Axisymmetric Flow and Heat Transfer in ${\mathbf{TiO}}_{\mathbf{2}}\mathbf{/}{\mathbf{H}}_{\mathbf{2}}\mathbf{O}$ Nanofluid over a Porous Stretching-Sheet with Slip Boundary Conditions via a Reliable Computational Strategy

Abstract

In this investigation, the motion of ${\mathrm{TiO}}_2/{\mathrm{H}}_2\mathrm{O}$ nano-structures towards heated and porous sheets by considering the MHD effect and partial slip at the boundary is inspected. The non-linear PDEs that correspond to the basic conservation laws are converted into ODEs with the help of suitable similarity transformation. Furthermore, the shooting method is used to solve these transformed ODEs and boundary conditions. The impact of thermophoresis properties has been shown graphically and the effect of these properties on the skin friction coefficient $\left(\mathrm{Cf}\right)$ and Nussetl number $\left(\mathrm{Nu}\right)$ are given in table form. The comparison between the present exploration and published work is carried out and validation among results is prepared. The enhancement in thermophysical parameters showed contrary results to the velocity profile of the ${\mathrm{TiO}}_2/{\mathrm{H}}_2\mathrm{O}$ nanofluid as compared with temperature profile. Moreover, it is observed that the higher estimation in the velocity slip parameter retards the flow and an enhancement in volume fraction increases the fluid’s temperature. Furthermore, it has been discovered that the geometry of nanoparticles has a major impact on the flow behaviour. The temperature distribution diminishes when the shape of the nanoparticles changes from platelet to spherical.

1. Introduction

It is a general fact that the solid has a greater thermal conductivity as compared to standard fluids just as ${\mathrm{C}}_2{\mathrm{H}}_6{\mathrm{O}}_2$, water, and mineral oil which causes the low heat transfer properties. Therefore, a contemporary way to improve the heat transportation in fluids is to produce a heat transfer medium by mixing the two substances. The easiest way is to suspend solid materials of micro-sized particles whose thermal conductivity of metal in the conventional fluids and new fluids are known as nanofluids. Naturally, the metals $\left(\mathrm{Cu},\mathrm{Al}\right),$ oxides $\left({\mathrm{TiO}}_2,{\mathrm{AL}}_2{\mathrm{O}}_3,\mathrm{CuO},{\mathrm{SiO}}_2\right),$ carbides $\left(\mathrm{SiC}\right)$, nitrides $\left(\mathrm{SiN},\mathrm{AlN}\right),$ or nanometals (graphite, carbon nanotubes) are the nanoparticles used in nanofluids. Maxwell [1] tried to calculate the solid-liquid mixture’s thermal conductivity. He did not consider the impact of size and shape factor of the nanoparticles in the analysis for the proposed correlation. Maxwell’s work was extended by Hamilton et al. [2] by introducing the new concept “Sphericity” in Maxwell’s proposed equations. However, this idea didn’t guarantee the clogging, sedimentation, and stability of the suspension. Choi et al. [3] introduced the idea of effective heat transfer in fluids during the study of mixing nanoparticles into traditional fluids in new coolants and cooling technologies. They opened the new era for the researcher in the field of nanofluids. Various disciplines and industries used nanofluids for their products such as in drug delivery [4,5], the automotive industry [6,7,8,9], heating and cooling processes [10,11,12,13], nuclear reactors [14], heat exchangers [15,16], oscillating heat pipes, micro-channel heat sinks, and heat transport of nanofluids in porous mediums [17,18,19,20], and the references mentioned therein.

Magnetic hydrodynamic heat and mass transmission of nanofluids has caught the interest of scientists in the last year. Sohail et al. [21] used the Cattaneo–Christov theory for heat and mass transfer to investigate the Sutterby nanofluid over a stretching cylinder. They discovered that raising the magnetic parameter improves the heat and concentration fields, whereas the reverse is true for the velocity field. Furthermore, as both solutal and thermal parameters increase, the thermal and concentration fields decrease. Naz et al. [22] explored the behavior of Carreau nanofluid past a flat cylinder in the occupancy of gyrotactic microorganisms and in the magnetic field. Behavior of physical thermophoresis parameters on velocity, thermal, mass transportation, impact of motile organism’s density number, and fluxes are shown graphically. It is analyzed that bio convection parameter and curvature enhances the mass transportation rate of microorganisms. Khan et al. [23] discussed entropy generation and heat absorption/generation of Sisko-nanomaterial over stretchable surface with MHD stagnation point flow. Different thermophoresis parameters and their effects on velocity, temperature, and concentration profiles are depicted graphically. They observed that the velocity profile increases monotonically against the velocity ratio parameter. Temperature rises as thermophoetic variables rise, while decay is measured as the Prandtl number rises. Concentration reduces with the increase for heterogeneous cases while the reverse behavior is noted for homogeneous cases. By using 2D MHD flow of Hybrid nanoparticles in the presence of forms of nanoparticles of micropolar dusty fluid, Ghadikolaei et al. [24] discovered that when the value of shape factors increases, the local Nusselt number in prescribed heat flux increases. Nadeem et al. [25] investigated MHD Williamson nanofluid flow over a hot surface and discovered that Williamson fluid had a lower thermal conductivity than MHD Williamson nanofluid.

