Time-Dependent Stagnation Point Flow of Water Conveying Titanium Dioxide Nanoparticle Aggregation on Rotating Sphere Object Experiencing Thermophoresis Particle Deposition Effects

Abstract

The notion of thermophoretic particle deposition is used in a number of applications, including thermal exchanger walls. It is important to identify the transport processes in action in systems such as thermal precipitators, exhaust devices, optical transmission fabrication processes, and so on. Based on these application points of view, the present work studies the performance of nanoparticle aggregation stagnation point flow over a rotating sphere during the occurrence of thermophoretic particle deposition. The nonlinear governing equations are transformed into the ordinary differential equation by utilizing suitable similarity variables. The numerical outcomes of the reduced equations along with boundary conditions are solved by the Runge–Kutta–Fehlberg 45 (RKF-45) order method with shooting procedure. The numerical results are shown with the assistance of graphs. The impacts of various dimensionless constraints on velocity, thermal, and concentration profiles are studied under the occurrence and absence of nanoparticle aggregation. The study reveals that the primary velocity is enhanced with increasing values of the acceleration parameter, but secondary velocity diminishes. The impressions of the rotation parameter will improve the primary velocity. The concentration profiles will diminish with an improvement in the thermophoretic parameter. The surface drag force is greater in nanoparticles with aggregation than nanoparticles without aggregation in the $C_{fx}$ case but a reverse behavior is seen in the $C_{fz}$ case. Further, the rate of heat distribution increases with a rise in the solid volume fraction, whereas the rate of mass distribution grows as the thermophoretic parameter grows.

1. Introduction

Nanofluids are a new category of nanotechnology-based thermal transfer liquids that are created by dissolving and suspending nanosized particles with distinctive diameters on the order of 10 nanometers. Base liquids often include water, motor oil, organic liquids, and other basic liquids. Choi [1] hypothesized about the possession of thermal conductivity and thermal transfer using nanofluids in 1995. A nanofluid is a developing liquid that has resulted in a number of scientific achievements in the nano industry. Choi, in his survey, discovered that a nanofluid is more efficient than any other fluid. The majority of nanofluid articles focus on understanding their behavior so that they may be used in situations where direct heat transfer improvement is critical, such as in many industrial applications, nuclear reactors, transportation, electronics, medicine, and food. Recently, in the context of a Casson nanofluid, Abu et al. [2] conducted a theoretical examination of a single vapor bubble in power-law fluid impacted by varying surface tension over Al₂O₃/H₂O nanofluids. Prasannakumara [3] used NaAlg-Cu nanofluid to conduct a quantitative investigation of the local thermal non-equilibrium conditions for a nano liquid (NaAlg-Cu) passing over a porous medium. Khan et al. [4] investigated the Cattaneo–Christov thermal flow model and the OHAM approach across three distinct fluids. The study demonstrated that a progressive increase in thermal relaxation parameters results in fluid temperature exhaustion and an increased heat transfer rate. Alzahrani et al. [5] explored the influence of thermal radiation with suction and slip conditions in the presence of a Casson nanofluid over a plane wall jet. Morad et al. [6] examined the thermophysical bubble movements in an N-dimensional Al₂O₃/H₂O nano liquid under two-phase turbulent flow. Bhowmik et al. [7] conducted a rod bundle thermal–hydraulic investigation with water and a water-based Al₂O₃ nanofluid for a tiny compact reactor. Computational assessment of the thermo-hydraulic performance of an Al₂O₃–water-based nanofluid in a hexagonal rod-bundle sub-channel was explored by Ahmed et al. [8].

When a group of nanoparticles stick together, they combine to form nanoparticle aggregation (N-Ag). The decrease in the contact surface of the fluid–solid interfaces results in a drop in the potential energy of the fluid system. Viscosity and thermal conductivity directly impact N-Ag. The aggregated nanoparticles tend to build flowing networks and continuous links to create a reduced thermal pathway. As a result, heat may be conveyed rather rapidly across the groupings, which, when combined with the augmented aggregation volume compared to nanoparticles, can increase the thermal capacity of the nanofluid. Thermal conductivity enhancement by the aggregation of nanoparticles with the occurrence of metal oxides and nanoparticles was studied by Wensel et al. [9]. Based on the nanolayer and nanoparticle aggregation, Xie et al. [10] determined the effective thermal conductivity of nanofluids. Swain and Mahanthesh et al. [11] investigated the N-Ag and Joule heating effect in a radiating magneto nano liquid flow over an exponential area. Rana et al. [12] inspected the Cattaneo–Christov (C-C) model and heat flux effects on a spinning disk with the Hall current effect in the presence of N-Ag. With sensitivity analysis, Sabu et al. [13] studied N-Ag dynamics on the quadratic convective magnetohydrodynamic flow of a nanomaterial across a slanted flat plate. Regarding the flow of a micropolar nanofluid across a stretching surface, Yu et al. [14] explored N-Ag and the thermophoretic particle deposition process. Radiative heat transmission and unsteady movement in an uneven channel with the aggregation kinematics of a nanofluid were explored by Jestine and Mahanthesh [15]. The aspects of a uniform horizontal magnetic field and N-Ag in the flow of a nanofluid with melting heat transfer were explored by Wang et al. [16].

