistance of the similarity technique. The first step of this technique is to introduce stream functions $\psi$ and subsequently, $\psi$ is used to express the velocity components in the $x$ and $y$ axes.

$$v_1=\frac{\partial \psi }{\partial y}, v_2=-\frac{\partial \psi }{\partial x}.$$

(8)

Then, similarity variables are formed as:

$${\rm Y}\left(x,y\right)=\sqrt{\frac{b}{{\nu }_f}}y, \psi \left(x,y\right)=\sqrt{{\nu }_{f}b}xf\left({\rm Y}\right), \theta \left({\rm Y}\right)=\frac{\yen -{\yen }_{\infty }}{{\yen }_w-{\yen }_{\infty }}.$$

(9)

When Equation (9) is substituted into Equations (2)–(5), then the following ODEs are obtained:

$$f^{‴}+{\varphi }_{\mathrm{a}}{\varphi }_{\mathrm{b}}\left[f{f}^{″}-f^{\prime 2}+{\beta }_1\left(2f{f}^{\prime }-f^2{f}^{‴}\right)\right]+{\beta }_2\left(f^{″2}-f{f}^{iv}\right)-\frac{1}{{\varphi }_{\mathrm{a}}}K{f}^{\prime }=0,$$

(10)

$${\theta }^{″}\left(1+ϵ\theta +\frac{1}{{\varphi }_{\mathrm{d}}}PrNr\right)+ϵ{\theta }^{\prime 2}+Pr\frac{{\varphi }_{\mathrm{c}}}{{\varphi }_{\mathrm{d}}}\left(f{\theta }^{\prime }-f^{\prime }\theta \right)=0,$$

(11)

with

$$\begin{array}{l}f\left(0\right)=S, f^{\prime }\left(0\right)=1+\Lambda f^{″}\left(0\right), {\theta }^{\prime }\left(0\right)=-B_{\varsigma }\left(1-\theta \left(0\right)\right)\hfill \\ f^{\prime }(\infty )\to 0, f^{″}(\infty )\to 0, \theta (\infty )\to 0,\hfill \end{array}\}$$

(12)

where the ${{\varphi }^{\prime }}_{i}s$ with $\mathrm{a}\le i\le \mathrm{d}$ in Equations (10)–(12) represent the following thermophysical properties for the Oldroyd-B nanofluid

$$\begin{array}{l}{\varphi }_{\mathrm{a}}={\left(1-\varphi \right)}^{2.5}, {\varphi }_{\mathrm{b}}=\left(1-\varphi +\varphi \frac{{\rho }_s}{{\rho }_f}\right), {\varphi }_{\mathrm{c}}=\left(1-\varphi +\varphi \frac{{(\rho C_p)}_s}{{(\rho C_p)}_f}\right)\hfill \\ {\varphi }_{\mathrm{d}}=\left(\frac{\left(k_s+2k_f\right)-2\varphi \left(k_f-k_s\right)}{\left(k_s+2k_f\right)+\varphi \left(k_f-k_s\right)}\right).\hfill \end{array}\}$$

(13)

As Equation (2) satisfies identity, the notation (${}^{\prime }$) is used for representing the derivatives for ${\rm Y}$. Here, ${\beta }_1=b{\lambda }_1$ (Deborah number-I), ${\beta }_2=b{\lambda }_2$ (Deborah number-II) defined along with $K=\frac{{\nu }_f}{bk}$ (porous medium) parameter and $Pr=\frac{{\nu }_f}{{\alpha }_f}$ (Prandtl number), ${\alpha }_f=\frac{{\kappa }_f}{{(\rho C_p)}_f}$ (thermal diffusivity), $S=-V_w\sqrt{\frac{1}{{\nu }_f b}}$ (mass transfer), $Nr=\frac{16}{3}\frac{{\sigma }^{*}{\yen }_{\infty }^3}{{\kappa }^{*}{\nu }_f{(\rho C_p)}_f}$ (thermal radiation), $\Lambda =\sqrt{\frac{b}{{\nu }_f}}N\nu \left(\mathrm{velocity} \mathrm{slip}\right)$, and $B_{\varsigma }=\frac{h_f}{k_0}\sqrt{\frac{{\nu }_f}{b}}$ (Biot number) parameters, respectively.