Khanafer et al. [26] discussed heat transportation performance of nanofluids by presenting the Nusselt number and heat transfer correlation of volume fraction for different Grashoff numbers. Oztop et al. [27] considered partially heated rectangular enclosures filled with nanofluids. They studied natural convection heat transportation with Cu nanoparticles and obtained the highest values of heat transfer. Hamad et al. [28] investigated fluid transportation across a porous membrane as well as heat transportation of an incompressible nanofluid over a semi-finite vertical stretching surface with a magnetic effect. They discussed that as the magnetic parameter increases, the boundary layer thickness also increases while the reverse behavior is observed for the momentum boundary layer thickness. Heris et al. [29] studied the convective heat transfer of oxide nanofluid and observed that there are different parameters to increase the heat transportation, as well as enhance the disorderly variation of nanoparticles, thermal conductivity, and fluctuations. Shah et al. [30] considered a thin film flow of nanofluid past a horizontal stretching disk by considering the effect of thermal radiation and magnetic properties. They discovered that as the magnetic parameters are increased, the thickness of the thin film nanofluid decreases.

The phenomenon of velocity slip has been investigated by taking into account a number of factors. The interface of slip velocity between the fluid and solid boundary appears as a result of nanoparticles, enticing researchers to explore the slip condition. Wang [31], Andersson [32], Ariel et al. [33], Ariel [34], Yang [35], and Abbas et al. [36] discussed problems related to flow considering slip boundary conditions. Mukhopadhyay [37] analyzed the fluid flow past a porous nonlinearly stretching surface with slip boundary condition and observed that with the enhancement of the slip parameter, the velocity profile decreases. Noghrehabadi et al. [38] studied partial slip at the boundary of nanofluid flow over a stretching sheet. They found that for increasing the value of the slip parameter, the Nusselt and Sherwood numbers decrease. Raisi et al. [39] numerically studied slip and non-slip conditions on nanofluids and found that the slip velocity coefficient increases only for greater Reynolds number which affects the rate of heat transfer. Yap et al. [40] analyzed the Williamson nanofluid flow with thermal and velocity slip conditions by taking the Buongiorno model and carried out the study of heat and mass transportation analysis. They showed that with the increase of thermal and velocity slip parameters, the thickness of boundary layer decreases.

The heat and mass transportation of MHD nanofluids with radiation was explored by Prasad et al. [41], who observed that the diffusion thermo/radiation absorption parameter increases velocity, temperature, and skin friction for Cu nanoparticles. This is because of Copper’s higher conductivity compared to TiO₂. Zangooee et al. [42] used AGM to study the MHD nanofluid $\left({\mathrm{TiO}}_2-\mathrm{GO}\right)$ between two radiative stretchable rotating disks and showed that the temperature increased with the concentration decreasing function of the Reynolds number. Islam et al. [43] studied nanofluids both theoretically and experimentally. They used water-ethylene glycol based ${\mathrm{TiO}}_2$ nanofluids as coolants in PEM fuel cells and observed that with the increase of temperature and concentration of nanoparticles, the increase in thermal and electrical conductivity of $50/50$ water-ethylene glycol based ${\mathrm{TiO}}_2$ nanofluids were almost linear. Bobbo et al. [44], considered water based SWCNH ${\mathrm{TiO}}_2$ nanoparticales and analyzed the viscosity experimentally as well as theoretically. They concluded that both have Newtonian behavior. Fedele et al. [45] investigated the viscosity and thermal conductivity of water-based nanofluids including TiO₂ nanoparticles and discovered that the ${\mathrm{TiO}}_2/{\mathrm{H}}_2\mathrm{O}$ nanofluid’s thermal characteristics made it appropriate for heat transfer applications. Fakour et al. [46] studied the heat transportation past an unsteady stretching elastic sheet by taking nanofluid thin film flow and demonstrated that the ${\mathrm{H}}_2\mathrm{O}-\mathrm{Al}$ nanofluid gave better heat transfer as compared to other types of nanofluids.

