1 Introduction

Heating and cooling applications of working fluids are critical in nuclear and chemical reactors, electronic devices, etc. Effective cooling procedures are necessary for cooling high-energy structures. The thermal performance of some common coolants (water, oil, ethylene glycol, etc.) is not significant and hence they are not able to perform as an efficient coolant. The last two decades have seen progressive research on enhancing the thermal efficiency of various fluids. More than two decades ago, it was discovered that the suspension of nanoparticles (NPs) in a working fluid contributes to increasing the thermal performance of the resultant fluid and these resultant fluids were named nanofluids (NFs) (Choi, 1995). The variation in size (diameter), substance (oxides of metal, nonmetal, metal, etc.), and dispersion of NPs, the different features of the NFs may be adjusted (Kumar et al., 2020; Sumithra and Sivaraj, 2022; Indumathi et al., 2022, and Ibrahim et al., 2021c). Because of this remarkable property, NFs are used in both traditional industries and solar technologies (Kumar et al., 2021; Vijayalakshmi and Sivaraj, 2022; Abdul Hakeem et al., 2016; Chu et al., 2021a; Madhukesh et al., 2021). Ibrahim et al. (Ibrahim et al., 2021a) studied the effect of SiO₂ NPs on the thermal conductivity of different base fluids. They concluded that for ethylene glycol and glycerol base fluid, SiO₂ NPs improve thermal conductivity by 12% and 6%, respectively.

The NF flows in various contexts are theoretically studied using the two-phase model (see Buongiorno, 2006) and the single-phase model (see Tiwari and Das, 2007). The NFs flow in various contexts is investigated using these models. Numerous authors have investigated different NFs flow over various geometries using various combinations of NPs and base fluid (BF). Santhosh et al. (Santhosh et al., 2023) explicated the thermal efficiency of Cu/water and Al₂O₃/water NF flow in a porous rectangular cavity with heat source/sink and Joule heating. Ganga et al. (Ganga et al., 2016) investigated the influence of heat source/sink and radiation on the magnetohydrodynamics flow of a nanofluid over a vertical plate. Mishra and Kumar (Mishra and Kumar, 2020) analyzed the flow of Ag/water over a stretching cylinder with self-heating and velocity slip effects. (Ragupathi et al. (Ragupathi et al., 2019) expounded on the heat transference mechanism of an NF flow with Fe₃O₄ and Al₂O₃ NPs with different base fluids past a Riga plate with a non-uniform heat source/sink. Shafiq et al. (Shafiq et al., 2020) explicated the thermal performance of three-dimensional NF flow in a rotating frame with convective heating and thermal slip effects. Adnan et al. (Adnan et al., 2020) studied the impacts of freezing temperature and the diameter of NPs on the heat transfer rate (HTR) over a curved Riga surface. Reddy et al. (Reddy et al., 2021) investigated the water-based NF flow with molybdenum disulfide and ferro sulfate NPs over a rotating disk with a magnetic field. Thirumalaisamy and Ramachandran (Thirumalaisamy and Ramachandran, 2023) analyzed the Fe₃O₄/water NF and Fe₃O₄-Cu/water hybrid NF flow in a square porous cavity with heat source/sink effects. Ganesh Kumar (Ganesh Kumar, 2022) and Gnaneswara Reddy et al. (Gnaneswara Reddy et al., 2020) expounded on the HTR of the water-based NF with AA7072-AA7075 alloys NPs over a stretching sheet and semi-infinite heated plate, respectively. Prakasha et al. (Prakasha et al., 2023) explicated the applications of non-linear radiation on ethylene glycol-based heat transfer rate of hybrid NF (Graphene, Magnesium oxide) and ternary hybrid NF (Graphene, Zirconium oxide, Magnesium oxide) over a plate. Nayak et al. (Nayak et al., 2019) studied the influence of a non-uniform heat source/sink and the Cattaneo-Christov heat flux model of third-grade nanofluid over an inclined and stretching Riga plate. Chu et al. (Chu et al., 2021b) studied the free convective flow of the Cu-Al₂O₃/H₂O NF in a microchannel. Khan et al. (Khan et al., 2021) investigated the bioconvection aspects in non-Newtonian nanofluid over periodically moving surface with chemical reaction and activation energy effects. Some other remarkable studies regarding the NF flow and their thermal performance can be referred to from Refs. (Hakeem et al., 2018; Ibrahim et al., 2021b; Khan et al., 2020; Puneeth et al., 2022; Thirumalaisamy et al., 2022).