When particles in a gas move from hot to cold zones as a result of a temperature profile, this is known as thermophoretic particle deposition (T-P-D). Settlement on cold surfaces, such as heat exchanger walls, might be aided by this movement. It is vital to understand the transport mechanisms at work in systems such as exhaust systems and thermal precipitators, and optical waveguide production procedures are critical. The practical application of the present concept was explained by Green and Lane [17] and Fuchs [18]. T-P-D was first studied by Goren [19]. Recently, Madhukesh et al. [20] inspected the mass and thermal transfer during the occurrence of a Casson fluid containing hybrid nanoparticles over a Riga surface in the context of the T-P-D effect. Waini et al. [21] investigated T-P-D and viscous dissipation effects in a micropolar hybrid nanofluid over a flat surface. Shankaralingappa et al. [22] investigated the T-P-D on a three-dimensional flow of Casson nano liquid containing sodium alginate across a stretched surface. Chen et al. [23] investigated T-P-D in the flow of a dual-layer Casson liquid with a magnetic dipole and extended Fourier’s and Fick’s equations. Madhukesh et al. [24] considered the thermophoretic impact and convective thermal conditions on the flow of a hybrid nanofluid over a moving thin needle. Gowda et al. [25] explored thermal transport in ferromagnetic liquid dynamics across a stretched sheet with the combined effects of T-P-D and a magnetic dipole.

Because all fluid flow and solid structure relationships are linked to stagnation point flow performance, stagnation point flow is a significant phenomenon. In engineering applications, the rate of fluctuation of the physical limitations around the flow is critical. The pressure and largest thermal exchange, as well as a reduction in velocity, occur near the flow’s environments in stagnation point motion. Electronic, hydrodynamic, and aerodynamic fields all play essential roles in flow at the stagnation point. The phenomenon of flow at the stagnation point has several applications in fluid flow and thermal transmission in a number of industries and technical domains. Zaib et al. [26] explored the creation of entropy and the dual branch solution in a mixed convection stagnation point flow over a Riga surface in the context of a micropolar nanofluid flow. Khan et al. [27] scrutinized the flow at the stagnation point in the presence of a hybrid nano liquid along a non-isothermal shrinking/stretching sheet with inertial and microstructure features. Gul et al. [28] explored the stagnation point flow of a blood-based hybrid nanofluid under mixed convection around a revolving sphere. Ghasemi and Hatami [29] investigated how solar radiation affects MHD stagnation point flow and heat conduction in a nanofluid over a stretched sheet.

The rotational flows and heat transfer properties of a driven flow stream over stationary spinning bodies of revolution are essential in a number of engineering applications, such as ballistic re-entry, fiber coatings, projectile motion, and rotational equipment design. The heat distribution between a fluid flow and a revolving sphere has several applications in technology, such as overcoming freezing system difficulties. Malvandi [30] examined the time-dependent flow of an unsteady spinning sphere’s stagnation point zone. Entropy analysis and unsteady MHD mixed convection stagnation point flow of a Casson nanofluid around a spinning sphere were examined by Madhy et al. [31]. Ahmed and Rashed [32] examined unsteady MHD mixed convection flow with the slip effect in a nanofluid in the stagnation region of an impulsively rotating sphere with the effects of thermal radiation and convective boundary conditions.

The present study was conducted to explore the behavior of nanoparticle aggregation and stagnation point flow over a rotating sphere in the presence of thermophoretic particle deposition.

The current study seeks to answer the following questions:

• What is the impact of the nanoparticle aggregation effect on the acceleration parameter in velocity, thermal, and concentration profiles?
• What is the thermal performance in the presence/absence of nanoparticle aggregation?
• What is the impact of the thermophoretic effect on the concentration profile in the presence/absence of nanoparticle aggregation?