Azeem et al. [47] explored unsteady, incompressible axisymmetric flow and heat transfer in a liquid film across a radially extending sheet using the magnetic effect. They came to the conclusion that raising the value of magnetic and unsteadiness parameters causes a decrease in the thickness of the film boundary layer. The velocity and thermal conductivity of a fluid are both increased when unsteadiness parameters are increased. They also noticed a decrease in temperature profile when the Prandtl and Eckert numbers increased. Jawad et al. [48] considered hybrid nanofluid (${\mathrm{Al}}_2{\mathrm{O}}_3-\mathrm{Cu}/{\mathrm{H}}_2\mathrm{O}$) with a magnetic field by using Tiwari and Das model in stagnation region. They analyzed that escalating the values of volume fraction enhances the flow profile across the stream and cross-stream direction. They showed that increasing the value the of nanoparticle’s shape and thermal radiation enhances the temperature profile. Sobia et al. [49] discussed the heat transportation of unsteady thin film flow of nanofluid $\left({\mathrm{SiO}}_2/{\mathrm{H}}_2\mathrm{O}\right)$ passed over stretching surface with convective condition at boundary. They came to the conclusion that platelet-shaped SiO₂ nanoparticles improve the flow and heat transfer rates. They also discovered that increasing the volume fraction and Biot number raises the temperature profile, but the slip parameter has the opposite effect. Sikdar et al. [50] completed an experimental study of ${\mathrm{TiO}}_2$ Nanofluid’s electrical conductivity and found that enhancement in electrical conductivity increases the temperature and concentration profile.

For more insight, there are many researchers who investigated the concept of heat transfer and fluid flow of nanofluid by using a single phase and two-phase model by taking different conditions and nanoparticles [51,52,53,54,55,56]. Sohail et al. [57] performed significant achievement in the field of thermal energy by using tri-hybrid nanoparticles over a stretching surface via a finite element method. Nazir et al. [58] discussed the thermal aspects’ enhancement into Hall and ion slip currents in Carreau Yasuda fluid over a porous cone using the finite element method. Sohail et al. [59] used variable properties in viscoelastic liquid in concentration and thermal energy effects.

To the author’s knowledge, no investigation has been described in literature investigating the analyzed axisymmetric flow with heat transportation of ${\mathrm{TiO}}_2/{\mathrm{H}}_2\mathrm{O}$ nanofluid passed a radially stretching porous surface with partial slip boundary condition also taking consideration of MHD effect and viscous dissipation. This work is organized as follows: the literature review is given in segment 1, the formulation of governing equations of the problem is presented in segment 2, the adopted method is listed and applied in segment 3, segment 4 mentions the physical interpretation, and a summary of the performed exploration is listed in segment 5.

2. Mathematical Description of the Problem

Assume the unsteady axisymmetric 2D motion of nanofluid on a radially stretched porous surface under role of magnetic field. The fluid flow begins when the sheet is radially stretched. U and T reveal wall velocity and temperature. Schematic presentation of nanofluid flow over a porous stretching sheet and the coordinate systemic is depicted in following Figure 1.

Base fluid water (H₂O), titanium dioxide (TiO₂) was considered as nanoparticles.

Table 1. Properties (Thermo-physical) regarding ${\mathrm{H}}_2\mathrm{O}$ and ${\mathrm{TiO}}_2$.

Nanoparticle/Based Fluid	$\mathit{\rho }\left(\mathbf{k}\mathbf{g}/\mathbf{m}³\right)$	$\mathit{C}\mathit{p}\left(\mathbf{J}/\mathbf{k}\mathbf{g}\mathbf{K}\right)$	$\mathit{K}\left({W}/\mathbf{m}\mathbf{K}\right)$	$\mathit{\sigma }{\left(\Omega \mathbf{m}\right)}^{-1}$
${\mathrm{TiO}}_2$	$4250$	$686.2$	$8.9538$	$0.125$
${\mathrm{H}}_2\mathrm{O}$	$997.1$	$4179$	$0.613$	$5.5$

illustrates the thermal properties [47]. Moreover, uniform structures of nanoparticles are addressed. Furthermore, the phase of nanofluid and nanoparticles are considered in thermal equilibrium with stretching velocity $U=br/\left(1-\alpha t\right),$ where $b$ and $\alpha$ are dimensional constants. $U\left(r,t\right),$ reflects the fixed elastic sheet at origin when radially stretched with force in positive $r$—direction and effective stretching rate $b/\left(1-\alpha t\right)$ increase with time as $0\le \alpha \le 1$. Temperature profile on the porous radially stretching sheet is formulated as

$$T=T_0-T_r\left[b{r}^2/2{\nu }_f\right]{\left(1-\alpha t\right)}^{-\frac{3}{2}},$$

(1)

where $T_0, T_r, \mathrm{and} {\nu }_f$ are the surface temperature, the constant reference temperature, such as $0\le T_r\le T_0,$ and kinematic viscosity of the water $\left({\mathrm{H}}_2\mathrm{O}\right)$, respectively. The variable magnetic field is assumed to be

$$B\left(r\right)=\frac{B_0}{{\left(1-\alpha r\right)}^{1/2}},$$

(2)

which is applied $z$—direction.