3.1. Nusselt Number

The local Nusselt number $\left(N{u}_x\right)$ is the physical parameter that controls the flow and can be expressed as [38,39]:

$$N{u}_x=\frac{x{q}_w}{k_f\left({\yen }_w-{\yen }_{\infty }\right)}$$

(14)

wherein $q_w$ indicates the heat flux, which is presented in Equation (15):

$$q_w=-k_{nf}\left(1+\frac{16}{3}\frac{{\sigma }^{*}{\yen }_{\infty }^3}{{\kappa }^{*}{\nu }_f{(\rho C_p)}_f}\right){\left(\frac{\partial \yen }{\partial y}\right)}_{y=0}$$

(15)

Applying the nondimensional transformations (9), one obtains:

$$N{u}_{x}R{e}_x^{-\frac{1}{2}}=-\frac{k_{nf}}{k_f}\left(1+Nr\right){\theta }^{\prime }\left(0\right),$$

(16)

where $N{u}_x$ and $C_f$ denote the Nusselt number and reduced skin friction, respectively. In addition, $R{e}_x=\frac{U_{w}x}{{\nu }_f}$ is the local $Re$ which relies on the stretching velocity $U_w\left(x\right)$.

3.2. Entropy Generation Analysis

Researchers in science and technology are always considering the amount of energy used and trying to minimize the wasted energy. Therefore, the reduction of entropy generation is a good decision, since entropy generation is one of the main factors that cause the irreversibility of convenient energy. A nonideal factor that contributes to an increment in the entropy of the system is the presence of penetrable materials. When penetrable materials affect the density of a nanofluid, the following effects arise: First, the medium’s permeability decreases, and then the thermal boundary layer and the system entropy increase. Therefore, the real entropy production in the nanofluids can be expressed by [38,39,40,41]:

$$E_G=\frac{k_{nf}}{{\yen }_{\infty }^2}\left\{{\left(\frac{\partial \yen }{\partial y}\right)}^2+\frac{16}{3}\frac{{\sigma }^{*}{\yen }_{\infty }^3}{{\kappa }^{*}{\nu }_f{(\rho C_p)}_f}{\left(\frac{\partial \yen }{\partial y}\right)}^2\right\}+\frac{{\mu }_{nf}}{{\yen }_{\infty }}{\left(\frac{\partial v_1}{\partial y}\right)}^2+\frac{{\mu }_{nf}{v}_1{}^2}{k{\yen }_{\infty }}.$$

(17)

Equation (17) is formed by three terms: the transfer of heat irreversibility, the fluid friction, and the porous media effect. The dimensionless entropy generation $N_G$ [42,43,44,45,46] is expressed as:

$$N_G=\frac{{\yen }_{\infty }{}^2{b}^2{E}_G}{k_f{\left({\yen }_w-T_{\infty }\right)}^2}.$$

(18)

By substituting Equation (9) into Equation (18), the final equation of dimensionless entropy generation is:

$$N_G=Re\left[{\varphi }_4\left(1+Nr\right)\theta {\prime }^2+\frac{1}{{\varphi }_1}\frac{Br}{\Omega }\left(f^{″2}+K{f}^{\prime 2}\right)\right].$$

(19)

4. Numerical Implementation

To tackle the nonlinear ordinary differential Equations (10) and (11) concerning the endpoint condition (12), we applied the Keller-box method [47] with the help of algebraic software MATLAB for various values of the involved parameters. The stepwise procedure of the Keller-box method is expressed in the flow chart from Figure 2.

Conversion of ODEs

We have introduced the subsequent substitutions for converting the higher-order ODEs to first-order ODEs. We introduce dependent variables $G{v}_1$, $G{v}_2,$ $G{v}_3,$ $G{v}_4,$ $G{v}_5$, and $G{v}_6$ such that:

$$G{v}_1=f, G{v}_2=f^{\prime }, G{v}_3=f^{″}, G{v}_4=f^{‴}, G{v}_5=\theta , G{v}_6={\theta }^{\prime }.$$

(20)

$$\frac{dG{v}_1}{d\mathsf{{\rm Y}}}=G{v}_2,$$

(21)

$$\frac{dG{v}_2}{d\mathsf{{\rm Y}}}=G{v}_3,$$

(22)

$$\frac{dG{v}_3}{d\mathsf{{\rm Y}}}=G{v}_4,$$

(23)

$$\frac{dG{v}_5}{d\mathsf{{\rm Y}}}=G{v}_6,$$

(24)

$$-{\beta }_{2}G{v}_1\frac{dG{v}_4}{d\mathsf{{\rm Y}}}+{\beta }_{2}G{v}_3^2+{\varphi }_a{\varphi }_{b}G{v}_{1}G{v}_3-{\varphi }_a{\varphi }_{b}G{v}_2^2+2{\varphi }_a{\varphi }_b{\beta }_{1}G{v}_{1}G{v}_2-{\varphi }_a{\varphi }_b{\beta }_{1}G{v}_1^{2}G{v}_4-\frac{1}{{\varphi }_a}KG{v}_2+G{v}_4=0,$$