The present manuscript is divided into five sections. The context of the study and implementation is discussed in the Section 1. Section 2 expands on the modeling and formulation of the problem. Section 3 describes the numerical technique and validation of the code. Section 4 delineates the results and physical perspectives of the problem. Section 5 summarizes the outcomes of the study.

2. Materials and Methods

Consider the incompressible boundary layer unsteady movement of nanoparticles towards the forward stagnation point section at a persistent temperature on a rotating sphere, as in Figure 1. The $x_1$ coordinate is represented on the surface, and the direction normal to the sphere is denoted by $y_1$. The sphere is rotating with angular velocity (time-dependent) $\mathsf{\Omega }(t)=B{t}^{-1}$, $B>0$. The temperature near the wall is assumed to be constant. Furthermore, the terms for viscous dissipation are negligible and the free stream and angular velocities vary with time in the aspect of $u_e(x_1,t)=A{x}_1{t}^{-1}$, $A>0$. The nanoparticle is in the thermal equilibrium state. The temperature at the wall and ambient temperature of the sphere are denoted by $T_W\&T_{\infty }$. Similarly, for concentration, the terms are $C_W\&C_{\infty }$. The mathematical model that represents the problem based on the assumptions stated above is as follows (see [24,28,30,31]).

$$\frac{\partial (r{v}_1)}{\partial y_1}=-\frac{\partial (r{u}_1)}{\partial x_1}$$

(1)

$$\frac{\partial u_1}{\partial t}+u_1\frac{\partial u_1}{\partial x_1}+v\frac{\partial u_1}{\partial y_1}=\left(\frac{w_1{}^2}{r_1}\right)\frac{d{r}_1}{d{x}_1}+u_e\frac{\partial u_e}{\partial x_1}+\frac{\partial u_e}{\partial t}+{\nu }_{nf}\frac{{\partial }^2{u}_1}{\partial y_1^2}$$

(2)

$$\frac{\partial w_1}{\partial t}+u_1\frac{\partial w_1}{\partial x_1}+v_1\frac{\partial w_1}{\partial y_1}={\nu }_{nf}\frac{{\partial }^2{w}_1}{\partial y_1^2}-\left(\frac{u_1{w}_1}{r_1}\right)\frac{d{r}_1}{d{x}_1}$$

(3)

$$\frac{\partial T_1}{\partial t}+u_1\frac{\partial T_1}{\partial x_1}+v_1\frac{\partial T_1}{\partial y_1}=\frac{k_{nf}}{{(\rho C_p)}_{nf}}\frac{{\partial }^2{T}_1}{\partial y_1^2}$$

(4)

$$\frac{\partial C_1}{\partial t}+u_1\frac{\partial C_1}{\partial x_1}+v_1\frac{\partial C_1}{\partial y_1}=D_B\frac{{\partial }^2{C}_1}{\partial y_1^2}-\frac{\partial (V_T(C_1-C_{\infty }))}{\partial y_1}$$

(5)

These are subjected to the initial and boundary conditions (see [30,31]):

$$u_1\left(0,x_1,y_1\right)=u_{1i}\left(x_1,y_1\right), v_1\left(0,x_1,y_1\right)=v_{1i}\left(x_1,y_1\right),$$

$$w_1\left(0,x_1,y_1\right)=w_{1i}\left(x_1,y_1\right), T_1\left(0,x_1,y_1\right)=T_{1i}\left(x_1,y_1\right),$$

$$C_1\left(0,x_1,y_1\right)=C_{1i}\left(x_1,y_1\right)$$

(6)

$$u_1\left(t,x_1,0\right)=u_1\left(t,x_1\right), v_1\left(t,x_1,0\right)=v_1\left(t,x_1\right), w_1\left(t,x_1,0\right)=\mathsf{\Omega }\left(t\right)r_1,$$

$$T_1\left(t,x_1,0\right)=T_W, C_1\left(t,x_1,0\right)=C_W.$$

$$u_1\left(t,x_1,w\right)=u_e\left(x_1,t\right)=A{x}_1/t, T_1\left(t,x_1,\infty \right)=T_{\infty }, w_1\left(t,x_1,0\right)=0,$$

$$C_1\left(t,x_1,\infty \right)=C_{\infty }$$

(7)

where $x_1\&y_1$ denote coordinates measured from the forward stagnation point and normal to the surface, respectively, and $z_1$ denotes the axis in the rotational direction. $r_1(x_1)$ is the radial distance between a symmetry axis and the surface element ($r_1(x_1)\simeq x_1$ in proximity to the point of stagnation, $u_1,v_1$ and $w_1$ are the velocity apparatuses along the $x_1,y_1$ and $z_1$ coordinates, $t$ signifies time, the subscript $i$ signifies the initial condition, thermal conductivity is $k$, heat capacitance is denoted by $(\rho C_p)$, $D_B$ denotes the Brownian diffusion coefficient, specific heat at constant temperature is $C_P$ and $T_1$ is the temperature.