The mathematical form of the physical model mentioned above is given as [47]

$$\frac{\partial }{\partial r}\left(u\right)+\frac{1}{r}\left(u\right)+\frac{\partial }{\partial z}\left(w\right)=0,$$

(3)

$$\frac{\partial u}{\partial t}+u\frac{\partial u}{\partial r}+w\frac{\partial u}{\partial z}={\nu }_{nf}\frac{{\partial }^{2}u}{\partial z^2}-\frac{{\sigma }_{nf}}{{\rho }_{nf}}{\left(B\left(r\right)\right)}^{2}u,$$

(4)

$$\frac{\partial T}{\partial t}+u\frac{\partial T}{\partial r}+w\frac{\partial T}{\partial z}={\alpha }_{nf}\frac{{\partial }^{2}T}{\partial z^2}+\frac{{\mu }_{nf}}{{\rho }_{nf}{C}_p}{\left(\frac{\partial u}{\partial z}\right)}^2,$$

(5)

where $u$ and $w$ are velocity components. The physical properties characterizing the ${\mathrm{H}}_2\mathrm{O}$ and ${\mathrm{TiO}}_2$, namely, density $\left({\rho }_{nf}\right),$ thermal diffusivity $\left({\alpha }_{nf}\right),$ viscosity $\left({\nu }_{nf}\right),$ electrical conductivity $\left({\sigma }_{nf}\right)$ and (volume fraction) $\phi$. For thermal conductivity $\left(K_{nf}\right)$ and dynamic viscosity $\left({\mu }_{nf}\right)$, Hamilton and Crosser’s models are used which are [2,60]

$${\alpha }_{nf}=\frac{K_{nf}}{{\left(\rho Cp\right)}_{nf}}, {\rho }_{nf}=\left(1-\phi \right){\rho }_f+\phi {\rho }_s, {\sigma }_{nf}=\left(1-\phi \right){\sigma }_f+\phi {\sigma }_s,$$

(6)

$${\mu }_{nf}={\mu }_f\left(1+A^1\phi +A^2{\phi }^2\right), \frac{K_{nf}}{K_f}=\frac{K_s+\left(nm-1\right)K_f+\left(nm-1\right)\left(K_s-K_f\right)\phi }{K_s+\left(nm-1\right)K_f-\left(K_s-K_f\right)\phi )}.$$

(7)

The shape factor $nm=\frac{3}{\phi }; \phi$ is the ratio of surface areas of sphere to particle.

Table 2. Shapes factor and viscosity co-efficient corresponding to five distinct nanoparticles.

Nanoparticle Shapes	Blade	Brick	Cylinder	Platelets	Sphere
$A_1$	$14.6$	$1.9$	$13.5$	$37.1$	$2.5$
$A_2$	$123.3$	$471.4$	$904.4$	$612.6$	$6.5$
$\mathrm{Shape} \mathrm{factor}\left(nm\right)$	$8.26$	$3.72$	$4.82$	$5.72$	$3.0$

presents the shape factor and viscosity coefficients corresponding to various shapes of nanoparticles which is utilized in this research.

The corresponding boundary conditions of the differential equations for the given system are

$$u=k\left(\frac{\partial u}{\partial z}\right)+U,0=v,T=T_s,\mathrm{at} z=0,u\to 0,T\to T_{\infty },\mathrm{as} z\to \infty .$$

(8)

The following similarity transformations are introduced for simplification of problem.

$$\psi =-r^{2}UR{e}^{-\frac{1}{2}}f\left(\eta \right),\theta \left(\eta \right)=\frac{\left(T-T_s\right)}{T_r\left[\frac{b{r}^2}{2{\nu }_f}\right]{\left(1-\alpha t\right)}^{-\frac{3}{2}}},\eta =\frac{z}{r}R{e}^{\frac{1}{2}},Re=\frac{rU}{{\nu }_f}.$$

(9)

