#### Abstract

The focus of the present paper is to analyze the shape effect of gold (Au) nanoparticles on squeezing nanofluid flow and heat transfer between parallel plates. The different shapes of nanoparticles, namely, column, sphere, hexahedron, tetrahedron, and lamina, have been examined using water as base fluid. The governing partial differential equations (PDEs) are transformed into ordinary differential equations (ODEs) by suitable transformations. As a result, nonlinear boundary value ordinary differential equations are tackled analytically using the homotopy analysis method (HAM) and convergence of the series solution is ensured. The effects of various parameters such as solid volume fraction, thermal radiation, Reynolds number, magnetic field, Eckert number, suction parameter, and shape factor on velocity and temperature profiles are plotted in graphical form. For various values of involved parameters, Nusselt number is analyzed in graphical form. The obtained results demonstrate that the rate of heat transfer is maximum for lamina shape nanoparticles and the sphere shape of nanoparticles has performed a considerable role in temperature distribution as compared to other shapes of nanoparticles.

#### 1. Introduction

Nanotechnology has recently emerged and has become a worldwide revolution to obtain exceptional qualities and features over the last few decades. It developed at such a fast pace and is still going through a revolutionary phase. Nanotechnology is coming together to play a crucial and commercial role in our future world. Gold nanoparticles are one of the utmost stable metal nanoparticles and their current fascinating features include assembly of several types in material science, individual nanoparticles behaviors, magnetic, nanocytotoxic, optical properties, size-related electronic, significant catalysis, and biological applications. Gold nanoparticles have attracted research attention due to their properties and various potential applications. This progression would go to the later generation of nanotechnology that requires products of gold nanoparticles with precise shape, controlled size, large production facilities, and pureness. Gold nanoparticles are widely used as preferred materials in numerous fields because of their unique optical and physical properties, that is, surface plasmon oscillation for labeling, sensing, and imaging. Recently, significant developments have been made in biomedical fields with superior biocompatibility in therapeutics and treatment of various diseases. Gold nanoparticles can be prepared and conjugate with numerous functionalizing agents such as dendrimers, ligands, surfactants, RNA, DNA, peptides, polymers, oligonucleotides, drugs, and proteins [1].

Squeezing nanofluid flow with the effect of thermal radiation and magnetohydrodynamics (MHD) has important uses in the development of the real world. It has gained the consideration of researchers due to its extensive usages. Squeezing flow has increasing usages in several areas, particularly in the food industry and chemical engineering. The undertakings and properties of the squeezing flow of nanofluid for industrial usages such as electronic, transportation, biomechanics, foods, and nuclear reactor have been explained in many publications in the open literature. There are various examples concerning squeezing flow but the most significant ones are injection, compression, and polymer preparation.

The squeezing flow of nanofluid has gained significant consideration due to the valuable verities of applications in the physical and biophysical fields [2]. Hayat et al. [3] discussed the MHD in squeezing flow by using two disks. Dib et al. [4] examined the analytical solution of squeezing nanofluid flow. Duwairi et al. [5] addressed the heat transfer on the viscous squeezed flow between parallel plates. Domairry and Hatami [6] examined the time-dependent squeezing of nanofluid flow between two surfaces by applying differential transformation techniques. Sheikholeslami and Ganji [7] studied the heat transfer in squeezed nanofluid flow based on homotopy perturbation method. The thermal radiation effect in two-dimensional and time-dependent squeezing flow was investigated using homotopy analysis method by Khan et al. [8]. Sheikholeslami et al. [9] presented the effect of MHD on squeezing nanofluid flow in a rotating system. Gupta and Saha Ray [10] investigated the unsteady squeezing nanofluid flow between two parallel plates by using the Chebyshev wavelet expansion. The effects of MHD on alumina-kerosene nanofluid and heat transfer within two horizontal plates were examined by Mahmood and Kandelousi [11].

In the fields of engineering and science, there are various mathematical problems to find but the exact solution is almost complicated. Homotopy analysis method (HAM) is a well-known and critical method for solving mathematics-related problems. The main advantage of the homotopy analysis method is finding the approximate solution to the nonlinear differential equation without linearization and discretization. Earlier time in 1992, Liao [12–16] introduced this technique to find out the analytical results of nonlinear problems. The author concluded that homotopy analysis method (HAM) quickly converges to an approximate solution. The homotopy analysis method gives us a series of solutions. The approximate solution by homotopy analysis method is quite perfect since it contained all the physical parameters involved in a problem. Due to the effectiveness and quick convergence of the solution, various researchers, namely, Rashidi et al. [17, 18] and Abbasbandy and Shirzadi [19, 20], used homotopy analysis method (HAM) to find the solutions of highly nonlinear and coupled equations. Hussain et al. [21] presented the bioconvection model for squeezing flow using homotopy analysis method with the effect of thermal radiation heat generation/absorption.