The below similarity parameters are introduced (see [30]):

$$\begin{array}{l}{r}_1\approx x_1, \frac{d{r}_1}{d{x}_1}=1, \eta =\frac{1}{\sqrt{{\nu }_{f}t}}{y}_1, w_1=\frac{B{x}_1}{t}s\left(\eta \right)\\ \psi =A{x}_1{\left(\frac{{\nu }_f}{t}\right)}^{1/2}f(\eta ), \theta (\eta )=\frac{T_1-T_{\infty }}{T_w-T_{\infty }}, \chi (\eta )=\frac{C_1-C_{\infty }}{C_w-C_{\infty }}\end{array}$$

(8)

The thermophoretic velocity is represented as $V_T=-\frac{{\nu }_f{K}_{1a}^{*}}{T_{r1}}\frac{\partial T_1}{\partial y_1}$ (see [24]), where $K_{1a}^{*}$ and $T_{r1}$ are the thermophoretic constant and reference temperature, respectively.

According to experimental and empirical investigations (see [30,31,32]), in the kinematics and thermal propagation of nanofluid movements, the nanoparticle aggregation factor is critical. The nanomaterial measurement findings agreed exactly when the aggregate component was considered. The nanoparticle volume fraction is determined by the aggregation kinetic factor, and it is given as

$${\varphi }_{a2}=\varphi {\left(\frac{r_{a2}}{r_{p2}}\right)}^{3-D_2}$$

(9)

As shown in

Table 1. Thermophysical properties of nanomaterials.

	Properties	Expression for Nanofluid
01	Viscosity	${\mu }_{nf}={\mu }_f{\left(1-\frac{{\varphi }_{a2}}{{\varphi }_{m2}}\right)}^{-2.5{\varphi }_{m2}}$
02	Density	${\rho }_{nf}={\rho }_f\left(1-{\varphi }_{a2}\right)+{\varphi }_{a2}\frac{{\rho }_{a2}}{{\rho }_f}$
03	Heat capacity	$\frac{{\left(\rho C_p\right)}_{nf}}{{\left(\rho C_p\right)}_f}={\varphi }_{a2}\left(\frac{{\left(\rho C_p\right)}_{a2}}{{\left(\rho C_p\right)}_f}\right)+\left(1-{\varphi }_{a2}\right)$
04	Thermal conductivity	$\frac{k_{nf}}{k_f}=\frac{2k_f+k_{a2}+2{\varphi }_{a2}(k_{a2}-k_f)}{2k_f+k_{a2}-{\varphi }_{a2}\left(k_{a2}-k_f\right)}$

(see [33]), the thermal conductivity is determined by mixing the modified Maxwell and Bruggeman models. Suitable expressions are given below (see [34,35,36]).

$$\frac{k_{a2}}{k_f}=\frac{2}{8}\left\{\begin{array}{l}\frac{k_{p1}}{k_f}\left(3{\varphi }_{i2}-1\right)+\left(3\left(1-{\varphi }_{i2}\right)-1\right)+\\ {\left[{\left(\frac{k_{p2}}{k_f}\left(3{\varphi }_{i2}-1\right)+\left(3\left(1-{\varphi }_{i2}\right)-1\right)\right)}^2+8\frac{k_{p2}}{k_f}\right]}^{0.5}\end{array}\right\}$$

$${\varphi }_{i2}={\left(\frac{r_{a2}}{r_{p2}}\right)}^{D_2-3}$$

$${\rho }_{a2}=\left(1-{\varphi }_{i2}\right){\rho }_f+{\varphi }_{i2}{\rho }_s,$$

$${\left(\rho C_p\right)}_{a2}=\left(1-{\varphi }_{i2}\right){\left(\rho C_p\right)}_f+{\varphi }_{i2}{\left(\rho C_p\right)}_s.$$

From the above-stated expressions, the subscripts $a2$ and $p2$ signify aggregates and nanoparticles, ${\varphi }_{m2}=0.605$ denotes particles that are spherical in shape, fractional index $D_2=1.8,$ $r_{a2}=3.34r_{p2}$ denotes the ratio of radii of aggregates to radii of nanoparticles and are the commonly accepted values.