A (stream function) $\psi$ defined by

$$u=-\frac{1}{r}\frac{\partial \psi }{\partial z}=U{f}^{\prime }\left(\eta \right),w=\frac{1}{r}\frac{\partial \psi }{\partial r}=-2Uf\left(\eta \right)R{e}^{-\frac{1}{2}}.$$

(10)

Using similarity variables (9), Equations (4) and (5) take the form

$${\epsilon }_{1}f‴-M{\epsilon }_{2}f\prime -S\left[f\prime +\frac{\eta }{2}f″\right]-{f^{\prime }}^2+2ff″=0,$$

(11)

$$\frac{{\epsilon }_2}{Pr}\theta ″+Ec{\epsilon }_1\left(f″\right)²-\frac{S}{2}\left[3\theta +\eta \theta \prime \right]-2f\prime \theta +2f\theta \prime =0,$$

(12)

subject to transform boundary conditions

$$f\left(0\right)=0,f\prime \left(0\right)=1+kf″\left(0\right),\theta \left(0\right)=1,f\prime (\infty )=0,\theta (\infty )=0.$$

(13)

The thermo-physical parameters $Pr, S, Ec, \mathrm{and} M$ appear in Equations (11) and (12) are mathematically defined as

$$Pr=\frac{\nu }{{\alpha }_{nf}},S=\frac{a}{b},Ec=\frac{U^2}{Cp\left(T_s-T\right)}, \mathrm{and} M=\frac{{\sigma }_f{B_0}^2}{{\rho }_{nf}b}.$$

(14)

Finally, ${\epsilon }_1,{\epsilon }_2,$ and ${\epsilon }_3$ are constants which are defined by

$${\epsilon }_1=\frac{\left(1+A^1\phi +A^2{\phi }^2\right)}{\left(1-\phi +\phi \frac{{\rho }_s}{{\rho }_f}\right)}, {\epsilon }_2=\frac{\frac{K_{nf}}{K}}{1-\phi +\phi \frac{{\left(\rho Cp\right)}_s}{{\left(\rho Cp\right)}_f}},{\epsilon }_3=\frac{1-\phi +\phi \frac{{\sigma }_s}{{\sigma }_f}}{1-\phi +\phi \frac{{\rho }_s}{{\rho }_f}}.$$

(15)

The desired temperature gradient and drag force coefficient are

$$\{\begin{array}{c}{C}_f=\frac{2{\mu }_{nf}{\left(\frac{\partial u}{\partial z}\right)}_{z=0}}{{\rho }_f{U}^2}=R{e}^{-\frac{1}{2}}\left(1+A^1\phi +A^2{\phi }^2\right)f^{″}\left(0\right), \\ Nu=\frac{-x{K}_{nf}{\left(\frac{\partial T}{\partial z}\right)}_{z=0}}{T_s-T_0}=R{e}^{\frac{1}{2}}\frac{K_{nf}}{K}\theta \prime \left(0\right). \end{array}$$

(16)

3. Numerical Scheme for Solution

Several numerical [27,39,52,55,58,59] and analytical schemes [4,5,21,22,28,31,57] are available to handle the nonlinear complex problems arising in mathematical physics. The exact solution is not possible for every problem due to highly nonlinearity and as well as due to the involvement of complex geometry and mixed boundary conditions. The solution of a nonlinear ODE system (11–12) subject to boundary conditions (13) is calculated using numerical technique known as BVP4C. In this case, the third- and second-order nonlinear ODEs are given by Equations (11) and (12), respectively, and have been reduced to first-order differential equations as follow and the better understanding of the used scheme is explained through Figure 2.

$$f=y_1,f\prime =y_2,f″=y_3,f‴=\frac{1}{{\epsilon }_1}\left[M{\epsilon }_3{y}_2+S\left\{y_2+\frac{\eta }{2}{y}_3\right\}+{y_2}^2-2y_1{y}_2\right],$$

(17)

$$\theta =y_4,\theta \prime =y_5,\theta ″=\frac{Pr}{{\epsilon }_2}\left[\frac{S}{2}\left\{3y_4+\eta {\epsilon }_0\right\}+2y_2{y}_4-2y_1{y}_5-Ec{\epsilon }_1{y_3}^2\right],$$

(18)

corresponding boundary condition becomes

$$y_1\left(0\right)=0,y_2\left(0\right)=1+k{y}_3\left(0\right),y_4\left(0\right)=1,y_2(\infty )=0,y_5(\infty )=0.$$

(19)

The iterative procedure will end with the appropriate level of precision.