The suspension of NPs in a working fluid is affected by various physical phenomena. The NPs behavior in the suspension and hence the properties of NFs are significantly affected by these physical factors. One such important factor in the suspension is the NPs aggregation. The resilient covalent and metallic bond between the NPs is the reason NPs aggregation occurs (Chen et al., 2007, 2020; Motevasel et al., 2018). The covalent and metallic bonds tend to hold the NPs together and hence they form the cluster known as NPs aggregates. As a result, rather than being present as individuals in a BF, NPs are present in clusters (see Fig. 1). Many studies (Chen et al., 2007, 2020; Motevasel et al., 2018) have experimentally validated the claim that the NPs aggregation severely affects the thermal performance and properties of NFs. Motevasel et al. (Motevasel et al., 2018) expounded on the significance of NPs aggregation effect on thermal conductivity (TC). They demonstrated that NPs aggregation has a significant impact on the physical features, thermal proficiency, rheology, and heat transference of BFs. They strenuously argued for the inclusion of the NPs aggregation effect even at low concentrations. Chen et al. (Chen et al., 2020) published a study based on experimental results to explicate the NPs aggregate influence on the thermal radiative properties of NFs. In their study, they also provided the theoretical models and correlations for the inclusion of NPs aggregation and validated the theoretical models and correlations with the experimental data. Chen et al. (Chen et al., 2007) published an experimental study that presents the rheology of ethylene glycol (EG) based NF with spherical Titania (TiO₂) NPs with NPs aggregates. They found from their experiments that the Maxwell model underrates thermal conductivity. Furthermore, they suggested that the Maxwell model could be modified to include the NPs aggregate influence on TC. For the modification, they suggested that the Bruggeman model can be used to calculate the TC of NPs aggregates and the value be substituted back in the Maxwell model for predicting the TC of NFs. They also validated that the shear viscosity of the NFs strongly depends on the NPs aggregates. Their modified models and correlations for TC and shear viscosity of the TiO₂/EG NF gave a soundly close precision with the data obtained through the experiment. In the second half of the last decade, few authors (Mackolil and Mahanthesh, 2021a, 2021b; Madhukesh et al., 2022; Mahanthesh and Thriveni, 2021a, 2021b; Rana et al., 2021; Wang et al., 2022) have studied the NPs aggregation effect on the NFs flows. Mackolil and Mahanthesh (Mackolil and Mahanthesh, 2021a, 2021b) studied the boundary layer flow of TiO₂/EG NF in presence of Marangoni convection and NPs aggregation effects. Rana et al. (Rana et al., 2021) investigated the influence of NPs aggregates on the HTR of TiO₂/EG NF flow over a plate with considerable buoyancy force. Madhukesh et al. (Madhukesh et al., 2022) explicated the significance of NPs thermophoretic deposition on the TiO₂/water NF over a rotating sphere with NPs aggregation effect. Wang et al. (Wang et al., 2022) expounded on the influence of chemical reactions on the HTR of TiO₂/EG NF flow over a rotating disk. Mahanthesh and Thriveni (Mahanthesh and Thriveni, 2021a, 2021b) studied the HTR of TiO₂/EG NF flow over a vertical cylinder with thermal radiation and the HTR of TiO₂/EG NF flow over a cylinder with heat generation and NPs aggregation effect, respectively. Some other remarkable investigations regarding the HTR of NF and their thermal performance with NPs aggregation effect can be referred from Refs. (Ali et al., 2022; Yaseen et al., 2022; Yu et al., 2022).

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Fig. 1

Physical Representation.