Heat transfer can be increased by using several methodologies and techniques such as increasing the heat transfer coefficient or heat transfer surface which allows for a higher heat transfer rate in small volume fraction. Cooling is a major technical challenge faced by increasing numbers of industries involving microelectronics, transportation, manufacturing, and solid-state lighting. So, there is an essential requirement for innovative coolant with a better achievement that would be employed for enhanced properties [22]. Recently, nanotechnology has contributed to improving the new and innovative class of heat transfer nanofluid. Base fluids are embedded with nanosize materials to obtain nanofluids (nanofibers, nanoparticles, nanotubes, nanorods, nanowires, droplet, or nanosheet) [23]. Significantly, nanofluids have the ability to enhance heat transfer rate in several areas like nuclear reactors, solar power plants, transportation industry (trucks, automobiles, and airplanes), electronics and instrumentation, biomedical applications, microelectromechanical system, and industrial cooling usages (cancer therapeutics, cryopreservation, and nanodrug delivery) [24]. There are several studies to show the applications of nanofluid heat transfer. Kristiawan et al. [25] studied the convective heat transfer in a horizontal circular tube using TiO_{2}-water nanofluid. Turkyilmazoglue and Pop discussed the heat and mass transfer of convection flow of nanofluids containing nanoparticles of Ag, Cu, TiO_{2}, Al_{2}O_{3}, and CuO [26]. Sheikholeslami and Ganji [7] presented the analytical results of heat transfer in water-Cu nanofluid. Qiang and Yimin [27] investigated the experimental studies of convective heat transfer in water-Cu nanofluid. Elgazery [28] examined the studies of Ag-Cu-Al_{2}O_{3}-TiO_{2}-water nanofluid over a vertical permeable stretching surface with a nonuniform heat source/sink. Rea et al. [29] studied the viscous pressure and convective heat transfer in a vertical heated tube of Al_{2}O_{3}-ZrO_{2}-water nanofluids. Salman et al. [30] discussed by using a numerical technique the concept of affecting convective heat transfer of nanofluid in microtube using different categories of nanoparticles such as Al_{2}O_{3}, Cuo, SiO_{2}, and ZnO. Sheikholeslami et al. [31, 32] studied hybrid nanofluid for heat transfer expansion. Hassan et al. [33] discussed convective heat transfer in Ag-Cu hybrid nanofluid flow. Bhatti et al. [34] examined numerically hall current and heat transfer effects on the sinusoidal motion of solid particles. Furthermore, many researchers did work on heat transfer and thermal radiation; see [35–39].

In light of the above literature study, it has been observed that Cu, Ag, Al_{2}O_{3}, SiO_{2}, Cuo, and ZnO are mostly used to find the heat transfer. The gold (Au) was rarely used to find the heat transfer rate due to mixed convection [40]. The shape of nanoparticles is very significant in the enhancement of heat transfer. It is necessary to find the heat transfer rate in nanofluid under the exact shapes of nanoparticles [41]. From the literature survey, it is observed that no effort has been made on gold (Au) nanoparticles shape effect on squeezing flow. The basic purpose of the present study is to analyze the shape effect of gold (Au) nanoparticles on squeezing nanofluid flow and heat transfer. Various types of nanoparticles are under deliberation: column, sphere, hexahedron, tetrahedron, and lamina. The effects of various physical parameters on velocity and temperature distributions are analyzed through plotted graphs.

#### 2. Problem Description

Consider heat transfer in the incompressible, two-dimensional, laminar, and stable squeezing nanofluid between two horizontal plates at and . The lower plate is fixed by two forces which are equal and opposite. Both the plates are separated by distance *h*. A uniform magnetic field is applied along *y*-axis. Moreover, the effect of nonlinear thermal radiation is also considered. The thermophysical properties of gold nanoparticles and water are presented in Table 1. The values of nanoparticles shapes-related parameters are presented in Table 2. The partial governing equations of the problem are modeled as [42]

The reverent boundary value conditions are

The following similarity variables are induced to nondimensionalize governing equations (1)–(4):

Equation (1) is identically satisfied. Eliminating the pressure and by using equation (6) into equations (2), (3), and (4), one has the following nonlinear coupled boundary value problems:

The dimensionless quantities are

Here, *A*, Pr, *M, R*, Ec, and Rd represent the suction parameter, Prandtl number, magnetic parameter, Reynolds number, Eckert number, and thermal radiation parameter, respectively. One has

Here, , and represent the ratio of density, viscosity, heat capacitances, and thermal conductivity, respectively.