4. Results and Discussion

This includes the simulation results of titanium dioxide-water nanofluid unsteady flow and heat transfer over a radially stretching surface (Figure 1). We set some fundamental parameters for this analysis, changed one of them and then composed a response. The results so obtained are compared for skin friction $\left(-f″\left(0\right)\right)$ and local Nusselt number $\left(-\theta \prime \left(0\right)\right)$, respectively, with different values of unsteadiness parameter (S), magnetic parameter (M), slip parameter (k), nanoparticles volume fraction $\left(\phi \right)$, Prandtl number $\left(Pr\right)$ and Eckert number $\left(Ec\right)$ by taking nanoparticle shapes like brick, cylinder, blade, sphere and platelets are depicted in

Table 3. The computed results for local skin friction.

$\mathit{S}$	$\mathit{M}$	$\mathit{k}$	$\mathit{\phi }$	$\mathit{P}\mathit{r}$	$\mathit{E}\mathit{c}$	$\left(-\mathit{f}″\left(0\right)\mathit{P}\mathit{r}\mathit{e}\mathit{s}\mathit{e}\mathit{n}\mathit{t}\right)$
$\mathbf{0.2}$	$\mathbf{1.0}$	$\mathbf{0.5}$	$\mathbf{0.02}$	$\mathbf{7.56}$	$\mathbf{1.0}$	$Blade$	$Brick$	$Cylinder$	$Platelets$	$Sphere$
$0.0$	$-$	$-$	$-$	$-$	$-$	$0.7155365$	$0.7347624$	$0.6740443$	$0.6335091$	$0.7680837$
$0.2$	$-$	$-$	$-$	$-$	$-$	$0.7350176$	$0.7545665$	$0.6928182$	$0.6515231$	$0.7884168$
$0.4$	$-$	$-$	$-$	$-$	$-$	$0.7535168$	$0.7733531$	$0.7106818$	$0.6686982$	$0.8076700$
$-$	$-$	$0.0$	$-$	$-$	$-$	$1.281081$	$1.339648$	$1.1615725$	$0.9274630$	$1.4460915$
$-$	$-$	$0.5$	$-$	$-$	$-$	$0.7350176$	$0.7545665$	$0.6928966$	$0.5906568$	$0.7884168$
$-$	$-$	$1.0$	$-$	$-$	$-$	$0.5254160$	$0.5357161$	$0.5027803$	$0.4414587$	$0.5532537$
$-$	$-$	$-$	$0.00$	$-$	$-$	$0.4317496$	$0.4317460$	$0.4317460$	$0.4317460$	$0.4317460$
$-$	$-$	$-$	$0.02$	$-$	$-$	$0.4113289$	$0.4177996$	$0.3968889$	$0.3821424$	$0.4287374$
$-$	$-$	$-$	$0.4$	$-$	$-$	$0.3912077$	$0.3891333$	$0.3520232$	$0.3406492$	$0.4255740$
$-$	$-$	$-$	$-$	$4.0$	$-$	$0.7350176$	$0.7545665$	$0.6928966$	$0.6516882$	$0.7884168$
$-$	$-$	$-$	$-$	$6.0$	$-$	$0.7350176$	$0.7545665$	$0.6928966$	$0.6516882$	$0.7884168$
$-$	$-$	$-$	$-$	$8.0$	$-$	$0.7350176$	$0.7545665$	$0.6928966$	$0.6516882$	$0.7884168$
$-$	$-$	$-$	$-$	$-$	$0.0$	$0.7350176$	$0.7545665$	$0.6928966$	$0.6516882$	$0.7884168$
$-$	$-$	$-$	$-$	$-$	$0.5$	$0.7350176$	$0.7545665$	$0.6928966$	$0.6516882$	$0.7884168$
$-$	$-$	$-$	$-$	$-$	$1.0$	$0.7350176$	$0.7545665$	$0.6928966$	$0.6516882$	$0.7884168$

and

Table 4. The computed results for local Nusselt number.