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NFs flow towards a wedge is an interesting topic to study due to the entrancing applications in engineering, such as in geothermal systems, raw oil extraction, storage of nuclear waste, insulating heat exchangers, etc. These applications have prompted scholars to investigate the NF flow across a wedge. Falkner and Skan (Falkner and Skan, 1931) were the first to use a wedge to configure fluid flow. Since then various researchers (Bano et al., 2020; Ibrahim and Tulu, 2019; Kudenatti and L., 2021; Mishra and Kumar, 2021; Raju and Sandeep, 2016; Sandeep and Reddy, 2017) have made many developments on the flow towards/over a wedge. The focus of the aforementioned studies was to study the HTR of a viscous or NFs but they have ignored the kinematics of NPs aggregation effect. Some studies (Ellahi et al., 2016; Rehman et al., 2022; Yaseen et al., 2022) have been published to investigate the NFs flow over a wedge with NPs aggregation effect, but the investigations have been done ignoring the various body and surface forces. Hence, the influence of NPs aggregation and the rheology of NPs aggregation kinematics on NF flow over a wedge needs resolute attention. TiO₂ NPs suspended in the base fluid (to increase their thermal characteristics) is a worthy combination because TiO₂ NPs have a more stable chemical nature compared to their other metallic state. Furthermore, TiO2 has a relatively cheap manufacturing cost, and TiO₂ NPs are manufactured on a big scale in industries. Authors of the present study have made a development on the Falkner-Skan problem and studied the NPs aggregation influence on the HTR of TiO₂/EG NF flow over a wedge (refer to Fig. 1). As a novelty, authors have considered the influence of mixed convection, thermal radiation, porous medium, and non-uniform heat source/sink together with the suction/injection effect. The author's goal with this investigation was to thoroughly analyze the influence of NPs aggregation. Furthermore, the following are the primary goals of the current study:

• Development of Falkner-Skan problem with NPs aggregation effect and important body and surface forces.

• Influence of NPs aggregation on the thermal field.

• Analyzing the HTR of TiO₂/EG NF flow with and without NPs aggregation.

The authors have specifically considered the TiO₂/EG NF for this investigation and modeling of the flow with NPs aggregation effect. Appropriate models for viscosity and TC were required to include the NPs aggregation effect and Chen et al. (Chen et al., 2007) have experimentally studied the values of viscosity and TC for the TiO₂/EG NF with NPs aggregation effect and have also provided the theoretical models for the same. Their modified models and correlations for TC and shear viscosity of the TiO₂/EG NF gave a soundly close agreement with the experimental data.

2 Mathematical model

Consider the TiO₂/EG flow (ethylene glycol-based NF with Titania NPs) and heat transference towards a wedge in a porous medium with permeability $K_o\left(x\right)=\frac{K}{x^{m-1}}$ under the influence of buoyancy force. The surface of the wedge is coincident with the x-axis and the y-axis is delineated normal to it (see Fig. 1). The free stream velocity of TiO₂/EG NF flow is $U_e=u_e{x}^m$ where u_e is constant. In addition, the wedge is moving with velocity $U_w=u_w{x}^m$ , where u_w is constant and the stretching and shrinking wedge are denoted by u_w > 0, and u_w < 0; and u_{w =} 0 for a stationary wedge. Here, m is the angle and $m={\beta }_w/\left(2-{\beta }_w\right)$ , where ${\beta }_w$ is the Hartree pressure gradient parameter. The total angle of the wedge is $\mathrm{\Omega }$ and $\mathrm{\Omega }={\beta }_w\pi$ . Moreover, the flow over a wedge is modeled when $m\in \left(0,1\right)$ . In addition, the flow past a flat surface along a horizontal axis is denoted by, $m=0$ $\left(i.e.{\beta }_w=0\right)$ , and $m=1$ $\left(i.e.{\beta }_w=1\right)$ denotes the stagnation point flow past a vertical flat surface. The HTR is analyzed in presence of thermal radiation and a non-uniform heat source/sink. The temperature of the surface is assumed as $T_f$ and the free stream temperature of TiO₂/EG NF flow is considered as $T_{\infty }$ . The boundary layer flow is subjected to suction/injection with $v_w$ as mass transfer velocity at the surface. Moreover, the last term in the energy equation (Eqn. (3)) stands for the non-uniform heat source/sink term $\left(Q\ast \ast \right)$ and it is explained later.