(25)

$$\frac{dG{v}_6}{d\mathsf{{\rm Y}}}+ϵG{v}_5\frac{dG{v}_6}{d\mathsf{{\rm Y}}}+\frac{1}{{\varphi }_d}PrNr\frac{dG{v}_6}{d\mathsf{{\rm Y}}}+ϵG{v}_6^2+Pr\frac{{\varphi }_c}{{\varphi }_d}G{v}_{1}G{v}_6-Pr\frac{{\varphi }_c}{{\varphi }_d}G{v}_{2}G{v}_5=0,$$

(26)

$$\begin{array}{ll}\hfill & G{v}_1\left(0\right)=S, G{v}_2\left(0\right)=1+\Lambda G{v}_3\left(0\right), G{v}_6\left(0\right)=-B_{\varsigma }\left(1-G{v}_5\left(0\right)\right),\hfill \\ \hfill & G{v}_2\left({\rm Y}\right)\to 0, G{v}_3\left({\rm Y}\right)\to 0, G{v}_5\left({\rm Y}\right)\to 0, as {\rm Y}\to \infty .\hfill \end{array}\}$$

(27)

The domain $\left[0, 1\right]$ has been converted into sub-domains utilizing a regular mesh with the subsequent grid points (Figure 3):

${\mathsf{{\rm Y}}}_0=0, {\mathsf{{\rm Y}}}_j={\mathsf{{\rm Y}}}_{j-1}+\Delta h_j, j=0,1,2,3\dots ,J, {\mathsf{{\rm Y}}}_J=1$ where, $\Delta h_j$ is the step-size (for further details, see Appendix A). For mathematical assessment, a mesh size of $\Delta h_j$ = 0.01 was figured out to be appropriate and the results are acquired with an error tolerance of ${10}^{-6}$.

5. Code Validation

In this section, our numerical results are validated by comparing with the findings from the literature [48,49,50,51]. The comparison of consistencies is tabulated in

Table 3. Comparison of $-{\theta }^{\prime }\left(0\right)$ with various values of $Pr$ when $\varphi =0$, $ϵ=0$, $\Lambda =0$, $Nr=0$, $S=0$, and $B_{\varsigma }=0$.

$$\mathit{P}\mathit{r}$$	Ref. [48]	Ref. [49]	Ref. [50]	Ref. [51]	Present
72 × 10−2	08086 × 10−4	08086 × 10−4	080863135 × 10−8	080876122 × 10−8	080876181 × 10−8
1 × 100	1 × 100	1 × 100	1 × 100	1 × 100	1 × 100
3 × 100	19237 × 10−4	19236 × 10−4	192368259 × 10−8	192357431 × 10−8	192357420 × 10−8
7 × 100	30723 × 10−4	30722 × 10−4	307225021 × 10−8	307314679 × 10−8	307314651 × 10−8
10 × 100	37207 × 10−4	37006 × 10−4	372067390 × 10−8	372055436 × 10−8	372055429 × 10−8

. It is shown that the present study displayed highly accurate outcomes for $-{\theta }^{\prime }\left(0\right)$ with various values of $Pr$ in the limiting cases.