$\mathit{S}$	$\mathit{M}$	$\mathit{k}$	$\mathit{\phi }$	$\mathit{P}\mathit{r}$	$\mathit{E}\mathit{c}$	$\left(-\mathit{\theta }\prime \left(0\right)\mathit{P}\mathit{r}\mathit{e}\mathit{s}\mathit{e}\mathit{n}\mathit{t}\right)$
$\mathbf{0.2}$	$\mathbf{1.0}$	$\mathbf{0.5}$	$\mathbf{0.02}$	$\mathbf{7.56}$	$\mathbf{1.0}$	$Blade$	$Brick$	$Cylinder$	$Platelets$	$Sphere$
$0.0$	$-$	$-$	$-$	$-$	$-$	$2.0944532$	$2.1456869$	$2.0959472$	$2.0502516$	$2.1668261$
$0.2$	$-$	$-$	$-$	$-$	$-$	$2.3907456$	$2.4500127$	$2.3922862$	$2.3423895$	$2.4767264$
$0.4$	$-$	$-$	$-$	$-$	$-$	$2.6594658$	$2.7258783$	$2.6612633$	$2.6074317$	$2.7571685$
$-$	$-$	$0.0$	$-$	$-$	$-$	$1.6787138$	$1.729614$	$1.6293238$	$2.0918449$	$1.7842033$
$-$	$-$	$0.5$	$-$	$-$	$-$	$2.3907456$	$2.4500127$	$2.391635$	$2.6116387$	$2.4767264$
$-$	$-$	$1.0$	$-$	$-$	$-$	$2.4113003$	$2.4565463$	$2.4523227$	$2.6497208$	$2.454599$
$-$	$-$	$-$	$0.00$	$-$	$-$	$2.4072763$	$2.4073228$	$2.4073228$	$2.4073237$	$2.4073254$
$-$	$-$	$-$	$0.02$	$-$	$-$	$2.3428233$	$2.3792423$	$2.4041469$	$2.4122462$	$2.3632343$
$-$	$-$	$-$	$0.4$	$-$	$-$	$2.2790864$	$2.3740099$	$2.3774746$	$2.3587298$	$2.319692$
$-$	$-$	$-$	$-$	$4.0$	$-$	$1.7602862$	$1.8001545$	$1.7720755$	$1.744038$	$1.8113522$
$-$	$-$	$-$	$-$	$6.0$	$-$	$2.1417315$	$2.1932347$	$2.1470323$	$2.1053742$	$2.2136552$
$-$	$-$	$-$	$-$	$8.0$	$-$	$2.4555829$	$2.5168879$	$2.4553144$	$2.4022715$	$2.5452625$
$-$	$-$	$-$	$-$	$-$	$0.0$	$3.2971464$	$3.3367739$	$3.4221096$	$3.4752914$	$3.2918987$
$-$	$-$	$-$	$-$	$-$	$0.5$	$2.843946$	$2.8933933$	$2.9068724$	$2.9081209$	$2.8843125$
$-$	$-$	$-$	$-$	$-$	$1.0$	$2.3907456$	$2.4500127$	$2.391635$	$2.3409504$	$2.4767264$

.

The effect of $S,k$ and $\phi$ on the velocity field is investigated here, taking into account the various shapes of titanium dioxide nanoparticles in base fluid water. $S$ has an effect on the velocity profile, as shown in Figure 3a–e. The velocity of nanofluid in region (boundary layer) declines using higher values of $S$. Thickness related to momentum is said to decrease the function against the values of $S.$ Therefore, fluid is known as thick when $S$ is increased. It is evident that the impact of $S$ on the velocity profile is more effective by using platelet-shaped ${\mathrm{TiO}}_2$ for the nanofluid as compared to the other shapes of the nanoparticles. It is found that the maximum velocity of ${\mathrm{TiO}}_2/{\mathrm{H}}_2\mathrm{O}$ nanofluid is attained with platelet-shaped particles than that of cylinder, blade, bricks, and sphere shaped particles. It is also seen that the minimum velocity of ${\mathrm{TiO}}_2/{\mathrm{H}}_2\mathrm{O}$ nanofluid is captured with sphere shaped nanoparticles than that of the other shapes of nanoparticles. Figure 4a–e represent the influence of velocity slip parameter $\left(k\right)$. It is evident from the figures that with the increase of $k$, the horizental velocity decreases. When platelets shape nanoparticles are employed, the velocity is determined to be the highest when compared to other shapes of nanoparticles used. Figure 5a–e shows the effect of volume fraction of the nanofluid on the velocity profile. It is analyzed that the enhancement in the velocity field occurs due to increase in nanoparticles volume fraction.