In this study, we consider where and represent thermal conductivity, dynamic viscosity, density, and specific heat of the fluid, respectively, whereas , and denote the thermal conductivity, dynamic viscosity, density, and specific heat of the solid, respectively. *m* and are the shape factor and volume fraction of nanoparticles, respectively.

The physical quantity of (Nusselt number) is defined as

#### 3. Solution by HAM

The auxiliary linear operators are selected as follows:

These auxiliary operators satisfy the following properties:

The initial guesses are chosen as

##### 3.1. Zeroth-Order Deformation

The corresponding zeroth deformation problem is defined as follows:

in which is called an embedding parameter and and are the convergence control parameters such that and ; it means that when *q* varies from 0 to 1, varies from initial guess to the final solution and varies from initial guesses .

The nonlinear operators and are given by

Expanding and with respect to *q* Maclaurin’s series and *q* = 0, we obtainwhere .

##### 3.2. Higher-Order Deformation Problem

The higher-order problems are as follows:where

The -order solutions arewhere are constants to be determined by using the boundary conditions.

##### 3.3. Convergence of Series Solutions

Zeroth- and higher-order deformation problems are given in equations (7) and (8), which clearly show that the series solutions contain nonzero auxiliary parameters and . The convergence of the solutions is checked through plotting -curves as displayed in Figures 1 and 2. It is evident that the series solutions (22) and (23) converge when and .

#### 4. Results and Discussion

The physical insight of the problem is discussed in this present portion. The schematic model of squeezing nanofluid is shown in Figure 3. The dynamics of heat transfer in the squeezing nanofluid fluid flow are described under the variation of dimensionless solid volume fraction, thermal radiation, Reynolds number, magnetic field, Eckert number, suction parameter, and shape factor. The analysis is carried out using the following range of parameters and . It is evident from Figures 4–7 that nanoparticles which participate in heat transfer are lamina > column > tetrahedron > hexahedron > sphere.

is a very important parameter for squeeze flow of nanofluid. From Figure 8, it is noted that the impact of on primary velocity seemed ineffective. It is also observed that the primary velocity is increased with increase of *R* as displayed in Figure 9. It is because that the inertia with the viscous ratio is dominant. From Figure 10, it is observed that the effect of *M* on the primary velocity is decreased due to the Lorentz force produced by *M*. The Lorentz force acts against the motion of squeeze flow of nanofluid. The variation of A on the dimensionless primary velocity is shown in Figure 11. From Figure 11, it can be seen that the primary velocity is intensifying with the increase of ; physically, the wall shear stress increases with the increase of .

The secondary velocity decreases in the half of the region as shown in Figure 12. In Figures 13 and 14, it is distinguished that secondary velocity changed in half of the region (the region above the central line between the plates). It happened due to the constraint of law of conservation of mass. The variation of is plotted in Figure 15; secondary velocity is also increased with the increase of

The shape effects of nanoparticles on dimensionless temperatures profiles are shown in Figure 16. It is noted from Figure 16 that sphere > hexahedron > tetrahedron > column > lamina. It is also observed that lamina nanoparticles have minimum temperature because of maximum viscosity while sphere shape nanoparticles have maximum temperature because of minimum viscosity. From Figure 17, it is observed that the temperature profile has a direct relation with ; the reason is that increasing the volume fraction causes enhanced thermal conductivity of the nanofluid which turns to increase the boundary layer thickness. From Figure 17, it is also observed that the sphere shape nanoparticles show a prominent role in temperature distribution. Figure 18 depicts the influence of *R* on the dimensionless temperature profile; with increasing *R*, the dimensionless temperature profile decreases because of decreasing the thermal layer thickness. Figure 18 showed that the effect of sphere shape nanoparticles is more significant than other shapes of nanoparticles under the influence of *R*. Figure 19 describes the impact of *M* on thermal boundary layer thickness. From Figure 19, it is noted that the temperature increases with the increase of *M*. The reason is that *M* tends to increase a dragging force which produces heat in temperature profile. Figure 19 depicts that the sphere shape nanoparticles in Au-water play a leading role in the temperature profile. Figure 20 shows that the temperature profile increases with A; physically, the heated nanofluid is pushed towards the wall, where the buoyancy forces can intensify the viscosity. That is why it decreases the wall shear stress. Figure 21 shows the effect of Rd on temperature profile; from this figure, it is illustrated that the Rd has an inverse relation with temperature profile. Due to this, the greater value of Rd corresponds to an increase in the dominance of conduction over radiation and hence reduction in the buoyancy force and the thermal boundary layer thickness. Under the effect of Rd, sphere shape nanoparticles have an important role in temperature distribution. Figure 22 displays that the squeezed nanofluid flow temperature increases with the increase of the Ec; the reason is that the frictional heat is deposited in squeezed nanofluid; however, thermal boundary layer thickness of sphere shape nanoparticles seems to be more animated in squeezed nanofluid by Ec effect.