6. Results and Discussion

This study aimed to analyze the thermal efficacy of OBF flowing over a stretching surface. The Keller-box numerical approach was applied to calculate numerical and graphical results. The impacts of physical parameters on the two-dimensional model of OBF are reported. These parameters are listed as the boundary layer thickness ${\rm Y}$, Deborah number ${\beta }_1$ and ${\beta }_2$, porous media parameter $K$, nanoparticle volume fraction parameter $\varphi$, velocity slip parameter $\Lambda$, variable thermal conductivity parameter thermal $ϵ,$radiation parameter$Nr$, Biot number $B_{\varsigma }$, suction parameter $S>0$, injection parameter $S<0$, $R_e$, and $B_r$. The characteristics of fluid swiftness, temperature, and entropy production are displayed in Figure 4, Figure 5, Figure 6, Figure 7, Figure 8, Figure 9, Figure 10, Figure 11, Figure 12, Figure 13, Figure 14, Figure 15, Figure 16, Figure 17, Figure 18, Figure 19, Figure 20, Figure 21, Figure 22, Figure 23, Figure 24, Figure 25, Figure 26, Figure 27, Figure 28, Figure 29, Figure 30 and Figure 31. The results are obtained for the Cu–EO– and MoS₂–EO–based non-Newtonian Oldroyd-B nanofluid. This numerical investigation was accomplished by setting up the following values for the related factors: ${\beta }_1=0.1$, ${\beta }_2=0.1$, $K=0.1$, $\varphi =0.1$, $\Lambda =0.2$, $Pr=6450$, $ϵ=0.2$, $Nr=0.3$, $B_{\varsigma }=0.1$, $S=0.1$, $Re=5$, and $Br=5$. The numerical measurements are presented in the form of tables and figures for two components of Oldroyd-B-nanofluid: Cu–EO and MoS₂–EO.

Figure 4, Figure 5, Figure 6, Figure 7, Figure 8 and Figure 9 describe a dimensionless number, known as Deborah number, on the variations of velocity, temperature, and entropy generation. Deborah number is defined as the relaxation time of non-Newtonian fluid, divided by the scale of time flow. The special effects of the Deborah numbers, which are denoted by ${\beta }_1$ and ${\beta }_2$ on velocity component $f^{\prime }\left({\rm Y}\right)$ are presented in Figure 4 and Figure 5, respectively. It is detected that $f^{\prime }\left({\rm Y}\right)$ profiles decrease with an increasing value of β₁, and it shows an opposite pattern when it is controlled by parameter ${\beta }_2$. Figure 6 and Figure 7 represent the effect of ${\beta }_1$and ${\beta }_2$ on temperature profile $\theta \left({\rm Y}\right)$, respectively. From Figure 6, it is observed that the values of $\theta \left({\rm Y}\right)$ and their associated thermal boundary layer thicknesses are increased when ${\beta }_1$ rises. A rise in the value of ${\beta }_1$ leads to an increase in reduction time. However, the effect of ${\beta }_2$ causes the temperature profile to drop, as shown in Figure 8. The results of entropy generation $N_G$ are presented in Figure 8 and Figure 9 for different values of ${\beta }_1$ and ${\beta }_2$, respectively. From these figures, the impact of the Deborah number ${\beta }_1$ and ${\beta }_2$ is to reduce the entropy generation profile.

Figure 10, Figure 11 and Figure 12 show the impact of porous media parameter $K$ on the variations of velocity, temperature, and entropy generation. The porous media parameter indicates the absorption rate of the fluid when passing through pores. Since a hybrid nanofluid contains more than one type of nanometer-sized particles, the absorption rate becomes slower, as shown in Figure 10. However, the role of porous media parameter $K$ is to increase the values of temperature (Figure 11) and entropy generation (Figure 12).

The impact of nanosolid particle size factor $\varphi$ on the variations of velocity (Figure 13), temperature (Figure 14), and entropy generation (Figure 15) are depicted against boundary layer thickness ${\rm Y}.$ The parameter $\varphi$ causes an increase in the variations of temperature $\theta \left({\rm Y}\right)$ and entropy generation $N_G$ because an increment in $\varphi$ causes the density of the nanofluid to increase. As a result, two effects co-occur: (a) a decreasing simultaneous velocity and (b) an increasing thermal conductivity and simultaneous temperature.

The graphical representations of velocity slip parameter $\Lambda$ can be seen in Figure 16, Figure 17 and Figure 18. These three figures display the variations of velocity, temperature, and entropy generation. The effect of the velocity slip factor is to decrease the values of velocity and entropy generation profiles. However, this parameter causes the temperature profile to increase because an increase in the slip factor produces frictional force, and the fluid velocity is reduced. Nevertheless, this slip parameter causes the temperature to shift upward.