The impact of $S, k, \phi , Pr, \mathrm{and} Ec$ on temperature field is predicted here by taking the different shapes of nanoparticles of titanium dioxide with base fluid water. Figure 6a–e indicate that $S$ had an impact on temperature profile. With increasing values of the steadiness parameter S, the nanofluid temperature decreases in the boundary layer region. Figure 7a–e demonstrate the effects of velocity slip parameter $\left(k\right)$ on the temperature; the graphical results showed that with the increase of k, the temperature decreases. It is estimated that heat energy becomes reduce against distribution in $k.$ This reducing impact into heat energy is due to role porosity of surface. Numbers of pores is known as reducing function into heat energy. Layers associated within heat energy are declined versus $k.$ Moreover, the impact of volume fraction of nanoparticles of different shapes on the temperature field $\theta \left(\eta \right)$ is predicted, in Figure 8a–e for ${\mathrm{TiO}}_2/{\mathrm{H}}_2\mathrm{O}$ nanofluid. It has been discovered that using nano-sized particles of various shapes in water increases thermal conductivity, resulting in an increase in the heat transfer rate. Figure 9a–e demonstrated that as the Prandtl number (Pr) varied, the thermal diffusion rate slowed, resulting in a lower temperature profile at all times. Thermal layers at boundary have decreasing function against Prandtl number. This decreasing function among heat energy and Prandtl number is developed due to using concept of Prandtl number. According to concept of Prandtl number, it is ratio among momentum and thermal layers at boundary of surface. Therefore, an increment in Prandtl number creates reduction in thermal energy. Momentum layers are based on distribution of Prandtl number. Thus, momentum layers at boundary are known as decreasing function versus an impact of Prandtl number. Physically, the fluids with larger values of Prandtl number have small thermal conductivity and high viscosity, due to which fluid become thick, and causes the decrease in the velocity field. It is concluded that with the increase of Eckert number causes the increases in temperature field as can be seen in Figure 10a–e. Figure 11a indicates that the Titanium dioxide/water nanofluid attains its maximum velocity when we used the platelets shape nanoparticles of Titanium dioxide than cylindrical than blade than brick shapes nanoparticles and then minimum in the case of sphere shape nanoparticle. Figure 11b depicts that the Titanium dioxide/water nanofluid attains its maximum temperature when we used the platelet-shaped nanoparticles of Titanium dioxide rather than cylindrical, blade or brick-shaped nanoparticles, and then the minimum in sphere shaped particles. It is noticed that the maximum production in heat energy occurs because of the maximum amount of viscous dissipation. Physically, viscous dissipation is defined as the rate of work done in view of movement of nanoparticles. Higher viscous dissipation brings an enhancement in the rate of work done of nanoparticles. It means that large viscous dissipation creates more heat energy in the particles. Moreover, a directly proportional relation is found between the Eckert number and thermal energy. Therefore, temperature profile increases in line with higher values of Eckert number.

Table 5. Impacts of pertinent parameters on Skin fraction coefficient.

$\mathit{M}$	$\mathit{S}$	$\mathit{P}\mathit{r}$	$\mathit{E}\mathit{c}$	$-\mathit{f}″\left(0\right)$	$-\mathit{f}″\left(0\right)$
				Reference [47]	Present
0	1.0	0.7	0.2	1.2784767	1.1353469
1	-	-	-	1.6237276	1.5146317
2	-	-	-	1.9067150	1.8155125
1	0.8	0.7	0.2	1.674472	1.6107074
-	1.0	-	-	1.6650959	1.6571311
-	1.2	-	-	1.6186823	1.7025145

present the comparative study. From the analysis, it is obvious that the obtained results are in good settlement with the findings mentioned in studies reported by [47].

5. Key Findings of Performed

Axisymmetric viscous flow in 2D of unsteady and incompressible ${\mathrm{TiO}}_2/{\mathrm{H}}_2\mathrm{O}$ nanofluid over a porous radially stretching sheet with velocity slip conditions at a boundary is analyzed here in this article. From the contemporary investigations, the main interpretations are

• Dimensionless velocity $f\prime \left(\eta \right)$ shows a decreasing nature for the unsteadiness parameter $\left(S\right)$ and velocity slip parameter $\left(k\right),$ while, the reverse behavior for volume fraction $\left(\phi \right);$
• Augmenting values of velocity slip and unsteadiness parameters retards the velocity field;
• Higher estimation of the volume fraction boosts the fluid’s temperature;
• For a rise in the volume fraction parameter $\left(\phi \right)$ causes the enhancement in temperature profile on the other hand the reverse behavior is observed in case of increasing the unsteadiness parameter $\left(S\right)$ and velocity slip parameter $\left(k\right);$
• Higher values of the Prandtl number $\left(Pr\right)$ results to decrease the temperature field and connected layer thickness;
• The increasing effect of the Eckert number $\left(Ec\right)$ on the temperature field is observed;
• It is concluded from the obtained results that the impact of the Eckert number and Prandtl number is opposite in nature;
• On the platelet and sphere-shaped nanoparticles, the velocity profile of ${\mathrm{TiO}}_2$ nanofluid reaches its maximum and minimum, respectively, while thermal conductivity follows a similar pattern.
• Maximum surface force is developed against higher values of the magnetic number, while the minimum production of surface force is generated with respect to higher values of the unsteadiness parameter;
• A significant performance of nanoparticles is observed for enhancement of thermal conductivity and production of heat energy.