The governing equations of the TiO₂/EG nanofluid flow over a wedge are written as follows (see Refs. (Anuar et al., 2021; Khan et al., 2022; Mishra and Kumar, 2021; Waini et al., 2020; Yaseen et al., 2022):

Continuity Equation:

Momentum Equation:

Energy Equation:

Boundary Conditions (BCs) (see Refs. Mishra and Kumar, 2021; Yaseen et al., 2022):

3 Engineering parameter

The physical quantity of significance in this study is the skin friction C_f and Nusselt number Nu_x (Yaseen et al., 2022):

Using Eqns. (11) and (16), the skin friction and Nusselt number in equation (15) is written as:

4 Methodology of numerical approach

This section focuses on the methodology used for deducing solutions as well as code validation. The equations are at first modeled as PDEs and later, altered into ODEs via similarity variables. The solution of the Eqns. (12)–(13) along with BCs (14) is deduced with the “bvp4c function” (a built-in package in MATLAB), the more specific details of the bvp4c function can be referred from the Shampine et al. (Shampine et al., 2003). The “bvp4c function” uses a finite difference scheme together with a precision of fourth order with the help of the “3-stage Lobatto IIIA formula” (Khan et al., 2020). The appropriate thickness of the boundary layer, point depicting the far field ${\eta }_{\infty }$ , and initial guess must be selected relying on the parameters applied, to obtain accurate solutions. To deduce the solution of the model, the ODEs obtained after similarity transformation are reduced into first-order ODEs by the following substitution (Yaseen et al., 2022):

Utilizing the new variables, the Eqns. (12)–(13) are reduced to first-order ODEs and the following MATLAB syntax is used:

The following MATLAB syntax is used for BCs at the surface and far-field:

Then, the numerical solutions are obtained by coding Eqns. (18)–(20) into the bvp4c solver. The output of bvp4c is a structure called “sol”. The syntax of the solver is given by “sol = bvp4c (@OdeBVP, @OdeBC, solinit, options)” which consists of several functions (Yaseen et al., 2022). The “@OdeBVP” function is where the Eqns. (18)–(19) are coded. The “@OdeBC” function is employed to code the BCs (20). The “solinit” function is used to code the initial mesh points and the initial solution approximation at the mesh points (Khan et al., 2020). Meanwhile, the “options” function is an optional argument for integration. The solver will then run and the outcomes will be printed out as numerical solutions and graphs. Furthermore, the values of missing conditions are guessed to initiate the process of finding the solution and other parameters present in the Eqns. (19) and (20) are set to find the desired solution. The process of iteration is repeated and the solution is accepted only when the conditions in Eqn. (20) are satisfied asymptotically. The process of finding the solution is shown via a flow chart in Fig. 2. To validate the model and the code used to find the numerical solution, an assessment in Table 3 is outlined with the published computations of Yacob et al. (Yacob et al., 2011) and Anuar et al. (Anuar et al., 2021) as a limiting case to validate the numerical code utilized to solve the current model. The comparative results are quite consistent, ensuring that the current conclusions are valid.

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Fig. 2

Flow chart.

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5 Results and discussion

This communication explores the impact of NPs aggregation on the flow of TiO₂/EG NF over a wedge. This section of the paper deals with the results and their analysis. The results are depicted graphically and through tabular values for TiO₂/EG NF over the wedge for two cases: with and without NPs aggregation. The discussion is focused on the velocity $f^{\prime }\left(\eta \right)$ , temperature $\theta \left(\eta \right)$ , skin friction, and Nusselt number of TiO₂/EG NF on the surface of the wedge. In the figures, the dotted lines depict the solution for TiO₂/EG NF without aggregation and solid lines depict the solution for TiO₂/EG NF flow with aggregation. During the derivation of the solution, authors have used the following general values of the parameters (present in Eqns. (19) and (20)) during the numerical computations: $m=0.5$ , $\mathit{Pr}=150.4583$ , $A^{\ast }=B^{\ast }=0.5$ , $\mathrm{\Pi }=0.5$ , $\lambda =-0.5$ , $\mathrm{\Lambda }=0.5$ , $R_d=2$ and $S=-0.2$ .