Figure 19 and Figure 20 show that the effect of the Biot number $B_{\varsigma }$ is to increase the variations of temperature $\theta \left({\rm Y}\right)$ and entropy generation $N_G.$ The Biot number is defined as a dimensionless number related to the convective boundary condition. Therefore, Figure 19 shows that the temperature profile and thermal boundary layer thickness are increased due to the effect of the Biot number. Moreover, the Biot number causes an enhancement in heat transfer rate. Thus, the entropy generation is also increased (Figure 20). The impact of variable thermal conductivity $\epsilon$ on temperature profiles can be seen in Figure 21. When the thermal conductivity increases, the thermal boundary layer thickness increases, and the hybrid nanofluid’s temperature rises. Meanwhile, the radiation parameter $Nr$ causes the temperature and entropy generation variations to be improved (Figure 22 and Figure 23, respectively). The radiation parameter is defined as the thermal radiation transfer divided by the conduction heat transfer. A higher amount of heat energy flows into the system is indicated by higher values of $Nr$. As a result, the temperature profile shifts upward.

The representation of the suction parameter and injection parameter is demonstrated in Figure 24, Figure 25 and Figure 26, and Figure 27, Figure 28 and Figure 29, respectively. The velocity profile under the influence of the suction parameter is depicted in Figure 24. Meanwhile, the same profile with the effect of the injection parameter is presented in Figure 27. Moreover, the temperature profiles due to the suction and injection parameters are depicted in Figure 25 and Figure 28, respectively. Figure 26 and Figure 29 show the entropy generation profiles against the variations of suction and injection parameters, respectively. The swiftness and temperature outlines are suppressed by the increment of the suction parameter, whereas the same parameter causes the entropy generation parameter to rise. This situation can be explained by the presence of suction causing the heated fluid to flow towards the sheet. Consequently, the buoyancy force slows down the fluid velocity due to the high viscosity. Instantaneously, the temperature is reduced, and the entropy generation increases. However, the opposite pattern has been observed for the same profiles when increasing the injection parameter. The role of the injection parameter is to give an opposite effect on the direction of heated fluid and buoyancy force. Hence, the effect of the injection parameter on all the outlines shows the reverse trend compared to the suction parameter.

The impact of the Reynolds number $Re$ and the Brinkman number $Br$ on entropy production $N_G$ are offered in Figure 30 and Figure 31, respectively. These figures show that $N_G$ increases as $Re$ and $Br$ increase. The Reynolds number is defined as the ratio of inertial forces (frictional forces) to viscous forces. The augmentation of the Reynolds number causes the enhancement of the following: frictional force at the fluid, heat, and mass transfer. Consequently, entropy production is produced. In addition, the increase of $Br$ causes an increase in the rate of viscous dissipation. The Brinkman number $Br$ is defined as the ratio between heat produced by viscous dissipation and heat transported by molecular conduction. This number is also known as a ratio of viscous heat generation to external heating. As a result, entropy generation $N_G$ is enhanced due to the change in viscous dissipation.

The numerical results of the local Nusselt number $=N_{u}R{e}_x^{\frac{-1}{2}}$ for various values of $B_{\varsigma },$ $\Lambda ,$ $S>0, \varphi ,$ $Nr,$ and $Pr$ are observed in

Table 4. Calculations of the Nusselt number $=N_{u}R{e}_x^{\frac{-1}{2}}$ for $Pr=6450$.

${\mathit{B}}_{\mathit{\varsigma }}$	$\mathbf{\Lambda }$	$\mathit{S}$	$\mathit{\varphi }$	$\mathit{N}\mathit{r}$	$\mathit{P}\mathit{r}$	${\mathit{N}}_{\mathit{u}}\mathit{R}{\mathit{e}}_{\mathit{x}}^{\frac{-1}{2}}$ Cu–EO	${\mathit{N}}_{\mathit{u}}\mathit{R}{\mathit{e}}_{\mathit{x}}^{\frac{-1}{2}}$ MoS2–EO	Relative% $\frac{\mathit{N}{\mathit{u}}_{\left(\mathit{C}\mathit{u}\right)}-\mathit{N}{\mathit{u}}_{\left(\mathit{M}\mathit{o}{\mathit{S}}_2\right)}}{\mathit{N}{\mathit{u}}_{\left(\mathit{C}\mathit{u}\right)}}\times 100$
0.1	0.2	0.1	0.1	0.3	0.2	0.0907	0.1209	25%
0.2						0.1735	0.2302	26%
0.3						0.2492	0.3291	27%
	0.1					0.0909	0.1212	24%
	0.2					0.0907	0.1209	23%
	0.3					0.0905	0.1207	22%
		0.1				0.0911	0.6010	29%
		0.2				0.0915	0.1219	28%
		0.3				0.0920	0.1225	27%
			0.01			0.1100	0.1079	1.9%
			0.05			0.0980	0.1209	20%
			0.1			0.0852	0.1388	40%
				0.1		0.0857	0.1400	38%
				0.2		0.0920	0.1513	39%
				0.3		0.0982	0.1625	39%
					10	0.0857	0.1513	41%
					15	0.0859	0.1519	42%
					20	0.0860	0.1523	43%