#### 5. Conclusion

In the present paper, the effect of gold (Au) nanoparticles on squeezing nanofluid flow has been thoroughly examined. The analytical solution was obtained by using homotopy analysis method (HAM) for a range of pertinent parameters such as shape factor, solid volume fraction, thermal radiation, Reynolds number, magnetic field, suction parameter, and Eckert number. The effects of various parameters have been illustrated through graphs. The keeps fixed at 6.2. In the view of results and discussions, the following deductions have arrived:(i)The nanoparticles of sphere shape show a remarkable role in the disturbance of temperature profiles(ii)The nanoparticles of tetrahedron shape show a moderate role in the disturbance of temperature profiles(iii)The nanoparticles of lamina shape play a lower role in the disturbance of temperature profiles(iv)The nanoparticles of lamina shape play a principal role in the heat transfer rate(v)The nanoparticles of tetrahedron shape show a moderate role in the heat transfer rate(vi)The nanoparticles of tetrahedron shape show a lower role in the heat transfer rate(vii)Performances of lamina and sphere shapes nanoparticles in the forms of disturbance on temperature profiles and the heat transfer are opposite to each other(viii)Performances of hexahedron and tetrahedron shapes nanoparticles in forms of the disturbance temperature profiles and the heat transfer are opposite to each other

#### Abbreviations

PDEs: | Partial differential equations |

ODEs: | Ordinary differential equations |

RNA: | Ribonucleic acid |

DNA: | Deoxyribonucleic acid |

MHD: | Magnetohydrodynamic |

HPM: | Homotopy perturbation method |

HAM: | Homotopy analysis method. |

#### Data Availability

No data were used to support this study.

#### Conflicts of Interest

The authors declare no conflicts of interest.

#### Authors’ Contributions

Umair Rashid, Azhar Iqbal, and Mohd. Junaid Siddiqui contributed to conceptualization; Umair Rashid, Thabet Abdeljawad, and Azhar Iqbal contributed to methodology; Umair Rashid, Haiyi Liang, Azhar Iqbal, and Muhammad Abbas contributed to software; Haiyi Liang, Umair Rashid, Thabet Abdeljawad, Azhar Iqbal, Mohd. Junaid Siddiqui, and Muhammad Abbas contributed to validation; Haiyi Liang, Umair Rashid, Thabet Abdeljawad, Azhar Iqbal, Mohd. Junaid Siddiqui, and Muhammad Abbas contributed to formal analysis; Haiyi Liang, Umair Rashid, Thabet Abdeljawad, Azhar Iqbal, Mohd. Junaid Siddiqui, and Muhammad Abbas contributed to investigation; Thabet Abdeljawad and Muhammad Abbas contributed to resources; Umair Rashid, Azhar Iqbal, and Muhammad Abbas are responsible for writing and original draft preparation; Haiyi Liang, Umair Rashid, Azhar Iqbal, and Muhammad Abbas are responsible for writing, review, and editing of the paper; Azhar Iqbal and Muhammad Abbas contributed to visualization; Haiyi Liang, Thabet Abdeljawad, and Muhammad Abbas contributed to supervision; Thabet Abdeljawad and Muhammad Abbas contributed to funding acquisition. All authors have read and agreed to the published version of the manuscript.

#### Acknowledgments

The authors thank Dr. Muhammad Kashif Iqbal, GC University, Faisalabad, Pakistan, for his assistance in proofreading the manuscript. The authors would like to acknowledge Prince Sultan University for funding this work through the research group Nonlinear Analysis Methods in Applied Mathematics (NAMAM) (Group no. RG-DES-2017-01-17).