5.1

5.1 Discussion of velocity profiles

Figs. 3-7 visualize the impact of emerging parameters on the velocity profile $f{}^{\prime }\left(\eta \right)$ . Fig. 3 displays the velocity $f{}^{\prime }\left(\eta \right)$ for different estimates of mixed convection parameter $\mathrm{\Lambda }$ . The negative values $\left(\mathrm{\Lambda }<0\right)$ and positive values $\left(\mathrm{\Lambda }>0\right)$ of the mixed convection parameter $\mathrm{\Lambda }$ denote the case of opposing flow and assisting flow, respectively. Fig. 3 demonstrates that TiO₂/EG NF velocity $f{}^{\prime }\left(\eta \right)$ increases with an increment in parameter $\mathrm{\Lambda }$ . The findings indicate that the velocity $f{}^{\prime }\left(\eta \right)$ of the TiO₂/EG NF rises for assisting flow and the magnitude of resistive forces escalates for opposing flow, hence the velocity $f{}^{\prime }\left(\eta \right)$ is low for the negative value of the parameter $\mathrm{\Lambda }$ . The convection phenomenon is the mechanism of transportation of heat. This phenomenon causes motion because of variation in fluid density, which occurs due to temperature gradients. An increment in the parameter $\mathrm{\Lambda }$ correlates to the increased buoyancy forces and indicates a favorable pressure gradient scenario throughout the flow, increasing the velocity profiles. Fig. 4 displays the velocity $f{}^{\prime }\left(\eta \right)$ for deviation in the porosity parameter $\mathrm{\Pi }$ . The findings indicate that the velocity $f{}^{\prime }\left(\eta \right)$ rises with an increment in the parameter $\mathrm{\Pi }$ . The increase in velocity is due to less resistance experienced by the TiO₂/EG NF during its movement in the boundary layer region. The increasing porosity parameter corresponds to a fall in the magnitude of surface resistive forces, and consequently, the velocity rises. Fig. 5 displays the velocity $f{}^{\prime }\left(\eta \right)$ for diverse estimates of suction/injection parameter S. The negative $\left(S<0\right)$ and positive $\left(S>0\right)$ values of parameter S symbolize the injection and suction, respectively. Furthermore $S=0$ symbolizes the non-existence of both. The graph demonstrates that the velocity of TiO₂/EG NF increases when parameter S is increased. The findings indicate that applying suction at the surface aids the velocity of the TiO₂/EG NF. The application of suction involves the removal of the layers with zero velocity. The removal of such layers reduces the friction provided to layers in motion. Consequently, the velocity rises. It confirms that the application of suction in the present model helps delay the boundary layer separation.

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Fig. 3

Effect of $\mathrm{\Lambda }$ on $f^{\prime }\left(\eta \right)$ .

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Fig. 4

Effect of $\mathrm{\Pi }$ on $f^{\prime }\left(\eta \right)$ .

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Fig. 5

Effect of S on $f^{\prime }\left(\eta \right)$ .

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Fig. 6

Effect of $\lambda$ on $f^{\prime }\left(\eta \right)$ .

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Fig. 7

Effect of $\phi$ on $f^{\prime }\left(\eta \right)$ .