. From this table, it is shown that the effect of $B_{\varsigma },$ $S>0, Nr,$ and $Pr$ are to enhance the wall heat flux $N_{u}R{e}_x^{\frac{-1}{2}}$ for both Cu–EO and MoS₂–EO nanofluids. In addition, $N_{u}R{e}_x^{\frac{-1}{2}}$ for both Cu–EO and MoS₂–EO is reduced due to the increment of $\Lambda .$ However, the effect of $\varphi$ is to reduce the rate of $N_{u}R{e}_x^{\frac{-1}{2}}$ for Cu–EO and to increase the local Nusselt number $N_{u}R{e}_x^{\frac{-1}{2}}$ for MoS₂–EO.

7. Final Outcomes

The numerical solutions are controlled by the following parameters: the first component of Deborah number ${\beta }_1$, second component of Deborah number ${\beta }_2$, porous media parameter $K$, volume friction parameter $\varphi$, velocity slip parameter $\Lambda$, Biot number $B_{\varsigma }$, variable thermal conductivity ε, radiation parameter $Nr$, suction parameter $S>0$, injection parameter $S<0$, Reynolds number Re, and Brinkman number $Br$. The previous section showed that the effect of the related parameters on the variations of velocity, temperature, and entropy generation was valid for both Cu–EO and MoS₂–EO. As a result, the main findings of this paper are summarized below:

(a)

An increase in velocity distribution was observed, under the influence of ${\beta }_2$ and $S>0$. Otherwise, this profile decreased due to the increase of ${\beta }_1$, $K$, $\varphi$, $\Lambda$ and $S>0$.

(b)

The velocity of molybdenum disulfide engine oil (MoS₂–EO) nanofluid was found to be higher than copper engine oil (Cu–EO) nanofluid. This comparison was depicted under the impact of the Deborah number, porous media parameter, nanoparticle volume fraction parameter, velocity slip parameter, suction, and injection.

(c)

The fluid temperature increased due to increasing values of ${\beta }_1$, $K$, $\varphi$, $\Lambda$, $B_{\varsigma }$, $\epsilon$, $Nr$, and $S<0$. However, the same profile decreased due to ${\beta }_2$ and $S>0$.

(d)

Temperature profiles showed that the graphs for MoS₂–EO were above those for the Cu–EO nanofluid. These two lines can be observed when varying controlling parameters such as the Deborah number, porous media parameter, velocity slip parameter, Biot number, variable thermal conductivity, radiation parameter, suction, and injection. However, the temperature measured for the Cu–EO nanofluid was greater than for MoS₂–EO, for various values of the nanoparticle volume fraction parameter.

(e)

The roles of Deborah number (${\beta }_1$ and ${\beta }_2$), velocity slip parameter, and injection parameter were to lessen the values of entropy generation. At the same time, this profile was enhanced when the following parameters increased: porous media parameter, volume friction parameter, Biot number, radiation parameter, suction parameter, Reynolds number, and Brinkman number.

(f)

At the small thickness boundary layer, the entropy generation of MoS₂–EO was higher than that of Cu–EO when this profile was affected by Deborah number. Moreover, Cu–EO entropy generation was greatest for increasing series of porous media parameter, nanoparticle volume fraction parameter, velocity slip parameter, Biot number, radiation parameter, suction, injection, Reynolds number, and Brinkman number.

(g)

The local Nusselt number was an ascending function of $B_{\varsigma },$ $S>0, Nr$, and $Pr$. Moreover, the local Nusselt number was a decreasing function of $\Lambda$. The effects of $B_{\varsigma },$ $S>0, Nr,$ $Pr,$ and $\Lambda$ are valid for both Cu–EO and MoS₂–EO. However, the effect of $\varphi$ on the local Nusselt number was described as follows: (i) decrease for Cu–EO; (ii) increase for MoS₂–EO.

8. Future Direction

In this work, we have employed the concept of entropy generation analysis and the effects of slip conditions on fluid flow. One can replace Equation (21) with another formula of entropy. For example, we suggest considering a fractional entropy such as the Tsallis entropy.