By using Equation (8), the system of Equations (1)–(5) reduces and boundary conditions are transformed into:

$$f”’+A_1{A}_2\left[f’-A{(f’)}^2+f”(\frac{\eta }{2}+fA)+A(\lambda s^2+1)-1\right]=0$$

(10)

$$s”+A_1{A}_2\left[s(1-2Af’)+s’(\frac{\eta }{2}+fA)\right]=0$$

(11)

$$\left(\frac{k_{nf}}{k_f}\right)\frac{\theta ”}{A_3}+\mathrm{Pr}\theta ’\left[\frac{\eta }{2}+fA)\right]=0$$

(12)

$$\chi ”+Sc \chi ’\left[\frac{\eta }{2}+fA\right]-{\tau }_{1}Sc(\chi \theta ”+\chi ’\theta ’)=0$$

(13)

and

$$\begin{array}{l}at \eta =0: f=0, f’=0, \theta =1, \chi =1, s=1\\ as \eta \to \infty : f’=1, \theta =0, \chi =0, s=0\end{array}$$

(14)

The above Equations (10)–(14) comprised different controlling parameters which are defined namely and mathematically in

Table 2. Flow controlling parameters.

	Name of the Parameter	Notation
01	Acceleration parameter	$A$
02	Rotation parameter	$\lambda =\frac{B^2}{A^2}$
03	Prandtl number	$\mathrm{Pr}=\frac{{\mu }_f{C}_p}{k_f}$
04	Schmidt number	$Sc=\frac{{\nu }_f}{D_B}$
05	Thermophoretic parameter	${\tau }_1=-K_{1a}^{*}(T_w-T_{\infty })T_{r1}^{-1}$
06	$A_1={\left(1-\frac{{\varphi }_{a2}}{{\varphi }_{m2}}\right)}^{2.5{\varphi }_{m2}}$
07	$A_2=\left(\left(1-{\varphi }_{a2}\right)+\left({\varphi }_{a2}\frac{{\rho }_{a2}}{{\rho }_f}\right)\right)$
08	$A_3=\left({\varphi }_{a2}\left(\frac{{\left(\rho C_p\right)}_{a2}}{{\left(\rho C_p\right)}_f}\right)+\left(1-{\varphi }_{a2}\right)\right)$

.

The important engineering coefficients $C_{fx}$, $C_{fz}$, $Nu$ and $Sh$ which are defined and simplified in

Table 3. Engineering coefficients and its reduced form.

	Engineering Coefficients	Definition	Reduced Form
01	Surface drag forceIn x-direction	$C_{fx}=\frac{{\left.{\mu }_{nf}\left(\frac{\partial u}{\partial y}\right)\right|}_{y=0}}{{\rho }_f{u}^2{}_e}$	$\sqrt{\mathrm{Re}}{C}_{fx}=\frac{f”\left(0\right)a^{\frac{-1}{2}}}{A_1}$
02	Surface drag forceIn z-direction	$C_{fz}=-\frac{{\left.{\mu }_{nf}\left(\frac{\partial w_1}{\partial y}\right)\right|}_{y=0}}{{\rho }_f{u}^2{}_e}$	$\sqrt{\mathrm{Re}}{C}_{fz}=-\frac{a^{\frac{-1}{2}}\sqrt{\lambda }s’\left(0\right)}{A_1}$
03	Nusselt number	$Nu=\frac{{\left.-K_{nf}x\left(\frac{\partial T}{\partial y}\right)\right|}_{y=0}}{K_f\left(T_w-T_{\infty }\right)}$	$\frac{Nu}{\sqrt{\mathrm{Re}}}=-\frac{K_{nf}}{K_f}\frac{{\theta }^{’}\left(0\right)}{\sqrt{a}}$
04	Sherwood number	$Sh=-\frac{{\left.x\left(\frac{\partial C_1}{\partial y}\right)\right|}_{y=0}}{\left(C_{W1}-C_{\infty }\right)}$	$\frac{Sh}{\sqrt{\mathrm{Re}}}=-x’\left(0\right)\sqrt{a}$

Where $\mathrm{Re}=\frac{u_{e}x}{{\nu }_f}$ as the local Reynolds number.