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Fig. 6 displays the velocity $f{}^{\prime }\left(\eta \right)$ with deviation in the stretching/shrinking parameter (of wedge) $\lambda$ . The negative $\left(\lambda <0\right)$ and positive values $\left(\lambda >0\right)$ of $\lambda$ symbolize the shrinking and stretching of the wedge, respectively. In addition, $\lambda =0$ symbolizes the static wedge. The findings indicate that the velocity $f{}^{\prime }\left(\eta \right)$ of TiO₂/EG NF is aided by the increase in the magnitude of stretching of the wedge and the flow is resisted by the increase in the shrinking of the wedge. Fig. 7 displays the velocity $f{}^{\prime }\left(\eta \right)$ for w.r.t volume fraction $\phi$ of TiO₂ NPs in EG, which is the BF. Adding more TiO₂ NPs to the base fluid increases the effective viscosity of TiO₂/EG NF. That being the case, the effective fluid is now denser and it restricts the movement of NF. Hence, the velocity $f{}^{\prime }\left(\eta \right)$ of TiO₂/EG NF decreases. Fig. 8 displays the 3-D visualization of velocity profile with NPs aggregation with variation in parameter $\mathrm{\Pi }$ and $\lambda$ . The trends in Fig. 8 are the same as seen in Figs. 4 and 6. Furthermore, for opposing flow $\left(\mathrm{\Lambda }<0\right)$ (see Fig. 3), the velocity $f{}^{\prime }\left(\eta \right)$ of TiO₂/EG NF with NPs aggregation is higher whereas, for assisting flow $\left(\mathrm{\Lambda }>0\right)$ the velocity $f{}^{\prime }\left(\eta \right)$ of TiO₂/EG NF without NPs aggregation is higher. For assisting flow, it is clear that the velocity boundary layer pattern and boundary layer thickness becomes denser and thinner in the presence of the nanoparticle aggregation effect. This is due to an increase in the effective viscosity caused by aggregate formation. Hence for assisting flow, the velocity decreases with the formation of NPs aggregates.

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Fig. 8

3D visualization of influence of $\mathrm{\Pi }$ and $\lambda$ on $f{}^{\prime }\left(\eta \right)$ .

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5.2

5.2 Discussion of temperature profiles

Fig. 9 displays the temperature $\theta \left(\eta \right)$ for variation in the mixed convection parameter $\mathrm{\Lambda }$ . The findings indicate that the temperature $\theta \left(\eta \right)$ of TiO₂/EG NF drops with an increment in the parameter $\mathrm{\Lambda }$ . The findings indicate that the temperature of the TiO₂/EG NF flow is aided by the opposing flow. The magnitude of resistive forces increases in case of opposing flow, hence the friction force increases, and the loss of energy increases due to friction. Thus the temperature $\theta \left(\eta \right)$ rises. Fig. 10 displays the temperature $\theta \left(\eta \right)$ for variation in heat source parameters $\left(A^{\ast },B^{\ast }>0\right)$ . The findings indicate that the thermal profile rises upon increasing the values of heat source parameters $B^{\ast }$ and $A^{\ast }$ . The heat source case denotes the generation of heat and hence it leads to a rise in temperature $\theta \left(\eta \right)$ . It is inferred that for higher heat source parameters, the thermal profile near the surface is higher than that at the surface; and this phenomenon is observed only for heat source parameters. Fig. 11 displays the temperature profile $\theta \left(\eta \right)$ for variation in the porosity parameter $\mathrm{\Pi }$ . The higher estimates of the porosity parameter $\mathrm{\Pi }$ lead to a reduction in the magnitude of resistive forces, which further reduces the loss of energy, and hence the temperature $\theta \left(\eta \right)$ falls.

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Fig. 9

Effect of $\mathrm{\Lambda }$ on $\theta \left(\eta \right)$ .

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Fig. 10

Effect of A* and B* on $\theta \left(\eta \right)$ .

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Fig. 11

Effect of $\mathrm{\Pi }$ on $\theta \left(\eta \right)$ .