. (see [30,31,32]).

3. Numerical Technique and Validation of the Code

The shooting methodology and RKF-45 method were utilized to find the solution of the reduced Equations (10)–(13) and boundary conditions (14) using well-known computational tools. The simplified equations are then transferred into a first-order system. Let us consider

$$\begin{array}{l}f=a_1,f’=a_2,f”=a_3,\\ s=a_4,s’=a_5,\\ \theta =a_6,\theta ’=a_7,\\ \chi =a_8,\chi ’=a_9\end{array}$$

As a result, we obtain

$$a_3{}^{’}=-A_1{A}_2\left[\left(a_2-A{a}_2^2\right)+a_3\left(\frac{\eta }{2}+a_{1}A\right)+A\left(\lambda a_4^2+1\right)-1\right]$$

(15)

$$a_5{}^{’}=-A_1{A}_2\left[a_4\left(1-2A{a}_2\right)+a_5\left(\frac{\eta }{2}+a_{1}A\right)\right]$$

(16)

$$a_7{}^{’}=-\frac{A_3\mathrm{Pr}{k}_f}{k_{nf}}{a}_7\left(\frac{\eta }{2}+a_{1}A\right)$$

(17)

$$a_9{}^{’}=-Sc\left[a_9\left(\frac{\eta }{2}+a_{1}A\right)-{\tau }_1\left(a_7^1{a}_8+a_9{a}_7\right)\right]$$

(18)

and

$$\left.\begin{array}{l}f\left(0\right)=0,f’\left(0\right)=0,f”\left(0\right)={\delta }_1,\\ S=1,S’={\delta }_2,\\ \theta =1,\theta ’={\delta }_3,\\ \chi =1,\chi ’={\delta }_4.\end{array}\right\}$$

(19)

The translated Equations (15)–(18) are numerically solved by estimating the missing boundary conditions as in Equation (19), using the shooting strategy and selecting the suitable value range for parameters as $A=\lambda =1,Sc=0.8,{\tau }_1=0.1$. The step size is set at 0.1, and the error tolerance is around 10⁻⁸. The present numerical scheme is validated with the existing literature, it is found that the results are best matched with the work of [30]; moreover, the error percentage is calculated with the work of [30].

4. Results and Discussion

The effect of different dimensionless restrictions that control the flow, temperature and concentration characteristics across a spinning sphere in the presence of a $Ti{O}_2/H_{2}O$-based nanofluid is discussed in this section. For all limitations, the investigation is performed in the presence/absence of nanoparticle aggregation. The numerical results are graphically illustrated to demonstrate how the individual dimensionless parameters ($A$, $\lambda$, $Sc$ and ${\tau }_1$) vary on the primary velocity, secondary velocity, temperature and concentration profiles. Table 1 shows the thermophysical characteristics of the nanomaterials. Table 2 displays the flow controlling parameters. The important engineering coefficients and its reduced form are tabulated in Table 3. The thermophysical characteristics of the base liquid and nanoparticles are shown in

Table 4. Thermophysical properties of base liquid and nanoparticles.

	$\mathit{\rho }\left(\mathbf{K}\mathbf{g}\mathbf{/}{\mathbf{m}}^{\mathbf{3}}\right)$	${\mathit{C}}_{\mathit{p}}\left({\mathbf{JK}}^{-\mathbf{1}}{\mathbf{kg}}^{\mathbf{-}\mathbf{1}}\right)$	$\mathit{k}\left({\mathbf{Wm}}^{\mathbf{-}\mathbf{1}}{\mathbf{K}}^{\mathbf{-}\mathbf{1}}\right)$	$\mathbf{Pr}$
${\mathrm{TiO}}_2$	4250	686.2	8.9538	-
${\mathrm{H}}_2\mathrm{O}$	997.1	4179	0.613	6.2

. The acquired numerical findings are compared to previously published literature (see

Table 5. Comparison of $f”\left(0\right)$ for several values of $A$ in the absence of $\varphi$.

	Ref. [37]	Ref. [30]	Present Result	Error %
	Quasilinearization	RK-4	RKF-45
$A$	$f”\left(0\right)$	$f”\left(0\right)$	$f”\left(0\right)$
0.5	0.79946	0.79913	0.79919	0.0075
1	1.28271	1.2828	1.28231	0.0389
2	1.91728	1.9172	1.91765	0.0234

,

Table 6. Comparison of $-s’\left(0\right)$ for several values of $A$ in the absence of $\varphi$.