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Fig. 12 displays the temperature $\theta \left(\eta \right)$ for changes in the radiation parameter R_d. The findings indicate that the temperature display transitioning behavior with radiation parameter R_d. Near the surface, the temperature falls for higher estimates of parameter R_d but after a reflection point, it rises for larger values of radiation parameter R_d. The higher values of the radiation parameter correspond to a higher amount of heat released in the medium. The energy released in the system with a rise in the radiation parameter is absorbed by the particles. Hence, the temperature of the nanofluid increases. The rising temperature $\theta \left(\eta \right)$ is accredited to the increased amount of radiation in the flow region. Fig. 13 displays the temperature profile $\theta \left(\eta \right)$ for variation in the stretching/shrinking parameter (of wedge) $\lambda$ . The results demonstrate that temperature $\theta \left(\eta \right)$ falls with an increment in parameter $\lambda$ . The findings indicate that the thermal profile is aided by the increase in the shrinking. The reason for this outcome is the resistance provided by the shrinking of the wedge to the flow. Fig. 14 displays the temperature profile $\theta \left(\eta \right)$ for changes in the volume fraction $\phi$ of TiO₂ NPs in EG, which is the BF. Adding more TiO₂ NPs in the BF increases the effective TC of the TiO₂/EG NF. That being the case, the operating fluid becomes highly conductive and the temperature of TiO₂/EG NF is enhanced. Furthermore, for opposing flow $\left(\mathrm{\Lambda }<0\right)$ (see Fig. 9), the temperature $\theta \left(\eta \right)$ of TiO₂/EG NF without NPs aggregation is higher whereas, for assisting flow $\left(\mathrm{\Lambda }>0\right)$ the temperature $\theta \left(\eta \right)$ of TiO₂/EG NF with NPs aggregation is higher. For opposing flow, the resistive force dominates and it reduces the interaction of the nanoparticle aggregates in the nanofluid and consequently, it acts to reduce the thermal conductivity of nanoparticles. Thus, with opposing flow the formation of nanoparticle aggregates reduces the thermal conductivity of the nanofluid, and hence the temperature of the nanofluid decreases. Figs. 15 and 16 display the 3-D visualization of the thermal profile with NPs aggregation for different values of the parameter $\mathrm{\Pi }$ and $\lambda$ ; and $A^{\ast },B^{\ast }$ and $\lambda$ . The trends in Figs. 15 and 16 are the same as seen in Figs. 9, 10, and 12.

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Fig. 12

Effect of R_d on $\theta \left(\eta \right)$ .

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Fig. 13

Effect of $\lambda$ on $\theta \left(\eta \right)$ .

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Fig. 14

Effect of $\phi$ on $\theta \left(\eta \right)$ .

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Fig. 15

3D visualization of influence of $\mathrm{\Pi }$ and $\lambda$ on $\theta \left(\eta \right)$ .

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Fig. 16

3D visualization of influence of A* &B* and $\lambda$ on $\theta \left(\eta \right)$ .

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5.3

5.3 Discussion of streamlines and Nusselt number

Fig. 17 visualizes the effect of the shrinking wedge $\left(\lambda =-0.5\right)$ , static wedge $\left(\lambda =0\right)$ , and stretching wedge $\left(\lambda =0.5\right)$ on streamlines of TiO₂/EG NF flow with NPs aggregation. The streamlines represent the path followed by particles and their movement along the stream. The streamline provides the velocity direction at any point with the help of tangent. The streamlines depict that the movement of the TiO₂/EG NF is restricted by the shrinking, whereas the stretching of the wedge aids the movement of the flow. Table 4 and Figs. 18-20 visualize the effect of emerging parameters (porosity parameter $\mathrm{\Pi }$ , mixed convection parameter $\mathrm{\Lambda }$ , stretching/shrinking parameter $\lambda$ , volume fraction $\phi$ , heat source parameters $A^{\ast },B^{\ast }$ , and radiation parameter R_d) on the skin friction and Nusselt number. The Nusselt number signifies the heat transmission rate on the surface of the wedge. The findings indicate that the increasing values of the mixed convection parameter $\mathrm{\Lambda }$ , porosity parameter $\mathrm{\Pi }$ , and radiation parameter R_d act to enhance the heat transmission rate at the surface of the wedge. Furthermore, the increasing values of the stretching/shrinking parameter $\lambda$ and heat source parameters $A^{\ast },B^{\ast }$ act to reduce the heat transmission rate. From Fig. 20, it is clear that the Nusselt number values are higher when NPs aggregation is taken into account. The reason for this outcome is that the thermal conductivity of aggregated NPs is greater than that of independent NPs. Hence, the heat transmission rate is higher for TiO₂/EG NF flow with aggregated NPs. It is observed that the increasing values of the radiation parameter R_d, porosity parameter $\mathrm{\Pi }$ , mixed convection parameter $\mathrm{\Lambda }$ , and volume fraction $\phi$ act to enhance the skin friction at the surface of the wedge. In addition, the increasing values of the heat source parameters $A^{\ast },B^{\ast }$ and stretching/shrinking parameter $\lambda$ act to reduce the skin friction at the surface of the wedge. It is seen that skin friction values are higher when NPs aggregation is taken into account. This is due to an increase in the effective viscosity caused by aggregate formation. Hence the skin friction at the surface rises with NPs aggregation effect.

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Fig. 17

Streamlines.

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Table 4 Numerical values of the skin friction and Nusselt number.

						With aggregation		Without aggregation
Rd	A*, B*	$\mathrm{\Pi }$	$\mathrm{\Lambda }$	$\lambda$	$\phi$	Cf*	$N{u}_x^{\ast }$	Cf*	$N{u}_x^{\ast }$
2	0.5	0.5	−0.5	−0.5	0.04	2.0269458	−0.05595535	1.31005339	−0.05851778
6						2.03075675	−0.08674277	1.31442874	−0.09873223
10						2.03478659	0.11739075	1.31915591	0.07715161
2	2.5					2.0204786	−0.28150822	1.30181487	−0.29443944
	4.5					2.01349328	−0.50979444	1.29280331	−0.53333156
	0.5	1.5				3.13255333	−0.05580724	2.06601183	−0.05864476
		2.5				3.95035702	−0.05227681	2.62025833	−0.05574282
		0.5	0			2.19631079	−0.05606718	1.50406111	−0.05872106
			1			2.50606328	−0.05617506	1.84048704	−0.05876718
			−0.5	0		1.57641145	0.02554052	1.04198727	0.01584842
				0.5		0.87447484	1.12159882	0.57952216	1.07844696
				−0.5	0.06	2.81265879	−0.05178458	1.39485217	−0.06251167
					0.08	4.2030547	−0.04281347	1.48466707	−0.06651996

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Fig. 18

3D visualization of influence of A* &B* and R_d on $N{u}_x^{\ast }$ .

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Fig. 19

3D visualization of influence of $\mathrm{\Lambda }$ and R_d on $N{u}_x^{\ast }$ .

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Fig. 20

Effect of (a) $\mathrm{\Pi }$ , (b) $\lambda$ , (c) A* & B*, and (d) R_d on $N{u}_x^{\ast }$ .

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6 Conclusions

This communication presents a development on the Falkner–Skan problem for a wedge with the influence of NPs aggregation in the presence of the non-uniform heat source/sink, porous medium, mixed convection, radiation, and suction/ injection effects. This model was developed using the combination of modified Maxwell and Bruggeman models for the inclusion of NPs aggregation influence in the flow of TiO₂/EG NF, which are already tested experimentally and predict true values for properties of TiO₂/EG NF. The mathematical model is solved via the “bvp4c function which is accessible by MATLAB software”. The outcomes of this study will be useful in many fields that utilize applications of flow over wedge such as in raw oil extraction, storage of nuclear waste, insulating heat exchangers, etc. The important results of this investigation are:

• The velocity of the TiO₂/EG nanofluid increases due to the stretching of the wedge and assisting flow.

• The temperature of the TiO₂/EG nanofluid increases due to the shrinking of the wedge and opposing flow.

• For assisting flow, the velocity is lower for nanofluid with nanoparticle aggregation due to an increase in the effective viscosity.

• For opposing flow, the formation of nanoparticle aggregates reduces the thermal conductivity of the nanofluid, and hence the temperature of the nanofluid decreases.

• Thermal profile rises due to increment in the heat source parameters.

• The Nusselt number and skin friction values are higher when NPs aggregation is taken into account.

• The Nusselt number is directly proportional to radiation and porosity parameters.