	Ref. [37]	Ref. [30]	Present Result	Error %
	Quasilinearization	RK-4	RKF-45
$A$	$-s’\left(0\right)$	$-s’\left(0\right)$	$-s’\left(0\right)$
0.5	0.30351	0.30339	0.30348	0.0296
1	0.64575	0.64579	0.64585	0.0092
2	1.05422	1.05415	1.05427	0.0113

and

Table 7. Comparison of $-\theta ’\left(0\right)$ for several values of $A$ in the absence of $\varphi$.

	Ref. [37]	Ref. [30]	Present Result	Error %
	Quasilinearization	RK-4	RKF-45
$A$	$-\theta ’\left(0\right)$	$-\theta ’\left(0\right)$	$-\theta ’\left(0\right)$
0.5	0.46743	0.467648	0.467656	0.0017
1	0.58957	0.589527	0.589536	0.0015
2	0.77954	0.779526	0.779538	0.0015

). The important engineering factors, such as $C_{fx}$, $C_{fz}$, $Nu$ and $Sh$, are calculated for various constraints in the presence and absence of the nanoparticle aggregation, as shown in

Table 8. Computational values of $C_{fx}$, $C_{fz}$, $Nu$ and $Sh$ for various dimensionless constraints in the presence of nanoparticle aggregation with $\varphi =0.01$.

$\mathit{\lambda }$	$\mathit{S}\mathit{c}$	${\mathit{\tau }}_1$	$\mathit{A}$	${\mathit{C}}_{\mathit{f}\mathit{x}}$	${\mathit{C}}_{\mathit{f}\mathit{z}}$	$\mathit{N}\mathit{u}$	$\mathit{S}\mathit{h}$
1	0.8	0.1	1	2.139808422	1.672304816	9.096392022	1.953593148
2				2.388841158	2.399686127	9.109856260	1.960978126
3				2.631209992	2.978918454	9.122719148	1.967982378
	0.8			2.139808422	1.672304816	9.096392022	0.511508126
	1.0			2.139808462	1.672304845	9.096392167	2.375204360
	1.2			2.139808462	1.672304845	9.096392167	2.796341524
		0.1		2.139808422	1.672304816	9.096392022	1.953593148
		0.3		2.139808462	1.672304845	9.096392167	2.971849166
		0.6		2.139808462	1.672304845	9.096392167	4.506727245
			0.5	1.136239692	0.897164782	8.985929840	1.846421159
			1.0	2.139808122	1.672304555	9.096391165	1.953593043
			1.5	3.091232933	2.389139595	9.212597417	2.049708182

and

Table 9. Computational values of $C_{fx}$, $C_{fz}$, $Nu$ and $Sh$ for various dimensionless constraints in the absence of nanoparticle aggregation with $\varphi =0.01$.

$\mathit{\lambda }$	$\mathit{S}\mathit{c}$	${\mathit{\tau }}_1$	$\mathit{A}$	${\mathit{C}}_{\mathit{f}\mathit{x}}$	${\mathit{C}}_{\mathit{f}\mathit{z}}$	$\mathit{N}\mathit{u}$	$\mathit{S}\mathit{h}$
1	0.8	0.1	1	2.150493768	1.707536159	9.346671153	2.106243350
2				2.386049271	2.445904509	9.356523653	2.112991274
3				2.615895707	3.031585469	9.365992932	2.119423484
	0.8			2.150493768	1.707536159	9.346671153	2.106243350
	1.0			2.150493746	1.707536179	9.346671124	2.564734427
	1.2			2.150493746	1.707536179	9.346671124	3.022504984
		0.1		2.150493768	1.707536159	9.346671153	2.106243350
		0.3		2.150493746	1.707536179	9.346671124	3.414321918
		0.6		2.150493746	1.707536179	9.346671124	5.384543102
			0.5	1.160159942	0.937705250	9.265662314	1.997594676
			1.0	2.150493777	1.707536205	9.346671237	2.106243364
			1.5	3.087013054	2.416943407	9.433920745	2.203222922

.

Figure 2, Figure 3, Figure 4 and Figure 5 display the influence of the acceleration parameter $\left(A\right)$ on the primary velocity profile $f’$, secondary velocity profile $s$, temperature profile $\theta$ and concentration profile $\chi$. Figure 2 shows that an inclination in the values of $A$ will increase the primary velocity $f’$. The reverse trend is observed in the secondary velocity profile $s$(see Figure 3). With the rise in the value of $A$, the $u_e$ also rises, which reduces the thickness of the boundary layer and increases the velocity in the lateral axis within the boundary layer. As a consequence, thermal distribution is decreased with a rise in $A$ (see Figure 4). An increase in the values of $A$ will diminish the thermal distribution due to a decrease in the thickness of the boundary layer. Similar behavior is exhibited by the concentration profile (see Figure 5). The velocity of the nanofluid with aggregation is lower when compared to nanoparticles without aggregation in the primary and secondary velocity profiles. Thermal dispersal is greater in the case of nanoparticles without aggregation than nanoparticles with aggregation. A similar trend is detected in the concentration profile.

The variation in the rotational parameter $\left(\lambda \right)$ on the primary velocity $f’$ profile is shown in Figure 6. The increase in $\lambda$ will accelerate the primary velocity $f’$. This is due to the rise in the $\lambda$, which includes additional momentum to the boundary layer, which results in a decline in the thickness of the boundary layer, causing the fluid to accelerate. It is also observed that nanoparticles without aggregation have lower velocity than nanoparticles with aggregation.

The impact of the Schmidt number $\left(Sc\right)$ and thermophoretic parameter $\left({\tau }_1\right)$ on the concentration profile $\chi$ is illustrated in Figure 7 and Figure 8. In the fluid flow, the ratio between mass and momentum diffusivity is called the Schmidt number. As the values of $Sc$ decrease, the molecular diffusion results in a decrease in the concentration profile. A similar observation is noted in the case of ${\tau }_1$ (see Figure 8). As ${\tau }_1$ increases, it results in an increase in the temperature gradient, which leads to the motion of the nanoparticles from the hot to cold region. The nanoparticles without aggregation show greater concentration than nanoparticles with aggregation.

Figure 9 and Figure 10 show the influence of $\lambda$ on $C_{fx}$ and $C_{fz}$ for a variation in $A$. From the figure, it is observed that the surface drag force is enhanced with an improvement in the values of $\lambda$ and $A$. A rise in the values of $A$ and $\lambda$ will decrease the thickness of the boundary layer, resulting in a high surface drag force. Surface drag force is greater in nanoparticles with aggregation than nanoparticles without aggregation in the $C_{fx}$ case but the reverse trend is seen in the $C_{fz}$ case.

Figure 11 shows the variation in $\varphi$ on $Nu$ for numerous values of $A$. The rate of thermal distribution enhances with inclined values of $\varphi$. As the values of the rotational parameter and solid volume fraction rise, the $Nu$ decreases; this is due to a decrease in the thickness due to the presence of $A$. It is clearly seen from the plot that nanoparticles without aggregation exhibit a higher rate of heat transfer than nanoparticles with aggregation. The rate of mass transfer on ${\tau }_1$ for numerous values of $A$ is illustrated in Figure 12. The growth in the values of ${\tau }_1$ and $A$ will increase the rate of mass transfer. The rate of mass transfer is greater in the case of nanoparticles without aggregation than nanoparticles with aggregation.

5. Conclusions

The current study is designed to examine the behavior of nanoparticle aggregation stagnation point flow over a rotating sphere with the occurrence of T-P-D. The numerical solutions to the flow governing equations are obtained with the help of the RKF-45 scheme by using the bvp5c package. The numerical data are graphically presented to show how the various dimensionless parameters fluctuate in their corresponding profiles. The following are the findings of the current investigation:

• Primary velocity enhances with increasing values of the acceleration parameter, but secondary velocity diminishes;
• Thermal distribution enhances in the case of nanoparticles in the absence of aggregation as compared to nanoparticles in the presence of aggregation over the acceleration parameter;
• Growth in the rotational parameter will escalate the primary velocity;
• For an inclination in the thermophoretic parameter, the concentration decreases more in the case of nanoparticle aggregation than in the absence of nanoparticle aggregation;
• The surface drag force is greater in nanoparticles with aggregation than nanoparticles without aggregation in the $C_{fx}$ case but the reverse trend is seen in the $C_{fz}$ case;
• The rate of heat distribution increases as the solid volume fraction rises, whereas the rate of mass transfer grows as the thermophoretic parameter increases.

The current work was limited to examining the behavior of nanoparticle aggregation stagnation point flow over a rotating sphere with the occurrence of thermophoretic particle deposition. The current study may be expanded to investigate shape factor behavior in heat and mass transfer in the presence of ternary nanofluid flow.